Global existence and non-uniqueness of 3D Euler equations perturbed by transport noise

Abstract

We construct Hölder continuous, global-in-time probabilistically strong solutions to 3D Euler equations perturbed by Stratonovich transport noise. Kinetic energy of the solutions can be prescribed a priori up to a stopping time, that can be chosen arbitrarily large with high probability. We also prove that there exist infinitely many Hölder continuous initial conditions leading to non-uniqueness of solutions to the Cauchy problem associated with the system. Our construction relies on a flow transformation reducing the SPDE under investigation to a random PDE, and convex integration techniques introduced in the deterministic setting by De Lellis and Székelyhidi, here adapted to consider the stochastic case. In particular, our novel approach allows to construct probabilistically strong solutions on $[0,\infty )$ directly.

1 Introduction

In the present work we consider the Euler equations on the three-dimensional torus $\mathbb {T}^3$ perturbed by Brownian noise of transport type:

$$\begin{aligned} {\left\{ \begin{array}{ll} d u + (u \cdot \nabla ) u \, dt + \sum _{k \in I} (\sigma _k \cdot \nabla ) u \circ dB^k + \nabla p \, dt = 0,\\ \mathord {\textrm{div}}\,u = 0, \end{array}\right. } \end{aligned}$$

(1.1)

where I is a finite set, $\{\sigma _k\}_{k \in I}$ is a collection of smooth divergence-free vector fields on the torus and $\{B^k\}_{k \in I}$ is a family of i.i.d. standard Brownian motions on a given filtered probability space $(\Omega ,{\mathcal {F}},\{{\mathcal {F}}_t\}_{t \ge 0}, \mathbb {P})$, with $\{{\mathcal {F}}_t\}_{t \ge 0}$ assumed to be complete and right-continuous. The notation $\cdot \,dB^k$ denotes the Stratonovich interpretation of the stochastic integral. The notion of solution to (1.1) we use throughout this paper is that of probabilistically strong, analytically weak solutions, here recalled.

Definition 1.1

Let $(\Omega ,{\mathcal {F}},\{{\mathcal {F}}_t\}_{t \ge 0}, \mathbb {P})$ and $\{B^k\}_{k \in I}$ be given as above. A progressively measurable stochastic process $(u,p):\Omega \rightarrow C_{loc}([0,\infty ), L^2(\mathbb {T}^3,\mathbb {R}^3 \times \mathbb {R}))$ almost surely is a probabilistically strong, analytically weak solution to (1.1) if for every $H \in C^\infty (\mathbb {T}^3)$ it holds almost surely

$$\begin{aligned} \int _{\mathbb {T}^3} u(x,t) \cdot \nabla H(x) dx&= 0, \quad \forall t \in [0,\infty ), \end{aligned}$$

and for every progressively measurable processes $H_0, \{H_k\}_{k \in I}: \Omega \rightarrow C_{loc}([0,\infty ),C^\infty (\mathbb {T}^3,\mathbb {R}^3))$ and semimartingale $h:\Omega \rightarrow C_{loc}([0,\infty ),C^\infty (\mathbb {T}^3,\mathbb {R}^3))$ satisfying

$$\begin{aligned} dh=H_0\,dt + \sum _{k \in I} H_k \circ dB^k, \end{aligned}$$

the process $t \mapsto \int _{\mathbb {T}^3} u(x,t) \cdot h(x,t) \,dx$ is a semimartingale and it holds almost surely

$$\begin{aligned} d\int _{\mathbb {T}^3} u \cdot h&= \int _{\mathbb {T}^3} u \cdot \left( H_0 + (u \cdot \nabla ) h \right) dt + \int _{\mathbb {T}^3} p \,\mathord {\textrm{div}}\,h \,dt \\&\quad + \sum _{k \in I} \int _{\mathbb {T}^3} u \cdot \left( H_k + (\sigma _k \cdot \nabla ) h \right) \circ dB^k. \end{aligned}$$

Notice that $L^2$ integrability in space of the velocity field u is required in order to make sense of the nonlinear term $(u \cdot \nabla ) u$ as a distribution. (Local) continuity of trajectories allows to uniquely identify the initial data $u_0 = u(\cdot ,0)$ and $p_0 = p(\cdot ,0)$ as random variables taking values in $L^2(\mathbb {T}^3,\mathbb {R}^3)$ and $L^2(\mathbb {T}^3,\mathbb {R})$ respectively, and consider the Cauchy problem associated to (1.1). For constant-in-time $h \in C^{\infty }(\mathbb {T}^3,\mathbb {R}^3)$, corresponding to $H_0=H_k=0$, this definition boils down to a more standard notion of distributional solutions to Euler equations, as for instance that used in [3]; and in fact the two definitions are equivalent, which can be shown using mollification and Itō formula as done in Appendix 1 of [17]. We decided to adopt this definition because it copes better with the flow transformation we are going to introduce in the next section, see Definition 1.2 and Proposition A.1.

In recent years, there have been several works aimed to justify the addition of transport noise in Euler equations, either by variational principles [37], homogenization techniques [11], conservation laws [20] or Wong-Zakai results [25]. More generally, a noise of transport type appears naturally in fluids when considering the effect that turbulence may have on the slow-varying, large-scale structures of the fluid itself, see [18, 26].

Moreover, regularization by transport noise has been shown to hold in many different instances, as in transport equation [22], point vortices dynamics [23], Vlasov-Poisson equations [19], Navier–Stokes equations [24] and other models [1, 21, 42, 43]. This perhaps suggests that a noise of transport type could help in proving well-posedness of Euler equations, or at least some partial result toward this direction. However, this simply turns out not to be the case. In this paper we are going to prove the following:

• (i) For every constants $\varkappa \in (0,1)$, $T\in (0,\infty )$, and for every suitable energy profile $e:[0,\infty ) \rightarrow (0,\infty )$ given a priori, there exist a stopping time ${\mathfrak {t}}$ satisfying $\mathbb {P}\{ {\mathfrak {t}} \ge T\} \ge \varkappa$ and a Hölder continuous, probabilistically strong, analytically weak solution to (1.1) having kinetic energy equal to e up to time ${\mathfrak {t}}$, namely

  $$\begin{aligned} \int _{\mathbb {T}^3} |u(x,t)|^2 dx = e(t), \quad \forall t \in [0,{\mathfrak {t}}]; \end{aligned}$$

• (ii) There exist infinitely many Hölder continuous divergence-free initial data $u(\cdot ,0)$ such that the Cauchy problem associated with (1.1) admits non-uniqueness in law.

More detailed statements of the previous results are given in Sect. 1.2 below.

It is worth mentioning the important progress in the existence theory of stochastic Euler equations implied by these results. Indeed, in three dimensions only local-in-time well-posedness of regular solutions was known so far, see [13] for transport noise and [30, 39] for other stochastic perturbations. On the other hand, our solutions are global-in-time, although we can not prescribe the initial datum as in the cited literature. From this point of view, the global existence of probabilistically strong solutions for given initial data is still an open problem. The more recent [36] and [35] provide global existence of analytically weak and probabilistically strong solution, but they are restricted to additive noise perturbations.

Those in the present paper are the first results of this kind dealing with Euler equations perturbed by transport noise.

1.1 Flow transformation and reformulation of the problem

In order to construct solutions to (1.1), it is convenient to rewrite the equation in a different form, so to “remove” the transport term $\sum _{k \in I} (\sigma _k \cdot \nabla ) u \circ dB^k$ from the equation, at least apparently.

Therefore, we can consider the flow generated by the noise, given by

$$\begin{aligned} \phi (x,t) = x + \sum _{k \in I}\int _0^t \sigma _k(\phi (x,s)) \circ dB^k(s). \end{aligned}$$

(1.2)

By our assumptions on the coefficients $\{\sigma _k\}_{k \in I}$, there exists a unique progressively measurable, probabilistically strong solution $\phi$ to (1.2) such that $\phi (\cdot ,t)$ takes values in the class of measure preserving $C^{\infty }$-diffeomorphisms of the torus for every $t \in [0,\infty )$, see [40]. Applying the change of variables¹

$$\begin{aligned} v(x,t)&= u(\phi (x,t),t), \quad q(x,t) = p(\phi (x,t),t), \end{aligned}$$

(1.3)

$$\begin{aligned} u(y,t)&= v(\phi ^{-1}(y,t),t), \quad p(y,t) = q(\phi ^{-1}(y,t),t), \end{aligned}$$

(1.4)

we can formally rewrite the SPDE (1.1) as a random PDE:

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t v + \mathord {\textrm{div}}^\phi (v \otimes v) + \nabla ^\phi q = 0,\\ \mathord {\textrm{div}}^\phi v = 0 , \end{array}\right. } \end{aligned}$$

(1.5)

where the symbols $\mathord {\textrm{div}}^\phi$, $\nabla ^\phi$ are abbreviations for the space-time dependent differential operators

$$\begin{aligned} \mathord {\textrm{div}}^\phi v = [\mathord {\textrm{div}}\,(v \circ \phi ^{-1})] \circ \phi , \quad \nabla ^\phi q = [\nabla (q \circ \phi ^{-1})] \circ \phi . \end{aligned}$$

It is worth mentioning at this point that, as already observed in [22, 23], a space-independent noise cannot regularize Euler equations since this would imply $\text{ div}^\phi =\text{ div }$ and $\nabla ^\phi =\nabla$, and therefore the stochastic system would be equivalent to the deterministic one (which notably admits non-unique solutions). However, in most of the examples mentioned above a genuinely space-dependent noise does in fact improve well-posedness, and this is why our analysis is not trivial.

For the system (1.5) we adopt the following notion of solution.

Definition 1.2

Let $(\Omega ,{\mathcal {F}},\{{\mathcal {F}}_t\}_{t \ge 0}, \mathbb {P})$ and $\{B^k\}_{k \in I}$ be given as above, and let $\phi$ be the unique stochastic flow of measure preserving diffeomorphisms given by (1.2). A progressively measurable stochastic process $(v,q){:}\Omega \rightarrow C_{loc}([0,\infty ), L^2(\mathbb {T}^3,\mathbb {R}^3 \times \mathbb {R}))$ almost surely is a probabilistically strong, analytically weak solution to (1.5) if for every $H \in C^\infty (\mathbb {T}^3)$ it holds almost surely

$$\begin{aligned} \int _{\mathbb {T}^3} v(x,t) \cdot \nabla ^\phi H(x,t) dx&= 0, \quad \forall t \in [0,\infty ), \end{aligned}$$

and for every progressively measurable processes $H_0, \{H_k\}_{k \in I}: \Omega \rightarrow C_{loc}([0,\infty ),C^\infty (\mathbb {T}^3,\mathbb {R}^3))$ and semimartingale $h:\Omega \rightarrow C_{loc}([0,\infty ),C^\infty (\mathbb {T}^3,\mathbb {R}^3))$ satisfying

$$\begin{aligned} dh=H_0\,dt + \sum _{k \in I} H_k \circ dB^k, \end{aligned}$$

the process $t \mapsto \int _{\mathbb {T}^3} v(x,t) \cdot h(x,t) \,dx$ is a semimartingale and it holds almost surely

$$\begin{aligned} d\int _{\mathbb {T}^3} v \cdot h&= \int _{\mathbb {T}^3} v \cdot \left( H_0 + (v \cdot \nabla ^\phi ) h \right) dt + \int _{\mathbb {T}^3} q \,\mathord {\textrm{div}}^\phi h \,dt + \sum _{k \in I} \int _{\mathbb {T}^3} v \cdot H_k \circ dB^k. \end{aligned}$$

The equivalence of systems (1.1) and (1.5) is shown in Proposition A.1. More precisely, we prove that a process (u, p) is a solution to (1.1) in the sense of Definition 1.1 if and only if (v, q) given by (1.3) is a solution to (1.5) in the sense of Definition 1.2. As a consequence, applying the inverse transformation (1.4) to a solution (v, q) of the modified equation (1.5) produces a solution (u, p) of the original equation (1.1). Like with (1.1), by continuity of trajectories one can uniquely determine the initial conditions $v_0=v(\cdot ,0)$ and $q_0=q(\cdot ,0)$, and consider the Cauchy problem associated to (1.5). Also, since $\phi (\cdot ,t)$ is measure preserving, it holds

$$\begin{aligned} \int _{\mathbb {T}^3} |v(x,t)|^2 dx = \int _{\mathbb {T}^3} |u(x,t)|^2 dx \end{aligned}$$

almost surely for every $t \in [0,\infty )$, and therefore we can state and prove our results in the framework of solutions to the random PDE (1.5).

1.2 Main results

The main result of the present paper is the following:

Theorem 1.3

(Global strong existence) Assume $\sigma _k$ smooth and divergence-free for every $k \in I$, I finite, and define $\phi$ by (1.2). Then there exist $\vartheta >0$ and, for any given $\varkappa \in (0,1)$ and $T\in (0,\infty )$, a stopping time ${\mathfrak {t}}$ satisfying $\mathbb {P}\{ {\mathfrak {t}} \ge T\} \ge \varkappa$ with the following property.

For every function e on $[0,\infty )$ satisfying ${\underline{e}} :=\inf _{t \in [0,\infty )} e(t)>0$ and ${\overline{e}} :=\Vert e\Vert _{C^2_t}<\infty$, there exists a global probabilistically strong, analytically weak solution (v, q) of (1.5) of class

$$\begin{aligned} v: \Omega \rightarrow C^{\vartheta }_{loc}([0,\infty ), C(\mathbb {T}^3,\mathbb {R}^3)) \cap C_{loc}([0,\infty ), C^\vartheta (\mathbb {T}^3,\mathbb {R}^3)),\\ q: \Omega \rightarrow C^{2\vartheta }_{loc}([0,\infty ), C(\mathbb {T}^3,\mathbb {R})) \cap C_{loc}([0,\infty ), C^{2\vartheta }(\mathbb {T}^3,\mathbb {R})), \end{aligned}$$

almost surely, such that v satisfies the almost sure identity

$$\begin{aligned} \int _{\mathbb {T}^3} |v(x,t)|^2 dx&= e(t), \quad \forall t \in [0,{\mathfrak {t}}]. \end{aligned}$$

As far as we know, this is the first result proving global existence of non-trivial, analytically weak, probabilistically strong solutions to Euler equations perturbed with transport noise. The only other example in the stochastic setting we are aware of is the very recent [35], dealing with the case of additive noise. We point out the fact that, in terms of space regularity of the solutions constructed, in the present paper we are able to produce even better solutions than those in [35] (Hölder continuous versus $H^\vartheta$ Sobolev regular). We also mention [30], where the authors prove that a suitable linear multiplicative noise yields global existence of smooth solutions with large probability.

As a matter of fact, the proof of Theorem 1.3 relies on a convex integration scheme, mostly inspired by the works [15, 16] on deterministic Euler equations. We shall build smooth approximate solutions $(v_n,q_n,\phi _n)$, $n \in \mathbb {N}$ to the random Euler system (1.5). We include a smooth approximating flow $\phi _n$ in the solution to take care of the lack of smoothness with respect to time of the Brownian flow $\phi$. To measure how far a given $(v_n,q_n,\phi _n)$ is from being a solution of (1.5) it is customary to introduce a system of differential equations called Euler-Reynolds system, that in our setting takes the form

$$\begin{aligned} \partial _t v + \mathord {\textrm{div}}^\phi (v \otimes v) + \nabla ^\phi q = \mathord {\textrm{div}}^\phi \mathring{R} \end{aligned}$$

(1.6)

where the Reynolds stress $\mathring{R}$ takes values in the space of $3 \times 3$ symmetric traceless matrices. In particular, we say that a quadruple $(v_n,q_n,\phi _n,\mathring{R}_n)$ is a solution of the Euler-Reynold system (1.6) if it is progressively measurable with respect to the filtration $\{{\mathcal {F}}_t\}_{t \ge 0}$ and (1.6) holds in strong analytical sense.

Notice that we are not imposing any divergence-free condition $\mathord {\textrm{div}}\,^{\phi _n} v_n = 0$ on the solution. The incompressibility condition on the limit will be restored prescribing the decay of some spatial Besov norm of $\mathord {\textrm{div}}\,^{\phi _n} v_n$ along the iteration.

However, since for technical reasons that will be clear later we are actually going to construct smooth approximate solutions $(v_n,q_n,\phi _n,\mathring{R}_n)$ also for negative times $t<0$, we require solutions also satisfy (1.6) in analytically strong sense for all times $t \in \mathbb {R}$. For negative times the flow $\phi _n$ will be defined simply as the identity on the torus, corresponding to absence of noise for $t<0$, and the process $(v_n,q_n,\mathring{R}_n)$ will be deterministic (thus preserving progressive measurability with respect to $\{{\mathcal {F}}_t\}_{\ge 0}$, extended to negative times identically equal to ${\mathcal {F}}_0$). In addition, we shall always work with solutions $v_n$ with zero spatial average: $\int _{\mathbb {T}^3} v_n = 0$ for every $t \in \mathbb {R}$. More details will be given in Sect. 2.

Then, given a sequence $(v_n,q_n,\phi _n,\mathring{R}_n)$ of solutions to the random Euler-Reynolds system (1.6), we intend to exhibit a solution (v, q) of (1.5) showing the convergences, with respect to suitable topologies:

$$\begin{aligned} v_n \rightarrow v, \quad q_n \rightarrow q, \quad \phi _n \rightarrow \phi , \quad \mathring{R}_n \rightarrow 0, \quad \mathord {\textrm{div}}\,^{\phi _n} v_n \rightarrow 0. \end{aligned}$$

Let us move to the non-uniqueness issue. Theorem 1.3 provides the existence of a solution to (1.5) with a prescribed energy profile up to time ${\mathfrak {t}}$. We do not claim this solution to be unique among solutions with the same energy. However, we can carry on the convex integration scheme outlined above in such a way that solutions given by Theorem 1.3 satisfy the following property.

Theorem 1.4

Let $e_1$, $e_2$ be energy profiles satisfying the hypotheses of Theorem 1.3 with the same ${\underline{e}}$, ${\overline{e}}$, and such that $e_1(t)=e_2(t)$ for every $t\in [0,T/2]$. Then the two global probabilistically strong solutions $(v_1,q_1)$ and $(v_2,q_2)$ given by Theorem 1.3 are such that $v_1(x,t)=v_2(x,t)$ for every $x \in \mathbb {T}^3$ and $t \in [0,T/2]$.

As a consequence, we disprove uniqueness-in-law for solutions (v, q) (with the aforementioned regularity) of the Cauchy problem associated with (1.5). Indeed, we can apply the previous theorem with $e_1, e_2$ such that $e_1(t)>e_2(t)$ for every $t \in (T/2,T]$, yielding $\int _{\mathbb {T}^3} |v_1(x,t)|^2 dx>\int _{\mathbb {T}^3} |v_2(x,t)|^2 dx$ for every $t \in (T/2,T]$ with probability greater than $\varkappa >0$. In particular, for every initial kinetic energy $e(0)>0$ there exists at least an initial datum $v(\cdot ,0):\Omega \rightarrow C^\vartheta (\mathbb {T}^3,\mathbb {R}^3)$ with $\int _{\mathbb {T}^3} |v(x,0)|^2 dx = e(0)$ almost surely such that the Cauchy problem associated with (1.5) is ill-posed.

Let us explain why Theorem 1.4 holds. We shall build iteratively (Proposition 2.4) a sequence of approximating solutions $(v_n,q_n,\phi _n,\mathring{R}_n)$, $n \in \mathbb {N}$ in such a way that the approximating solution at level $n+1$, evaluated at time $t \in [0,\infty )$, only depends on the restriction to times less or equal than t of the energy profile, the flow (1.2), and the approximating solutions at levels $k \le n$. In formulae, for fixed $x \in \mathbb {T}^3$ and $t \in [0,\infty )$, the quantities $v_{n+1}(x,t)$, $q_{n+1}(x,t)$, $\phi _{n+1}(x,t)$, and $\mathring{R}_{n+1}(x,t)$ only depend upon e(s), $\phi (y,s)$, $v_k(y,s)$, $q_k(y,s)$, $\phi _k(y,s)$, $\mathring{R}_k(y,s)$, for arbitrary $s \le t$, $k \le n$, and $y \in \mathbb {T}^3$. We point out that doing so is almost forced in the stochastic setting, in order to preserve progressive measurability of solutions constructed via the convex integration scheme. Moreover, we always start the iteration with the same quadruple $(v_0,q_0,\phi _0,\mathring{R}_0)=(0,0,\phi _0,0)$, where $\phi _0$ depends only on $\phi$, ${\underline{e}}$, ${\overline{e}}$.

This means, in the setting of Theorem 1.4 above, that the two solutions $v_1$ and $v_2$, evaluated at any time $t \in [0,T/2]$, are obtained as the limit of the same sequence (since $e_1(t)=e_2(t)$ for every $t \in [0,T/2]$); therefore, the two solutions coincide. Taking $t=0$, we deduce in addition that the initial conditions $v_1(\cdot ,0)$ and $v_2(\cdot ,0)$ coincide as well. Since the two solutions can not have the same law for times $t \in (T/2,T]$ (they have different energy with positive probability), this gives non-uniqueness in law for the Cauchy problem.

The property above can be checked looking at the construction outlined in Sect. 3.

Finally, notice that the convenient $C^\infty$-regularity assumption on the coefficients $\{\sigma _k\}_{k \in I}$ can be relaxed to $\sigma _k$ of class $C^\kappa$ for some $\kappa$ sufficiently large, and also we can replace the Brownian motions $\{B^k\}_{k \in I}$ with more general paths, for instance fractional Brownian motions with Hurst parameter $H>1/4$, reinterpreting the equation and the stochastic integral therein in the proper way (cf. Remark 2.3 below). We omit details for the sake of simplicity.

1.3 Bibliographic discussion

Literature on Euler equations is extremely vast and impossible to sum up exhaustively here. In two spatial dimensions, well-posedness of (deterministic) Euler equations with initial vorticity (i.e. the curl of the velocity field) in $L^\infty (\mathbb {T}^2)$ is known since the work of Yudovich [52]. Uniqueness of solutions may fail in non-regular settings, as proved for instance in [14, 47, 48]; actually, this phenomenon is generic, in the sense that non-uniqueness holds for every initial datum within the class of square-integrable solutions [49] or in the case of vorticities given by the sum of delta Dirac masses [31]. As for the stochastic case, namely with noise of transport type put into the dynamics, well-posedness was proved in [3] for initial vorticity in $L^\infty (\mathbb {T}^2)$, and in [4] existence of solutions with vorticity in $H^{-1}(\mathbb {T}^2)$ with definite sign was proved.

In dimension three, one has only local-in-time well-posedness of regular solutions, both in the deterministic case (see [44] and reference therein) and with transport noise [13]. Other stochastic perturbations of 3D Euler equations have been considered in [30, 39]. Less regular deterministic solutions are not unique by the aforementioned results in two dimensions. Moreover, non-uniqueness holds true in a class of relatively regular solutions (almost 1/3-Hölder continuous) as a consequence of the series of papers [7, 14,15,16, 38] finally solving the Onsager’s conjecture. More recently, in [36] the authors proved ill-posedness (i.e. global existence and non-uniqueness of generalized solutions) even when additive noise is put into the equations. Their result has been successively refined in [35] to analytically weak solutions.

1.4 Main novelties

It is worth comparing the present paper with previous works using convex integration techniques, both deterministic and stochastic. The strategy of the proof of Proposition 2.4 is similar to that in [16], which is the first result using convex integration techniques to produce Hölder solutions to (deterministic) Euler equations. Of course, the construction presented here needs several non-trivial adjustments with respect to that in [16], due to the operators $\mathord {\textrm{div}}^\phi$, $\nabla ^\phi$ present in (1.5):

• (1) They depend also on time, thus compromising some geometric properties of the Euler equations; in addition, the operators $\mathord {\textrm{div}}^\phi$, $\nabla ^\phi$ do not commute with space-time mollifications, and this produces an additional mollification error in the Reynold stress along the iteration;

• (2) The flow $\phi$ is non-smooth in time, thus requiring mollification of the flow and ad hoc control over the error made in doing so;

• (3) Finally, we cannot impose the divergence-free condition on the approximation $v_n$, since we would lose control over the time derivative of the compressibility corrector and the associated estimates on the compressibility error in the Reynold stress. Thus, we just aim at reducing iteratively the size of $\mathord {\textrm{div}}\,^{\phi _n} v_n$, without requiring it to be zero at every step of the iteration.

In order to deal with the flow $\phi$, and in particular with its growth for large times, in the present paper we introduce a sequence of stopping times ${\mathfrak {t}}_L \rightarrow \infty$ as $L \rightarrow \infty$. However, we point out that contrary to other works introducing stopping time in the convex integration scheme [33, 34, 36, 41, 46, 50, 51], here we do not produce local-in-time solutions, successively extended to global solutions à la Leray (which is practically impossible due to lack of compactness) or gluing together another convex integration solution (which we cannot do since we are not able to solve the Cauchy problem associated to Euler equations for every initial datum $v_0$). Rather, we develop a genuinely stochastic convex integration procedure and construct directly global solutions; we use the stopping times only to control the growth of solutions in suitable Hölder spaces. This is more similar in the spirit to the recent works [9, 35], where global solutions are produced by controlling the growth in expectation of the norms of the solution.

1.5 Frequently used notation

We adopt the following notation throughout the paper. Hölder spaces of functions of time $t \in \mathbb {R}$ or space $x \in \mathbb {T}^3$ with values in some Banach space E are denoted respectively $C^\vartheta _t E$ and $C^\vartheta _x E$, $\vartheta >0$. When confusion may not arise we simply denote $C^\vartheta _t=C^\vartheta _t E$ and $C^\vartheta _x=C^\vartheta _x E$. By convention $C^1_t$ and $C^1_x$ denote the spaces of Lipschitz functions (and not the space of continuously differentiable ones). A similar convention holds for $C^n_t$ and $C^n_x$, $n \in \mathbb {N}$, $n > 1$. Hölder seminorms are denoted $[\,\,\cdot \,\,]_{C^\vartheta _t}$ and $[\,\,\cdot \,\,]_{C^\vartheta _x}$, and Hölder norms $\Vert \cdot \Vert _{C^\vartheta _t}$ and $\Vert \cdot \Vert _{C^\vartheta _x}$, respectively.

For a stopping time ${\mathfrak {t}}$, we denote $C^\vartheta _{\le {\mathfrak {t}}}$ the space of Hölder functions $f\,{:}\,(-\infty ,{\mathfrak {t}}] \rightarrow E$, endowed with associated seminorm $[\,\,\cdot \,\,]_{C^\vartheta _{\le {\mathfrak {t}}}}$ and norm $\Vert \cdot \Vert _{C^\vartheta _{\le {\mathfrak {t}}}}$. Hölder (semi)norms in $C^\vartheta _{\le {\mathfrak {t}}}$ are defined for a generic $f:\mathbb {R}\rightarrow E$ upon restriction to $(-\infty ,{\mathfrak {t}}]$.

For functions of space and time we denote $\Vert \cdot \Vert _{C^\vartheta _{t,x}} :=\Vert \cdot \Vert _{C^{\vartheta }_t C_x} + \Vert \cdot \Vert _{C_t C^{\vartheta }_x}$ and $\Vert \cdot \Vert _{C^\vartheta _{\le {\mathfrak {t}},x}} :=\Vert \cdot \Vert _{C^{\vartheta }_{\le {\mathfrak {t}}} C_x} + \Vert \cdot \Vert _{C_{\le {\mathfrak {t}}}C^{\vartheta }_x}$ the space-time $\vartheta$-Hölder norms, $\vartheta \in (0,1]$.

We shall sometimes need to work with Besov spaces $B^{\alpha }_{p,q} :=B^{\alpha }_{p,q}(\mathbb {T}^3)$, $\alpha \in \mathbb {R}$, $p,q \in [1,\infty ]$ defined as the subset of distributions $u \in {\mathcal {S}}'(\mathbb {T}^3)$ such that

$$\begin{aligned} \Vert u \Vert _{B^\alpha _{p,q}} := \left\| \left( 2^{j \alpha } \Vert \Delta _j u \Vert _{L^p(\mathbb {T}^3)}\right) _{j \ge -1} \right\| _{\ell ^q} < \infty . \end{aligned}$$

In the previous line, $\{\Delta _j\}_{j \ge -1}$ denotes the Littlewood–Paley blocks corresponding to a dyadic partition of unity, as used for instance in [32] or [45]. For $p,q<\infty$ the Besov space $B^\alpha _{p,q}$ is separable and coincides with the closure of $C^\infty (\mathbb {T}^3)$ with respect to $\Vert \cdot \Vert _{B^\alpha _{p,q}}$. Notably $B^{\alpha }_{p,q} = (B^{-\alpha }_{p',q'})^*$, with equivalence of norms, when $1/p+1/p'=1/q+1/q'=1$, and $B^\alpha _{\infty ,\infty }=C^\alpha _x$ for every non-integer $\alpha >0$, again with equivalence of norms. Some useful lemmas on Hölder and Besov spaces are collected in Appendix 1.

2 Preliminaries and main iterative proposition

In this section we give more details about the strategy of the proof of Theorem 1.3. We have been strongly inspired by the construction of [16] on deterministic Euler equations.

As already mentioned in the previous section, we need to introduce a sequence $\phi _n$ of smooth approximations of the flow $\phi$, successively extended to negative times via the formula $\phi _n(t)=\phi _n(0)=Id_{\,\mathbb {T}^3}$ for every $t<0$. This extension is only for technical reasons, and will be only needed in next Sect. 3. The smooth approximations are obtained via mollification of the noise, and the procedure is described in details in next Sect. 2.1. The idea is simply to mollify the driving Brownian motion B with a mollification parameter $\varsigma _n$, and then define the approximating flow $\phi _n$ pathwise as a Riemann–Stieltjes integral with respect to the mollified B. In Sects. 2.2 and 2.3, making use of some result from Rough Paths theory, we provide estimates on the distance (in suitable spaces) between $\phi _n$ and $\phi$, in terms of the mollification parameter $\varsigma _n$ (here $\phi$ is extended as well to negative times as the identity on the torus). With these estimates in hand, we can introduce a sequence of stopping times ${\mathfrak {t}}_L$, $L \in \mathbb {N}$ to control the approximating flow uniformly in $\omega \in \Omega$ and localize the problem, see Sect. 2.4.

After this preliminary preparation, we state our main Proposition 2.4 in Sect. 2.5. The idea of the proposition is the same as in [16], and it consists in collecting a series of iterative estimates that, if verified, allow to show convergence of the sequence $(v_n,q_n,\phi _n,\mathring{R}_n)$ as $n \rightarrow \infty$ towards a limit $(v,q,\phi ,0)$ solution of (1.5) with desired Hölder regularity. This is done by controlling, iteratively, the norms in $C_{\le {\mathfrak {t}}_L} C_x$, $C_{\le {\mathfrak {t}}_L} C^1_x$ and $C^1_{\le {\mathfrak {t}}_L,x}$, and using interpolation.

2.1 Mollification of the noise

We intend to work pathwise, and thus we assume for simplicity that every realisation of the $\mathbb {R}^{|I|}$-valued driving noise $B=(B^k)_{k \in I}$ has $C^\alpha _{loc}$ time regularity, $\alpha \in (1/3,1/2)$. Since B is of class $C^\alpha _{loc}$ in time, the associated flow $\phi$ is expected to have the same time regularity. Unfortunately, for our purposes this is not sufficient, as we shall need to take time derivatives of the flow. Therefore, we introduce a sequence of smooth approximations $\{\phi _n\}_{n \in \mathbb {N}}$ as follows.

Fix $n \in \mathbb {N}$ and let $\varsigma _n>0$ be a constant to be properly chosen below, in accordance with other parameters of the convex integration procedure. For the time being, we shall assume $\varsigma _n$ monotonically decreasing in n and $(n+1)\varsigma _n^{\alpha -\beta } \rightarrow 0$ for every $\beta \in (0,\alpha )$.

Next, let $\theta :\mathbb {R}\rightarrow \mathbb {R}$ be a smooth mollifier with support contained in (0, 1), and define for $t \in \mathbb {R}$:

$$\begin{aligned} \theta _n(t) :=\varsigma _n^{-1} \theta (t\varsigma _n^{-1}), \quad B_n(t) :=(B *\theta _n) (t) = \int _\mathbb {R}B(t-s) \theta _n(s) ds, \end{aligned}$$

with the convention that $B(t-s)=B(0)=0$ whenever $t-s$ is negative, and the second equation is intended to hold component-wise, in particular $(B_n)^k = B^k_n = (B^k *\theta _n)$ for every $k \in I$. Notice that $B_n$ is smooth at every time $t \in \mathbb {R}$ and it is identically zero for negatives times, being the mollification one-sided. Finally, define $\phi _n$ as the unique solution of the integral equation

$$\begin{aligned} \phi _n(x,t) :=x + \sum _{k \in I}\int _0^t \sigma _k(\phi _n(x,s)) \,dB^k_n(s), \quad x \in \mathbb {T}^3, \,t \in \mathbb {R}, \end{aligned}$$

(2.1)

where the integral is understood pathwise as a Riemann-Stieltjes integral, and notice that $\phi _n(x,t)= \phi _n(x,0) = x$ for $t<0$. We extend the flow $\phi$ defined by (1.2) to $t<0$ similarly, imposing $\phi (x,t)= \phi (x,0) = x$ (the resulting flow is locally $C^\alpha$ in time). With probability one, for every fixed $t \in \mathbb {R}$ the map $\phi _n(\cdot ,t)$ is measure preserving since $\sigma _k$ is divergence-free and $dB^k_n = {\dot{B}}^k_n ds$ is space-independent, as a consequence of Liouville Theorem.

Let us also mention that $\phi _n$ is indeed an approximation of $\phi$ by Wong–Zakai Theorem, and the rate of convergence is made explicit in the next Lemma 2.2.

2.2 Rough paths preliminaries

We need to introduce some auxiliary concepts from the theory of Rough Paths. We keep the background at minimum, referring to the books [28] and [29] when needed.

Let $\alpha \in (1/3,1/2)$, and for any given stopping time ${\mathfrak {t}}$ let ${\mathscr {C}}^\alpha _{g,{\mathfrak {t}}} :={\mathscr {C}}^\alpha _g([0,{\mathfrak {t}}],\mathbb {R}^{|I|})$ be the space of $\alpha$-Hölder geometric rough paths ${\textbf{X}}=(X,\mathbb {X})$ endowed with the metric

$$\begin{aligned} \varrho _{\alpha ,{\mathfrak {t}}}({\textbf{X}}, {\textbf{Y}}) :=\sup _{0\le s<t \le {\mathfrak {t}}} \frac{\Vert X_{s,t}-Y_{s,t}\Vert }{|t-s|^\alpha } + \sup _{0\le s<t \le {\mathfrak {t}}} \frac{\Vert \mathbb {X}_{s,t}-\mathbb {Y}_{s,t}\Vert }{|t-s|^{2\alpha }}. \end{aligned}$$

For any smooth path $X:\mathbb {R}\rightarrow \mathbb {R}^{|I|}$, let us define the step-2 Lyons lift of (the restriction on $[0,{\mathfrak {t}}]$ of) X as the rough path ${\textbf{X}}=(X,\mathbb {X}) \in {\mathscr {C}}^\alpha _{g,{\mathfrak {t}}}$ given by the Riemann–Stieltjes integral

$$\begin{aligned} \mathbb {X}_{s,t} :=\int _s^t X_{s,r} \otimes dX_r, \quad s,t \in [0,{\mathfrak {t}}]. \end{aligned}$$

It is well known that the Stratonovich lift of B is almost surely an $\alpha$-Hölder geometric rough path, which we denote with the symbol ${\textbf{B}}=(B,\mathbb {B})$. For any $n \in \mathbb {N}$, let us denote ${\textbf{B}}_n=(B_n,\mathbb {B}_n) \in {\mathscr {C}}^\alpha _{g,{\mathfrak {t}}}$ the step-2 Lyons lift of $B_n$.

Lemma 1.5

For every $K > 0$ and $\beta \in (0,\alpha )$, $\alpha \in (1/3,1/2)$ there exists a stopping time ${\mathfrak {s}}$ with ${\mathfrak {s}} \le K$ almost surely and for every $n,m \in \mathbb {N}$, $n \le m$:

$$\begin{aligned} \varrho _{\beta ,{\mathfrak {s}}}({\textbf{B}}_m,{\textbf{B}}_n) \le K (n+1) \varsigma _n^{\alpha -\beta },\\ \varrho _{\beta ,{\mathfrak {s}}}({\textbf{B}},{\textbf{B}}_n) \le K (n+1) \varsigma _n^{\alpha -\beta }. \end{aligned}$$

Moreover, ${\mathfrak {s}}$ can be chosen so that ${\mathfrak {s}} \rightarrow \infty$ almost surely as $K \rightarrow \infty$.

Proof

First of all, notice that up to replacing K with K/2, the first inequality descends from the second one and triangle inequality, thus we only focus on the latter.

We apply [28, Corollary 15.32] on the time interval [0, T], $T<\infty$ arbitrary, with ${\textbf{X}}={\textbf{B}}_m$, ${\textbf{Y}}={\textbf{B}}_n$ and $\varepsilon$ given by (cf. Remark 15.33 therein)

$$\begin{aligned} \varepsilon ^\frac{2\alpha }{\alpha -\beta }&= \max _{i,j \in I}\sup _{s,t \in [0,T]} \mathbb {E}\left[ (B_m^i(t)-B_n^i(t))(B_m^j(s)-B_n^j(s))\right] \\&\le C \varsigma _n^{2\alpha } \mathbb {E}\left[ \Vert B\Vert ^2_{C^\alpha _{\le T}}\right] \le C \varsigma _n^{2\alpha }. \end{aligned}$$

We point out that the expectation in the line above is finite and bounded by some constant C depending only on T and $\alpha$, see [29, Proposition 3.5]. Thus, by the aforementioned [28, Corollary 15.32] and this choice of $\varepsilon$ there exists $C=C(T,\beta ,\alpha )$ such that for every finite $q \ge 1$ it holds²

$$\begin{aligned} \left\| \varrho _{\beta ,T}({\textbf{B}}_m,{\textbf{B}}_n) \right\| _{L^q(\Omega )} \le C q^{1/2} \varsigma _n^{\alpha -\beta }. \end{aligned}$$

Therefore the sequence $\{{\textbf{B}}_n\}_{n\in \mathbb {N}}$ is Cauchy in $L^q(\Omega ,{\mathscr {C}}^\beta _{g,T})$, and since the space ${\mathscr {C}}^\beta _{g,T}$ is complete with respect to the distance $\varrho _{\beta ,T}$ by [28, Theorem 8.13], there exists a limit ${\textbf{B}}_\infty \in L^q(\Omega ,{\mathscr {C}}^\beta _{g,T})$ such that the previous inequality holds when ${\textbf{B}}_m$ is replaced by ${\textbf{B}}_\infty$. In addition, observe that ${\textbf{B}}_\infty = {\textbf{B}}$ by well-known Wong Zakai results, see for instance [27]. Hence we have proved

$$\begin{aligned} \left\| \varrho _{\beta ,T}({\textbf{B}},{\textbf{B}}_n) \right\| _{L^q(\Omega )} \le C q^{1/2} \varsigma _n^{\alpha -\beta } \end{aligned}$$

for every finite $q \ge 1$.

Standard estimates now imply that the random variable $c(T,n,\omega ) :=\varrho _{\beta ,T}({\textbf{B}},{\textbf{B}}_n)\, \varsigma _n^{\beta -\alpha }$ has Gaussian tails for every $n \in \mathbb {N}$ and thus, by Borel-Cantelli theorem, for every $T \in [0,\infty )$ and $\omega \in \Omega$ there exists $N=N(T,\omega )$ such that $c(T,n,\omega )\le n+1$ for every $n \ge N$. In particular, for every T the random variable

$$\begin{aligned} {\tilde{c}}(T,\omega ) :=\sup _{n \in \mathbb {N}} \frac{c(T,n,\omega )}{n+1} < \infty \quad \text{ almost } \text{ surely }, \end{aligned}$$

and it is almost surely non-decreasing and lower-semicontinuous with respect to T as a supremum of non-decreasing and lower-semicontinuous functions.

Therefore, recalling that the filtration $\{{\mathcal {F}}_t\}_{t \ge 0}$ is complete and right-continuous, we can define the stopping time³${\mathfrak {s}}$ as

$$\begin{aligned} {\mathfrak {s}}&:=\inf \left\{ s \ge 0 : {\tilde{c}}(s,\cdot ) > K \right\} \wedge K, \end{aligned}$$

and then for every n we have almost surely:

$$\begin{aligned} \varrho _{\beta ,{\mathfrak {s}}}({\textbf{B}},{\textbf{B}}_n) \le {\tilde{c}}({\mathfrak {s}}) (n+1) \varsigma _n^{\alpha -\beta } \le K (n+1) \varsigma _n^{\alpha -\beta }. \end{aligned}$$

Finally, ${\mathfrak {s}} \rightarrow \infty$ almost surely as $K \rightarrow \infty$ because ${\tilde{c}}$ is almost surely finite and non-decreasing. $\square$

2.3 Choice of the stopping time

For an arbitrary constant $K>0$ to be chosen later, define the stopping time

$$\begin{aligned} {\mathfrak {t}} :={\mathfrak {s}} \wedge \inf \left\{ s \ge 0 : \Vert B \Vert _{C^\alpha _{\le s}}> K \right\} \wedge \inf \left\{ s \ge 0 : \Vert \mathbb {B}\Vert _{C^{2\alpha }_{\le s}} > K \right\} , \end{aligned}$$

(2.2)

where ${\mathfrak {s}}$ is given by Lemma 2.1 (and depends on K), $\Vert B \Vert _{C^\alpha _{\le s}}$ denotes the $\alpha$-Hölder norm of B on the time interval $(-\infty ,s]$ (or equivalently on [0, s] since $B(t)=0$ for $t<0$) and $\Vert \mathbb {B}\Vert _{C^{2\alpha }_{\le s}}$ denotes the $2\alpha$-Hölder norm of the two-indices process $\mathbb {B}$, restricted on the time interval $[0,s]\times [0,s]$.

Observe that ${\mathfrak {t}} \le {\mathfrak {s}} \le K$ almost surely. The constant K can be taken large enough so that ${\mathfrak {t}}$ is large with high probability: namely, given parameters $\varkappa ,T$ as in the statement of Theorem 1.3, there exists $K_0=K_0(\varkappa ,T)$ sufficiently large such that $\mathbb {P}\{ {\mathfrak {t}} \ge T\} \ge \varkappa$. This is because both $\Vert B \Vert _{C^\alpha _{\le s}}$ and $\Vert \mathbb {B}\Vert _{C^{2\alpha }_{\le s}}$ are almost surely finite for every $s<\infty$, and ${\mathfrak {s}} \rightarrow \infty$ almost surely as $K \rightarrow \infty$ by construction.

Now we state a crucial lemma describing rates of convergence of the approximating flow $\phi _n \rightarrow \phi$, as well as its inverse $\phi _n^{-1} \rightarrow \phi ^{-1}$. Since $\phi _n$, $\phi$ equal the identity map on the torus for negative times, the convergence is only interesting for positive times. For technical reasons we must restrict ourselves to time intervals of the form $[0,{\mathfrak {t}}]$, where ${\mathfrak {t}}$ is given by (2.2) and depends on $K>0$. In particular, we shall see that the approximation becomes worse and worse as $K \rightarrow \infty$.

Lemma 1.6

For every $n \in \mathbb {N}$ the map $\phi _n(\cdot ,t)$ defined by (2.1) is almost surely a $C^\infty$-diffeomorphism of the torus for every $t \in \mathbb {R}$, and the following hold true.

• i) For every $K > 0$, $\kappa \in \mathbb {N}$ and $\beta \in (0,\alpha )$ there exist constants $C_1,C_2$ such that for every $n \in \mathbb {N}$ it holds

  $$\begin{aligned} \Vert \phi _{n+1} - \phi _n \Vert _{C^\beta _{\le {\mathfrak {t}}} C^\kappa _x} \le C_1 (n+1) \varsigma _n^{\alpha -\beta }, \quad \Vert \phi _n \Vert _{C^\alpha _{\le {\mathfrak {t}}} C^\kappa _x} \le C_2,\\ \Vert \phi _{n+1}^{-1} - \phi _n^{-1} \Vert _{C^\beta _{\le {\mathfrak {t}}} C^\kappa _x} \le C_1 (n+1) \varsigma _n^{\alpha -\beta }, \quad \Vert \phi _n^{-1} \Vert _{C^\alpha _{\le {\mathfrak {t}}} C^\kappa _x} \le C_2; \end{aligned}$$

  Moreover, the same inequalities hold with $\phi _{n+1}$ replaced by $\phi$.

• ii) For every $K > 0$ and $\kappa ,r \in \mathbb {N}$ there exists a constant $C_3$ such that for every $n \in \mathbb {N}$ it holds

  $$\begin{aligned} \Vert \phi _n \Vert _{C^r_{\le {\mathfrak {t}}} C^\kappa _x} \le C_3 \varsigma _n^{\alpha -r}, \quad \Vert \phi _n^{-1} \Vert _{C^r_{\le {\mathfrak {t}}} C^\kappa _x} \le C_3 \varsigma _n^{\alpha -r}. \end{aligned}$$

Proof of Lemma 2.2

First, notice that we can replace the time interval $(-\infty ,{\mathfrak {t}}]$ with $[0,{\mathfrak {t}}]$ in all the norms on the left-hand-sides. By [29, Theorem 8.5] (cf. also [12, Lemma 13]), for every $K>0$, $\kappa \in \mathbb {N}$ and $\beta \in (0,\alpha )$ there exist constant $C_1,C_2$ such that for every n

$$\begin{aligned} \Vert \phi _{n+1} - \phi _n \Vert _{C^\beta _{\le {\mathfrak {t}}} C^\kappa _x} \le C_1 \varrho _{\beta ,{\mathfrak {t}}}({\textbf{B}}_{n+1},{\textbf{B}}_n), \quad \Vert \phi _n \Vert _{C^\alpha _{\le {\mathfrak {t}}} C^\kappa _x} \le C_2,\\ \Vert \phi _{n+1}^{-1} - \phi _n^{-1} \Vert _{C^\beta _{\le {\mathfrak {t}}} C^\kappa _x} \le C_1 \varrho _{\beta ,{\mathfrak {t}}}({\textbf{B}}_{n+1},{\textbf{B}}_n), \quad \Vert \phi _n^{-1} \Vert _{C^\alpha _{\le {\mathfrak {t}}} C^\kappa _x} \le C_2, \end{aligned}$$

and the same estimates hold true when ${\textbf{B}}_{n+1}$ is replaced by ${\textbf{B}}$. Notice that the mentioned result is stated on finite time intervals [0, K]; the case $[0,{\mathfrak {t}}] \subset [0,K]$ is handled by considering the stopped processes ${\textbf{B}}_n(\cdot \wedge {\mathfrak {t}})$. Also, it is worth mentioning that we use the bound on $\Vert \mathbb {B} \Vert _{C^{2\alpha }_{\le {\mathfrak {t}}}}$ coming from the definition of ${\mathfrak {t}}$ to verify the assumptions of the aforementioned theorem: indeed, the constants in the thesis may depend also on $\Vert \mathbb {B} \Vert _{C^{2\alpha }_{\le {\mathfrak {t}}}}$ and $\Vert \mathbb {B}_n \Vert _{C^{2\alpha }_{\le {\mathfrak {t}}}}$⁴ (as well as on $\Vert B \Vert _{C^{\alpha }_{\le {\mathfrak {t}}}}$ and $\Vert B_n \Vert _{C^{\alpha }_{\le {\mathfrak {t}}}}$).

Part i) of the thesis then follows by Lemma 2.1 and ${\mathfrak {t}} \le {\mathfrak {s}}$.

For part ii) we proceed iteratively. The equations satisfied by the time derivatives of $\phi _n$ for positive times $t>0$ are

$$\begin{aligned} \phi _n(x,t)&= x + \sum _{k \in I} \int _0^t \sigma _k(\phi _n(x,s)) {\dot{B}}^k_n(s) ds,\\ {\dot{\phi }}_n(x,t)&= \sum _{k \in I} \sigma _k(\phi _n(x,t)) {\dot{B}}^k_n(t),\\ \ddot{\phi }_n(x,t)&= \sum _{k \in I} {\dot{\phi }}_n(x,t) \cdot \nabla \sigma _k(\phi _n(x,t)) {\dot{B}}^k_n(t) + \sigma _k(\phi _n(x,t)) {\ddot{B}}^k_n(t),\\&\cdots \\ \partial ^r_t \phi _n(x,t)&= \sum _{k \in I} \sum _{m=0}^{r-1} C(r,m) \partial ^m_t \sigma _k(\phi _n(x,t)) \partial ^{r-1-m}_t {\dot{B}}^k_n(t), \end{aligned}$$

giving estimates (use [16, Proposition 4.1] to bound derivatives of $\sigma _k \circ \phi _n$)

$$\begin{aligned}{}[ \phi _n ]_{C^1_{\le {\mathfrak {t}}}C^\kappa _x}&\le \sum _{k \in I} \Vert \sigma _k \circ \phi _n \Vert _{C_{\le {\mathfrak {t}}}C^\kappa _x} \Vert {\dot{B}}^k_n\Vert _{C_{\le {\mathfrak {t}}}} \le CK \varsigma _n^{\alpha -1},\\ [ \phi _n ]_{C^2_{\le {\mathfrak {t}}}C^\kappa _x}&\le \sum _{k \in I} \Vert \nabla \sigma _k \circ \phi _n \Vert _{C_{\le {\mathfrak {t}}}C^\kappa _x} [ \phi _n ]_{C^1_{\le {\mathfrak {t}}}C^\kappa _x} \Vert {\dot{B}}^k_n\Vert _{C_{\le {\mathfrak {t}}}} + \Vert \sigma _k \circ \phi _n \Vert _{C_{\le {\mathfrak {t}}}C^\kappa _x} \Vert {\ddot{B}}^k_n\Vert _{C_{\le {\mathfrak {t}}}}\\&\le C K^2 \varsigma _n^{2\alpha -2} + C K \varsigma _n^{\alpha -2} \le C K^2 \varsigma _n^{\alpha -2},\\&\cdots \\ [ \phi _n ]_{C^r_{\le {\mathfrak {t}}}C^\kappa _x}&\le C K^r \varsigma _n^{\alpha -r}. \end{aligned}$$

Finally, estimates for the inverse flow are obtained similarly, noticing that $\phi _n^{-1}$ is driven by $B_n(t-\cdot )$, cf. also the proof of [28, Proposition 11.13], namely

$$\begin{aligned} \phi _n^{-1}(x,t) = x - \sum _{k \in I} \int _0^t \sigma _k(\phi _n^{-1}(x,s)) {\dot{B}}^k_n(t-s) ds. \end{aligned}$$

$\square$

2.4 Localization

Let us fix $\beta \in (0,\alpha )$. A close inspection of the proof of Lemma 2.2 guarantees that the constants $C_1, C_2, C_3$ can be chosen to be increasing with respect to $K, \kappa , r$. Since throughout our construction we will need only a finite number of derivatives with respect to space and time, we can assume $\kappa , r$ bounded from above by a universal constant, so that we can suppose $C_1, C_2, C_3$ only depend on K (at fixed $\beta$, $\alpha$). In view of this, for every integer $L \ge 1$ we can find a parameter $K_L$ such that Lemma 2.2 holds with $C_1 = C_2 = C_3 = C L$, for some universal constant C. Moreover, without loss of generality we can assume the sequence $K_L$ to be increasing and diverging to $\infty$, and $K_0=K_1 \le K_L$ for every $L \ge 1$ (possibly taking a larger value of C if necessary; recall that $K_0$ was such that $\mathbb {P} \{ {\mathfrak {t}} \ge T \} \ge \varkappa$). Thus, given ${\mathfrak {s}}_L$ as in Lemma 2.1 with $K=K_L$ and defining

$$\begin{aligned} {\mathfrak {t}}_L :={\mathfrak {s}}_L \wedge \inf \left\{ s \ge 0 : \Vert B \Vert _{C^\alpha _{\le s}}> K_L \right\} \wedge \inf \left\{ s \ge 0 : \Vert \mathbb {B}\Vert _{C^{2\alpha }_{\le s}} > K_L \right\} , \end{aligned}$$

(2.3)

we obtain a sequence of non-decreasing stopping times ${\mathfrak {t}} = {\mathfrak {t}}_1 \le \dots \le {\mathfrak {t}}_L \le \dots$ such that ${\mathfrak {t}}_L \rightarrow \infty$ almost surely as $L \rightarrow \infty$ and

$$\begin{aligned} \Vert \phi _{n+1} - \phi _n \Vert _{C^\beta _{\le {\mathfrak {t}}_L} C^\kappa _x} \le CL (n+1) \varsigma _n^{\alpha -\beta }, \quad \Vert \phi _n \Vert _{C^\alpha _{\le {\mathfrak {t}}_L} C^\kappa _x} \le CL,\\ \Vert \phi _{n+1}^{-1} - \phi _n^{-1} \Vert _{C^\beta _{\le {\mathfrak {t}}_L} C^\kappa _x} \le CL (n+1) \varsigma _n^{\alpha -\beta }, \quad \Vert \phi _n^{-1} \Vert _{C^\alpha _{\le {\mathfrak {t}}_L} C^\kappa _x} \le CL, \end{aligned}$$

as well as

$$\begin{aligned} \Vert \phi _n \Vert _{C^r_{\le {\mathfrak {t}}_L} C^\kappa _x} \le CL \varsigma _n^{\alpha -r}, \quad \Vert \phi _n^{-1} \Vert _{C^r_{\le {\mathfrak {t}}_L} C^\kappa _x} \le CL \varsigma _n^{\alpha -r}, \end{aligned}$$

for some constant C depending only on $K_0$, $\alpha$ and $\beta$.

Remark 2.3

Estimates similar to those given above hold more generally when the collection of Brownian motions $\{B^k\}_{k \in I}$ is replaced by fractional Brownian motions $\{B^{H,k}\}_{k \in I}$ with Hurst parameter $H>1/4$. The lower threshold comes from the fact that we are using that iterated integrals of the mollified fractional Brownian motions satisfy a Wong-Zakai-type result (cf. [27]), and therefore one can use this limit as a “canonical" enhancement of $\{B^{H,k}\}_{k \in I}$ when interpreting the integral (1.2) defining the flow $\phi$. The space of geometric rough paths and the distance on it must be modified accordingly, including the step-3 iterated integral of the path, and the exponent $\alpha$ must be taken smaller than H. As a consequence, with suitable interpretation of the stochastic integral in (1.1) and the notion of solution to (1.1) and (1.5), we can prove the analogue of our main results also in the fractional case, with minor modifications in the construction and in the choice of parameters carried on in the remainder of the paper. We omit additional details.

2.5 Main proposition

Throughout the scheme we will make the following choice of parameters:

$$\begin{aligned} \delta _n :=a^{1-b^n}, \quad D_n :=a^{cb^n}, \quad L_n :=L^{m^{n+1}}, \end{aligned}$$

(2.4)

where

$$\begin{aligned} a \ge 2, \quad b=m+\varepsilon , \quad c=\frac{b^4(1+\varepsilon )-1/2}{b-1-\varepsilon }>0, \quad \varepsilon > 0, \end{aligned}$$

and $m\ge 4$ is given by Proposition 2.4 below. We point out that differently from [16] here $\varepsilon$ cannot be taken arbitrarily small, and it shall be fixed only at the end of Sect. 5. Notice that, at fixed $\varepsilon$ and for large values of m, the parameter c is approximately equal to $(1+\varepsilon )b^3$; in particular $D_n$ increases in n as a negative power of $\delta _{n+3}$ (with exponent depending on $\varepsilon$). The choice of the parameter $\varsigma _n$, upon which the definition of the mollified flow $\phi _n$ depends, will be given in (3.1).

Our goal is to prove the following:

Proposition 1.8

Let e be a smooth positive function on $\mathbb {R}$ satisfying ${\underline{e}} :=\inf _{t \in \mathbb {R}} e(t)>0$ and ${\overline{e}} :=\Vert e\Vert _{C^2_t}<\infty$. Then there exist constants $\varepsilon , m$ as above, $a \ge 2$ depending on ${\underline{e}},{\overline{e}},K_0$, a constant $\eta \in (0,1)$ depending on ${\underline{e}},{\overline{e}}$, constants $M_v,M_q$ depending only on ${\overline{e}}$ and a constant $A \in (0,\infty )$ depending on ${\underline{e}},{\overline{e}},K_0$ with the following property.

Fix $\phi _n$ as in (2.1) and let $(v_n,q_n,\phi _n,\mathring{R}_n)$, $n \in \mathbb {N}$, be a solution of (1.6) satisfying the inductive estimates⁵

$$\begin{aligned} \left| e(t) (1-\delta _n)- \int _{\mathbb {T}^3} |v_n(x,t)|^2 dx \right| \le \tfrac{1}{4} \delta _n e(t), \qquad \forall t \le {\mathfrak {t}}, \end{aligned}$$

(2.5)

and for every $L \in \mathbb {N}, L \ge 1$

$$\begin{aligned} \Vert \mathring{R}_n\Vert _{C_{\le {\mathfrak {t}}_L}C_x}\le & {} \eta L_n \delta _{n+1}, \end{aligned}$$

(2.6)

$$\begin{aligned} \Vert q_n\Vert _{C_{\le {\mathfrak {t}}_L}C_x}\le & {} M_q L_n \sum _{k=0}^{n-1} \delta _{k}, \end{aligned}$$

(2.7)

$$\begin{aligned} \Vert \mathord {\textrm{div}}\,^{\phi _n} v_n \Vert _{C_{\le {\mathfrak {t}}_L} B^{-1}_{\infty ,\infty }}\le & {} L_n \delta _{n+2}^{5/4}, \end{aligned}$$

(2.8)

$$\begin{aligned} \max \left\{ \Vert v_n\Vert _{C^1_{\le {\mathfrak {t}}_L,x}}, \Vert q_n\Vert _{C^1_{\le {\mathfrak {t}}_L,x}}, \Vert \mathring{R}_n\Vert _{C_{\le {\mathfrak {t}}_L}C^1_x} \right\}\le & {} L_n D_n. \end{aligned}$$

(2.9)

Then there exists a second quadruple $(v_{n+1},q_{n+1},\phi _{n+1},\mathring{R}_{n+1})$ solution of (1.6) satisfying (2.1) and the inductive estimates (2.5), (2.6), (2.7), (2.8) with n replaced by $n+1$, and for every $L \in \mathbb {N}, L \ge 1$

$$\begin{aligned} \Vert v_{n+1} - v_n \Vert _{C_{\le {\mathfrak {t}}_L}C_x}&\le M_v L_n^4 \delta _n^{1/2}, \\ \Vert q_{n+1}-q_n \Vert _{C_{\le {\mathfrak {t}}_L}C_x}&\le M_q L_n \delta _n, \\ \max \left\{ \Vert v_{n+1}\Vert _{C^1_{\le {\mathfrak {t}}_L,x}}, \Vert q_{n+1}\Vert _{C^1_{\le {\mathfrak {t}}_L,x}}, \Vert \mathring{R}_{n+1}\Vert _{C_{\le {\mathfrak {t}}_L}C^1_x} \right\}&\le A L_{n+1} \delta _n^{1/2} \left( \frac{D_n}{\delta _{n+4}} \right) ^{1+\varepsilon }. \end{aligned}$$

Moreover, the quadruple $(v_{n+1},q_{n+1},\phi _{n+1},\mathring{R}_{n+1})$ evaluated at time $t \in [0,\infty )$ only depends upon e(s), $\phi (y,s)$, $v_k(y,s)$, $q_k(y,s)$, $\phi _k(y,s)$, $\mathring{R}_k(y,s)$, for arbitrary $s \le t$, $k \le n$, and $y \in \mathbb {T}^3$.

We point out that, in the statement of the previous proposition, the parameters a and A are allowed to depend on ${\underline{e}},{\overline{e}},K_0$, whereas $\varepsilon$ and m are universal. We shall see that the Hölder exponent $\vartheta$ of the solutions we can construct will in fact depend only on $\varepsilon$ and m, and therefore is universal as well. However, the local-in-time $\vartheta$-Hölder norm of the solutions does depend on the parameter a (as well as $M_v$ and $M_q$); in particular it may vary as the energy profile e and the constant $K_0$ change.

Also, we recall that approximate solutions $(v_n,q_n,\phi _n,\mathring{R}_n)$ in the iteration are defined for all times $t \in \mathbb {R}$. For negative times, they correspond to deterministic approximate solutions of deterministic Euler equations and noise is only inserted for $t>0$ (we shall restrict them to positive times when proving Theorem 1.3).

Next we show that Proposition 2.4 above implies Theorem 1.3.

Proof of Theorem 1.3

The proof is mostly inspired by [16].

Step 1. We define the initial step of the iteration $(v_0,q_0,\mathring{R}_0)$ to be identically zero, which is admissible since (2.6), (2.7), (2.8) and (2.9) are trivially satisfied and (2.5) holds true because by definition $\delta _0=1$. Applying iteratively the proposition we get a sequence $\{v_n\}_{n \in \mathbb {N}}$ satisfying

$$\begin{aligned} \Vert v_n \Vert _{C_{\le {\mathfrak {t}}_L}C_x}&\le M_v \sum _{k \le n} L_k a^{\frac{1}{2}(1-b^k)} \le M_v L_n \sum _{n \in \mathbb {N}} 2^{\frac{1}{2}(1-4^n)} \le 2 M_v L_n, \\ \Vert v_n \Vert _{C^1_{\le {\mathfrak {t}}_L,x}}&\le L_n D_n, \end{aligned}$$

where in the second line we have used

$$\begin{aligned} A \delta _n^{1/2} \left( \frac{D_n}{\delta _{n+4}} \right) ^{1+\varepsilon }&= A a^{1/2-(1+\varepsilon )} a^{(-1/2+c(1+\varepsilon )+b^4(1+\varepsilon ))b^n} \\&= A a^{1/2-(1+\varepsilon )} a^{cb^{n+1}} \le D_{n+1}, \end{aligned}$$

since $(-1/2+c(1+\varepsilon )+b^4(1+\varepsilon ))=cb$ by the very definition of c and taking $a \ge A^{\frac{1}{(1+\varepsilon )-1/2}}$.

Step 2. By assumption, for every $L,n \in \mathbb {N}$, $L \ge 1$ we have

$$\begin{aligned} \Vert v_{n+1} - v_n \Vert _{C_{\le {\mathfrak {t}}_L}C_x}&\le M_v L_n^4 \delta _n^{1/2} = M_v L^{4 m^{n+1}} a^{\frac{1}{2} (1-b^n)}, \\ \Vert v_{n+1} - v_n \Vert _{C^1_{\le {\mathfrak {t}}_L,x}}&\le 2 L_{n+1} D_{n+1} = 2 L^{m^{n+2}} a^{cb^{n+1}}, \end{aligned}$$

and therefore by interpolation

$$\begin{aligned} \Vert v_{n+1} - v_n \Vert _{C^\vartheta _{\le {\mathfrak {t}}_L,x}}&\le 2^\vartheta M_v^{1-\vartheta } L^{(1-\vartheta )4m^{n+1}}L^{\vartheta m^{n+2}} a^{\frac{1-\vartheta }{2}} a^{(\vartheta cb - \frac{(1-\vartheta )}{2})b^n} \\&\le M_v L^{m^{n+2}} a^{\frac{1-\vartheta }{2}} a^{(\vartheta cb - \frac{(1-\vartheta )}{2})b^n}, \end{aligned}$$

where we have assumed without any loss of generality $M_v \ge 2$. Now choose

$$\begin{aligned} 0<\vartheta&< \frac{1}{2cb+1} = \frac{m-1}{2(1+\varepsilon )(m+\varepsilon )^5-1-\varepsilon }, \end{aligned}$$

(2.10)

so that $\vartheta cb-\frac{(1-\vartheta )}{2} =:-\gamma <0$, and define

$$\begin{aligned} n_0 = \max \left\{ n \in \mathbb {N}: L^{m^{n+2}} > a^{\gamma b^{n-1}} \right\} . \end{aligned}$$

The maximum above exists since $b>m$ and $\gamma >0$, and simple algebraic manipulations show that $n_0$ satisfies

$$\begin{aligned} n_0&< \frac{\log \log L + 2 \log m - \log \gamma + \log b - \log \log a}{\log b - \log m} \\&\le C (1+\log \log L) \end{aligned}$$

for some constant C not depending on L. With this choice of $n_0$ we estimate

$$\begin{aligned} \sum _{n \in \mathbb {N}} \Vert v_{n+1} - v_n \Vert _{C^\vartheta _{\le {\mathfrak {t}}_L,x}}&\le \sum _{n \le n_0} \Vert v_{n+1} - v_n \Vert _{C^\vartheta _{\le {\mathfrak {t}}_L,x}} + \sum _{n > n_0} \Vert v_{n+1} - v_n \Vert _{C^\vartheta _{\le {\mathfrak {t}}_L,x}} \\&\le C M_v a^{1/2} (L^{m^{n_0+3}}+1) \le C M_v a^{1/2} L^{m^{C(1+\log \log L)}}, \end{aligned}$$

where the constant C may vary from line to line. Thus $v_n$ converges in $C^{\vartheta }_{\le {\mathfrak {t}}_L} C_x \cap C_{\le {\mathfrak {t}}_L} C^\vartheta _x$ towards a limit v with uniform bound in $\omega \in \Omega$

$$\begin{aligned} \Vert v\Vert _{C^\vartheta _{\le {\mathfrak {t}}_L,x}} \le C M_v a^{1/2} L^{m^{C(1+\log \log L)}}. \end{aligned}$$

Moreover, it is easy to check that v restricted to $(-\infty ,{\mathfrak {t}}_{L-1}]$ is simply the limit of $v_n$ in $C^{\vartheta }_{\le {\mathfrak {t}}_{L-1}} C_x \cap C_{\le {\mathfrak {t}}_{L-1}} C^\vartheta _x$ for every $L>1$. This identifies uniquely a limit object in $C^{\vartheta }_{loc} C_x \cap C_{loc} C^\vartheta _x$, denoted again by v, by simply gluing together limits at different values of L.

Since $v_n(\cdot \wedge {\mathfrak {t}}_L)$ is progressively measurable for every $n \in \mathbb {N}$ and $L \ge 1$, the limit process $v(\cdot \wedge {\mathfrak {t}}_L)$ is also progressively measurable; so also v is. Moreover, because of (2.5) it satisfies

$$\begin{aligned} \int _{\mathbb {T}^3}|v(x,t)|^2 dx = e(t), \quad t \le {\mathfrak {t}}. \end{aligned}$$

(2.11)

Step 3. Convergence of the pressure $q_n$, as well as the regularity of the limit q, can be carried on exactly as in the previous step. The only difference comes down to the fact that $\Vert q_{n+1} - q_n \Vert _{C_{\le {\mathfrak {t}}_L}C_x} \le M_q L_n \delta _n = M_q L^{m^{n+1}} a^{1-b^n}$ and thus

$$\begin{aligned} \Vert q_{n+1} - q_n \Vert _{C^\vartheta _{\le {\mathfrak {t}}_L,x}} \le M_q L^{m^{n+2}} a^{1-\vartheta } a^{(\vartheta cb - (1-\vartheta ))b^n}. \end{aligned}$$

As a consequence, one can produce solutions q with Hölder regularity $\vartheta '$ such that

$$\begin{aligned} 0< \vartheta ' < \frac{1}{cb+1}, \end{aligned}$$

(2.12)

which is approximately twice the regularity of v for large values of b. Up to choosing a smaller $\vartheta$, we can actually suppose $\vartheta '=2\vartheta$ and thus the limit v satisfies the uniform bound (with respect to $\omega \in \Omega$)

$$\begin{aligned} \Vert q\Vert _{C^{2\vartheta }_{\le {\mathfrak {t}}_L,x}} \le C M_q a L^{m^{C(1+\log \log L)}}. \end{aligned}$$

Step 4. Finally, notice that for every L we have $\phi _n \rightarrow \phi$, $\phi _n^{-1} \rightarrow \phi ^{-1}$ almost surely in $C_{\le {\mathfrak {t}}_L} C_x^2$ (by Lemma 2.2), $\mathring{R}_n \rightarrow 0$ almost surely in $C_{\le {\mathfrak {t}}_L} C_x$ and $\mathord {\textrm{div}}\,^{\phi _n} v_n \rightarrow 0$ almost surely in $C_{\le {\mathfrak {t}}_L} B^{-1}_{\infty ,\infty }$ (by iterative assumptions (2.6) and (2.8), having chosen $b>m$), and therefore the couple (v, q) is a solution to (1.5) on the time interval $[0,{\mathfrak {t}}_L]$. Indeed, for every $H \in C^\infty (\mathbb {T}^3)$ and $h \in C_{loc}([0,\infty ),C^\infty (\mathbb {T}^3,\mathbb {R}^3))$ we have $\nabla ^{\phi _n} H \rightarrow \nabla ^\phi H$ and $(v_n \cdot \nabla ^{\phi _n}) h \rightarrow (v \cdot \nabla ^\phi )h$, $\mathord {\textrm{div}}\,^{\phi _n} h \rightarrow \mathord {\textrm{div}}^\phi h$ in $C_{[0,{\mathfrak {t}}_L]} C_x$ (cf. also the estimates in Sect. 4.4). Since L is arbitrary and ${\mathfrak {t}}_L \rightarrow \infty$ as $L \rightarrow \infty$, the proof is complete. $\square$

Remark 2.5

(On the Hölder exponent $\vartheta$) From (2.10) one obtains an upper bound for the Hölder regularity of the solution (v, q). In [16] $\vartheta$-Hölder solutions to the deterministic Euler equations are constructed for every $\vartheta <1/10$; in the series of papers [5, 7, 8], culminating with [6, 38], the space regularity threshold has been successively extended up to exponent 1/3, which is optimal since solutions to Euler equations more regular than this necessarily preserve their kinetic energy [10].

Here, due to the presence of the noise and because m will be taken large, we have to restrict ourselves to much smaller values of $\vartheta$. The choice of parameters done in Sect. 5 only allows values of $\vartheta$ smaller than approximately $2.76 \times 10^{-9}$, which is an extremely low threshold when compared with the deterministic literature on the Onsager’s conjecture. Although we have not optimized our choice of parameters—that is, our threshold is not sharp - we believe one cannot do much better than this following the construction of the present paper, not to mention reaching the same threshold as in [16] $\vartheta <1/10$. It is interesting to understand if technical refinements could improve regularity of solutions up to critical exponent 1/3 (at least when considering space regularity), as it was done in the deterministic case. However, the most stringent constraint here is the requirement $b>m$, that is a genuinely stochastic problem (the constant $L_n$ is the estimates comes from the growth of the flow $\phi$); thus, it seems not possible to take ideas from more refined deterministic convex integration schemes (as for instance that of [38]) in order to improve the resulting regularity.

Remark 2.6

(On the local-in-time Hölder norm of solutions) Also, we would like to mention the following fact. The estimates

$$\begin{aligned} \Vert v\Vert _{C^\vartheta _{\le {\mathfrak {t}}_L,x}}&\le C M_v a^{1/2} L^{m^{C(1+\log \log L)}}, \quad \Vert q\Vert _{C^{2\vartheta }_{\le {\mathfrak {t}}_L,x}} \le C M_q a L^{m^{C(1+\log \log L)}}, \end{aligned}$$

involve the constants $M_v$ and $M_q$, which in turn only depend on ${\overline{e}}$, and the parameter a, that however depends also on ${\underline{e}}$ and $K_0$. The latter dependence on a comes from the fact that we started the iteration from the ansatz $(v_0,q_0,\mathring{R}_0)$ identically zero, and therefore we needed $\delta _0=1$ in order to satisfy (2.5). If we were able to cook up an initial $(v_0,q_0,\mathring{R}_0)$ with “better" energy profile to start the iteration, we could redefine $\delta _n=a^{-b^n}$ and lose the dependence on a in the Hölder norms of the solution (v, q).

To conclude this section, let us mention that Theorem 1.4 follows readily from the previous construction and the arguments of Sect. 1.2, using the property that $(v_{n+1},q_{n+1},\phi _{n+1},\mathring{R}_{n+1})(x,t)$ depends on e(s), $\phi (y,s)$, $v_k(y,s)$, $q_k(y,s)$, $\phi _k(y,s)$, $\mathring{R}_k(y,s)$, for arbitrary $s \le t$, $k \le n$, and $y \in \mathbb {T}^3$. This last property will be implicitly checked in Sect. 3 below.

3 Convex integration scheme

Recall that our main aim is to construct the quadruple $(v_{n+1},q_{n+1},\phi _{n+1},\mathring{R}_{n+1})$ solution of the Euler–Reynolds system, given $(v_n,q_n,\phi _n,\mathring{R}_n)$ as in the statement of Proposition 2.4. Of course, the candidate approximating flow $\phi _n$ is already given by (2.1), so we shall focus on the definition of $v_{n+1}$, $q_{n+1}$ and $\mathring{R}_{n+1}$ only. Let us work at fixed $n \in \mathbb {N}$. The construction follows closely that of [16].

3.1 Choice of parameters

For future reference, here we collect all the parameters necessary to the convex integration scheme. We recall the quantities from Sect. 2.5

$$\begin{aligned} \delta _n :=a^{1-b^n}, \quad D_n :=a^{cb^n}, \quad L_n :=L^{m^{n+1}}, \end{aligned}$$

where

$$\begin{aligned} b=m+\varepsilon , \quad c=\frac{b^4(1+\varepsilon )-1/2}{b-1-\varepsilon }>0, \end{aligned}$$

and the coefficients $a \ge 2$, $m\ge 4$ and $\varepsilon >0$ are still to be determined and will be fixed only in Sect. 5. According to Proposition 2.4, the parameter $\delta _n$ dictates the decay, along the iteration, of the following quantities: the energy error, the Reynolds stress, the divergence of the velocity field, and the velocity and pressure increments. On the other hand, the parameter $D_n$ is used to keep track of the growth of derivatives of $v_n$, $q_n$, and $\mathring{R}_n$. $L_n$ serves to control how the iterative estimates deteriorate on larger and larger time intervals of the form $[0,{\mathfrak {t}}_L]$, $L\in \mathbb {N}$, $L \ge 1$.

Let us also discuss the choice of the parameter $\eta \in (0,1)$. There are two upper bounds that we need to be satisfied: $\eta \le \frac{{\underline{e}}}{8C({\overline{e}}^{1/2}+1)}$ from Lemma 3.1 (C is a constant defined therein), and $\eta \le \frac{r_0 {\underline{e}}}{40}$ (where $r_0$ is defined in Sect. 3.6). We shall fix $\eta \in (0,1)$ according to these constraints.

Next, we need to introduce two mollification parameters for the convex integration scheme. Fix $\alpha \in (1/3,1/2)$ close to 1/2. Let us define the parameters $\ell$ and $\varsigma _n$ by the relations

$$\begin{aligned} \ell ^\alpha :=\frac{c_{n,\ell }}{C_\ell } \frac{\delta _{n+3}^{4/3}}{D_n}, \qquad \varsigma _n^{\alpha _\star } :=\frac{1}{C_\varsigma } \frac{\delta _{n+3}^{4/3}}{n+1}, \end{aligned}$$

(3.1)

for some $C_\ell ,C_\varsigma >1$ sufficiently large and $\alpha _\star \in (0,\alpha )$ to be chosen sufficiently close to $\alpha$, and all independent of n, whereas $c_{n,\ell } \in (1,2)$ is such that $\ell ^{-1}$ is an integer power of 2 (it will be needed in Sect. 4.5 to apply Lemma C.7). The introduction of the additional parameter $\alpha _\star$ is useful to simplify the estimates in Sect. 4.4, where we need to take an auxiliary parameter $\alpha ' \in (\alpha _\star ,\alpha )$, see Proposition 4.8. Without loss of generality we may suppose $D_n \ell ^{\alpha }\le \eta \delta _n$ and $D_n \le \varsigma _{n+1}^{\alpha -1}$.

Given a standard mollifier $\chi \in C^{\infty }_c([-1,1]^3 \times [0,1))$ define $\chi _\ell (x,t) :=\ell ^{-4} \chi (x\ell ^{-1},t\ell ^{-1})$ and

$$\begin{aligned} v_\ell :=v_n *\chi _\ell , \quad q_\ell :=q_n *\chi _\ell , \quad \mathring{R}_\ell :=\mathring{R}_n *\chi _\ell . \end{aligned}$$

Here it is important that $v_n,q_n,\mathring{R}_n$ are defined also for negative times $t<0$ and do satisfy (1.6) in analytically strong sense. Indeed, we shall need in the following that $v_\ell ,q_\ell ,\mathring{R}_\ell$ satisfy

$$\begin{aligned} \partial _t v_\ell + \chi _\ell *\mathord {\textrm{div}}\,^{\phi _n}\left( v_n \otimes v_n\right) + \chi _\ell *\nabla ^{\phi _n} q_n = \chi _\ell *\mathord {\textrm{div}}\,^{\phi _n} \mathring{R}_n, \end{aligned}$$

which would not be generally true for times $t \in (0,\ell )$ otherwise.

Also, let $\lambda ,\mu \in \mathbb {N}$ be large parameters (possibly depending on n) such that $\lambda /\mu \in \mathbb {N}$. Notice that the latter condition implies in particular $\lambda \ge \mu$. These parameters dictate the frequency of space-time oscillations of the building block of the convex integration scheme. In order to simplify the computations in Sect. 4, we shall assume $\mu ^2 \varsigma _{n+1}^{\alpha -2} \le \lambda \le D_{n+1}$ and for some $r_\star \in \mathbb {N}$ to be suitably chosen (and independent of n):

$$\begin{aligned} \lambda ^{r}&\ge \mu ^{r+5} \varsigma _{n+1}^{(r+5)(\alpha -1)-2} \left( D_n \ell ^{-r-4} + \varsigma _{n+1}^{\alpha -1} \right) , \quad \forall r \ge r_\star . \end{aligned}$$

(3.2)

In order to satisfy the previous conditions on $\lambda , \mu$ we can choose $\alpha _\star , \alpha$ sufficiently close to 1/2, $r_\star \ge 7$ and the parameter $\mu$ as

$$\begin{aligned} \mu :=c_{n,\mu } C_\mu \ell ^{-r_\star } \end{aligned}$$

for some constant $C_\mu >1$, possibly large but finite and independent of n, and $c_{n,\mu } \in (1,2)$ so that $\mu \in \mathbb {N}$, and

$$\begin{aligned} \lambda :=c_{n,\lambda } \mu ^2 \varsigma _{n+1}^{\alpha -2} \end{aligned}$$

where $c_{n,\lambda } \in (1,2)$ satisfies $c_{n,\lambda } \varsigma _{n+1}^{\alpha -2} \in \mathbb {N}$, implying $\lambda \in \mathbb {N}$ and $\lambda / \mu \in \mathbb {N}$. Notice that the condition $\lambda \le D_{n+1}$ requires the parameters A and $\varepsilon$ in Proposition 2.4 to be sufficiently large. More details will be given in Sect. 5.

3.2 Outline of the construction

As usual in convex integration schemes, the idea is to define $v_{n+1}$ as a perturbation of $v_n$. However, in order to guarantee smoothness throughout the construction, we in fact construct $v_{n+1}$ as a perturbation of $v_\ell$.

The main building block used in this paper to define the principal perturbation $w_o$ is a modified version of Beltrami flows E (see Sect. 3.3 below), obtained via composition with the flow $\phi _{n+1}$. This implies, among other things, that E is not steady-state and not even a solution to the stochastic Euler Eq. (1.5). Because of this time dependence, we need to introduce an additional dependence on ${{\dot{\phi }}}_{n+1}$ in the amplitude coefficients to compensate for the time derivative of E. The key feature of Beltrami flows is that they oscillate at frequency $\lambda \gg 1$ in space, producing cancellations in space integrals of the perturbation. By stationary phase Lemma [15, Proposition 5.2] and its stochastic counterpart Proposition C.5, these cancellations help in controlling Hölder norms of the new velocity field $v_{n+1}$ and Reynold stress $\mathring{R}_{n+1}$. On top of that, a fine tuning of the amplitude of E permits us to control the amount of kinetic energy introduced in the system at the n-th step of the construction, cf. the term $\rho _\ell$ defined in Sect. 3.6. We take advantage of this to aim at the desired energy profile e when iterating the construction. However, because our iterative estimates deteriorate for large values of t (corresponding to large values of L) and we want to define $\rho _\ell$ independent of L to preserve adaptedness of solutions, we can only reach the desired energy profile up to time ${\mathfrak {t}}$. For larger times we content ourselves to inject enough energy at each step so that the perturbation $w_o$ is well-defined (cf. the term under square root in the definition of $\rho _\ell$ and (3.12) below).

Finally, since we want to control the quantity $\mathord {\textrm{div}}^{\phi _{n+1}}v_{n+1}$ we need to add a corrector term $w_c$ to make sure the latter decreases fast enough at every step of the iteration (Sect. 3.7). This step is far more involved than usual, since we cannot choose $w_c$ simply as the orthogonal Leray projection of $v_\ell + w_o$ because of a lack of control on $\partial _t w_c$.

Once $v_{n+1} = v_\ell + w_o + w_c$ is defined with the procedure above, the Reynolds stress $\mathring{R}_{n+1}$ and pressure $q_{n+1}$ can be determined inverting the operator $\mathord {\textrm{div}}^{\phi _{n+1}}$, similarly to what has been done for the deterministic case in [15], see Sect. 3.8.

3.3 Modified Beltrami flows

The following construction follows that of [15, Proposition 3.1]. For any given vector $k \in \mathbb {Z}^3 {\setminus } \{{\textbf{0}}\}$, denote

$$\begin{aligned} M_k :=Id - \frac{k}{|k|} \otimes \frac{k}{|k|}. \end{aligned}$$

Let $\lambda _0\ge 1$ and let $A_k\in \mathbb {R}^3$ be such that $A_k\cdot k=0$, $|A_k|=\tfrac{1}{\sqrt{2}}$ and $A_{-k}=A_k$ for $k\in \mathbb {Z}^3$ with $|k|=\lambda _0$. Furthermore, let us define

$$\begin{aligned} E_k :=A_k+i\frac{k}{|k|}\times A_k\in \mathbb {C}^3. \end{aligned}$$

For any collection $\{a_k\}_{k \in \mathbb {Z}^3, |k|=\lambda _0}$ of complex numbers $a_k\in \mathbb {C}$ satisfying $\overline{a_k} = a_{-k}$ for every k, the vector field

$$\begin{aligned} E(x,t) :=\sum _{|k|=\lambda _0}a_k E_ke^{ik\cdot \phi _{n+1}(x,t)} \end{aligned}$$

is real valued and satisfies

$$\begin{aligned} \mathord {\textrm{div}}^{\phi _{n+1}}E = 0, \quad \mathord {\textrm{div}}^{\phi _{n+1}}(E\otimes E)=\nabla ^{\phi _{n+1}}\left( \frac{|E|^2}{2} \right) . \end{aligned}$$

(3.3)

Furthermore, since $\phi _{n+1}$ is measure preserving:

$$\begin{aligned} \frac{1}{(2\pi )^3}\int _{\mathbb {T}^3} E\otimes E\,dx = \frac{1}{2} \sum _{|k|=\lambda _0} |a_k|^2 M_k. \end{aligned}$$

We shall call the field E modified Beltrami wave. This is the natural adaptation to our stochastic setting of classical Beltrami waves, that are steady-state solutions of Euler equations. However, our E is not a solution to (1.5) because of its dependence of t. We shall overcome this issue taking suitable time dependent amplitude coefficients $a_k=a_k(t)$, see next subsections for details.

3.4 The transport coefficients $\psi _k^{(j)}$

Recall the following construction from [15]. Let $c_1$, $c_2$ be such that $\frac{\sqrt{3}}{2}< c_1< c_2 < 1$ and fix a non negative function $\varphi \in C^\infty _c (B_{c _2} (0))$ which is identically equal to 1 on the ball $B_{c_1} (0)$. Next, consider the lattice $\mathbb {Z}^3$ and its quotient by $(2\mathbb {Z})^3$, and denote by ${\mathcal {C}}_j$, $j=1, \ldots , 8$ the eight equivalence classes of $\mathbb {Z}^3/(2\mathbb {Z})^3$. For each $k\in \mathbb {Z}^3$ denote by $\varphi _k$ the function $\varphi _k (x):= \varphi (x-k)$. Observe that, if $k\ne l \in {\mathcal {C}}_j$, then $|k-l|\ge 2 > 2 c_2$; hence, $\varphi _k$ and $\varphi _l$ have disjoint supports.

On the other hand, the function

$$\begin{aligned} \varphi _\Sigma :=\sum _{k\in \mathbb {Z}^3} \varphi _k^2 \end{aligned}$$

is smooth, bounded and bounded away from zero. We then define, for $v \in \mathbb {R}^3$ and $\tau \in \mathbb {R}$:

$$\begin{aligned} \alpha _k(v)&:=\frac{\varphi _k(v)}{\sqrt{\varphi _\Sigma (v)}}, \quad k \in \mathbb {Z}^3, \\ \psi ^{(j)}_k(v,\tau )&:=\sum _{l\in {\mathcal {C}}_j}\alpha _l(\mu v)e^{-ik\cdot (\frac{l}{\mu })\tau }, \quad k \in \mathbb {Z}^3, \, j=1,\dots ,8. \end{aligned}$$

Since $\alpha _l$ and $\alpha _{{{\tilde{l}}}}$ have disjoint supports for $l\ne {{\tilde{l}}}\in {\mathcal {C}}_j$, it follows that for all $v,\tau$ as above and $k \in \mathbb {Z}^3$

$$\begin{aligned} \sum _{j=1}^8|\psi ^{(j)}_k(v,\tau )|^2&= \sum _{j=1}^8 \sum _{l\in {\mathcal {C}}_j}\alpha _l(\mu v)^2 =1, \end{aligned}$$

and

$$\begin{aligned} \sup _{v,\tau }|D^r_v\psi ^{(j)}_k(v,\tau )|&\le C\mu ^r, \quad j=1,\dots ,8, \end{aligned}$$

(3.4)

with the constant $C=C(r,|k|)$ depending only on $r \in \mathbb {N}$ and |k|. More generally, in [16, Proposition 4.2] it is proved that for any $k \in \mathbb {Z}^3$ the derivatives of $\psi _k^{(j)}$ with respect to $\tau$ are controlled on the set $|v| \le V$ by

$$\begin{aligned} \sup _{|v| \le V,\tau } \left| D^r_v \partial _{\tau }^h\psi ^{(j)}_k \right|&\le CV^h\mu ^r, \end{aligned}$$

(3.5)

where $C=C(r,h,|k|)$ and $V>0$ is any given constant.

Next, we recall the following estimate from [15] on the material derivative of $\psi _k^{(j)}$. For any $r \in \mathbb {N}$ and $k \in \mathbb {Z}^3$ there exists a constant $C=C(r,|k|)$ such that for every $j=1,\dots ,8$ it holds

$$\begin{aligned} \sup _{v,\tau } \left| D^r_v(\partial _{\tau }\psi ^{(j)}_k+i(k \cdot v) \psi ^{(j)}_k )\right|&\le C\mu ^{r-1} . \end{aligned}$$

(3.6)

3.5 The energy pumping term

Next, set for $t \in \mathbb {R}$

$$\begin{aligned} {\tilde{e}}(t) :=\frac{1}{3 (2\pi )^3} \left( e (t) \left( 1-\delta _{n+1}\right) - \int _{\mathbb {T}^3} |v_\ell (x,t)|^2 \, dx\right) . \end{aligned}$$

Lemma 1.11

There exists a finite constant $C=C(\chi )$ depending only on the mollifier $\chi$ such that the following inequality holds true almost surely for every $t \le {\mathfrak {t}}$:

$$\begin{aligned} {\tilde{e}}(t)&\le \frac{\delta _n}{3(2\pi )^3} \left( \tfrac{5}{4} {\overline{e}} + C \eta \left( {\overline{e}}^{1/2}+1 \right) \right) . \end{aligned}$$

Moreover, if $\eta \le \frac{{\underline{e}}}{8C({\overline{e}}^{1/2}+1)}$ then we also have almost surely for every $t \le {\mathfrak {t}}$:

$$\begin{aligned} {\tilde{e}}(t)&\ge \frac{\delta _n {\underline{e}}}{24(2\pi )^3}. \end{aligned}$$

Proof

By triangle inequality, we have $||v_\ell |^2-|v_n|^2| \le |v_\ell -v_n|^2 + 2|v_n||v_\ell -v_n|$; moreover, $|v_\ell -v_n| \le C D_n \ell \le C \eta \delta _n$ for every $t \le {\mathfrak {t}}$, by assumption on $\ell$. In the previous inequality, $C=C(\chi )$ is a finite constant depending only on the mollifier $\chi$. By assumption (2.5) we have $\int _{\mathbb {T}^3}|v_n(x,t)|^2 dx \le (1-\tfrac{3}{4} \delta _n) e(t) \le {\overline{e}}$ almost surely for every $t \le {\mathfrak {t}}$, therefore

$$\begin{aligned} \int _{\mathbb {T}^3} \left| |v_\ell (x,t)|^2-|v_n(x,t)|^2\right| dx&\le C \eta \delta _n \left( {\overline{e}}^{1/2}+1 \right) , \end{aligned}$$

almost surely for every $t \le {\mathfrak {t}}$. As a consequence we have the estimate

$$\begin{aligned} {\tilde{e}}(t)&= \frac{1}{3 (2\pi )^3} \left( e (t) \left( 1-\tfrac{5}{4}\delta _n \right) - \int _{\mathbb {T}^3} |v_n|^2 dx + e (t)\left( \tfrac{5}{4}\delta _n-\delta _{n+1}\right) + \int _{\mathbb {T}^3} (|v_n|^2-|v_\ell |^2) \, dx\right) \\&\le \frac{\delta _n}{3(2\pi )^3} \left( \tfrac{5}{4} {\overline{e}} + C \eta \left( {\overline{e}}^{1/2}+1 \right) \right) \end{aligned}$$

almost surely for every $t \le {\mathfrak {t}}$. In a similar fashion, one can rewrite

$$\begin{aligned} {\tilde{e}}(t)&= \frac{1}{3 (2\pi )^3} \left( e (t) \left( 1-\tfrac{3}{4}\delta _n \right) - \int _{\mathbb {T}^3} |v_n|^2 dx + e (t)\left( \tfrac{3}{4}\delta _n-\delta _{n+1}\right) + \int _{\mathbb {T}^3} (|v_n|^2-|v_\ell |^2) \, dx\right) \\&\ge \frac{\delta _n}{3(2\pi )^3} \left( \tfrac{1}{4} {\underline{e}} - C \eta \left( {\overline{e}}^{1/2}+1 \right) \right) , \end{aligned}$$

where we have used $\tfrac{3}{4}\delta _n - \delta _{n+1} \ge \tfrac{1}{4}\delta _n$ (recall that the parameter a, b are chosen both greater or equal than 2). In order to obtain the thesis it is sufficient to take $\eta$ such that

$$\begin{aligned} \tfrac{1}{4} {\underline{e}} - C \eta \left( {\overline{e}}^{1/2}+1 \right) \ge \tfrac{1}{8} {\underline{e}}, \end{aligned}$$

which is always possible since we have assumed ${\underline{e}}>0$, and this corresponds exactly to the condition in the statement of the lemma. $\square$

Hereafter, we shall always assume $\eta$ satisfies the condition of previous Lemma 3.1. Let $\gamma _n:\mathbb {R}\rightarrow (0,\infty )$ be an adapted process with twice-differentiable trajectories satisfying almost surely

$$\begin{aligned} \gamma _n (t)&= {\tilde{e}}(t), \quad \forall t \le {\mathfrak {t}}, \end{aligned}$$

(3.7)

$$\begin{aligned} \tfrac{1}{2} {\tilde{e}}({\mathfrak {t}})&\le \gamma _n (t) \le \tfrac{3}{2} {\tilde{e}}({\mathfrak {t}}), \quad \forall t \ge {\mathfrak {t}}, \end{aligned}$$

(3.8)

$$\begin{aligned} \Vert \gamma _n\Vert _{C^k_t}&\le C \Vert {\tilde{e}}_{\le {\mathfrak {t}}}\Vert _{C^k_t}, \quad \forall k \le 2, \end{aligned}$$

(3.9)

for some unimportant constant C, and for some arbitrary $r_0>0$ define

$$\begin{aligned} \rho _\ell (x,t)&:=\frac{2}{r_0} \sqrt{\eta ^2 \delta _{n+1}^2 + |\mathring{R}_\ell (x,t)|^2} + \gamma _n(t), \\ R_\ell (x,t)&:=\rho _\ell (x,t) Id - \mathring{R}_\ell (x,t). \end{aligned}$$

The parameter $r_0$ will be fixed in next Sect. 3.6 according to a geometric lemma from [15]. Condition (3.7) above is imposed to reduce the error (up to time ${\mathfrak {t}}$) between the kinetic energy of $v_{n+1}$ and the desired energy profile e. After time ${\mathfrak {t}}$, we are no longer able to control ${\tilde{e}}$ from below for large times (in particular if $t>{\mathfrak {t}}_L$ with L large), and thus we inject energy in the system according to $\gamma _n(t)$ and not aiming for any prescribed energy profile. Notice that $\gamma _n$ does not depend on the parameter L, in particular we do not use any stopping time of the form ${\mathfrak {t}}_L$, $L>1$ in the definition of $\gamma _n$, see Remark 3.2 below. Conditions (3.8) and (3.9) are needed to show convergence of the so-obtained iterative scheme for times $t \ge {\mathfrak {t}}$. Let us also mention that by (3.7), (3.8) and Lemma 3.1 we have $c \delta _n \le \gamma _n(t) \le C \delta _n$ for every $t \in \mathbb {R}$, for some constants $c=c({\underline{e}})>0$ and $C=C({\overline{e}})<\infty$.

Remark 3.2

We point out that it is always possible to construct a process $\gamma _n$ enjoying the properties (3.7), (3.8) and (3.9). For instance, an explicit construction goes as follows. Let $F:\mathbb {R}\rightarrow (-1,1)$ be given by $F(t):=\frac{2}{\pi }\arctan (t)$. For every $k\in \{0,1,2\}$ let ${\tilde{e}}^{(k)}({\mathfrak {t}})$ denote the derivative of ${\tilde{e}}$ of order k evaluated at time $t={\mathfrak {t}}$. For every $\omega \in \Omega$, let $f=f_\omega$ be the unique polynomial of order 2 such that $f({\mathfrak {t}})={\tilde{e}}({\mathfrak {t}})$ and

$$\begin{aligned} \left. \frac{{\tilde{e}}({\mathfrak {t}})}{2} \frac{d^k}{dt^k} F\left( \frac{f-{\tilde{e}}({\mathfrak {t}})}{\Vert {\tilde{e}}_{\le {\mathfrak {t}}}\Vert _{C^2_t}} \right) \right| _{t={\mathfrak {t}}} = {\tilde{e}}^{(k)}({\mathfrak {t}}), \qquad k=1,2, \end{aligned}$$

which is uniquely determined by

$$\begin{aligned} \left. \frac{d}{dt} F \left( \frac{f-{\tilde{e}}({\mathfrak {t}})}{\Vert {\tilde{e}}_{\le {\mathfrak {t}}}\Vert _{C^2_t}} \right) \right| _{t={\mathfrak {t}}}&= \frac{F'(0) f'({\mathfrak {t}}) }{\Vert {\tilde{e}}_{\le {\mathfrak {t}}}\Vert _{C^2_t}}, \\ \left. \frac{d^2}{dt^2} F \left( \frac{f-{\tilde{e}}({\mathfrak {t}})}{\Vert {\tilde{e}}_{\le {\mathfrak {t}}}\Vert _{C^2_t}} \right) \right| _{t={\mathfrak {t}}}&= \frac{F'(0) f''({\mathfrak {t}}) }{\Vert {\tilde{e}}_{\le {\mathfrak {t}}}\Vert _{C^2_t}} + \frac{F''(0)f'({\mathfrak {t}})^2}{\Vert {\tilde{e}}_{\le {\mathfrak {t}}}\Vert _{C^2_t}^2}. \end{aligned}$$

Notice that the coefficients of f are ${\mathcal {F}}_{\mathfrak {t}}$-measurable real valued random variables, and defining

$$\begin{aligned} \gamma _n(t) :={\left\{ \begin{array}{ll} {\tilde{e}}(t), &{} \text{ if } t \le {\mathfrak {t}}, \\ {\tilde{e}}({\mathfrak {t}}) \left( 1 + \frac{1}{2} F\left( \frac{f(t)-{\tilde{e}}({\mathfrak {t}})}{\Vert {\tilde{e}}_{\le {\mathfrak {t}}}\Vert _{C^2_t}} \right) \right) , &{} \text{ if } t \ge {\mathfrak {t}}, \end{array}\right. } \end{aligned}$$

we have a $\gamma _n$ satisfying (3.7), (3.8) and (3.9) with C depending only on F. Moreover, $\gamma _n$ is adapted since $v_n$ (and hence ${\tilde{e}}$) is so. In particular, $\gamma _n$ is deterministic for negative times.

3.6 The oscillatory term $w_o$

We recall (cf. [15, Lemma 3.2]) that for every $N\in \mathbb {N}$ there exist $r_0>0$ and $\lambda _0>1$ with the following property: there exist pairwise disjoint subsets $\Lambda _j\subset \{k\in \mathbb {Z}^3:\,|k|=\lambda _0\}$, for $j\in \{1, \ldots , N\},$ and smooth positive functions

$$\begin{aligned} \gamma ^{(j)}_k\in C^{\infty }\left( B_{r_0} (Id)\right) , \quad j\in \{1,\ldots , N\}, \, k\in \Lambda _j, \end{aligned}$$

such that $k\in \Lambda _j$ implies both $-k\in \Lambda _j$ and $\gamma ^{(j)}_k = \gamma ^{(j)}_{-k}$, and for each $R\in B_{r_0} (Id)$ we have the identity

$$\begin{aligned} R = \frac{1}{2} \sum _{k\in \Lambda _j} \left( \gamma ^{(j)}_k(R)\right) ^2 M_k, \quad \forall R\in B_{r_0}(Id). \end{aligned}$$

We apply the previous lemma with $N=8$ to obtain $\lambda _0>1$, $r_0>0$ and pairwise disjoint families $\Lambda _j$ together with corresponding functions $\gamma ^{(j)}_k\in C^{\infty }\left( B_{r_0}(Id)\right)$.

Finally, let us define

$$\begin{aligned} w_o(x,t) \nonumber&:={W(x,t,\lambda \phi _{n+1}(x,t),\lambda t), } \\ W(y,s,\xi ,\tau )&:=\sum _{k \in \Lambda } a_k(y,s,\tau ) E_ke^{ik\cdot \xi }, \end{aligned}$$

(3.10)

where $\Lambda :=\cup _{j} \Lambda _j$ and the amplitude coefficients $a_k$, $k \in \Lambda$ are defined by

$$\begin{aligned} a_k(y,s,\tau )&:={\textbf{1}}_{\{k \in \Lambda _j\}} \sqrt{\rho _\ell (y,s)}\, \gamma _k^{(j)}\left( \frac{R_\ell (y,s)}{\rho _\ell (y,s)}\right) \psi _k^{(j)} \left( {\tilde{v}}(y,s),\tau \right) , \\ \nonumber {\tilde{v}}(y,s)&:={v_\ell (y,s) + {\dot{\phi }}_{n+1}(y,s).} \end{aligned}$$

(3.11)

Notice that the maps $\gamma _k^{(j)}\left( \frac{R_\ell (y,s)}{\rho _\ell (y,s)}\right)$ are well-defined because (recall $\gamma _n \ge c \delta _n \ge 0$)

$$\begin{aligned} \left\| Id - \frac{R_\ell (y,s)}{\rho _\ell (y,s)} \right\| _{C_t C_x } = \left\| \frac{\mathring{R}_\ell (y,s)}{\rho _\ell (y,s)} \right\| _{C_t C_x} \le \frac{r_0}{2}. \end{aligned}$$

(3.12)

Remark 3.3

The auxiliary velocity ${\tilde{v}}$ is needed because, in order to control the transport error below (see (3.17) and Proposition 4.3), we will need cancellations for the quantities

$$\begin{aligned} \partial _\tau a_k(x,t,\lambda t) + i k \cdot [v(x,t)+{\dot{\phi }}_{n+1}(x,t)]\, a_k(x,t,\lambda t), \quad k \in \Lambda . \end{aligned}$$

However, some care is needed since ${{\dot{\phi }}}_{n+1}$ is typically large, in particular it diverges as $\varsigma _{n+1}^{\alpha -1}$ as $n \rightarrow \infty$, thus affecting both space and time regularity of the coefficients $a_k$.

Also, as a consequence of (3.7), (3.8), Lemma 3.1 and the iterative assumption (2.6), there exists a constant C depending only on ${\overline{e}}$ such that for every $x \in \mathbb {T}^3$ and $t \le {\mathfrak {t}}_L$

$$\begin{aligned} |w_o(x,t)| \le C L_n^{1/2} \delta _n^{1/2}. \end{aligned}$$

(3.13)

In particular, from (3.13) we also deduce

$$\begin{aligned} |w_o(x,t)| \le C \delta _n^{1/2}, \quad \forall t \le {\mathfrak {t}}. \end{aligned}$$

3.7 The correction $w_c$

Let ${\mathcal {P}} = Id - {\mathcal {Q}}$ be the classical Leray projector on zero-average, divergence free velocity fields. Define ${\mathcal {P}}^{\phi _{n+1}}, {\mathcal {Q}}^{\phi _{n+1}}$ as the operators acting on a given $v \in C^\infty (\mathbb {T}^3,\mathbb {R}^3)$ as

$$\begin{aligned} {\mathcal {P}}^{\phi _{n+1}} v :=[{\mathcal {P}}(v \circ \phi _{n+1}^{-1})] \circ \phi _{n+1}, \quad {\mathcal {Q}}^{\phi _{n+1}} v :=[{\mathcal {Q}}(v \circ \phi _{n+1}^{-1})] \circ \phi _{n+1}. \end{aligned}$$

It holds

$$\begin{aligned} \mathord {\textrm{div}}^{\phi _{n+1}}{\mathcal {P}}^{\phi _{n+1}} v = [\mathord {\textrm{div}}\,({\mathcal {P}}^{\phi _{n+1}} v \circ \phi _{n+1}^{-1})] \circ \phi _{n+1} = [\mathord {\textrm{div}}\,({\mathcal {P}}(v \circ \phi _{n+1}^{-1})] \circ \phi _{n+1} = 0, \end{aligned}$$

and

$$\begin{aligned} \mathord {\textrm{div}}^{\phi _{n+1}}{\mathcal {Q}}^{\phi _{n+1}} v&= [\mathord {\textrm{div}}\,({\mathcal {Q}}^{\phi _{n+1}} v \circ \phi _{n+1}^{-1})] \circ \phi _{n+1}\\ {}&= [\mathord {\textrm{div}}\,({\mathcal {Q}}(v \circ \phi _{n+1}^{-1})] \circ \phi _{n+1} = \mathord {\textrm{div}}^{\phi _{n+1}}v. \end{aligned}$$

Moreover, denoting $\Delta ^{\phi _{n+1}} v :=\mathord {\textrm{div}}^{\phi _{n+1}}\nabla ^{\phi _{n+1}} v = [\Delta (v \circ \phi _{n+1}^{-1})] \circ \phi _{n+1}$ and introducing the zero-mean solution $\psi$ of the Poisson equation $\Delta ^{\phi _{n+1}} \psi = \mathord {\textrm{div}}^{\phi _{n+1}}v$ we have the alternative expression

$$\begin{aligned} {\mathcal {Q}}^{\phi _{n+1}} v = \nabla ^{\phi _{n+1}} \psi + \frac{1}{(2\pi )^3}\int _{\mathbb {T}^3} v, \end{aligned}$$

provided $v \in C^\infty (\mathbb {T}^3,\mathbb {R}^3)$.

Recall that we want a control on $\mathord {\textrm{div}}^{\phi _{n+1}}v_{n+1}$, where $v_{n+1}=v_\ell + w_o + w_c$ and $w_c$ is still to be found. Let us compute

$$\begin{aligned} \mathord {\textrm{div}}^{\phi _{n+1}}v_\ell&= \mathord {\textrm{div}}^{\phi _{n+1}}v_\ell - \left( \mathord {\textrm{div}}^{\phi _{n+1}}v_n \right) *\chi _\ell \\&\quad + \nonumber \left( \mathord {\textrm{div}}^{\phi _{n+1}}v_n \right) *\chi _\ell - \left( \mathord {\textrm{div}}\,^{\phi _n}v_n\right) *\chi _\ell \\&\quad + \nonumber \left( \mathord {\textrm{div}}\,^{\phi _n}v_n\right) *\chi _\ell . \end{aligned}$$

(3.14)

We shall see that the first two terms on the right-hand-side of (3.14) are already small and compatible with the inductive assumption on the divergence (2.8), without any compressibility correction needed. The only term to really “compensate" for in (3.14) is $\left( \mathord {\textrm{div}}\,^{\phi _n}v_n\right) *\chi _\ell$, and the idea is to compensate with some $w^1_c$ such that $\partial _t w^1_c$ is easy to control. This happens when $w^1_c$ is a convolution, because one can put time derivatives on the mollification kernel. Therefore, let us define

$$\begin{aligned} w^1_c :=-({\mathcal {Q}}^{\phi _n}v_n) *\chi _\ell , \end{aligned}$$

which is such that

$$\begin{aligned} \left( \mathord {\textrm{div}}\,^{\phi _n}v_n\right) *\chi _\ell + \mathord {\textrm{div}}^{\phi _{n+1}}w^1_c&= \left( \mathord {\textrm{div}}\,^{\phi _n} {\mathcal {Q}}^{\phi _n} v_n\right) *\chi _\ell - \mathord {\textrm{div}}\,^{\phi _n} \left( ({\mathcal {Q}}^{\phi _n}v_n) *\chi _\ell \right) \nonumber \\&\quad + \mathord {\textrm{div}}\,^{\phi _n} \left( ({\mathcal {Q}}^{\phi _n}v_n) *\chi _\ell \right) - \mathord {\textrm{div}}^{\phi _{n+1}}\left( ({\mathcal {Q}}^{\phi _n}v_n) *\chi _\ell \right) \nonumber \\ \end{aligned}$$

(3.15)

can be controlled as the first two lines in (3.14). As for the principal perturbation $w_o$, we can take care of its divergence by adding

$$\begin{aligned} w^2_c :=- {\mathcal {Q}}^{\phi _{n+1}} w_o \end{aligned}$$

so that $w_o+w^2_c = {\mathcal {P}}^{\phi _{n+1}} w_o$. We shall see that there is no problem in controlling $\partial _t w^2_c$ because of the particular geometric structure of the perturbation $w_o$, cf. Sect. 4.5 for details.

The vector field $v_{n+1}$ is then defined as the sum

$$\begin{aligned} v_{n+1} :=v_\ell + w_o + w_c :=v_\ell + w_o + w^1_c + w^2_c, \end{aligned}$$

and has zero space average by construction.

3.8 The Reynold stress and the new pressure $q_{n+1}$

Let us preliminarily recall the following result from [15] on the left inverse of the operator $\text{ div }$, here adapted to deal with the operator $\mathord {\textrm{div}}^{\phi _{n+1}}$.

Lemma 1.14

Let $v \in C^\infty (\mathbb {T}^3,\mathbb {R}^3)$ and ${\mathcal {R}}v$ be the matrix-valued function defined in [15, Definition 4.2], so that ${\mathcal {R}}v$ takes values in the space of symmetric trace-free matrices and $\mathord {\textrm{div}}\,{\mathcal {R}}v = v - \frac{1}{(2\pi )^3}\int _{\mathbb {T}^3}v$. Then the operator ${\mathcal {R}}^{\phi _{n+1}}$ defined as

$$\begin{aligned} {\mathcal {R}}^{\phi _{n+1}} v :=[{\mathcal {R}}(v \circ \phi _{n+1}^{-1})] \circ \phi _{n+1} \end{aligned}$$

satisfies $\mathord {\textrm{div}}^{\phi _{n+1}}({\mathcal {R}}^{\phi _{n+1}} v) = v - \frac{1}{(2\pi )^3}\int _{\mathbb {T}^3}v$.

Suppose we are given the new pressure $q_{n+1}$. Since we have defined $v_{n+1} = v_\ell + w_o + w_c$ and we are looking for a solution to the Euler–Reynolds system, we shall choose the new Reynold stress $\mathring{R}_{n+1}$ such that

$$\begin{aligned} \mathring{R}_{n+1} :={\mathcal {R}}^{\phi _{n+1}} \left( \partial _t v_{n+1} + \mathord {\textrm{div}}^{\phi _{n+1}}(v_{n+1} \otimes v_{n+1}) + \nabla ^{\phi _{n+1}}q_{n+1} \right) . \end{aligned}$$

(3.16)

Notice that $\partial _t v_{n+1} + \mathord {\textrm{div}}^{\phi _{n+1}}(v_{n+1} \otimes v_{n+1}) + \nabla ^{\phi _{n+1}}q_{n+1}$ has average zero whatever the choice of $q_{n+1}$. Indeed, $v_{n+1}$ has average zero by construction, and thus so does $\partial _t v_{n+1}$; on the other hand, the vector field $\mathord {\textrm{div}}^{\phi _{n+1}}(v_{n+1} \otimes v_{n+1}) + \nabla ^{\phi _{n+1}}q_{n+1}$ has average zero because $\phi _{n+1}$ is measure preserving and thus

$$\begin{aligned}&\int _{\mathbb {T}^3} \mathord {\textrm{div}}^{\phi _{n+1}}(v_{n+1} \otimes v_{n+1}) + \nabla ^{\phi _{n+1}}q_{n+1}\\&\quad = \int _{\mathbb {T}^3} \mathord {\textrm{div}}^{\phi _{n+1}}\left( v_{n+1} \otimes v_{n+1} + q_{n+1} Id \right) \\&\quad = \int _{\mathbb {T}^3} \mathord {\textrm{div}}\,\left[ \left( v_{n+1} \otimes v_{n+1} + q_{n+1} Id \right) \circ \phi _{n+1}^{-1} \right] = 0. \end{aligned}$$

Therefore from (3.16) and Lemma 3.4 it follows that $\mathring{R}_{n+1}$ is symmetric and trace-free, and the following identity holds

$$\begin{aligned} \partial _t v_{n+1} + \mathord {\textrm{div}}^{\phi _{n+1}}(v_{n+1}\otimes v_{n+1}) + \nabla ^{\phi _{n+1}} q_{n+1} = \mathord {\textrm{div}}^{\phi _{n+1}}\mathring{R}_{n+1}; \end{aligned}$$

otherwise said, $(v_{n+1},q_{n+1},\phi _{n+1},\mathring{R}_{n+1})$ is a solution of the Euler-Reynolds system at level $n+1$. Let us now see how to choose the new pressure term $q_{n+1}$. Recalling

$$\begin{aligned} \partial _t v_\ell + \chi _\ell *\mathord {\textrm{div}}\,^{\phi _n}\left( v_n \otimes v_n\right) + \chi _\ell *\nabla ^{\phi _n} q_n = \chi _\ell *\mathord {\textrm{div}}\,^{\phi _n} \mathring{R}_n, \end{aligned}$$

and $v_{n+1} = v_\ell + w_o + w_c$ we have for any ${\tilde{\rho }}_\ell$

$$\begin{aligned}&\partial _t v_{n+1} + \mathord {\textrm{div}}^{\phi _{n+1}}(v_{n+1}\otimes v_{n+1}) + \nabla ^{\phi _{n+1}} q_\ell - \nabla ^{\phi _{n+1}} \frac{1}{2}\left( |w_o|^2-{\tilde{\rho }}_\ell \right) \nonumber \\&\quad = \nonumber \underbrace{\left[ \partial _t w_o + \mathord {\textrm{div}}^{\phi _{n+1}}(w_o \otimes v_\ell ) - w_o\, \mathord {\textrm{div}}^{\phi _{n+1}}v_\ell \right] }_{=\,transport\ error} \\&\qquad + \nonumber \underbrace{\left[ \mathord {\textrm{div}}^{\phi _{n+1}}(v_\ell \otimes v_\ell - (v_n \otimes v_n) *\chi _\ell )\right] }_{=\,mollification\ error I} \\&\qquad + \nonumber \underbrace{\left[ \mathord {\textrm{div}}\,^{\phi _n}\left( \left( v_n \otimes v_n\right) *\chi _\ell + q_\ell Id - \mathring{R}_\ell \right) - \left( \mathord {\textrm{div}}\,^{\phi _n} \left( v_n \otimes v_n + q_n Id - \mathring{R}_n \right) \right) *\chi _\ell \right] }_{=\,mollification\ error\ II} \\&\qquad + \nonumber \underbrace{\left[ \mathord {\textrm{div}}^{\phi _{n+1}}\left( w_o \otimes w_o - \frac{1}{2}\left( |w_o|^2-{\tilde{\rho }}_\ell \right) Id +\mathring{R}_\ell \right) \right] }_{=\,oscillation\ error} \\&\qquad + \nonumber \underbrace{\left[ \left( \mathord {\textrm{div}}^{\phi _{n+1}}-\mathord {\textrm{div}}\,^{\phi _n}\right) ((v_n \otimes v_n) *\chi _\ell - \mathring{R}_\ell + q_\ell Id) + w_o\, \mathord {\textrm{div}}^{\phi _{n+1}}v_\ell \right] }_{=\,flow\ error} \\&\qquad + \underbrace{\left[ \partial _t w_c + \mathord {\textrm{div}}^{\phi _{n+1}}(v_{n+1} \otimes w_c + w_c \otimes v_{n+1} - w_c \otimes w_c +v_\ell \otimes w_o) \right] }_{=\,compressibility\ error}. \end{aligned}$$

(3.17)

The oscillation error above can be controlled when we take ${\tilde{\rho }}_\ell$ equal to the energy pumping term defined in Sect. 3.5, cf. the computations in Sect. 4.3. This suggests the choice

$$\begin{aligned} q_{n+1}&:=q_\ell - \frac{1}{2} \left( |w_o|^2-{\tilde{\rho }}_\ell \right) , \\ {\tilde{\rho }}_\ell (x,t)&:=\frac{2}{r_0} \sqrt{\eta ^2 \delta _{n+1}^2 + |\mathring{R}_\ell (x,t)|^2}. \end{aligned}$$

Notice that the part with $\gamma _n$ in the definition of $\rho _\ell$ does not appear here since it is x-independent. Incidentally, in view of (3.16) and since the operator ${\mathcal {R}}^{\phi _{n+1}}$ is linear, we can split the Reynolds stress into five parts. Each terms will be controlled separately in Sect. 4.

4 Iterative estimates

In the present section we collect all the necessary estimates on the several error terms from (3.17). Recall the definition of ${\tilde{v}}$ (3.11); as a preliminary step, we need to give good bounds for Hölder norms of the amplitude coefficients $a_k$, their material derivatives $\partial _\tau a_k + i(k \cdot {\tilde{v}})a_k$, and time derivatives (i.e. with respect to both the variables s and $\tau$) of both of them. This is done in forthcoming Proposition 4.1, whose proof is given in Appendix 1. As a consequence of that we have Corollary 4.2, describing Hölder regularity of the Fourier coefficients (with respect to $\xi \in \mathbb {T}^3$) of the matrix-valued field $W \otimes W$.

By interpolation between Hölder spaces $C^r_x$, $r \in [1,r_\star +5]$ the results of Proposition 4.1 and Corollary 4.2, here stated for $r\ge 1$, $r \in \mathbb {N}$ still hold true when r is not an integer. That will be used in the forthcoming subsections to control, respectively, the transport error (Sect. 4.1), the oscillation error (Sect. 4.3), the compressibility error (Sect. 4.5), as well as to give bounds on the pressure (Sect. 4.7) and the energy (Sect. 4.8).

Proposition 1.15

Let $a_k$ be given by (3.11), $k \in \Lambda$, and let $s \le {\mathfrak {t}}_L$, $\tau \le \lambda {\mathfrak {t}}_L$ be fixed, $L \ge 1$. Then there exists a constant C depending only on ${\overline{e}},K_0,\eta ,M_v$ such that for every $r \in [0,1]$:

$$\begin{aligned} \Vert a_k(\cdot ,s,\tau )\Vert _{C^r_x}&\le C L_n^{7/2} \delta _n^{1/2} \mu ^r \varsigma _{n+1}^{r(\alpha -1)}, \end{aligned}$$

(4.1)

$$\begin{aligned} \Vert \partial _\tau a_k(\cdot ,s,\tau )\Vert _{C^r_x}&\le C L_n^{7/2} \delta _n^{1/2} \mu ^r \varsigma _{n+1}^{(r+1)(\alpha -1)}, \end{aligned}$$

(4.2)

$$\begin{aligned} \Vert (\partial _\tau a_k + i(k \cdot {\tilde{v}})a_k)(\cdot ,s,\tau )\Vert _{C^r_x}&\le C L_n^{7/2} \delta _n^{1/2} \mu ^{r-1} \varsigma _{n+1}^{r(\alpha -1)} , \end{aligned}$$

(4.3)

$$\begin{aligned} \Vert \partial _s a_k(\cdot ,s,\tau )\Vert _{C_x}&\le CL_n^{7/2} \mu \delta _n^{1/2} \varsigma _{n+1}^{\alpha -2}. \end{aligned}$$

(4.4)

Moreover, there exists a constant C depending only on ${\overline{e}},K_0,\eta ,M_v,r_\star$ such that for every $r \in \mathbb {N}$, $2 \le r \le r_\star + 5$:

$$\begin{aligned} \Vert a_k(\cdot ,s,\tau )\Vert _{C^r_x}&\le C L_n^{3r} \delta _n^{1/2} \mu ^r \varsigma _{n+1}^{(r-1)(\alpha -1)} (D_n \ell ^{1-r} + \varsigma _{n+1}^{\alpha -1}), \end{aligned}$$

(4.5)

$$\begin{aligned} \Vert \partial _\tau a_k(\cdot ,s,\tau )\Vert _{C^r_x}&\le C L_n^{3r} \delta _n^{1/2} \mu ^r \varsigma _{n+1}^{r(\alpha -1)} (D_n \ell ^{1-r} + \varsigma _{n+1}^{\alpha -1}), \end{aligned}$$

(4.6)

$$\begin{aligned} \Vert (\partial _\tau a_k + i(k \cdot {\tilde{v}})a_k)(\cdot ,s,\tau )\Vert _{C^r_x}&\le C L_n^{3r} \delta _n^{1/2} \mu ^{r-1} \varsigma _{n+1}^{(r-1)(\alpha -1)} (D_n \ell ^{1-r} + \varsigma _{n+1}^{\alpha -1}) , \end{aligned}$$

(4.7)

$$\begin{aligned} \Vert \partial _s a_k(\cdot ,s,\tau )\Vert _{C^r_x}&\le C L_n^{3r+5/2} \delta _n^{1/2} \mu ^{r+1} \varsigma _{n+1}^{r(\alpha -1)-1} (D_n \ell ^{1-r}+\varsigma _{n+1}^{\alpha -1}). \end{aligned}$$

(4.8)

Let us denote ${\mathcal {S}}^{3 \times 3}$ the space of symmetric $3 \times 3$ matrices. Arguing as in [15, Proposition 6.1] we can also deduce the following:

Corollary 1.16

Let $W=W(y,s,\xi ,\tau )$ be defined by (3.10). Then the matrix-valued field $W \otimes W$ can be written as

$$\begin{aligned} W \otimes W (y,s,\xi ,\tau ) = R_\ell (y,s) + \sum _{1 \le |k| \le 2\lambda _0} U_k(y,s,\tau ) e^{i k \cdot \xi }, \end{aligned}$$

with $U_k \in C^\infty _{loc}(\mathbb {T}^3 \times \mathbb {R}^2, {\mathcal {S}}^{3 \times 3})$, $k \in \Lambda$, satisfying $U_k k = \tfrac{1}{2} Tr(U_k)k$. In addition, for every fixed $s \le {\mathfrak {t}}_L$, $\tau \le \lambda {\mathfrak {t}}_L$, $L \ge 1$, and for every $r \in (0,1]$ there exists a constant C depending only on ${\overline{e}},K_0,\eta ,M_v$ such that:

$$\begin{aligned} \Vert U_k(\cdot ,s,\tau )\Vert _{C^r_x}&\le C L_n^4 \delta _n \mu ^r \varsigma _{n+1}^{r(\alpha -1)} . \end{aligned}$$

(4.9)

Moreover, there exists C depending on ${\overline{e}},K_0,\eta ,M_v,r_\star$ such that for every $r \in \mathbb {N}$, $2 \le r \le r_\star +5$:

$$\begin{aligned} \Vert U_k(\cdot ,s,\tau )\Vert _{C^r_x}&\le C L_n^{3r+5/2} \delta _n \mu ^r \varsigma _{n+1}^{(r-1)(\alpha -1)} (D_n \ell ^{1-r} + \varsigma _{n+1}^{\alpha -1}) . \end{aligned}$$

(4.10)

In the remaining part of the present section we prove the iterative estimates necessary to the proof of our main Proposition 2.4. The underlying rationale consists in giving individual Hölder bounds to the error terms appearing in the decomposition (3.17). Some of these errors—namely, the transport error, the oscillation error and the compressibility error—are controlled via the same stationary phase Lemma (cf. [16, Proposition 4.4]) previously used in the deterministic setting, or a slight modification of it detailed in Proposition C.5. Worth to mention is the presence of the mollification error II and flow error terms, respectively due to the fact that the space-time dependent differential operator $\mathord {\textrm{div}}\,^{\phi _n}$ does not commute with convolutions and the mollification of the noise $\phi _n$. These terms are very specific to our construction and impose a relatively fast decay of $\varsigma _n$ to be dealt with. Since we are going to use Proposition 4.1 and Corollary 4.2 in the following, the constants C in the remainder of this section may depend on parameters ${\overline{e}},K_0,\eta ,M_v,r_\star$ (without mentioning explicitly).

4.1 Estimate on the transport error

The introduction of the modified Beltrami flows, as well as the exact form of the amplitude coefficients $a_k$, $k \in \mathbb {N}$ in previous subsections, is justified by the following observation. Let us rewrite the transport error in (3.17) as

$$\begin{aligned} \partial _t w_o + \mathord {\textrm{div}}^{\phi _{n+1}}(w_o \otimes v_\ell ) - w_o\, \mathord {\textrm{div}}^{\phi _{n+1}}v_\ell&= \partial _t w_o + (v_\ell \cdot \nabla ^{\phi _{n+1}}) w_o. \end{aligned}$$

Denoting $\Omega _k (\xi ) :=E_k e^{ik\cdot \xi }$ and recalling the expression of $w_o$:

$$\begin{aligned} w_o(x,t)&= \sum _{k \in \Lambda } {a_k(x,t,\lambda t)} \, \Omega _k (\lambda \phi _{n+1}(x,t)), \end{aligned}$$

we can compute explicitly

$$\begin{aligned} \partial _t w_o (x,t)&= \sum _{k \in \Lambda } \partial _s a_k(x,t,\lambda t) \, \Omega _k(\lambda \phi _{n+1}(x,t)) \\&\quad + \lambda \sum _{k \in \Lambda } \partial _\tau a_k(x,t,\lambda t) \, \Omega _k(\lambda \phi _{n+1}(x,t)) \\&\quad + \lambda \sum _{k \in \Lambda } (i k \cdot {\dot{\phi }}_{n+1}(x,t)) a_k (x,t,\lambda t) \, \Omega _k(\lambda \phi _{n+1}(x,t)), \end{aligned}$$

and

$$\begin{aligned} (v_\ell \cdot \nabla ^{\phi _{n+1}}) w_o (x,t)&= \sum _{k \in \Lambda } v_\ell (x,t) \cdot \nabla ^{\phi _{n+1}}a_k (x,t,\lambda t) \,\Omega _k(\lambda \phi _{n+1}(x,t)) \\&\quad + \lambda \sum _{k \in \Lambda } (i k \cdot v_\ell (x,t)) a_k(x,t,\lambda t)\,\Omega _k(\lambda \phi _{n+1}(x,t)). \end{aligned}$$

Therefore, recalling the definition of ${\tilde{v}} :=v_\ell + {\dot{\phi }}_{n+1}$ we have

$$\begin{aligned} \partial _t w_o + (v_\ell \cdot \nabla ^{\phi _{n+1}}) w_o&= \lambda \sum _{k \in \Lambda } \left( \partial _\tau a_k + i (k \cdot {\tilde{v}}) a_k \right) \Omega _k + \sum _{k \in \Lambda } (v_\ell \cdot \nabla ^{\phi _{n+1}}a_k + \partial _s a_k ) \Omega _k , \end{aligned}$$

where in the expression above the left-hand-side is evaluated at (x, t), whereas the right-hand-side is evaluated at the quadruple $(y,s,\xi ,\tau )=(x,t, \lambda \phi _{n+1}(x,t),\lambda t)$.

Proposition 1.17

Let us denote $\mathring{R}^{trans} :={\mathcal {R}}^{\phi _{n+1}} (\partial _t w_o + (v_\ell \cdot \nabla ^{\phi _{n+1}}) w_o)$. Then for every $r \ge r_\star +2$ and $\delta >0$ sufficiently small there exists a constant C such that almost surely for every $L \in \mathbb {N}$, $L \ge 1$

$$\begin{aligned} \Vert \mathring{R}^{trans}\Vert _{C_{\le {\mathfrak {t}}_L} C_x}&\le CL_n^{3r+5}\lambda ^{\delta } \mu ^{-1} \delta _n^{1/2}, \\ \Vert \mathring{R}^{trans}\Vert _{C_{\le {\mathfrak {t}}_L} C^1_x}&\le C L_n^{3r+8} \lambda ^{\delta } \mu \delta _n^{1/2} \varsigma _{n+1}^{\alpha -1} . \end{aligned}$$

Proof

It is convenient to divide $\mathring{R}^{trans}$ into three terms:

$$\begin{aligned} \mathring{R}^{trans} = \mathring{R}^{trans}_1 + \mathring{R}^{trans}_2 + \mathring{R}^{trans}_3, \end{aligned}$$

where, denoting $b_k :=(\partial _\tau a_k + i(k\cdot {\tilde{v}})a_k) \circ \phi _{n+1}^{-1}$ and $\Omega _k^\lambda :=\Omega _k(\lambda \, \cdot )$, with a slight abuse of notation we have denoted:

$$\begin{aligned} \mathring{R}^{trans}_1&:=\lambda {\mathcal {R}}^{\phi _{n+1}} \left( \sum _{k \in \Lambda } (b_k \,\Omega _k^\lambda ) \circ \phi _{n+1} \right) , \\ \mathring{R}^{trans}_2&:={\mathcal {R}}^{\phi _{n+1}} \left( \sum _{k \in \Lambda } (v_\ell \circ \phi _{n+1}^{-1} \cdot \nabla _y (a_k\circ \phi _{n+1}^{-1}) \,\Omega _k^\lambda ) \circ \phi _{n+1} \right) , \\ \mathring{R}^{trans}_3&:={\mathcal {R}}^{\phi _{n+1}} \left( \sum _{k \in \Lambda } (\partial _s a_k \circ \phi _{n+1}^{-1} \,\Omega _k^\lambda ) \circ \phi _{n+1} \right) . \end{aligned}$$

Let us control each term separately, starting from $\mathring{R}^{trans}_1$. Proposition 4.1 will be implicitly used throughout this section. Applying Lemma C.1 and Proposition C.5 with $r \ge r_\star +2$ and $\delta$ sufficiently small, we obtain for every $L \in \mathbb {N}$, $L \ge 1$

$$\begin{aligned} \Vert \mathring{R}^{trans}_1\Vert _{C_{\le {\mathfrak {t}}_L}C^\delta _x}&\le CL^\delta \sum _{k \in \Lambda } \left( \lambda ^{\delta } \Vert b_k \Vert _{C_x} + \lambda ^{1+\delta -r} [b_k]_{C^r_x} + \lambda ^{1-r} [b_k]_{C^{r+\delta }_x} \right) \\&\le C L^\delta L_n^{7/2} \lambda ^{\delta } \mu ^{-1} \delta _n^{1/2} \\&\quad + C L^{r+\delta } L_n^{3r} \lambda ^{1+\delta -r} \mu ^{r-1} \delta _n^{1/2} \varsigma _{n+1}^{(r-1)(\alpha -1)} (D_n \ell ^{1-r} + \varsigma _{n+1}^{\alpha -1}) \\&\quad + C L^{r+2\delta } L_n^{3r+3\delta } \lambda ^{1-r} \mu ^{r+\delta -1} \delta _n^{1/2} \varsigma _{n+1}^{(r+\delta -1)(\alpha -1)} (D_n \ell ^{1-r-\delta } + \varsigma _{n+1}^{\alpha -1}) \\&\le C L^{r+2\delta } L_n^{3r+3\delta } \lambda ^{\delta } \mu ^{-1} \delta _n^{1/2}, \end{aligned}$$

where we have used the relation $\lambda ^{r-2} \ge \mu ^r \varsigma _{n+1}^{r(\alpha -1)} (D_n \ell ^{-r} + \varsigma _{n+1}^{\alpha -1})$ coming from (3.2) and the choice of r.

To estimate the $C^1_x$ norm, we need to take the space derivative of $\mathring{R}^{trans}_1$ in the first place. With a slight abuse on notation, we can write

$$\begin{aligned} \mathring{R}^{trans}_1&= \lambda {\mathcal {R}}^{\phi _{n+1}} \left( \sum _{k \in \Lambda } (b_k \Omega _k^\lambda ) \circ \phi _{n+1} \right) = \lambda \left( {\mathcal {R}} \sum _{k \in \Lambda } b_k \Omega _k^\lambda \right) \circ \phi _{n+1}, \end{aligned}$$

(4.11)

where the second identity comes from the very definition of ${\mathcal {R}}^{\phi _{n+1}}$, ${\mathcal {R}}$ being the inverse divergence operator defined in [15]. Therefore, by Lemma C.1 the following inequality holds true:

$$\begin{aligned} \Vert \mathring{R}^{trans}_1 \Vert _{C_{\le {\mathfrak {t}}_L}C^1_x} \le C L \lambda \left\| {\mathcal {R}} \sum _{k \in \Lambda } b_k \Omega _k \right\| _{C_{\le {\mathfrak {t}}_L}C^1_x}, \end{aligned}$$

and since ${\mathcal {R}}$ commutes with every directional derivative⁶$\partial _{x_i}$, $i=1,2,3$:

$$\begin{aligned} \partial _{x_i} {\mathcal {R}} \sum _{k \in \Lambda } b_k \Omega _k^\lambda&= {\mathcal {R}} \sum _{k \in \Lambda } \partial _{x_i}(b_k \Omega _k^\lambda ) = {\mathcal {R}} \sum _{k \in \Lambda } (\partial _{x_i}b_k) \Omega _k^\lambda + i \lambda k_i {\mathcal {R}} \sum _{k \in \Lambda } b_k \Omega _k^\lambda , \end{aligned}$$

we can use the stationary phase Lemma [16, Proposition 4.4] to get

$$\begin{aligned} \Vert \mathring{R}^{trans}_1\Vert _{C_{\le {\mathfrak {t}}_L}C^1_x}&\le CL \lambda \Vert \mathring{R}^{trans}_1 \Vert _{C_{\le {\mathfrak {t}}_L}C_x} + CL \lambda \max _{i=1,2,3}\left\| {\mathcal {R}} \sum _{k \in \Lambda } (\partial _{x_i} b_k) \Omega _k^\lambda \right\| _{C_{\le {\mathfrak {t}}_L}C_x} \\&\le CL \lambda \Vert \mathring{R}^{trans}_1 \Vert _{C_{\le {\mathfrak {t}}_L}C_x} \\&\quad + CL \sum _{k \in \Lambda } \left( \lambda ^{\delta } [ b_k ]_{C^1_x} + \lambda ^{1+\delta -r} [b_k]_{C^{r+1}_x} + \lambda ^{1-r} [b_k]_{C^{r+1+\delta }_x} \right) \\&\le C L^{r+1+2\delta } L_n^{3r+3\delta } \lambda ^{\delta } \mu ^{-1} \delta _n^{1/2} \\&\quad + CL^2L_n^3 \lambda ^{\delta } \delta _n^{1/2} \varsigma _{n+1}^{\alpha -1} + CL^{r+2}L_n^{3r+3} \lambda ^{1+\delta -r} \mu ^{r} \delta _n^{1/2} \varsigma _{n+1}^{r(\alpha -1)} (D_n \ell ^{-r} + \varsigma _{n+1}^{\alpha -1}) \\&\quad + CL^{r+2+\delta }L_n^{3r+3\delta +3} \lambda ^{1-r}\mu ^{r+\delta } \delta _n^{1/2} \varsigma _{n+1}^{(r+\delta )(\alpha -1)} (D_n \ell ^{-r-\delta } + \varsigma _{n+1}^{\alpha -1}) \\&\le CL^{r+2+\delta }L_n^{3r+3\delta +3} \lambda ^{\delta } \delta _n^{1/2} \varsigma _{n+1}^{\alpha -1}. \end{aligned}$$

Let us move to the term $\mathring{R}^{trans}_2$. Spatial Hölder norms are dealt with using again the stationary phase Lemma. We have, omitting details for the sake of brevity (estimates on $a_k$ are as usual given by Proposition 4.1)

$$\begin{aligned} \Vert \mathring{R}^{trans}_2 \Vert _{C_{\le {\mathfrak {t}}_L}C^\delta _x}&\le CL^{1+\delta } \sum _{k \in \Lambda } \left( \lambda ^{\delta -1} \Vert v_\ell \cdot \nabla _y a_k \Vert _{C_x} \right. \\&\quad \left. + L^r\lambda ^{\delta -r} [v_\ell \cdot \nabla _y a_k]_{C^r_x} + L^{r+\delta }\lambda ^{-r} [v_\ell \cdot \nabla _y a_k]_{C^{r+\delta }_x} \right) \\&\le C L^{r+2+2\delta }L_n^{3r+3\delta +4} \lambda ^{\delta -1} \mu \delta _n^{1/2} \varsigma _{n+1}^{\alpha -1}, \end{aligned}$$

as well as (the space derivative is taken as for the term $\mathring{R}^{trans}_1$)

$$\begin{aligned} \Vert \mathring{R}^{trans}_2\Vert _{C_{\le {\mathfrak {t}}_L}C^1_x}&\le CL^2 \lambda \Vert \mathring{R}^{trans}_2 \Vert _{C_{\le {\mathfrak {t}}_L}C_x} \\&\quad + C L^{2+\delta }\sum _{k \in \Lambda } \left( \lambda ^{\delta -1} [v_\ell \cdot \nabla _y a_k]_{C^1_x} + L^r \lambda ^{\delta -r} [v_\ell \cdot \nabla _y a_k]_{C^{r+1}_x}\right. \\&\quad \left. + L^{r+\delta }\lambda ^{-r} [v_\ell \cdot \nabla _y a_k]_{C^{r+1+\delta }_x} \right) \\&\le C L^{r+4+2\delta }L_n^{3r+3\delta +7} \lambda ^{\delta } \mu \delta _n^{1/2} \varsigma _{n+1}^{\alpha -1}. \end{aligned}$$

We have used relations $\lambda ^{r-1} \ge \mu ^{r+3} \varsigma _{n+1}^{(r+3)(\alpha -1)} (D_n \ell ^{-r-2} + \varsigma _{n+1}^{\alpha -1})$ and $\lambda \ge \mu (D_n \ell ^{-1} + \varsigma _{n+1}^{\alpha -1})$ to highlight one single term on the right-hand-side of each of the previous inequalities.

Finally, for $\mathring{R}^{trans}_3$ we have

$$\begin{aligned} \Vert \mathring{R}^{trans}_3 \Vert _{C_{\le {\mathfrak {t}}_L}C^\delta _x}&\le CL^\delta \sum _{k \in \Lambda } \left( \lambda ^{\delta -1} \Vert \partial _s a_k \Vert _{C_x} + L^r \lambda ^{\delta -r} [\partial _s a_k]_{C^r_x} + L^{r+\delta }\lambda ^{-r} [\partial _s a_k]_{C^{r+\delta }_x} \right) \\&\le C L^{r+2\delta } L_n^{3r+3\delta +5/2} \lambda ^{\delta -1} \mu \delta _n^{1/2} \varsigma _{n+1}^{\alpha -2}, \end{aligned}$$

and

$$\begin{aligned}&\Vert \mathring{R}^{trans}_3 \Vert _{C_{\le {\mathfrak {t}}_L}C^1_x}\\&\quad \le CL \lambda \Vert \mathring{R}^{trans}_3 \Vert _{C_{\le {\mathfrak {t}}_L}C_x} \\&\qquad + CL^{1+\delta } \sum _{k \in \Lambda } \left( \lambda ^{\delta -1} [ \partial _s a_k ]_{C^1_x} + L^r \lambda ^{\delta -r} [\partial _s a_k]_{C^{r+1}_x} + L^{r+\delta } \lambda ^{-r} [\partial _s a_k]_{C^{r+\delta +1}_x} \right) \\&\quad \le C L^{r+1+2\delta } L_n^{3r+3\delta +11/2} \lambda ^{\delta -1} \mu ^{2} \delta _n^{1/2} \varsigma _{n+1}^{2\alpha -3}. \end{aligned}$$

Let us recollect what we have proved. For $\Vert \mathring{R}^{trans}\Vert _{C_{\le {\mathfrak {t}}_L}C_x}$ we have for $\delta$ sufficiently small and $m \ge 2r$

$$\begin{aligned} \Vert \mathring{R}^{trans}\Vert _{C_{\le {\mathfrak {t}}_L}C_x}&\le C L^{r+2\delta } L_n^{3r+3\delta } \lambda ^{\delta } \mu ^{-1} \delta _n^{1/2} + C L^{r+2+2\delta }L_n^{3r+3\delta +4} \lambda ^{\delta -1} \mu \delta _n^{1/2} \varsigma _{n+1}^{\alpha -1} \\&\quad + C L^{r+2\delta } L_n^{3r+3\delta +5/2} \lambda ^{\delta -1} \mu \delta _n^{1/2} \varsigma _{n+1}^{\alpha -2} \\&\le CL_n^{3r+5}\lambda ^{\delta } \mu ^{-1} \delta _n^{1/2}, \end{aligned}$$

where we have used the relation $\lambda \ge \mu ^2 \varsigma _{n+1}^{\alpha -2}$. Moreover, the same inequality also implies

$$\begin{aligned} \Vert \mathring{R}^{trans}\Vert _{C_{\le {\mathfrak {t}}_L}C^1_x}&\le CL^{r+2+\delta }L_n^{3r+3\delta +3} \lambda ^{\delta } \delta _n^{1/2} \varsigma _{n+1}^{\alpha -1} + C L^{r+4+2\delta }L_n^{3r+3\delta +7} \lambda ^{\delta } \mu \delta _n^{1/2} \varsigma _{n+1}^{\alpha -1} \\&\quad + C L^{r+1+2\delta } L_n^{3r+3\delta +11/2} \lambda ^{\delta -1} \mu ^{2} \delta _n^{1/2} \varsigma _{n+1}^{2\alpha -3} \\&\le C L_n^{3r+8} \lambda ^{\delta } \mu \delta _n^{1/2} \varsigma _{n+1}^{\alpha -1}. \end{aligned}$$

$\square$

4.2 Estimate on the mollification error

The mollification error in (3.17) is divided into two contributions. The first one is due to the fact that we have replaced $v_n$ with $v_\ell$ in the construction of $v_{n+1}$, but convolution with $\chi _\ell$ does not commute with the tensor product. The second one, instead, comes from the fact that the space-time differential operator $\mathord {\textrm{div}}\,^{\phi _n}$ does not commute with the convolution with $\chi _\ell$, which is strikingly different from what happens in the deterministic case.

Let us denote $\mathring{R}^{moll} :=\mathring{R}^{moll}_1 +\mathring{R}^{moll}_2$, where we define

$$\begin{aligned} \mathring{R}^{moll}_1&:={\mathcal {R}}^{\phi _{n+1}}\mathord {\textrm{div}}^{\phi _{n+1}}(v_\ell \otimes v_\ell - (v_n \otimes v_n) *\chi _\ell ), \\ \mathring{R}^{moll}_2&:={\mathcal {R}}^{\phi _{n+1}} \left( \mathord {\textrm{div}}\,^{\phi _n}\left( \left( v_n \otimes v_n\right) *\chi _\ell + q_\ell Id - \mathring{R}_\ell \right) \right. \\&\quad \left. - \left( \mathord {\textrm{div}}\,^{\phi _n} \left( v_n \otimes v_n + q_n Id - \mathring{R}_n \right) \right) *\chi _\ell \right) . \end{aligned}$$

To better control the second term we will need the following.

Lemma 1.18

There exists a constant C depending only on $K_0$ and $\chi$ with the following property. Let $n \in \mathbb {N}$ and $L \in \mathbb {N}$, $L\ge 1$ be fixed and let $f:\mathbb {T}^3 \times \mathbb {R}\rightarrow \mathbb {R}^3$ be of class $C_{\le {\mathfrak {t}}_L} C^1_x$. Then, denoting $G :=\mathord {\textrm{div}}\,^{\phi _n} (f*\chi _\ell ) - (\mathord {\textrm{div}}\,^{\phi _n} f)*\chi _\ell$ it holds

$$\begin{aligned} \Vert G \Vert _{C_{\le {\mathfrak {t}}_L}C_x}&\le CL^2 \Vert f\Vert _{C_{\le {\mathfrak {t}}_L}C^1_x} \ell ^{\alpha }, \\ \Vert G \Vert _{C_{\le {\mathfrak {t}}_L}C^1_x}&\le CL^2 \Vert f\Vert _{C_{\le {\mathfrak {t}}_L}C^1_x} \ell ^{-1}. \end{aligned}$$

Proof

For fixed $x \in \mathbb {T}^3$, $t\in \mathbb {R}$ we have (here we denote $\text{ div}_y$, $\partial _{y_k}$ etc. the derivatives with respect to the space variable)

$$\begin{aligned} \left( \mathord {\textrm{div}}\,^{\phi _n} f \right) (x,t)&:=\left( \mathord {\textrm{div}}\,_y(f(\phi _{n}^{-1}(y,t),t)) \right) \mid _{y=\phi _n(x,t)} \\&= \left( \sum _{k=1}^3 \partial _{y_k}(f(\phi _{n}^{-1}(y,t),t)) \right) \mid _{y=\phi _n(x,t)} \\&= \left( \sum _{k,j=1}^3 (\partial _{y_j}f)(\phi _{n}^{-1}(y,t),t) (\partial _{y_k} (\phi _n^{-1})^j)(y,t) \right) \mid _{y=\phi _n(x,t)} \\&= \sum _{k,j=1}^3 (\partial _{y_j}f)(x,t) (\partial _{y_k} (\phi _n^{-1})^j)(\phi _n(x,t),t) . \end{aligned}$$

Therefore we can compute

$$\begin{aligned} \left( (\mathord {\textrm{div}}\,^{\phi _n} f)*\chi _\ell \right) (x,t)&= \int _{\mathbb {T}^3 \times \mathbb {R}} \left( \mathord {\textrm{div}}\,^{\phi _n} f \right) (z,s) \chi _\ell (x-z,t-s) dzds \\&= \int _{\mathbb {T}^3 \times \mathbb {R}} \sum _{k,j=1}^3 (\partial _{y_j}f)(z,s) (\partial _{y_k}\\&\quad (\phi _n^{-1})^j)(\phi _n(z,s),s) \chi _\ell (x-z,t-s) dzds, \end{aligned}$$

whereas on the other hand

$$\begin{aligned}&\left( \mathord {\textrm{div}}\,^{\phi _n} (f*\chi _\ell ) \right) (x,t) \\ {}&\quad = \sum _{k,j=1}^3 (\partial _{y_j}(f*\chi _\ell ))(x,t) (\partial _{y_k} (\phi _n^{-1})^j)(\phi _n(x,t),t) \\&\quad = \sum _{k,j=1}^3 (\partial _{y_j}f*\chi _\ell )(x,t) (\partial _{y_k} (\phi _n^{-1})^j)(\phi _n(x,t),t) \\&\quad = \int _{\mathbb {T}^3 \times \mathbb {R}} \sum _{k,j=1}^3 (\partial _{y_j}f)(z,s) (\partial _{y_k} (\phi _n^{-1})^j)(\phi _n(x,t),t) \chi _\ell (x-z,t-s) dzds. \end{aligned}$$

Thus, it holds for every $x\in \mathbb {T}^3$ and $t \in \mathbb {R}$

$$\begin{aligned} | G(x,t) |&\le \int _{\mathbb {T}^3 \times \mathbb {R}} \sum _{k,j=1}^3 |(\partial _{y_j}f)(z,s)| \left| (\partial _{y_k} (\phi _n^{-1})^j)(\phi _n(x,t),t)-(\partial _{y_k} (\phi _n^{-1})^j)(\phi _n(z,s),s) \right| \chi _\ell (x-z,t-s) dzds \\&\le C \Vert f\Vert _{C_{\le {\mathfrak {t}}_L}C^1_x} \int _{\mathbb {T}^3 \times \mathbb {R}} \sum _{k,j=1}^3 \left| (\partial _{y_k} (\phi _n^{-1})^j)(\phi _n(x,t),t)-(\partial _{y_k} (\phi _n^{-1})^j)(\phi _n(z,t),t) \right| \chi _\ell (x-z,t-s) dzds \\&\quad + C \Vert f\Vert _{C_{\le {\mathfrak {t}}_L}C^1_x} \int _{\mathbb {T}^3 \times \mathbb {R}} \sum _{k,j=1}^3 \left| (\partial _{y_k} (\phi _n^{-1})^j)(\phi _n(z,t),t)-(\partial _{y_k} (\phi _n^{-1})^j)(\phi _n(z,s),s) \right| \chi _\ell (x-z,t-s) dzds \\&\le C \Vert f\Vert _{C_{\le {\mathfrak {t}}_L}C^1_x} \int _{\mathbb {T}^3 \times \mathbb {R}} \sum _{k,j=1}^3 \left\| (\partial _{y_k} (\phi _n^{-1})^j) \circ \phi _n \right\| _{C_{\le {\mathfrak {t}}_L}C^1_x} |x-z| \chi _\ell (x-z,t-s) dzds \\&\quad + C \Vert f\Vert _{C_{\le {\mathfrak {t}}_L}C^1_x} \int _{\mathbb {T}^3 \times \mathbb {R}} \sum _{k,j=1}^3 \left\| (\partial _{y_k} (\phi _n^{-1})^j) \circ \phi _n \right\| _{C^{\alpha }_{\le {\mathfrak {t}}_L}C_x} |t-s|^\alpha \chi _\ell (x-z,t-s) dzds. \end{aligned}$$

By Lemmas C.1, C.3 and 2.2 we have

$$\begin{aligned} \left\| (\partial _{y_k} (\phi _n^{-1})^j) \circ \phi _n \right\| _{C_{\le {\mathfrak {t}}_L}C^1_x}&\le CL \left\| \partial _{y_k} (\phi _n^{-1})^j \right\| _{C_{\le {\mathfrak {t}}_L}C^1_x} \le CL \left\| \phi _n^{-1} \right\| _{C_{\le {\mathfrak {t}}_L}C^2_x} \le CL^2, \\ \left\| (\partial _{y_k} (\phi _n^{-1})^j) \circ \phi _n \right\| _{C^{\alpha }_{\le {\mathfrak {t}}_L}C_x}&\le CL \left\| \partial _{y_k} (\phi _n^{-1})^j \right\| _{C_{\le {\mathfrak {t}}_L}C^1_x} + \left\| \partial _{y_k} (\phi _n^{-1})^j \right\| _{C^{\alpha }_{\le {\mathfrak {t}}_L}C_x} \\&\le CL \left\| \phi _n^{-1} \right\| _{C_{\le {\mathfrak {t}}_L}C^2_x} + \left\| \phi _n^{-1} \right\| _{C^{\alpha }_{\le {\mathfrak {t}}_L}C^1_x} \le CL^2, \end{aligned}$$

and since $|x-z|,|t-s| \le \ell$ in the support of $\chi _\ell$ and $\int \chi _\ell =1$ we get the first inequality.

Let us bound G in $C_{\le {\mathfrak {t}}_L}C^1_x$. It holds for every $x,y \in \mathbb {T}^3$ and $t \in \mathbb {R}$

$$\begin{aligned}&G(x,t)-G(y,t) \\&\quad = \int _{\mathbb {T}^3 \times \mathbb {R}} \sum _{k,j=1}^3 (\partial _{y_j}f)(z,s) \left( (\partial _{y_k} (\phi _n^{-1})^j)(\phi _n(x,t),t)-(\partial _{y_k} (\phi _n^{-1})^j)(\phi _n(z,s),s) \right) \chi _\ell (x-z,t-s) dzds \\&\qquad - \int _{\mathbb {T}^3 \times \mathbb {R}} \sum _{k,j=1}^3 (\partial _{y_j}f)(z,s) \left( (\partial _{y_k} (\phi _n^{-1})^j)(\phi _n(y,t),t)-(\partial _{y_k} (\phi _n^{-1})^j)(\phi _n(z,s),s) \right) \chi _\ell (y-z,t-s) dzds \\&\quad = \int _{\mathbb {T}^3 \times \mathbb {R}} \sum _{k,j=1}^3 (\partial _{y_j}f)(z,s) \left( (\partial _{y_k} (\phi _n^{-1})^j)(\phi _n(x,t),t)-(\partial _{y_k} (\phi _n^{-1})^j)(\phi _n(y,t),t) \right) \chi _\ell (x-z,t-s) dzds \\&\qquad - \int _{\mathbb {T}^3 \times \mathbb {R}} \sum _{k,j=1}^3 (\partial _{y_j}f)(z,s) \left( (\partial _{y_k} (\phi _n^{-1})^j)(\phi _n(y,t),t)\right. \\&\qquad \left. -(\partial _{y_k} (\phi _n^{-1})^j)(\phi _n(z,s),s) \right) (\chi _\ell (y-z,t-s)-\chi _\ell (x-z,t-s) )dzds, \end{aligned}$$

implying, since the measure of the support of $\chi _\ell (y-z,t-s)-\chi _\ell (x-z,t-s)$ is of order $\ell ^4$,

$$\begin{aligned} |G(x,t)-G(y,t)|&\le C\Vert f\Vert _{C_{\le {\mathfrak {t}}_L}C^1_x} \left\| (\partial _{y_k} (\phi _n^{-1})^j) \circ \phi _n \right\| _{C_{\le {\mathfrak {t}}_L}C^1_x} |x-y| \\&\quad + C\Vert f\Vert _{C_{\le {\mathfrak {t}}_L}C^1_x} \left\| (\partial _{y_k} (\phi _n^{-1})^j) \circ \phi _n \right\| _{C_{\le {\mathfrak {t}}_L}C_x} \Vert \chi _\ell \Vert _{C_{\le {\mathfrak {t}}_L}C^1_x} \ell ^4 |x-y| \\&\le CL^2\Vert f\Vert _{C_{\le {\mathfrak {t}}_L}C^1_x} |x-y| + CL\Vert f\Vert _{C_{\le {\mathfrak {t}}_L}C^1_x} \ell ^{-1} |x-y|. \end{aligned}$$

Thus $[ G ]_{C_{\le {\mathfrak {t}}_L}C^1_x} \le CL^2 \Vert f\Vert _{C_{\le {\mathfrak {t}}_L}C^1_x} \ell ^{-1}$ and the proof is complete. $\square$

We are now ready to prove the following:

Proposition 1.19

For every $\delta \in (0,1)$ there exists a constant C such that almost surely for every $L \in \mathbb {N}$, $L \ge 1$

$$\begin{aligned} \Vert \mathring{R}^{moll}\Vert _{C_{\le {\mathfrak {t}}_L} C_x}&\le C L_n^3 D_n \ell ^{\alpha } , \\ \Vert \mathring{R}^{moll}\Vert _{C_{\le {\mathfrak {t}}_L} C^1_x}&\le C L_n^3 D_n \ell ^{-\delta } . \end{aligned}$$

Proof

By (C.5) for every $\delta >0$ there exists a constant C such that almost surely

$$\begin{aligned} \Vert \mathring{R}^{moll}_1\Vert _{C_{\le {\mathfrak {t}}_L} C_x}&\le CL^{2\delta } \Vert v_\ell \otimes v_\ell - (v_n \otimes v_n) *\chi _\ell \Vert _{C_{\le {\mathfrak {t}}_L} C^{\delta }_x} \\&\le CL^{2\delta } \ell ^{1-\delta } \Vert v_n \otimes v_n\Vert _{C^1_{\le {\mathfrak {t}}_L,x}} \\&\le CL^{2\delta }\ell ^{1-\delta } \Vert v_n\Vert _{C_{\le {\mathfrak {t}}_L}C_x}\Vert v_n\Vert _{C^1_{\le {\mathfrak {t}}_L,x}} \\&\le CL^{2\delta }L_n^2 \ell ^{1-\delta } D_n, \\ \Vert \mathring{R}^{moll}_1\Vert _{C_{\le {\mathfrak {t}}_L} C^1_x}&\le CL^{2+2\delta } \Vert v_\ell \otimes v_\ell - (v_n \otimes v_n) *\chi _\ell \Vert _{C_{\le {\mathfrak {t}}_L}C^{1+\delta }_x} \\&\le CL^{2+2\delta } \ell ^{-\delta } \Vert v_n \otimes v_n\Vert _{C^1_{\le {\mathfrak {t}}_L,x}} \\&\le CL^{2+2\delta }L_n^2 \ell ^{-\delta } D_n . \end{aligned}$$

To control $\mathring{R}^{moll}_2$, we apply Lemma 4.4 with $f:=v_n \otimes v_n + q_n Id - \mathring{R}_n$ (or rather the rows of the same matrix field) and use (C.7), the inequality $\Vert G\Vert _{C_{\le {\mathfrak {t}}_L} B^{\beta /\alpha -1}_{\infty ,\infty }} \le C \Vert G\Vert _{C_{\le {\mathfrak {t}}_L} C_x}$ (see [45, Remark A.3]), and (C.4) respectively to get

$$\begin{aligned} \Vert \mathring{R}^{moll}_2 \Vert _{C_{\le {\mathfrak {t}}_L}C^\delta _x}&\le CL^{5+4\delta } \Vert G\Vert _{C_{\le {\mathfrak {t}}_L}B^{\delta -1}_{\infty ,\infty }} \le CL^{5+4\delta } \Vert G\Vert _{C_{\le {\mathfrak {t}}_L}C_x} \le CL^{7+4\delta } L_n^2 D_n \ell ^{\alpha }, \\ \Vert \mathring{R}^{moll}_2 \Vert _{C_{\le {\mathfrak {t}}_L}C^{1+\delta }_x}&\le CL^{1+2\delta } \Vert G\Vert _{C_{\le {\mathfrak {t}}_L}C^\delta _x} \le CL^{3+2\delta } L_n^2 D_n . \end{aligned}$$

$\square$

4.3 Estimate on the oscillation error

In the decomposition (3.17) above, the introduction of the incremental pressure term $- \tfrac{1}{2}\nabla ^{\phi _{n+1}}(|w_o|^2-{\tilde{\rho }}_\ell ) = -\tfrac{1}{2}\mathord {\textrm{div}}^{\phi _{n+1}}((|w_o|^2-{\tilde{\rho }}_\ell )Id)$ may seem arbitrary. We shall see now that this choice contributes to produce better estimates for the (inverse divergence of the) oscillation error term, thus justifying our choice of the pressure $q_{n+1}$.

Let $W=W(y,s,\xi ,\tau )$ be defined by (3.10), and recall from Corollary 4.2 the decomposition

$$\begin{aligned} W \otimes W (y,s,\xi ,\tau ) = R_\ell (y,s) + \sum _{1 \le |k| \le 2\lambda _0} U_k(y,s,\tau ) e^{i k \cdot \xi }, \end{aligned}$$

with $U_k \in C^\infty _{loc}(\mathbb {T}^3 \times \mathbb {R}^2, {\mathcal {S}}^{3 \times 3})$, $k \in \Lambda$, satisfying $U_k k = \tfrac{1}{2} Tr(U_k)k$. Using the identity $w_o(x,t) = W(x,t,\lambda \phi _{n+1}(x,t),\lambda t)$ we can rewrite the previous line as

$$\begin{aligned} w_o \otimes w_o (x,t) = R_\ell (x,t) + \sum _{1 \le |k| \le 2\lambda _0} U_k(x,t,\lambda t)\, \Omega _k( \lambda \phi _{n+1}(x,t)). \end{aligned}$$

Therefore,

$$\begin{aligned} \mathord {\textrm{div}}^{\phi _{n+1}}&\left( w_o \otimes w_o - \frac{1}{2}\left( |w_o|^2-{\tilde{\rho }}_\ell \right) Id +\mathring{R}_\ell \right) \\&= \mathord {\textrm{div}}^{\phi _{n+1}}\left( w_o \otimes w_o - \frac{1}{2}\left( |w_o|^2-{\rho }_\ell \right) Id +\mathring{R}_\ell \right) \\&= \mathord {\textrm{div}}^{\phi _{n+1}}\left( w_o \otimes w_o - R_\ell - \frac{1}{2}\left( |w_o|^2-Tr(R_\ell ) \right) Id \right) \\&= \sum _{1 \le |k| \le 2\lambda _0} \mathord {\textrm{div}}^{\phi _{n+1}}\left( U_k - \tfrac{1}{2} Tr(U_k) Id \right) (x,t,\lambda t) \Omega _k( \lambda \phi _{n+1}(x,t)), \end{aligned}$$

where in the last line we have used the relation $U_k k = \tfrac{1}{2} Tr(U_k)k$.

Proposition 1.20

Let us denote $\mathring{R}^{osc} :={\mathcal {R}}^{\phi _{n+1}} \mathord {\textrm{div}}^{\phi _{n+1}}(w_o \otimes w_o - \frac{1}{2}\left( |w_o|^2-{\tilde{\rho }}_\ell \right) Id +\mathring{R}_\ell )$. Then for every $r \ge r_\star +1$ and $\delta >0$ sufficiently small there exists a constant C such that almost surely for every $L \in \mathbb {N}$, $L \ge 1$

$$\begin{aligned} \Vert \mathring{R}^{osc} \Vert _{C_{\le {\mathfrak {t}}_L} C_x}&\le CL_n^{3r+6} \lambda ^{\delta -1} \mu \delta _n \varsigma _{n+1}^{\alpha -1}, \\ \Vert \mathring{R}^{osc} \Vert _{C_{\le {\mathfrak {t}}_L} C^1_x}&\le C L_n^{3r+9} \lambda ^{\delta } \mu \delta _n \varsigma _{n+1}^{\alpha -1} . \end{aligned}$$

Proof

Let us start with the estimate in $C_{\le {\mathfrak {t}}_L}C_x$. We apply Proposition C.5 with $r \ge r_\star +1$ and $\delta >0$ sufficiently small to obtain for every $L \in \mathbb {N}$, $N \ge 1$

$$\begin{aligned} \Vert \mathring{R}^{osc} \Vert _{C_{\le {\mathfrak {t}}_L}C^\delta _x}&\le CL^{\delta +1} \\ {}&\quad \sum _{1 \le |k| \le 2\lambda _0} \left( \lambda ^{\delta -1} [U_k]_{C^1_x} + L^r \lambda ^{\delta -r} [U_k]_{C^{r+1}_x} + L^{r+\delta } \lambda ^{-r} [U_k]_{C^{r+1+\delta }_x} \right) , \end{aligned}$$

where we denote $[U_k]_{C^\cdot _x}=[U_k(\cdot ,t,\lambda t)]_{C^\cdot _x}$ the spatial Hölder seminorms at fixed $t \le {\mathfrak {t}}_L$. By Corollary 4.2 and interpolation inequality

$$\begin{aligned} \Vert \mathring{R}^{osc} \Vert _{C_{\le {\mathfrak {t}}_L}C^\delta _x}&\le CL^{r+2\delta +1} L_n^{3r+3\delta +11/2} \lambda ^{\delta -1} \mu \delta _n \varsigma _{n+1}^{\alpha -1}. \end{aligned}$$

The previous bound makes use of $\lambda ^{r-1} \ge \mu ^{r+1} \varsigma _{n+1}^{r(\alpha -1)} (D_n \ell ^{-r-1} + \varsigma _{n+1}^{\alpha -1})$.

The same arguments used to control the $C^1_x$ norm of $\mathring{R}^{trans}$ also imply (use that for our choice of r it holds $\lambda ^{r-1} \ge \mu ^{r+3} \varsigma _{n+1}^{(r+3)(\alpha -1)}(D_n \ell ^{1-r}+\varsigma _{n+1}^{\alpha -1}) \ge \mu ^{r+2} \varsigma _{n+1}^{(r+2)(\alpha -1)}(D_n \ell ^{1-r}+\varsigma _{n+1}^{\alpha -1})^2$ )

$$\begin{aligned} \Vert \mathring{R}^{osc} \Vert _{C_{\le {\mathfrak {t}}_L}C^1_x}&\le CL \lambda \Vert \mathring{R}^{osc} \Vert _{C_{\le {\mathfrak {t}}_L}C_x} \\&\quad + CL^2 \sum _{1 \le |k| \le 2\lambda _0}\left( \lambda ^{\delta -1} [U_k]_{C^2_x} + L^r \lambda ^{\delta -r} [U_k]_{C^{r+2}_x} + L^{r+\delta } \lambda ^{-r} [U_k]_{C^{r+2+\delta }_x} \right) \\&\le C L_n^{3r+9} \lambda ^{\delta } \mu \delta _n \varsigma _{n+1}^{\alpha -1}. \end{aligned}$$

$\square$

4.4 Estimate on the flow error

The flow error in the decomposition (3.17) is peculiar of the particular construction carried on in the present paper. It is due to the fact that the Euler-Reynolds systems (1.6) at level n and $n+1$ are obtained composing with flows $\phi _n$ and $\phi _{n+1}$, respectively.

The key lemma we are going to use in this subsection is Lemma 4.7 below.

Lemma 1.21

For every $\delta \in (0,1)$, $\alpha ' \in (0,\alpha )$ there exists a constant C with the following property. For every $n \in \mathbb {N}$, given any smooth vector field $v \in C^\infty (\mathbb {T}^3,\mathbb {R}^3)$ on the torus and denoting $G :=\left( \mathord {\textrm{div}}^{\phi _{n+1}}-\mathord {\textrm{div}}\,^{\phi _n}\right) v$, almost surely for every $L \in \mathbb {N}$, $L \ge 1$ it holds

$$\begin{aligned} \Vert G\Vert _{C_{\le {\mathfrak {t}}_L} B^{\delta -1}_{\infty ,\infty }}&\le C L^3 (n+1)\varsigma _n^{\alpha '} \Vert v \Vert _{C_{\le {\mathfrak {t}}_L} C^{\delta }_x}, \\ \Vert G\Vert _{C_{\le {\mathfrak {t}}_L} C^\delta _x}&\le C L^3 (n+1)\varsigma _n^{\alpha '} \Vert v \Vert _{C_{\le {\mathfrak {t}}_L} C^{1+\delta }_x} , \\ \Vert G\Vert _{C_{\le {\mathfrak {t}}_L} C^{1+\delta }_x}&\le C L^4 (n+1)\varsigma _n^{\alpha '} \Vert v \Vert _{C_{\le {\mathfrak {t}}_L} C^{2+\delta }_x} . \end{aligned}$$

Proof

For $\phi =\phi _n$ or $\phi =\phi _{n+1}$ it holds

$$\begin{aligned} \mathord {\textrm{div}}^\phi v = \sum _{j=1}^3 \partial _{x_j} (v^{j} \circ \phi ^{-1}) \circ \phi = \sum _{j,k=1}^3 (\partial _{x_k} v^{j}) (\partial _{x_j} (\phi ^{-1})^k \circ \phi ). \end{aligned}$$

Therefore

$$\begin{aligned} G&= \sum _{j,k=1}^3 (\partial _{x_k} v^{j}) \left( \partial _{x_j} (\phi _{n+1}^{-1})^k \circ \phi _{n+1} - \partial _{x_j} (\phi _{n}^{-1})^k \circ \phi _{n+1} \right) \\&\quad + \sum _{j,k=1}^3 (\partial _{x_k} v^{j}) \left( \partial _{x_j} (\phi _{n}^{-1})^k \circ \phi _{n+1} - \partial _{x_j} (\phi _{n}^{-1})^k \circ \phi _{n} \right) . \end{aligned}$$

Thus by paraproduct estimates in Besov spaces [45, Proposition A.7], Lemma C.1 and Lemma 2.2 we have

$$\begin{aligned} \Vert G \Vert _{C_{\le {\mathfrak {t}}_L} B^{\delta -1}_{\infty ,\infty }}&\le C \Vert v \Vert _{C_{\le {\mathfrak {t}}_L} C^{\delta }_x} \Vert \partial _{x_j} (\phi _{n+1}^{-1})^k \circ \phi _{n+1} - \partial _{x_j} (\phi _{n}^{-1})^k \circ \phi _{n+1} \Vert _{C_{\le {\mathfrak {t}}_L} C^1_x} \\&\quad + C \Vert v \Vert _{C_{\le {\mathfrak {t}}_L} C^{\delta }_x} \Vert \partial _{x_j} (\phi _{n}^{-1})^k \circ \phi _{n+1} - \partial _{x_j} (\phi _{n}^{-1})^k \circ \phi _{n} \Vert _{C_{\le {\mathfrak {t}}_L} C^1_x} \\&\le C \Vert v \Vert _{C_{\le {\mathfrak {t}}_L} C^{\delta }_x} L\Vert \phi ^{-1}_{n+1}-\phi ^{-1}_n \Vert _{C_{\le {\mathfrak {t}}_L} C^{2}_x} \\&\quad +C \Vert v \Vert _{C_{\le {\mathfrak {t}}_L} C^\delta _x} \left( L\Vert \phi _{n+1}-\phi _n \Vert _{C_{\le {\mathfrak {t}}_L} C^{1}_x} + L^2\Vert \phi ^{-1}_{n+1}-\phi ^{-1}_n \Vert _{C_{\le {\mathfrak {t}}_L} C_x} \right) , \end{aligned}$$

where in the first line we have used that $\Vert \partial _{x_k}v\Vert _{C_{\le {\mathfrak {t}}_L} B^{\delta -1}_{\infty ,\infty }} \le C \Vert v\Vert _{C_{\le {\mathfrak {t}}_L} B^{\delta }_{\infty ,\infty }} = C \Vert v\Vert _{C_{\le {\mathfrak {t}}_L} C^{\delta }_x}$. To conclude, recall that again by Lemma 2.2 it holds

$$\begin{aligned} \Vert \phi _{n+1}^{-1}-\phi _n^{-1}\Vert _{C_{\le {\mathfrak {t}}_L} C^2_x}&\le CL (n+1) \varsigma _n^{\alpha '}, \\ \Vert \phi _{n+1}-\phi _n\Vert _{C_{\le {\mathfrak {t}}_L} C^2_x}&\le CL (n+1) \varsigma _n^{\alpha '}. \end{aligned}$$

The inequalities concerning the $C_{\le {\mathfrak {t}}_L} C^{\delta }_x$ and $C_{\le {\mathfrak {t}}_L} C^{1+\delta }_x$ norms are completely analogous, and we omit their proof. $\square$

Let us denote $F_n :=v_n \otimes v_n - \mathring{R}_n + q_n Id$ and

$$\begin{aligned} F_\ell :=F_n *\chi _\ell = (v_n \otimes v_n) *\chi _\ell - \mathring{R}_\ell + q_\ell Id. \end{aligned}$$

Notice that for every $r \ge 0$ and almost surely for every $L \in \mathbb {N}$, $L \ge 1$ it holds:

$$\begin{aligned} \Vert F_\ell \Vert _{C_{\le {\mathfrak {t}}_L}C_x^r} \le C \ell ^{-r} \Vert F_n \Vert _{C_{\le {\mathfrak {t}}_L} C_x} \le C L_n^2 \ell ^{-r}, \end{aligned}$$

where the constant C depends only on the mollifier $\chi$ and r.

Finally, define $G_\ell :=\left( \mathord {\textrm{div}}^{\phi _{n+1}}-\mathord {\textrm{div}}\,^{\phi _n}\right) F_\ell$ and let us split $\mathring{R}^{flow} :=\mathring{R}^{flow}_1 + \mathring{R}^{flow}_2$, where

$$\begin{aligned} \mathring{R}^{flow}_1&:={\mathcal {R}}^{\phi _{n+1}} G_\ell , \\ \mathring{R}^{flow}_2&:={\mathcal {R}}^{\phi _{n+1}} \left( w_o \,\mathord {\textrm{div}}^{\phi _{n+1}}v_\ell \right) . \end{aligned}$$

Proposition 1.22

Let $\mathring{R}^{flow} :=\mathring{R}^{flow}_1 + \mathring{R}^{flow}_2$ be defined as above. Then for every $r \ge r_\star +2$, $\delta \in (0,1)$ sufficiently small and $\alpha ' \in (\alpha _\star ,\alpha )$ there exists a constant C such that almost surely for every $L \in \mathbb {N}$, $L \ge 1$

$$\begin{aligned} \left\| \mathring{R}^{flow} \right\| _{C_{\le {\mathfrak {t}}_L} C_x}&\le C L_n^{3r+5} \ell ^{-\delta } (n+1)\varsigma _n^{\alpha '} \\ \left\| \mathring{R}^{flow} \right\| _{C_{\le {\mathfrak {t}}_L} C^1_x}&\le CL_n^{3r+8} \ell ^{-1-\delta } (n+1)\varsigma _n^{\alpha '} . \end{aligned}$$

Proof

Let us focus first on the term $\mathring{R}^{flow}_1$. By (C.7) and Lemma 4.7, for every $\delta >0$ one has

$$\begin{aligned} \Vert \mathring{R}^{flow}_1 \Vert _{C_{\le {\mathfrak {t}}_L} C^\delta _x}&\le CL^{5+4\delta } \Vert G_\ell \Vert _{B^{\delta -1}_{\infty ,\infty }} \\&\le CL^{8+4\delta } (n+1)\varsigma _n^{\alpha '} \Vert F_\ell \Vert _{C_{\le {\mathfrak {t}}_L} C^\delta _x} \\&\le CL^{8+4\delta }L_n^2 \ell ^{-\delta } (n+1)\varsigma _n^{\alpha '}, \end{aligned}$$

and similarly, using (C.4) instead of (C.7)

$$\begin{aligned} \Vert \mathring{R}^{flow}_1 \Vert _{C_{\le {\mathfrak {t}}_L} C^{1+\delta }_x}&\le CL^{9+3\delta } (n+1)\varsigma _n^{\alpha '} \Vert F_\ell \Vert _{C_{\le {\mathfrak {t}}_L} C^{1+\delta }_x} \\&\le CL^{9+3\delta } L_n^2 \ell ^{-1-\delta } (n+1)\varsigma _n^{\alpha '}. \end{aligned}$$

Moving to the second term $\mathring{R}^{flow}_2$, we notice that

$$\begin{aligned} w_o\, \mathord {\textrm{div}}^{\phi _{n+1}}v_\ell&= \left( \sum _{k \in \Lambda } a_k\circ \phi _{n+1}^{-1}\, \text{ div }\left( v_\ell \circ \phi _{n+1}^{-1}\right) \Omega _k^\lambda \right) \circ \phi _{n+1}, \end{aligned}$$

(4.12)

so that by Proposition C.5, Lemma C.1 and taking $r \ge r_\star + 2$

$$\begin{aligned} \Vert \mathring{R}^{flow}_2 \Vert _{C_{\le {\mathfrak {t}}_L}C^{\delta }_x}&\le C L^\delta \sum _{k \in \Lambda } \lambda ^{\delta -1} \Vert a_k\Vert _{C_x} \Vert v_\ell \Vert _{C^1_x} \\&\quad + C L^{r+1+\delta } \sum _{k \in \Lambda } \sum _{r_1+r_2=r} \lambda ^{\delta -r} \Vert a_k\Vert _{C^{r_1}_x} \Vert v_\ell \Vert _{C^{r_2+1}_x} \\&\quad + C L^{r+2+\delta } \sum _{k \in \Lambda } \sum _{r_1+r_2=r+1} \lambda ^{-r} \Vert a_k\Vert _{C^{r_1}_x} \Vert v_\ell \Vert _{C^{r_2+1}_x} \\&\le CL^{r+3}L_n^{3r+4} \lambda ^{\delta -1} \delta _n^{1/2} D_n. \end{aligned}$$

In the line above we have used, as usual, the factors $\lambda ^{-r}$ to compensate, up to a power of $L_n$, for the quantities $\Vert a_k\Vert _{C^{r_1}_x} \Vert v_\ell \Vert _{C^{r_2+1}_x}$. Taking the space derivative in (4.12) as usual, we obtain in a similar fashion

$$\begin{aligned} \Vert \mathring{R}^{flow}_2 \Vert _{C_{\le {\mathfrak {t}}_L}C^{1+\delta }_x}&\le CL \lambda \Vert \mathring{R}^{flow}_2 \Vert _{C_{\le {\mathfrak {t}}_L}C^{\delta }_x} \\&\quad + CL^3 \sum _{k \in \Lambda } \left( \lambda ^{\delta -1} [a_k]_{C^1_x} [v_\ell ]_{C^1_x} + \lambda ^{\delta -1} \Vert a_k\Vert _{C_x} [v_\ell ]_{C^2_x} \right) \\&\quad + C L^{r+2+\delta } \sum _{k \in \Lambda } \sum _{r_1+r_2=r+1} \lambda ^{\delta -r} \Vert a_k\Vert _{C^{r_1}_x} \Vert v_\ell \Vert _{C^{r_2+1}_x} \\&\quad + C L^{r+3+\delta } \sum _{k \in \Lambda } \sum _{r_1+r_2=r+2} \lambda ^{-r} \Vert a_k\Vert _{C^{r_1}_x} \Vert v_\ell \Vert _{C^{r_2+1}_x} \\&\le CL^{r+4}L_n^{3r+7} \lambda ^{\delta } \delta _n^{1/2} D_n. \end{aligned}$$

$\square$

4.5 Estimate on the compressibility error

Let us finally move to the last term to control in the decomposition (3.17), namely the compressibility error term. It is due to the fact that we have added a compressibility corrector $w_c$ to the oscillatory term $w_o$ when defining the new velocity $v_{n+1} = v_\ell + w_o + w_c$, in order to satisfy the condition on the divergence (2.8) at level $n+1$. In particular, recall we have defined $w_c = w^1_c + w^2_c$, where $w^1_c = - ({\mathcal {Q}}^{\phi _n} v_n ) *\chi _\ell$ and $w^2_c = - {\mathcal {Q}}^{\phi _{n+1}} w_o$. We will need the following preliminary:

Lemma 1.23

For every $\delta \in (0,1)$ sufficiently small there exists a constant C such that, almost surely for every $L \in \mathbb {N}$, $L \ge 1$:

$$\begin{aligned} \Vert w_c^1\Vert _{C_{\le {\mathfrak {t}}_L}C^{\delta }_x}&\le CL^{7} L_n \delta _{n+2}^{6/5}, \\ \Vert w_c^1\Vert _{C_{\le {\mathfrak {t}}_L}C^{1+\delta }_x}&\le CL^{7} L_n \delta _{n+2}^{6/5} \ell ^{-1}, \\ \Vert w^2_c \Vert _{C_{\le {\mathfrak {t}}_L}C^\delta _x}&\le C L^{1+2\delta } L_n^{3+3\delta } \lambda ^{\delta -1} \mu \delta _n^{1/2} \varsigma _{n+1}^{\alpha -1}, \\ \Vert w^2_c \Vert _{C_{\le {\mathfrak {t}}_L}C^{1+\delta }_x}&\le CL^{3+2\delta } L_n^{6+3\delta }\lambda ^{\delta } \mu \delta _n^{1/2} \varsigma _{n+1}^{\alpha -1}. \end{aligned}$$

Proof

First, notice that $\Vert w_c^1\Vert _{C_{\le {\mathfrak {t}}_L}C^{\delta }_x} \le \Vert {\mathcal {Q}}^{\phi _n}v_n\Vert _{C_{\le {\mathfrak {t}}_L}C^{\delta }_x}$ and $\Vert w_c^1\Vert _{C_{\le {\mathfrak {t}}_L}C^{1+\delta }_x} \le \ell ^{-1}\Vert {\mathcal {Q}}^{\phi _n}v_n\Vert _{C_{\le {\mathfrak {t}}_L}C^{\delta }_x}$. In addition, since $v_n$ is zero mean it holds $Q^{\phi _n}v_n = \nabla ^{\phi _n} \psi$, where $\psi$ solves $\Delta ^{\phi _n} \psi = \mathord {\textrm{div}}\,^{\phi _n} v_n$ and $\int _{\mathbb {T}^3} \psi = 0$. Therefore, assuming $\delta$ sufficiently small, by iterative assumption (2.8), Schauder estimates in Besov spaces and interpolation we have

$$\begin{aligned} \Vert Q^{\phi _n}v_n\Vert _{C_{\le {\mathfrak {t}}_L}C^\delta _x}&\le CL^{1+2\delta } \Vert \psi \Vert _{C_{\le {\mathfrak {t}}_L}C^{1+\delta }_x}\nonumber \\&\le CL^{6+2\delta } \Vert \mathord {\textrm{div}}\,^{\phi _n} v_n \Vert _{C_{\le {\mathfrak {t}}_L}B^{\delta -1}_{\infty ,\infty }}\nonumber \\&\le CL^{6+2\delta } \Vert \mathord {\textrm{div}}\,^{\phi _n} v_n \Vert ^{1-2\delta }_{C_{\le {\mathfrak {t}}_L}B^{-1}_{\infty ,\infty }} \Vert \mathord {\textrm{div}}\,^{\phi _n} v_n \Vert ^{2\delta }_{C_{\le {\mathfrak {t}}_L}B^{-1/2}_{\infty ,\infty }}\nonumber \\&\le CL^{6+10\delta } L_n \delta _{n+2}^{5(1-2\delta )/4} D_n^\delta \le CL^{7} L_n \delta _{n+2}^{6/5}, \end{aligned}$$

(4.13)

provided $\delta$ is taken sufficiently small.

As for the inequalities involving $w_c^2$, the proof is similar to [15, Lemma 6.2] and [16, Proposition 6.1]. Let us define the field $u_c: \mathbb {T}^3 \times \mathbb {R}\rightarrow \mathbb {R}^3$ as

$$\begin{aligned} u_c(x,t)&:=i \sum _{k \in \Lambda } {\nabla ^{\phi _{n+1}}a_k (x,t,\lambda t)} \times \frac{k}{|k|^2} \times \Omega _k(\lambda \phi _{n+1}(x,t)); \end{aligned}$$

since $k \cdot E_k = 0$ (recall that $\Omega _k(\xi ) = E_k e^{ik \cdot \xi }$) it is easy to check by direct verification

$$\begin{aligned} w_c^2&= \frac{1}{\lambda } {\mathcal {Q}}^{\phi _{n+1}} u_c + \frac{1}{\lambda } {\mathcal {Q}}^{\phi _{n+1}} \nabla ^{\phi _{n+1}}\times \left( \sum _{k \in \Lambda } -i a_k \frac{k}{|k|^2}\times (\Omega _k^\lambda \circ \phi _{n+1}) \right) \\&= \frac{1}{\lambda } {\mathcal {Q}}^{\phi _{n+1}} u_c = \frac{1}{\lambda } {\mathcal {Q}}(u_c \circ \phi _{n+1}^{-1}) \circ \phi _{n+1}, \end{aligned}$$

where the last line is justified by the identity ${\mathcal {Q}}^{\phi _{n+1}} (\nabla ^{\phi _{n+1}} \times v)= 0$ for every smooth vector field v. Schauder estimate then gives for $r=0,1$ and $\delta \in (0,1)$

$$\begin{aligned} \Vert w_c^2 \Vert _{C_{\le {\mathfrak {t}}_L}C^{r+\delta }_x}&\le CL^{r+\delta }\lambda ^{-1} \Vert u_c \circ \phi _{n+1}^{-1} \Vert _{C_{\le {\mathfrak {t}}_L}C^{r+\delta }_x}, \end{aligned}$$

and thus, recalling Proposition 4.1 we get

$$\begin{aligned} \Vert w_c^2 \Vert _{C_{\le {\mathfrak {t}}_L}C^{\delta }_x}&\le CL^{1+\delta }\lambda ^{-1} \sum _{k \in \Lambda } \left( [a_k]_{C^1_x} \Vert \Omega _k^\lambda \Vert _{C^\delta _x} + L^{\delta }\Vert a_k\Vert _{C^{1+\delta }_x} \Vert \Omega _k^\lambda \Vert _{C_x}\right) \\&\le CL^{1+2\delta } L_n^{3+3\delta }\lambda ^{\delta -1} \mu \delta _n^{1/2} \varsigma _{n+1}^{\alpha -1}, \end{aligned}$$

and

$$\begin{aligned} \Vert w_c^2 \Vert _{C_{\le {\mathfrak {t}}_L}C^{1+\delta }_x}&\le CL^{2+\delta }\lambda ^{-1} \sum _{k \in \Lambda } \left( [a_k]_{C^1_x} \Vert \Omega _k^\lambda \Vert _{C^{1+\delta }_x} + L^{1+\delta }\Vert a_k\Vert _{C^{2+\delta }_x} \Vert \Omega _k^\lambda \Vert _{C_x}\right) \\&\le CL^{3+2\delta }L_n^{6+3\delta }\lambda ^{\delta } \mu \delta _n^{1/2} \varsigma _{n+1}^{\alpha -1}. \end{aligned}$$

$\square$

We will also need the following:

Lemma 1.24

Fix $r\ge r_\star + 2$. Then there exists a constant C such that, almost surely for every $L \in \mathbb {N}$, $L \ge 1$:

$$\begin{aligned} \Vert w_o \Vert _{C_{\le {\mathfrak {t}}_L}C^\delta _x}&\le C L^{2\delta } L_n^{7/2} \lambda ^\delta \delta _n^{1/2}, \\ \Vert w_o \Vert _{C_{\le {\mathfrak {t}}_L}C^{1+\delta }_x}&\le C L^{2+2\delta } L_n^{3+3\delta } \lambda ^{1+\delta } \delta _n^{1/2} \mu ^\delta \varsigma _{n+1}^{\delta (\alpha -1)} , \\ \Vert \partial _t w_o \Vert _{C_{\le {\mathfrak {t}}_L} C_x}&\le C L_n^{3r+9} \lambda ^{1+\delta } \delta _n^{1/2} \mu ^\delta \varsigma _{n+1}^{\delta (\alpha -1)} . \end{aligned}$$

Proof

It holds by the very definition $w_o \circ \phi _{n+1}^{-1}= \sum _{k \in \Lambda } (a_k \circ \phi _{n+1}^{-1}) \Omega _k^\lambda$, and thus by Lemma C.1 and Proposition 4.1

$$\begin{aligned} \Vert w_o \Vert _{C_{\le {\mathfrak {t}}_L}C^\delta _x}&\le C L^\delta \sum _{k \in \Lambda } \left( \Vert a_k\Vert _{C_x} \Vert \Omega _k^\lambda \Vert _{C^\delta _x} + L^\delta \Vert a_k\Vert _{C^\delta _x} \Vert \Omega _k^\lambda \Vert _{C_x} \right) \\&\le C L^{2\delta } L_n^{7/2} \lambda ^\delta \delta _n^{1/2}, \\ \Vert w_o \Vert _{C_{\le {\mathfrak {t}}_L}C^{1+\delta }_x}&\le C L^{1+\delta } \sum _{k \in \Lambda } \left( L^\delta \Vert a_k\Vert _{C^\delta _x} \Vert \Omega _k^\lambda \Vert _{C^{1+\delta }_x} + L^{1+\delta }\Vert a_k\Vert _{C^{1+\delta }_x} \Vert \Omega _k^\lambda \Vert _{C^\delta _x} \right) \\&\le C L^{2+2\delta } L_n^{3+3\delta } \lambda ^{1+\delta } \delta _n^{1/2} \mu ^\delta \varsigma _{n+1}^{\delta (\alpha -1)}. \end{aligned}$$

This proves the first two bounds. As for the second one, recall the decomposition of the transport error $\mathring{R}^{trans}$ from Sect. 4.1. By Lemma 3.4 it holds

$$\begin{aligned} \mathord {\textrm{div}}^{\phi _{n+1}}\mathring{R}^{trans}&= \partial _t w_o + (v_\ell \cdot \nabla ^{\phi _{n+1}})w_o - \int _{\mathbb {T}^3} \partial _t w_o - \int _{\mathbb {T}^3} (v_\ell \cdot \nabla ^{\phi _{n+1}})w_o, \end{aligned}$$

implying (use (C.8) with $r=1$ to control the average of $\partial _t w_o$)

$$\begin{aligned} \Vert \partial _t w_o \Vert _{C_{\le {\mathfrak {t}}_L} C_x}&\le \Vert \mathord {\textrm{div}}^{\phi _{n+1}}\mathring{R}^{trans}\Vert _{C_{\le {\mathfrak {t}}_L} C_x} + C\Vert (v_\ell \cdot \nabla ^{\phi _{n+1}})w_o \Vert _{C_{\le {\mathfrak {t}}_L} C_x}\\ {}&\quad + \left\| \int _{\mathbb {T}^3} \partial _t w_o \right\| _{C_{\le {\mathfrak {t}}_L}} \\&\le CL \Vert \mathring{R}^{trans} \Vert _{C_{\le {\mathfrak {t}}_L} C^1_x} + CL_n\Vert w_o \Vert _{C_{\le {\mathfrak {t}}_L} C^1_x} + \left\| \int _{\mathbb {T}^3} \partial _t w_o \right\| _{C_{\le {\mathfrak {t}}_L}} \\&\le CL_n^{3r+9} \lambda ^{1+\delta } \delta _n^{1/2} \mu ^\delta \varsigma _{n+1}^{\delta (\alpha -1)}, \end{aligned}$$

completing the proof of the lemma. $\square$

We are finally ready to prove the following:

Proposition 1.25

Let us denote

$$\begin{aligned} \mathring{R}^{comp} :={\mathcal {R}}^{\phi _{n+1}} (\partial _t w_c + \mathord {\textrm{div}}^{\phi _{n+1}}(v_{n+1} \otimes w_c + w_c \otimes v_{n+1} - w_c \otimes w_c +v_\ell \otimes w_o)). \end{aligned}$$

Then for every $r \ge r_\star + 1$, $\delta >0$ sufficiently small there exists a constant C such that almost surely for every $L\in \mathbb {N}$, $L \ge 1$

$$\begin{aligned} \Vert \mathring{R}^{comp} \Vert _{C_{\le {\mathfrak {t}}_L} C_x}&\le C L_n^{3r+6} \lambda ^\delta \delta _n^{1/2} \delta _{n+2}^{6/5}, \\ \Vert \mathring{R}^{comp} \Vert _{C_{\le {\mathfrak {t}}_L} C^1_x}&\le CL_n^{3r+10} \lambda ^{1+\delta } \mu ^\delta \delta _n^{1/2} \delta _{n+2}^{6/5} \varsigma _{n+1}^{\delta (\alpha -1)} . \end{aligned}$$

Proof

We split $\mathring{R}^{comp}$ into four terms:

$$\begin{aligned} \mathring{R}^{comp} = \mathring{R}^{comp}_1 + \mathring{R}^{comp}_2 + \mathring{R}^{comp}_3 + \mathring{R}^{comp}_4, \end{aligned}$$

where we define

$$\begin{aligned} \mathring{R}^{comp}_1&:={\mathcal {R}}^{\phi _{n+1}} \partial _t w^1_c , \\ \mathring{R}^{comp}_2&:={\mathcal {R}}^{\phi _{n+1}} \partial _t w^2_c , \\ \mathring{R}^{comp}_3&:={\mathcal {R}}^{\phi _{n+1}}\mathord {\textrm{div}}^{\phi _{n+1}}\left( v_{n+1} \otimes w_c + w_c \otimes v_{n+1} - w_c \otimes w_c \right) , \\ \mathring{R}^{comp}_4&:={\mathcal {R}}^{\phi _{n+1}} \mathord {\textrm{div}}^{\phi _{n+1}}(v_\ell \otimes w_o). \end{aligned}$$

We shall provide estimates for each term separately, starting from $\mathring{R}^{comp}_1$. Since $\chi _\ell$ is supported in $[0,\ell ]^3 \times [0,\ell ] \subset [0,2\pi ]^3 \times [0,\ell ]$ and $Q^{\phi _n}v_n$ has zero spatial average at every fixed time, one has the alternative expression

$$\begin{aligned} -\partial _t w^1_c=(Q^{\phi _n}v_n) *\partial _t\chi _\ell&= \int _{\mathbb {R}^3 \times \mathbb {R}} (Q^{\phi _n}v_n)(\cdot -y,\cdot -s) \partial _t\chi _\ell (y,s) dyds \nonumber \\&= \int _{[0,2\pi ]^3 \times [0,\ell ]} (Q^{\phi _n}v_n)(\cdot -y,\cdot -s) \partial _t\chi _\ell (y,s) dyds \nonumber \\&= \int _{[0,2\pi ]^3 \times [0,\ell ]} (Q^{\phi _n}v_n)(\cdot -y,\cdot -s) \partial _t\chi ^0_\ell (y,s) dyds, \end{aligned}$$

(4.14)

where we have denoted $\partial _t\chi ^0_\ell :=\partial _t\chi _\ell - (2\pi )^{-3 }\int _{\mathbb {T}^3} \partial _t\chi _\ell$ the zero-mean version (on the torus) of the convolution kernel $\partial _t\chi _\ell$. We denote the spatial convolution on the torus by the symbol $*_{\mathbb {T}^3}$, i.e. for every fixed $t \in \mathbb {R}$

$$\begin{aligned}&\int _0^\ell (Q^{\phi _n}v_n)(\cdot ,t-s) *_{\mathbb {T}^3} \partial _t\chi ^0_\ell (\cdot ,s) ds \\ {}&\quad :=\int _{[0,2\pi ]^3 \times [0,\ell ]} (Q^{\phi _n}v_n)(\cdot -y,t-s) \partial _t\chi ^0_\ell (y,s) dyds. \end{aligned}$$

Using (C.7) we compute

$$\begin{aligned} \Vert \mathring{R}^{comp}_1 \Vert _{C_{\le {\mathfrak {t}}_L}C^\delta _x}&= \Vert {\mathcal {R}}^{\phi _{n+1}} \partial _t w^1_c \Vert _{C_{\le {\mathfrak {t}}_L}C^\delta _x} \le C L^{5+4\delta } \Vert \partial _t w^1_c \Vert _{C_{\le {\mathfrak {t}}_L}B^{\delta -1}_{\infty ,\infty }}, \end{aligned}$$

and by (4.14) above and Lemma C.6 we have for any $p>1$

$$\begin{aligned} \Vert \partial _t w^1_c \Vert _{C_{\le {\mathfrak {t}}_L}B^{\delta -1}_{\infty ,\infty }}&= \sup _{t \le {\mathfrak {t}}_L} \int _0^\ell \Vert (Q^{\phi _n}v_n)(\cdot ,t-s) *_{\mathbb {T}^3} \partial _t\chi ^0_\ell (\cdot ,s) \Vert _{B^{\delta -1}_{\infty ,\infty }} ds \\&\le C\ell \Vert Q^{\phi _n}v_n\Vert _{C_{\le {\mathfrak {t}}_L}L^\infty _x} \Vert \partial _t\chi ^0_\ell \Vert _{C_{\le {\mathfrak {t}}_L}B^{2\delta -1}_{p,\infty }}. \end{aligned}$$

Now observe that $\partial _t \chi ^0_\ell (\cdot ,s)= \ell ^{-2} \left( \ell ^{-3}\partial _t \chi (\cdot /\ell ,s/\ell ) - (2\pi )^{-3}\int _{\mathbb {T}^3} \partial _t \chi (y,s/\ell ) dy\right)$ for every $s \in \mathbb {R}$. Therefore by Lemma C.7 and taking p sufficiently close to 1 we have

$$\begin{aligned} \Vert \partial _t \chi ^0_\ell (\cdot ,s)\Vert _{B^{2\delta -1}_{p,\infty }}&= \ell ^{-2} \left\| \ell ^{-3}\partial _t \chi (\cdot /\ell ,s/\ell ) - (2\pi )^{-3}\int _{\mathbb {T}^3} \partial _t \chi (y,s/\ell ) dy\right\| _{B^{2\delta -1}_{p,\infty }} \\&= \ell ^{-2}\ell ^{3(1/p-1)+1-2\delta }\Vert \partial _t \chi (\cdot ,s/\ell )\Vert _{B^{2\delta -1}_{p,\infty }} \\&\le \ell ^{-1-3\delta }\Vert \partial _t \chi \Vert _{C_{\le {\mathfrak {t}}_L}B^{2\delta -1}_{p,\infty }} \le C \ell ^{-1-3\delta }. \end{aligned}$$

Recalling (4.13), we arrive to

$$\begin{aligned} \Vert \mathring{R}^{comp}_1 \Vert _{C_{\le {\mathfrak {t}}_L}C^\delta _x} \le C L^{12+4\delta } L_n \delta _{n+2}^{6/5} \ell ^{-3\delta }. \end{aligned}$$

As for the $C_{\le {\mathfrak {t}}_L} C^1_x$ norm of $\mathring{R}^{comp}_1$ we use (C.5) and similar arguments to get

$$\begin{aligned} \Vert \mathring{R}^{comp}_1 \Vert _{C_{\le {\mathfrak {t}}_L} C^1_x}&\le CL^{1+2\delta } \Vert \partial _t w^1_c \Vert _{C_{\le {\mathfrak {t}}_L} C^\delta _x} \le C L^{8+2\delta } L_n \delta _{n+2}^{6/5} \ell ^{-1-3\delta }. \end{aligned}$$

As for the term involving $\partial _t w^2_c$, the analysis is more involved since $w_c^2$ is not a convolution, and we first need the following observation. Recall the definition of $u_c$ from Lemma 4.9. We have

$$\begin{aligned} \partial _t w^2_c&= \frac{1}{\lambda } [{\mathcal {Q}} \partial _t (u_c \circ \phi _{n+1}^{-1})] \circ \phi _{n+1} + \frac{1}{\lambda } {\dot{\phi }}_{n+1}\cdot [(\nabla {\mathcal {Q}}(u_c \circ \phi _{n+1}^{-1})) \circ \phi _{n+1}] \\&= \frac{1}{\lambda } [{\mathcal {Q}} \partial _t (u_c \circ \phi _{n+1}^{-1})] \circ \phi _{n+1} + {\dot{\phi }}_{n+1}\cdot \nabla ^{\phi _{n+1}} w^2_c. \end{aligned}$$

As a consequence, applying the operator ${\mathcal {R}}^{\phi _{n+1}}$ to both sides of the previous equation we get

$$\begin{aligned} \mathring{R}^{comp}_2 = \frac{1}{\lambda } [{\mathcal {R}}{\mathcal {Q}} \partial _t (u_c \circ \phi _{n+1}^{-1})] \circ \phi _{n+1} + {\mathcal {R}}^{\phi _{n+1}} \left( {\dot{\phi }}_{n+1}\cdot \nabla ^{\phi _{n+1}} w^2_c \right) . \end{aligned}$$

Now we proceed as usual; first, by Lemma C.1, (C.7), the paraproduct estimates for Besov spaces [45, Proposition A.7], and Lemma 4.9

$$\begin{aligned} \Vert \mathring{R}^{comp}_2 \Vert _{C_{\le {\mathfrak {t}}_L} C^\delta _x}&\le CL^\delta \lambda ^{-1} \Vert {\mathcal {R}} {\mathcal {Q}} \partial _t(u_c\circ \phi _{n+1}^{-1}) \Vert _{C_{\le {\mathfrak {t}}_L} C^\delta _x} + CL^{5+4\delta } \left\| {\dot{\phi }}_{n+1}\cdot \nabla ^{\phi _{n+1}} w^2_c \right\| _{C_{\le {\mathfrak {t}}_L} B^{\delta -1}_{\infty ,\infty }} \\&\le CL^\delta \lambda ^{-1} \Vert {\mathcal {R}} {\mathcal {Q}} \partial _t(u_c\circ \phi _{n+1}^{-1}) \Vert _{C_{\le {\mathfrak {t}}_L} C^\delta _x} + CL^{11+4\delta } \varsigma _{n+1}^{\alpha -1} \left\| w^2_c \right\| _{C_{\le {\mathfrak {t}}_L} C^{\delta }_x} \\&\le CL^\delta \lambda ^{-1} \Vert {\mathcal {R}} {\mathcal {Q}} \partial _t(u_c\circ \phi _{n+1}^{-1}) \Vert _{C_{\le {\mathfrak {t}}_L} C^\delta _x} + CL^{12+6\delta }L_n^{3+3\delta } \lambda ^{\delta -1} \mu \delta _n^{1/2} \varsigma _{n+1}^{2\alpha -2}. \end{aligned}$$

Since

$$\begin{aligned} \partial _t(u_c\circ \phi _{n+1}^{-1})&= i \sum _{k \in \Lambda } {\nabla _y \partial _t (a_k\circ \phi _{n+1}^{-1})} \times \frac{k}{|k|^2}\times \Omega _k^\lambda , \end{aligned}$$

the stationary phase Lemma implies, together with the assumption (3.2)

$$\begin{aligned} \Vert \mathring{R}^{comp}_2 \Vert _{C_{\le {\mathfrak {t}}_L}C^\delta _x}&\le C L^{r+1+2\delta } \sum _{k \in \Lambda } \left( \lambda ^{\delta -2}[\partial _s a_k]_{C^1_x} + \lambda ^{\delta -r-1}[\partial _s a_k]_{C^{r+1}_x} + \lambda ^{-r-1}[\partial _s a_k]_{C^{r+1+\delta }_x} \right) \\&\quad + C L^{r+1+2\delta } \sum _{k \in \Lambda } \left( \lambda ^{\delta -1}[\partial _\tau a_k]_{C^1_x} + \lambda ^{\delta -r}[\partial _\tau a_k]_{C^{r+1}_x} + \lambda ^{-r}[\partial _\tau a_k]_{C^{r+1+\delta }_x} \right) \\&\quad + {C L^{r+3+2\delta } \varsigma _{n+1}^{\alpha -1} \sum _{k \in \Lambda } \left( \lambda ^{\delta -1}[a_k]_{C^2_x} + \lambda ^{\delta -r}[a_k]_{C^{r+2}_x} + \lambda ^{-r}[a_k]_{C^{r+2+\delta }_x} \right) } \\&\quad + CL^{12+6\delta }L_n^{3+3\delta } \lambda ^{\delta -1} \mu \delta _n^{1/2} \varsigma _{n+1}^{2\alpha -2} \\&\le CL^\delta L_n^{3r+3\delta +11/2} \lambda ^{\delta -1} \mu \delta _n^{1/2} \varsigma _{n+1}^{2\alpha -2}, \end{aligned}$$

and with usual arguments one deduces also, for $\lambda ^{r-1} \ge \mu ^{r+4} \varsigma _{n+1}^{(r+4)(\alpha -1)}(D_n \ell ^{-r-3} + \varsigma _{n+1}^{\alpha -1})$:

$$\begin{aligned} \Vert \mathring{R}^{comp}_2 \Vert _{C_{\le {\mathfrak {t}}_L}C^1_x}&\le CL \lambda \Vert \mathring{R}^{comp}_2 \Vert _{C_{\le {\mathfrak {t}}_L}C_x} \\&\quad + C L^{r+3+\delta } \sum _{k \in \Lambda } \left( \lambda ^{\delta -2}[\partial _s a_k]_{C^2_x} + \lambda ^{\delta -r-1}[\partial _s a_k]_{C^{r+2}_x} + \lambda ^{-r-1}[\partial _s a_k]_{C^{r+2+\delta }_x} \right) \\&\quad + C L^{r+3+\delta } \sum _{k \in \Lambda } \left( \lambda ^{\delta -1}[\partial _\tau a_k]_{C^2_x} + \lambda ^{\delta -r}[\partial _\tau a_k]_{C^{r+2}_x} + \lambda ^{-r}[\partial _\tau a_k]_{C^{r+2+\delta }_x} \right) \\&\quad + C L^{r+5+2\delta } \varsigma _{n+1}^{\alpha -1} \sum _{k \in \Lambda } \left( \lambda ^{\delta -1}[a_k]_{C^3_x} + \lambda ^{\delta -r}[a_k]_{C^{r+3}_x} + \lambda ^{-r}[a_k]_{C^{r+3+\delta }_x} \right) \\&\quad + CL^{15+6\delta }L_n^{6+3\delta } \lambda ^\delta \mu \delta _{n}^{1/2} \varsigma _{n+1}^{2\alpha -2} \\&\le CL L_n^{3r+3\delta +17/2} \lambda ^{\delta } \mu \delta _n^{1/2} \varsigma _{n+1}^{2\alpha -2}. \end{aligned}$$

Moving to $\mathring{R}^{comp}_3$, we have by Lemmas C.4, 4.9 and 4.10

$$\begin{aligned} \Vert \mathring{R}^{comp}_3 \Vert _{C_{\le {\mathfrak {t}}_L}C^\delta _x}&\le C L^{2\delta } \Vert v_{n+1} \otimes w_c + w_c \otimes v_{n+1} - w_c \otimes w_c \Vert _{C_{\le {\mathfrak {t}}_L}C^\delta _x} \\&\le C L^{2\delta }\left( \Vert v_{n+1}\Vert _{C_{\le {\mathfrak {t}}_L}C^\delta _x} \Vert w_c\Vert _{C_{\le {\mathfrak {t}}_L}C^\delta _x} + \Vert w_c\Vert _{C_{\le {\mathfrak {t}}_L}C^\delta _x}^2 \right) \\&\le C L^{2\delta }\left( \Vert v_\ell \Vert _{C_{\le {\mathfrak {t}}_L}C^\delta _x} \Vert w_c\Vert _{C_{\le {\mathfrak {t}}_L}C^\delta _x} + \Vert w_o\Vert _{C_{\le {\mathfrak {t}}_L}C^\delta _x} \Vert w_c\Vert _{C_{\le {\mathfrak {t}}_L}C^\delta _x} + \Vert w_c\Vert _{C_{\le {\mathfrak {t}}_L}C^\delta _x}^2 \right) \\&\le C L_n^{13/2+4\delta } \lambda ^\delta \delta _n^{1/2} \delta _{n+2}^{6/5}, \end{aligned}$$

and

$$\begin{aligned} \Vert \mathring{R}^{comp}_3 \Vert _{C_{\le {\mathfrak {t}}_L}C^{1+\delta }_x}&\le C L^{2+2\delta } \Vert v_{n+1} \otimes w_c + w_c \otimes v_{n+1} - w_c \otimes w_c \Vert _{C_{\le {\mathfrak {t}}_L}C^{1+\delta }_x} \\&\le C L^{2+2\delta }\left( \Vert v_{n+1}\Vert _{C_{\le {\mathfrak {t}}_L}C^{\delta }_x} \Vert w_c\Vert _{C_{\le {\mathfrak {t}}_L}C^{1+\delta }_x} + \Vert v_{n+1}\Vert _{C_{\le {\mathfrak {t}}_L}C^{1+\delta }_x} \Vert w_c\Vert _{C_{\le {\mathfrak {t}}_L}C^{\delta }_x} \right) \\&\quad + CL^{2+2\delta } \Vert w_c\Vert _{C_{\le {\mathfrak {t}}_L}C^{\delta }_x} \Vert w_c\Vert _{C_{\le {\mathfrak {t}}_L}C^{1+\delta }_x} \\&\le C L^{2+2\delta }\left( \Vert v_\ell \Vert _{C_{\le {\mathfrak {t}}_L}C^{\delta }_x} \Vert w_c\Vert _{C_{\le {\mathfrak {t}}_L}C^{1+\delta }_x} + \Vert v_\ell \Vert _{C_{\le {\mathfrak {t}}_L}C^{1+\delta }_x} \Vert w_c\Vert _{C_{\le {\mathfrak {t}}_L}C^{\delta }_x} \right) \\&\quad + C L^{2+2\delta }\left( \Vert w_o\Vert _{C_{\le {\mathfrak {t}}_L}C^{\delta }_x} \Vert w_c\Vert _{C_{\le {\mathfrak {t}}_L}C^{1+\delta }_x} + \Vert w_o\Vert _{C_{\le {\mathfrak {t}}_L}C^{1+\delta }_x} \Vert w_c\Vert _{C_{\le {\mathfrak {t}}_L}C^{\delta }_x} \right) \\&\quad + CL^{2+2\delta } \Vert w_c\Vert _{C_{\le {\mathfrak {t}}_L}C^{\delta }_x} \Vert w_c\Vert _{C_{\le {\mathfrak {t}}_L}C^{1+\delta }_x} \\&\le C L_n^{10} \lambda ^{1+\delta } \mu ^\delta \delta _n^{1/2} \delta _{n+2}^{6/5} \varsigma _{n+1}^{\delta (\alpha -1)}. \end{aligned}$$

We have nothing left but $\mathring{R}^{comp}_4$. To better control this term, we rewrite

$$\begin{aligned} {\mathcal {R}}^{\phi _{n+1}} \mathord {\textrm{div}}^{\phi _{n+1}}(v_\ell \otimes w_o)&= {\mathcal {R}}^{\phi _{n+1}} \left( v_\ell \, \mathord {\textrm{div}}^{\phi _{n+1}}(w_o) \right) + {\mathcal {R}}^{\phi _{n+1}} \left( (w_o \cdot \nabla ^{\phi _{n+1}}) v_\ell \right) \\&= {\mathcal {R}} \left( \, \sum _{k \in \Lambda } (v_\ell \circ \phi _{n+1}^{-1})\, \text{ div } (a_k\circ \phi _{n+1}^{-1}) \, \Omega _k \right) \circ \phi _{n+1} \\&\quad + {\mathcal {R}} \left( \, \sum _{k \in \Lambda } ((a_k\circ \phi _{n+1}^{-1}) \cdot \nabla ) (v_\ell \circ \phi _{n+1}^{-1}) \Omega _k \right) \circ \phi _{n+1}, \end{aligned}$$

so that we can apply stationary phase Lemma and Lemma C.1 to get (details omitted):

$$\begin{aligned} \Vert \mathring{R}^{comp}_4 \Vert _{C_{\le {\mathfrak {t}}_L}C^\delta _x}&\le CL_n^{3r+6} \lambda ^{\delta -1} \mu \delta _n^{1/2} \varsigma _{n+1}^{\alpha -1}, \end{aligned}$$

and

$$\begin{aligned} \Vert \mathring{R}^{comp}_4 \Vert _{C_{\le {\mathfrak {t}}_L}C^{1+\delta }_x}&\le CL_n^{3r+7} \lambda ^{\delta -1} \mu ^2 \delta _n^{1/2} \varsigma _{n+1}^{\alpha -1} (D_n \ell ^{-1} + \varsigma _{n+1}^{\alpha -1}). \end{aligned}$$

$\square$

4.6 Estimate on the divergence

Recall that by construction it holds $\mathord {\textrm{div}}^{\phi _{n+1}}v_{n+1} = \mathord {\textrm{div}}^{\phi _{n+1}}v_\ell + \mathord {\textrm{div}}^{\phi _{n+1}}w^1_c$, and recalling (3.14) and (3.15)

$$\begin{aligned} \mathord {\textrm{div}}^{\phi _{n+1}}v_{n+1}&= \mathord {\textrm{div}}^{\phi _{n+1}}v_\ell - \left( \mathord {\textrm{div}}^{\phi _{n+1}}v_n \right) *\chi _\ell \nonumber \\&\quad + \left( \mathord {\textrm{div}}^{\phi _{n+1}}v_n \right) *\chi _\ell - \left( \mathord {\textrm{div}}\,^{\phi _n}v_n\right) *\chi _\ell \nonumber \\&\quad + \left( \mathord {\textrm{div}}\,^{\phi _n}Q^{\phi _n}v_n\right) *\chi _\ell - \mathord {\textrm{div}}\,^{\phi _n} \left( (Q^{\phi _n}v_n) *\chi _\ell \right) \nonumber \\&\quad + \mathord {\textrm{div}}\,^{\phi _n} \left( (Q^{\phi _n}v_n) *\chi _\ell \right) - \mathord {\textrm{div}}^{\phi _{n+1}}\left( (Q^{\phi _n}v_n) *\chi _\ell \right) . \end{aligned}$$

(4.15)

Proposition 1.26

For every $\delta$ sufficiently small and $\alpha ' \in (0,\alpha )$ there exists a constant C such that for all $L \in \mathbb {N}$, $L \ge 1$ it holds almost surely

$$\begin{aligned} \Vert \mathord {\textrm{div}}^{\phi _{n+1}}v_{n+1}\Vert _{C_{\le {\mathfrak {t}}_L} B^{-1}_{\infty ,\infty }} \le C L^{10} L_n (D_n^{1+2\delta }\ell ^\alpha + D_n^{\delta } (n+1) \varsigma _n^{\alpha '}). \end{aligned}$$

Proof

We refer to decomposition (4.15) above. By Lemmas 4.4, C.4 and iterative assumption (2.9) it holds

$$\begin{aligned} \left\| \mathord {\textrm{div}}^{\phi _{n+1}}v_\ell - \left( \mathord {\textrm{div}}^{\phi _{n+1}}v_n \right) *\chi _\ell \right\| _{C_{\le {\mathfrak {t}}_L}C_x}&\le CL^2 \Vert v_n\Vert _{C_{\le {\mathfrak {t}}_L}C^1_x} \ell ^{\alpha } \\&\le CL^2L_n D_n \ell ^{\alpha }, \\ \left\| \left( \mathord {\textrm{div}}\,^{\phi _n}Q^{\phi _n}v_n\right) *\chi _\ell - \mathord {\textrm{div}}\,^{\phi _n} \left( (Q^{\phi _n}v_n) *\chi _\ell \right) \right\| _{C_{\le {\mathfrak {t}}_L}C_x}&\le CL^2 \Vert Q^{\phi _n} v_n\Vert _{C_{\le {\mathfrak {t}}_L}C^1_x} \ell ^{\alpha } \\&\le CL^{4+2\delta } \Vert v_n\Vert _{C_{\le {\mathfrak {t}}_L}C^{1+\delta }_x} \ell ^{\alpha } \\&\le CL^{4+2\delta }L_n D_n^{1+2\delta } \ell ^{\alpha }. \end{aligned}$$

In the last line above we have used the bound $\Vert v_n\Vert _{C_{\le {\mathfrak {t}}_L}C^{1+\delta }_x} \le L_n D_n^{4/3}$, justified by the following observation. If $n=0$ we have defined $v_n=0$, thus let us assume $n \ge 1$ without any loss of generality. Then, $v_n$ was constructed from $v_{n-1},q_{n-1},\mathring{R}_{n-1}$ by the formula $v_n = v_{n-1} *\chi _{\ell _{n-1}} + w_{o,n-1} + w_{c,n-1}$, and Lemmas 4.9, 4.10 and the assumption $\lambda _{n-1} \le D_n$ imply for $\delta$ sufficiently small

$$\begin{aligned}&\Vert v_n\Vert _{C_{\le {\mathfrak {t}}_L}C^{1+\delta }_x} \\ {}&\quad \le C L^{1+\delta } L_{n-1}^{6+3\delta } \lambda _{n-1}^{1+\delta } \delta _{n-1}^{1/2} \mu _{n-1}^\delta \varsigma _n^{\delta (\alpha -1)} \le C L_n \lambda _{n-1}^{1+2\delta } \le C L_n D_n^{1+2\delta }. \end{aligned}$$

Let us move to the remaining terms in (4.15). Using Lemma 4.7 and the convolution inequality Lemma C.6

$$\begin{aligned} \left\| \left( \mathord {\textrm{div}}^{\phi _{n+1}}v_n \right) *\chi _\ell - \left( \mathord {\textrm{div}}\,^{\phi _n}v_n\right) *\chi _\ell \right\| _{C_{\le {\mathfrak {t}}_L}B^{-1}_{\infty ,\infty }}&\le C \left\| \left( \mathord {\textrm{div}}^{\phi _{n+1}}- \mathord {\textrm{div}}\,^{\phi _n}\right) v_n \right\| _{C_{\le {\mathfrak {t}}_L}B^{\delta -1}_{\infty ,\infty }} \\&\le CL^3\Vert v_n\Vert _{C_{\le {\mathfrak {t}}_L}C^\delta _x}(n+1)\varsigma _n^{\alpha '} \\&\le CL^3L_nD_n^\delta (n+1)\varsigma _n^{\alpha '}, \end{aligned}$$

and

$$\begin{aligned}&\left\| \mathord {\textrm{div}}\,^{\phi _n} \left( (Q^{\phi _n}v_n) *\chi _\ell \right) - \mathord {\textrm{div}}^{\phi _{n+1}}\left( (Q^{\phi _n}v_n) *\chi _\ell \right) \right\| _{C_{\le {\mathfrak {t}}_L}B^{-1}_{\infty ,\infty }} \\&\qquad \le CL^3 \Vert (Q^{\phi _n}v_n) *\chi _\ell \Vert _{C_{\le {\mathfrak {t}}_L} C^\delta _x}(n+1)\varsigma _n^{\alpha '} \\&\qquad \le CL^{10} L_n \delta _{n+2}^{6/5}(n+1) \varsigma _n^{\alpha '}. \end{aligned}$$

$\square$

4.7 Estimate on the pressure

In the previous subsections we have collected several estimates that together allow to control iteratively the Reynold stress and the velocity field. In order to prove our main iterative proposition we still need to provide suitable bounds on the pressure term, which we intend to do in this subsection.

Recall the definition of the new pressure $q_{n+1}$ and the energy pumping term ${\tilde{\rho }}_\ell$

$$\begin{aligned} q_{n+1}&= q_\ell - \frac{1}{2} \left( |w_o|^2-{\tilde{\rho }}_\ell \right) , \\ {\tilde{\rho }}_\ell (x,t)&= \frac{2}{r_0} \sqrt{\eta ^2 \delta _{n+1}^2 + |\mathring{R}_\ell (x,t)|^2}. \end{aligned}$$

We have:

Proposition 1.27

There exists a constant $M_q$ as in the statement of Proposition 2.4 such that for every $L \in \mathbb {N},$ $L \ge 1$ it holds almost surely

$$\begin{aligned} \Vert q_{n+1}-q_n \Vert _{C_{\le {\mathfrak {t}}_L}C_x}&\le M_q L_n \delta _n. \end{aligned}$$

Moreover, Let $r\ge r_\star + 2$. Then, for every $\delta >0$ there exists a constant C such that for every $L \in \mathbb {N},$ $L \ge 1$ it holds almost surely

$$\begin{aligned} \Vert q_{n+1}-q_n \Vert _{C^1_{\le {\mathfrak {t}}_L,x}}&\le C L_n^{3r+10} \lambda ^{1+\delta } \delta _n \mu ^\delta \varsigma _{n+1}^{\delta (\alpha -1)}. \end{aligned}$$

Proof

Let us rewrite $q_{n+1}-q_n = q_\ell - q_n - \frac{1}{2} \left( |w_o|^2-{\tilde{\rho }}_\ell \right)$. Standard mollification estimates and the iterative assumption (2.9) yield

$$\begin{aligned} \Vert q_\ell -q_n \Vert _{C_{\le {\mathfrak {t}}_L}C_x}&\le CL_n \ell D_n , \\ \Vert q_\ell -q_n \Vert _{C^1_{\le {\mathfrak {t}}_L,x}}&\le C L_n D_n. \end{aligned}$$

The previous inequalities hold with a constant C depending only on the mollifier $\chi$, which however can be thought of as fixed. Notice that $\ell D_n \le \delta _n$ by assumption.

In addition, the energy pumping term ${\tilde{\rho }}_\ell$ is easily controlled with estimates

$$\begin{aligned} \Vert {\tilde{\rho }}_\ell \Vert _{C_{\le {\mathfrak {t}}_L}C_x}&\le \tfrac{3}{r_0} L_n \delta _{n+1}, \\ \Vert {\tilde{\rho }}_\ell \Vert _{C^1_{\le {\mathfrak {t}}_L,x}}&\le C L_n^2 \delta _{n+1} \ell ^{-1}, \end{aligned}$$

where $r_0$ has been defined in Sect. 3.6 and we have used $\eta <1$. Also, by (3.13) we have

$$\begin{aligned} \Vert |w_o|^2 \Vert _{C_{\le {\mathfrak {t}}_L}C_x}&\le C L_n \delta _n, \end{aligned}$$

where the constant C may depend on ${\overline{e}}$. Thus, we have proved

$$\begin{aligned} \Vert q_{n+1}-q_n \Vert _{C_{\le {\mathfrak {t}}_L}C_x}&\le M_q L_n \delta _n \end{aligned}$$

for some constant $M_q$ depending only on ${\overline{e}}$. Finally, by Lemma 4.10 it holds

$$\begin{aligned} \Vert |w_o|^2 \Vert _{C^1_{\le {\mathfrak {t}}_L,x}}&\le C L_n^{3r+10} \lambda ^{1+\delta } \delta _n \mu ^\delta \varsigma _{n+1}^{\delta (\alpha -1)}, \end{aligned}$$

completing the proof. $\square$

4.8 Estimate on the energy

Finally, we have to check the iterative condition on the energy. This is the content of the following:

Proposition 1.28

Recall the definition of $r_0$ from Sect. 3.6. Up to choosing $C_\varsigma , C_\mu$ large enough, the following holds true almost surely:

$$\begin{aligned} \left| e(t)(1-\delta _{n+1}) - \int _{\mathbb {T}^3} |v_{n+1}(x,t)|^2 dx \right| \le \frac{10\, \eta }{r_0} \delta _{n+1}, \quad \forall t \le {\mathfrak {t}}. \end{aligned}$$

Proof

Rewrite

$$\begin{aligned} |v_{n+1}|^2&= |v_\ell |^2 + |w_o|^2 + |w_c|^2 + 2 v_\ell \cdot w_o + 2 v_\ell \cdot w_c + 2 w_o \cdot w_c. \end{aligned}$$

(4.16)

Since $\chi _\ell \ge 0$ and by assumption (2.5), Hölder inequality yields

$$\begin{aligned} \int _{\mathbb {T}^3} |v_\ell (x,t)|^2 dx&= \int _{\mathbb {T}^3} \left| \int _{\mathbb {T}^3 \times \mathbb {R}} v_n(x-y,t-s) \chi _\ell (y,s) dy ds\right| ^2 dx \\&\le \int _{\mathbb {T}^3} \int _{\mathbb {T}^3 \times \mathbb {R}} |v_n(x-y,t-s)|^2 \chi _\ell (y,s) dy ds dx \\&\le \int _{\mathbb {T}^3 \times \mathbb {R}} e(t-s) \chi _\ell (y,s) dy ds \le {\overline{e}}, \end{aligned}$$

implying by Lemma 4.9

$$\begin{aligned} \left| \int _{\mathbb {T}^3} v_\ell (x,t) \cdot w_c(x,t) dx \right| \le C {\overline{e}}^{1/2} \Vert w_c\Vert _{C_{\le {\mathfrak {t}}}C_x} \le C \delta _{n+2}^{6/5}, \quad \forall t \le {\mathfrak {t}}, \end{aligned}$$

where the constant C depends only on ${\overline{e}}$. Moreover, recalling

$$\begin{aligned} w_o(x,t) = \sum _{|k|=\lambda _0} a_k(x,t,\lambda t) E_ke^{ik\cdot \lambda \phi _{n+1}(x,t)} \end{aligned}$$

and applying (C.8) and Lemma C.1 we obtain almost surely for every $t \le {\mathfrak {t}}$

$$\begin{aligned} \left| \int _{\mathbb {T}^3} v_\ell (x,t) \cdot w_o(x,t) dx \right|&\le C\lambda ^{-1}\sum _{|k|=\lambda _0} [v_\ell \cdot a_k]_{C^1_x} \\&\le C\lambda ^{-1}\sum _{|k|=\lambda _0} \left( \Vert v_\ell \Vert _{C_{\le {\mathfrak {t}}} C_x} [a_k]_{C_{\le {\mathfrak {t}}} C^1_x} + [v_\ell ]_{C_{\le {\mathfrak {t}}} C^1_x} \Vert a_k\Vert _{C_{\le {\mathfrak {t}}} C_x} \right) \\&\le C\lambda ^{-1}\sum _{|k|=\lambda _0} \left( [a_k]_{C_{\le {\mathfrak {t}}} C^1_x} + D_n \Vert a_k\Vert _{C_t C_x} \right) \\&\le C \lambda ^{-1} \mu \delta _n^{1/2} ( D_n + \varsigma _{n+1}^{\alpha -1} ). \end{aligned}$$

We also point out that the previous bounds do not contain any factor L, since we have purposefully restricted ourselves to times $t \le {\mathfrak {t}} = {\mathfrak {t}}_1$. Therefore, recalling (4.16) and the bounds just obtained, we have for $\delta$ sufficiently small

$$\begin{aligned}&\left| \int _{\mathbb {T}^3} \left( |v_{n+1}(x,t)|^2 - |v_\ell (x,t)|^2 - |w_o(x,t)|^2 \right) dx \right| \\ {}&\quad \le C \Vert w_c \Vert _{C_x} \left( \Vert w_c \Vert _{C_x} + \Vert w_o\Vert _{C_x} \right) + C \delta _{n+2}^{6/5} + C \lambda ^{-1} \mu \delta _n^{1/2} ( D_n + \varsigma _{n+1}^{\alpha -1} ) \\&\quad \le C \lambda ^\delta \delta _n^{1/2} \delta _{n+2}^{6/5}. \end{aligned}$$

Moreover, recalling Corollary 4.2,

$$\begin{aligned} \int _{\mathbb {T}^3} |w_o|^2 dx&= \int _{\mathbb {T}^3} Tr(R_\ell ) dx + \sum _{1 \le |k| \le 2\lambda _0}\int _{\mathbb {T}^3} Tr(U_k) e^{ik \lambda \phi _{n+1}} dx, \end{aligned}$$

and

$$\begin{aligned} \int _{\mathbb {T}^3} Tr(R_\ell ) dx&= 3\int _{\mathbb {T}^3} \rho _\ell dx = e(t)(1-\delta _{n+1}) - \int _{\mathbb {T}^3} |v_\ell |^2 dx + 3\int _{\mathbb {T}^3} {{\tilde{\rho }}}_\ell dx , \end{aligned}$$

we can rearrange terms to get

$$\begin{aligned}&\left| e(t)(1-\delta _{n+1}) - \int _{\mathbb {T}^3} |v_{n+1}|^2 dx\right| \\ {}&\le \left| \int _{\mathbb {T}^3} \left( |v_{n+1}|^2 - |v_\ell |^2 - |w_o|^2 \right) dx \right| + 3\left| \int _{\mathbb {T}^3} {{\tilde{\rho }}}_\ell dx\right| \\&\quad + \sum _{1 \le |k| \le 2\lambda _0} \left| \int _{\mathbb {T}^3} Tr(U_k) e^{ik \lambda \phi _{n+1}} dx \right| \\&\le C \lambda ^\delta \delta _n^{1/2} \delta _{n+2}^{6/5} + 9 r_0^{-1} \eta \delta _{n+1} + C \lambda ^{-1} \mu \delta _n \varsigma _{n+1}^{\alpha -1}.\\&\le 10 r_0^{-1} \eta \delta _{n+1}. \end{aligned}$$

In the expression above we have taken $\delta$ sufficiently small, $\alpha _\star <\alpha$ sufficiently close to 1/2, and $C_\varsigma$, $C_\mu$ sufficiently large, so that the factor $10 r_0^{-1} \eta$ appears on the right-hand-side. The proof is complete. $\square$

5 Proof of Proposition 2.4

We are finally ready to prove our main proposition. Progressive measurability of the approximate solution at level $n+1$ descends directly from the definition of $v_{n+1},q_{n+1},\mathring{R}_{n+1}$.

We verify first the iterative assumption on the energy (2.5) and the $C_{\le {\mathfrak {t}}_L} C_x$ norms, namely (2.6) and (2.7). As for the former, we have proved in Proposition 4.14 that almost surely for every $t \le {\mathfrak {t}}$

$$\begin{aligned} \left| e(t)(1-\delta _{n+1}) - \int _{\mathbb {T}^3} |v_{n+1}(x,t)|^2 dx \right| \le \tfrac{10\,\eta }{r_0} \delta _{n+1} \le \tfrac{1}{4} \delta _{n+1}{\underline{e}} \le \tfrac{1}{4} \delta _{n+1}e(t) , \end{aligned}$$

where the second inequality holds true if we take $\eta$ small enough. Recall also the upper bound on $\eta$ given by Lemma 3.1. The value of $\eta$ will be fixed hereafter according to these constraints.

Let us move to (2.6). By (3.16) and (3.17) it holds

$$\begin{aligned} \mathring{R}_{n+1} = \mathring{R}^{trans} + \mathring{R}^{moll} + \mathring{R}^{osc} + \mathring{R}^{flow} + \mathring{R}^{comp}. \end{aligned}$$

Putting together Propositions 4.3, 4.5, 4.6, 4.8 and 4.11 from Sect. 4 we can bound, almost surely for every $t \le {\mathfrak {t}}_L$, $L \in \mathbb {N}$, $L \ge 1$:

$$\begin{aligned} \Vert \mathring{R}_{n+1}\Vert _{C_{\le {\mathfrak {t}}_L} C_x}&\le CL_n^{3r+5}\lambda ^{\delta } \mu ^{-1} \delta _n^{1/2} + CL_n^3 D_n \ell ^\alpha + CL_n^{3r+6} \lambda ^{\delta -1} \mu \delta _n \varsigma _{n+1}^{\alpha -1} \\&\quad + CL_n^{3r+5} \ell ^{-\delta } (n+1)\varsigma _n^{\alpha '} + C L_n^{3r+6} \lambda ^\delta \delta _n^{1/2} \delta _{n+2}^{6/5} , \end{aligned}$$

where $r \ge r_\star +2$, $\delta >0$ is small, $\alpha ' \in (0,\alpha )$ is close to 1/2, and C is an unimportant constant possibly depending on ${\overline{e}},K_0, \eta , M_v$ and $r_\star$. Now we can choose $\mu ,\lambda$ large such that the first and third line on the right-hand-side of the inequality above are small; then, $D_n \ell ^\alpha$ and $(n+1)\varsigma _{n+1}^{\alpha '}$ are small by definition of $\ell ,\varsigma _n$. More precisely, up to choosing $\delta$ small enough, $\alpha _\star< \alpha ' < \alpha$ sufficiently close to 1/2, and $C_\varsigma ,C_\mu$ large enough, the expression above can be rewritten as

$$\begin{aligned} \Vert \mathring{R}_{n+1}\Vert _{C_{\le {\mathfrak {t}}_L} C_x}&\le \eta L_n^{3r+6} \delta _{n+2}. \end{aligned}$$

Thus the desired estimate holds true as soon as we choose

$$\begin{aligned} m \ge 3r + 6 \ge 3r_\star + 12. \end{aligned}$$

Iterative assumption (2.7) follows immediately from Proposition 4.13, whereas the bound (2.8) on $\mathord {\textrm{div}}^{\phi _{n+1}}v_{n+1}$ comes from Proposition 4.12, up to choosing $C_\ell ,C_\varsigma$ large enough.

Let us check

$$\begin{aligned} \Vert v_{n+1}-v_n \Vert _{C_{\le {\mathfrak {t}}_L}C_x} \le M_v L_n^4 \delta _n^{1/2}. \end{aligned}$$

It holds $v_{n+1}-v_n = v_\ell - v_n + w_o + w_c$, and by mollification and (3.13) we have

$$\begin{aligned} \Vert v_\ell - v_n \Vert _{C_{\le {\mathfrak {t}}_L}C_x}&\le C L_n \ell D_n \le C L_n \delta _{n}, \\ \Vert w_o \Vert _{C_{\le {\mathfrak {t}}_L}C_x}&\le C L_n^{1/2} \delta _n^{1/2}, \end{aligned}$$

for some constant C depending only on ${\overline{e}}$. Moreover, by Lemma 4.9 it holds

$$\begin{aligned} \Vert w_c \Vert _{C_{\le {\mathfrak {t}}_L}C_x}&\le CL_n^{3+4\delta } \delta _{n+2}^{6/5}, \end{aligned}$$

for some universal constant C, and thus the desired estimate holds true for some $M_v$ depending only on ${\overline{e}}$.

Finally, we need to prove

$$\begin{aligned} \max \left\{ \Vert v_{n+1}\Vert _{C^1_{\le {\mathfrak {t}}_L,x}}, \Vert q_{n+1}\Vert _{C^1_{\le {\mathfrak {t}}_L,x}}, \Vert \mathring{R}_{n+1}\Vert _{C_{\le {\mathfrak {t}}_L}C^1_x} \right\}&\le A L_{n+1} \delta _n^{1/2} \left( \frac{D_n}{\delta _{n+4}} \right) ^{1+\varepsilon }. \end{aligned}$$

Let us start with $\Vert \mathring{R}_{n+1}\Vert _{C_{\le {\mathfrak {t}}_L}C^1_x}$. Collecting the results of Sect. 4, we have

$$\begin{aligned} \Vert \mathring{R}_{n+1}\Vert _{C_{\le {\mathfrak {t}}_L}C^1_x} \le C L_n^{3r+10} \delta _n^{1/2} \lambda ^{1+2\delta } . \end{aligned}$$

Therefore, we can choose for instance

$$\begin{aligned} r_\star = 7, \qquad m \ge 3r +10&\ge 3r_\star + 16 = 37, \qquad \varepsilon \ge 15, \end{aligned}$$

and A sufficiently large so that $\lambda ^{1+2\delta } \le A \left( \frac{D_n}{\delta _{n+4}} \right) ^{1+\varepsilon } \le D_{n+1}$. Indeed, up to multiplicative constants depending on $C_\ell , C_\varsigma$ and $C_\mu$ it holds for $\alpha _\star$ sufficiently close to 1/2 and $\delta$ small enough

$$\begin{aligned} \lambda ^{1+2\delta } \lesssim \frac{D_n^{15}}{\delta _{n+3}^{20} \delta _{n+4}^5}, \end{aligned}$$

and we can use A to absorb any multiplicative constant in the inequality above.

Next, by Proposition 4.13 and the iterative assumption (2.9)

$$\begin{aligned} \Vert q_{n+1} \Vert _{C^1_{\le {\mathfrak {t}}_L,x}} \le CL_n D_n + CL_n^{3r+10} \lambda ^{1+2\delta } \delta _n \le A L_{n+1} \delta _n^{1/2} \left( \frac{D_n}{\delta _{n+4}} \right) ^{1+\varepsilon }. \end{aligned}$$

As for the spatial norm of $v_{n+1}$, we recall by Lemmas 4.9 and 4.10

$$\begin{aligned} \Vert v_{n+1}\Vert _{C_{\le {\mathfrak {t}}_L}C^1_x}&\le C L_n^7 \delta _n^{1/2} \lambda ^{1+2\delta }, \end{aligned}$$

whereas for the time derivative of $v_{n+1}$ we take advantage of

$$\begin{aligned} \partial _t v_{n+1} = \mathord {\textrm{div}}^{\phi _{n+1}}\left( \mathring{R}_{n+1} - v_{n+1} \otimes v_{n+1} - q_{n+1} Id \right) \end{aligned}$$

and the bounds on the $C_{\le {\mathfrak {t}}_L}C^1_x$ norms of $v_{n+1}$, $q_{n+1}$ and $\mathring{R}_{n+1}$ just proved, up to choosing $m \ge 3r+11$ to compensate for an additional factor $L_n$.

To conclude, take for instance $r_\star =7$, $m=38$ and $\varepsilon =15$. The resulting Hölder exponent $\vartheta$ is given by (2.10) and (2.12): we can choose any

$$\begin{aligned}&\vartheta < \frac{1}{2cb+2} = \frac{m-1}{2(1+\varepsilon )(m+\varepsilon )^5+m-2-\varepsilon } \\ {}&\quad = \frac{38-1}{2(1+15)(38+15)^5+38-2-15}. \end{aligned}$$

The upper threshold is approximatively given by $2.76 \times 10^{-9}$, which is extremely low and far from the Onsager’s critical exponent 1/3 of the deterministic case. Since we have used a suboptimal choice of parameters during our construction for the only sake of simplicity, this exponent could be slightly improved taking more care of that. However, we do not believe that a significant improvement on the exponent is within the reach of the techniques used in this paper. Finally, if we restrict ourselves to local solutions, namely on the time interval $t \le {\mathfrak {t}}_1$, we do not need to impose $m \ge 3r+11 \ge 3r_\star +17$, and we can just take $m=4$. This gives upper threshold for local solutions approximatively equal to $3.78 \times 10^{-8}$.

Appendices

Appendix A: Equivalence between (1.1) and (1.5)

Proposition A.1

A process (u, p) is a solution to (1.1) in the sense of Definition 1.1 if and only if (v, q) given by (1.3) is a solution to (1.5) in the sense of Definition 1.2.

Proof

It is clear that progressive measurability and regularity of trajectories are preserved by the changes of coordinates (1.3) and (1.4), as well as the divergence-free conditions. Therefore, we only need to check the conditions on the time evolution of the coupling between solutions and test functions. Let (v, q) be a solution to (1.5) in the sense of Definition 1.2, and let $h:\Omega \rightarrow C_{loc}([0,\infty ), C^\infty (\mathbb {T}^3,\mathbb {R}^3)$ be a square integrable semimartingale satisfying

$$\begin{aligned} dh=H_0\,dt + \sum _{k \in I} H_k \circ dB^k, \end{aligned}$$

for some progressively measurable processes $H_0, \{H_k\}_{k \in I}: \Omega \rightarrow C_{loc}([0,\infty ),C^\infty (\mathbb {T}^3,\mathbb {R}^3))$. Since $\phi$ is measure preserving, and recalling $v=u\circ \phi$ we have $\int _{\mathbb {T}^3} u \cdot h =\int _{\mathbb {T}^3} v \cdot (h\circ \phi )$. Moreover, the process $h \circ \phi$ is a semimartingale satisfying

$$\begin{aligned} d(h\circ \phi ) = (H_0 \circ \phi ) \,dt + \sum _{k \in I} ((H_k + (\sigma _k\cdot \nabla ) h) \circ \phi ) \circ dB^k(t), \end{aligned}$$

and therefore

$$\begin{aligned} d\left( \int _{\mathbb {T}^3} u \cdot h \right)&= \int _{\mathbb {T}^3} v \cdot (H_0 \circ \phi + (v \cdot \nabla ^\phi )(h \circ \phi )) dt + \int _{\mathbb {T}^3} q \, \mathord {\textrm{div}}^\phi (h\circ \phi ) \,dt \\&\quad + \sum _{k \in I}\int _{\mathbb {T}^3} v \cdot \left( (H_k+(\sigma _k\cdot \nabla ) h)\circ \phi \right) \circ dB^k \\&= \int _{\mathbb {T}^3} u \cdot (H_0 + (u \cdot \nabla ) h) dt + \int _{\mathbb {T}^3} p \, \mathord {\textrm{div}}\,h \,dt \\&\quad + \sum _{k \in I}\int _{\mathbb {T}^3} u \cdot (H_k + (\sigma _k \cdot \nabla ) h) \circ dB^k. \end{aligned}$$

The converse implication can be proved in a similar fashion, noticing that

$$\begin{aligned} d(h \circ \phi ^{-1})&= (H_0 \circ \phi ^{-1})\,dt + \sum _{k \in I} (H_k \circ \phi ^{-1} - (\sigma _k \cdot \nabla )(h \circ \phi ^{-1})) \circ dB^k. \end{aligned}$$

$\square$

Appendix B: Proof of Proposition 4.1

In this section all the multiplicative constants C may depend on ${\overline{e}},\eta ,M_v,r_\star$ without explicit mentioning, but will be always independent of $s,\tau$. In particular, when no ambiguity may occur we drop the symbol $C_{\le {\mathfrak {t}}_L}$ in the uniform-in-time spatial Hölder norms $C_{\le {\mathfrak {t}}_L}C_x^r$ below.

We first recall the following estimate for the Hölder norm of products. Let f, g be smooth functions on the torus and take $r \in [0,1]$; then it is immediate to see

$$\begin{aligned} \Vert fg \Vert _{C^r_x} \le \Vert f\Vert _{C^r_x} \Vert g\Vert _{C_x}+\Vert f\Vert _{C_x} \Vert g\Vert _{C^r_x}. \end{aligned}$$

For $r \in \mathbb {N}$, $r > 1$, using the Leibniz rule for the derivative of the product, we have for every integer $k \le r$ a constant $C=C(k)$ such that

$$\begin{aligned}{}[fg]_{C^{k}_x}&\le C \sum _{r_1+r_2=k} [ f ]_{C^{r_1}_x} [ g ]_{C^{r_2}_x} \le C \sum _{r_1+r_2=k} \Vert f \Vert _{C^{r_1}_x} \Vert g \Vert _{C^{r_2}_x}, \qquad r_1,r_2 \in \mathbb {N}, \end{aligned}$$

and taking the sum over $k \le r$ we get a constant $C=C(r)$, increasing in r, such that

$$\begin{aligned} \Vert fg \Vert _{C^{r}_x}&\le C \sum _{r_1+r_2=r} \Vert f \Vert _{C^{r_1}_x} \Vert g \Vert _{C^{r_2}_x}, \qquad r_1,r_2 \in \mathbb {N}. \end{aligned}$$

Therefore, recalling (3.11),

$$\begin{aligned} a_k(y,s,\tau )&= {\textbf{1}}_{\{k \in \Lambda _j\}} \sqrt{\rho _\ell (y,s)} \, \underbrace{\gamma _k^{(j)}\left( \frac{R_\ell (y,s)}{\rho _\ell (y,s)}\right) }_{=:\Gamma (y,s)} \, \underbrace{\psi _k^{(j)} \left( {\tilde{v}}(y,s),\tau \right) }_{=:\Psi (y,s,\tau )}, \end{aligned}$$

for any $s \le {\mathfrak {t}}_L$ and $\tau \le \lambda {\mathfrak {t}}_L$ we have

$$\begin{aligned} \Vert a_k(\cdot ,s,\tau )\Vert _{C^r_x}&\le \Vert \sqrt{\rho _\ell }\Vert _{C^r_x} \Vert \Gamma \Vert _{C_x} \Vert \Psi \Vert _{C_x}\\&\quad + \nonumber \Vert \sqrt{\rho _\ell }\Vert _{C_x} \Vert \Gamma \Vert _{C^r_x} \Vert \Psi \Vert _{C_x}\\&\quad + \nonumber \Vert \sqrt{\rho _\ell }\Vert _{C_x} \Vert \Gamma \Vert _{C_x} \Vert \Psi \Vert _{C^r_x},&r \in [0,1],\quad \\ \Vert a_k(\cdot ,s,\tau )\Vert _{C^r_x}&\le C \sum _{r_1+r_2+r_3=r} \Vert \sqrt{\rho _\ell }\Vert _{C^{r_1}_x} \Vert \Gamma \Vert _{C^{r_2}_x} \Vert \Psi \Vert _{C^{r_3}_x},&r \in \mathbb {N}, r > 1. \end{aligned}$$

(B.1)

Also, notice that since $a_k$ depends on $\tau$ only via $\psi _k^{(j)}$,

$$\begin{aligned} \partial _\tau a_k&= {\textbf{1}}_{\{k \in \Lambda _j\}} \sqrt{\rho _\ell } \, \Gamma \, \partial _\tau \Psi , \end{aligned}$$

and thus

$$\begin{aligned} \Vert \partial _\tau a_k(\cdot ,s,\tau )\Vert _{C^r_x}&\le \Vert \sqrt{\rho _\ell }\Vert _{C^r_x} \Vert \Gamma \Vert _{C_x} \Vert \partial _\tau \Psi \Vert _{C_x} \end{aligned}$$

(B.2)

$$\begin{aligned}&\quad + \nonumber \Vert \sqrt{\rho _\ell }\Vert _{C_x} \Vert \Gamma \Vert _{C^r_x} \Vert \partial _\tau \Psi \Vert _{C_x} \\&\quad + \nonumber \Vert \sqrt{\rho _\ell }\Vert _{C_x} \Vert \Gamma \Vert _{C_x} \Vert \partial _\tau \Psi \Vert _{C^r_x},&r \in [0,1],\quad \end{aligned}$$

(B.3)

$$\begin{aligned} \Vert \partial _\tau a_k(\cdot ,s,\tau )\Vert _{C^r_x}&\le C \sum _{r_1+r_2+r_3=r} \Vert \sqrt{\rho _\ell }\Vert _{C^{r_1}_x} \Vert \Gamma \Vert _{C^{r_2}_x} \Vert \partial _\tau \Psi \Vert _{C^{r_3}_x},&r \in \mathbb {N}, r > 1. \end{aligned}$$

(B.4)

Similarly, for $r \in [0,1]$

$$\begin{aligned}&\Vert (\partial _\tau a_k + i(k \cdot {\tilde{v}})a_k)(\cdot ,s,\tau )\Vert _{C^r_x} \nonumber \\ {}&\le \Vert \sqrt{\rho _\ell }\Vert _{C^r_x} \Vert \Gamma \Vert _{C_x} \Vert \partial _\tau \Psi + i(k \cdot {\tilde{v}})\Psi \Vert _{C_x} \\&\quad + \nonumber \Vert \sqrt{\rho _\ell }\Vert _{C_x} \Vert \Gamma \Vert _{C^r_x} \Vert \partial _\tau \Psi + i(k \cdot {\tilde{v}})\Psi \Vert _{C_x} \\&\quad + \nonumber \Vert \sqrt{\rho _\ell }\Vert _{C_x} \Vert \Gamma \Vert _{C_x} \Vert \partial _\tau \Psi + i(k \cdot {\tilde{v}})\Psi \Vert _{C^r_x}, \end{aligned}$$

(B.5)

whereas for $r \in \mathbb {N}, r > 1$

$$\begin{aligned}&\Vert (\partial _\tau a_k + i(k \cdot {\tilde{v}})a_k)(\cdot ,s,\tau )\Vert _{C^r_x} \nonumber \\ {}&\le C \sum _{r_1+r_2+r_3=r} \Vert \sqrt{\rho _\ell }\Vert _{C^{r_1}_x} \Vert \Gamma \Vert _{C^{r_2}_x} \Vert \partial _\tau \Psi + i(k \cdot {\tilde{v}})\Psi \Vert _{C^{r_3}_x}. \end{aligned}$$

(B.6)

Also, notice that $\Vert \rho _\ell \Vert _{C_x} \le C L_n \delta _n$ and $\Vert \Gamma \Vert _{C_x},\Vert \Psi \Vert _{C_x} \le C$ by construction.

Proof of (4.1). In view of (B.1) we need to estimate the quantities $\sqrt{\rho _\ell }, \Gamma , \Psi$ in the Hölder space $C^r_x$, $r \in [0,1]$. For $r=0$ the bounds obtained are sub-optimal, but still sufficient for our future purposes and we do not treat this case separately for the sake of simplicity.

Bound on $\Vert \sqrt{\rho _\ell }\Vert _{C^r_x}$. Since $c\delta _n \le \rho _\ell \le CL_n \delta _n$ and ${\tilde{\rho }}_\ell \ge \eta \delta _{n+1}$ by construction, for every $s \le {\mathfrak {t}}_L$ it holds

$$\begin{aligned} \Vert \sqrt{\rho _\ell (\cdot ,s)}\Vert _{C^r_x}&\le C L_n^{1/2} \delta _n^{1/2} + C \delta _n^{-1/2} \Vert \rho _\ell (\cdot ,s)\Vert _{C^r_x} \\&\le C L_n^{1/2} \delta _n^{1/2} + C \delta _n^{-1/2} L_n \Vert \mathring{R}_\ell (\cdot ,s)\Vert _{C^r_x} \\&\le C L^2_n \delta _n^{-1/2} \delta _{n+1}^{1-r} D_n^r. \end{aligned}$$

Bound on $\Vert \Gamma \Vert _{C^r_x}$. By definition of $\rho _\ell$, the range of the function $R_\ell / \rho _\ell$ is contained in $B_{\frac{r_0}{2}(Id)}$; since $\gamma _k^{(j)}$ is smooth on $B_{r_0(Id)}$ we can bound

$$\begin{aligned} \Vert \Gamma (\cdot ,s)\Vert _{C^r_x} \le C \left\| \frac{R_\ell (\cdot ,s)}{\rho _\ell (\cdot ,s)} \right\| _{C^r_x} \le C + C \left\| \frac{\mathring{R}_\ell (\cdot ,s)}{\rho _\ell (\cdot ,s)} \right\| _{C^r_x}, \end{aligned}$$

with the constant C depending only on $\gamma _{k}^{(j)}$ and $r_0$. Therefore for every $s \le {\mathfrak {t}}_L$ we have

$$\begin{aligned} \Vert \Gamma (\cdot ,s)\Vert _{C^r_x}&\le C+C \Vert \mathring{R}_\ell (\cdot ,s)\Vert _{C_x} \Vert \rho _\ell ^{-1}(\cdot ,s)\Vert _{C^r_x} + C \Vert \mathring{R}_\ell (\cdot ,s)\Vert _{C^r_x} \Vert \rho _\ell ^{-1}(\cdot ,s)\Vert _{C_x}. \\&\le C L_n^3 \delta _n^{-1}\delta _{n+1}^{1-r} D_n^r. \end{aligned}$$

Bound on $\Vert \Psi \Vert _{C^r_x}$. Recall that $r\in [0,1]$, thus (3.4) implies

$$\begin{aligned} \Vert \Psi (\cdot ,s,\tau )\Vert _{C^r_x}&\le \Vert \psi _k^{(j)}(\cdot ,\tau )\Vert _{C^r_v} \Vert {\tilde{v}}(\cdot ,s)\Vert _{C^1_x}^r \le C \mu ^r \Vert {\tilde{v}}(\cdot ,s)\Vert _{C^1_x}^r. \end{aligned}$$

By Lemma C.1, for every $s \le {\mathfrak {t}}_L$ and $\tau \le \lambda {\mathfrak {t}}_L$ it holds (recall that the very definition of $D_n,\varsigma _n$ implies $D_n \le \varsigma _{n+1}^{\alpha -1}$)

$$\begin{aligned} \Vert {\tilde{v}}(\cdot ,s)\Vert _{C^1_x}&\le \Vert v_\ell (\cdot ,s)\Vert _{C^1_x} + \Vert {\dot{\phi }}_{n+1}(\cdot ,s)\Vert _{C^1_x} \\&\le \nonumber C L_n D_n + CL \varsigma _{n+1}^{\alpha -1} \\&\le \nonumber CL_n \varsigma _{n+1}^{\alpha -1}. \end{aligned}$$

(B.7)

Putting all together we recover the desired bound for $\Vert a_k(\cdot ,s,\tau )\Vert _{C^r_x}$, $r \in [0,1]$. $\square$

Proof of (4.3). Thanks to (B.5) and the estimates on $\Vert \sqrt{\rho _\ell (\cdot ,s)}\Vert _{C^r_x}$, $\Vert \Gamma (\cdot ,s)\Vert _{C^r_x}$ shown above, we only need to control the quantity $\Vert \partial _\tau \Psi + i(k \cdot {\tilde{v}})\Psi \Vert _{C^r_x}$, $r \in [0,1]$. Since $\partial _\tau \Psi + i(k \cdot {\tilde{v}})\Psi$ depends on y only through its argument ${\tilde{v}}$, it holds

$$\begin{aligned} \Vert \partial _\tau \Psi + i(k \cdot {\tilde{v}})\Psi \Vert _{C^r_x}&\le \Vert \partial _\tau \Psi + i(k \cdot {\tilde{v}})\Psi \Vert _{C^r_v} \Vert {\tilde{v}} \Vert _{C^1_x}^r \le C \mu ^{r-1} \Vert {\tilde{v}} \Vert _{C^1_x}^r, \end{aligned}$$

where we have used (3.6) with $r=0,1$ and interpolation. Using (B.7) and collecting all the necessary bounds, we get the desired inequality. $\square$

Proof of (4.5). Let us now move to the estimate for $\Vert a_k(\cdot ,s,\tau )\Vert _{C^r_x}$, $r \in \mathbb {N}$, $r > 1$. By (B.2) we only need to bound $\sqrt{\rho _\ell }, \Gamma , \Psi$ in the Hölder spaces $C^{r_i}_x$, $r_i \in \mathbb {N}$, $r_i \ge 1$, $i=1,2,3$.

Bound on $\Vert \sqrt{\rho _\ell }\Vert _{C^{r_1}_x}$. By [16, Proposition 4.1] we have for every $s \le {\mathfrak {t}}_L$

$$\begin{aligned} \Vert \sqrt{\rho _\ell } \Vert _{C^{r_1}_x}&\le C L_n^{1/2} \delta _n^{1/2} + C \sum _{j=1}^{r_1} \delta _n^{1/2-j} \Vert \rho _\ell \Vert _{C_x}^{j-1} \Vert \rho _\ell \Vert _{C^{r_1}_x} \\&\le CL_n^{1/2} \delta _n^{1/2} + C \delta _n^{-1/2} L_n^{{r_1}-1} \Vert \rho _\ell \Vert _{C^{r_1}_x}, \\ \Vert \rho _\ell \Vert _{C^{r_1}_x}&\le CL_n \delta _n + C \sum _{j=1}^{r_1} \delta _{n+1}^{1-j} L_n^j \Vert \mathring{R}_\ell \Vert _{C_x}^{j-1} \Vert \mathring{R}_\ell \Vert _{C^{r_1}_x} \\&\le C L_n^{2{r_1}} D_n \ell ^{1-{r_1}}, \end{aligned}$$

from which we deduce

$$\begin{aligned} \Vert \sqrt{\rho _\ell } \Vert _{C^{r_1}_x}&\le C L_n^{3{r_1}-1} \delta _n^{-1/2} D_n \ell ^{1-{r_1}}. \end{aligned}$$

Bound on $\Vert \Gamma \Vert _{C^{r_2}_x}$. Again by Leibniz rule for the derivative of a product,

$$\begin{aligned} \Vert \Gamma \Vert _{C^{r_2}_x}&\le C \sum _{h_1+h_2={r_2}} \Vert \mathring{R}_\ell \Vert _{C^{h_1}_x} \Vert \rho _\ell ^{-1}\Vert _{C^{h_2}_x} , \end{aligned}$$

and [16, Proposition 4.1] implies for $h_2 \ge 1$ and for every $s \le {\mathfrak {t}}_L$

$$\begin{aligned} \Vert \rho _\ell ^{-1}\Vert _{C^{h_2}_x}&\le C \sum _{j=1}^{h_2} \delta _n^{-1-j} \Vert \rho _\ell \Vert _{C_x}^{j-1} \Vert \rho _\ell \Vert _{C^{h_2}_x} \le C L_n^{3{h_2}-1} \delta _n^{-2} D_n \ell ^{1-{h_2}}. \end{aligned}$$

Therefore, since $\Vert \mathring{R}_\ell \Vert _{C_x} \le L_n \delta _{n+1}$ and $\Vert \mathring{R}_\ell \Vert _{C^{h_1}_x} \le L_n D_n \ell ^{1-h_1}$ when $h_1 \ge 1$, for every $s \le {\mathfrak {t}}_L$ it holds

$$\begin{aligned} \Vert \Gamma \Vert _{C^{r_2}_x}&\le C L_n^{3{r_2}} \delta _n^{-1} D_n \ell ^{1-{r_2}} . \end{aligned}$$

Bound on $\Vert \Psi \Vert _{C^{r_3}_x}$. As usual, we estimate for given $s \le {\mathfrak {t}}_L$ and $\tau \le \lambda {\mathfrak {t}}_L$

$$\begin{aligned} \Vert \Psi \Vert _{C^{r_3}_x}&\le C+C \sum _{j=1}^{r_3} \mu ^j \Vert {\tilde{v}} \Vert _{C_x}^{j-1} \Vert {\tilde{v}}\Vert _{C^{r_3}_x} \\&\le C+C \sum _{j=1}^{r_3} \mu ^j ( L_n^{1/2} + L \varsigma _{n+1}^{\alpha -1} )^{j-1} ( L_n D_n \ell ^{1-{r_3}} + L\varsigma _{n+1}^{\alpha -1}) \\&\le C L_n^{({r_3}+1)/2} \mu ^{r_3} \varsigma _{n+1}^{({r_3}-1)(\alpha -1)}(D_n \ell ^{1-{r_3}} + \varsigma _{n+1}^{\alpha -1}). \end{aligned}$$

We have used Lemma 2.2 in the second inequality. Taking into account the bounds above we arrive to

$$\begin{aligned} \Vert a_k(\cdot ,s,\tau )\Vert _{C^r_x}&\le C L_n^{3r} \delta _n^{1/2} \mu ^r \varsigma _{n+1}^{(r-1)(\alpha -1)} (D_n \ell ^{1-r} + \varsigma _{n+1}^{\alpha -1}). \end{aligned}$$

$\square$

Proof of (4.7). As for the estimate (4.7), we only need to control $\Vert \partial _\tau \Psi + i(k \cdot {\tilde{v}})\Psi \Vert _{C^{r_3}_x}$, ${r_3} \in \mathbb {N}$, ${r_3} \ge 1$. We proceed as follows. Using [16, Proposition 4.1] and (3.6), we have for given $s \le {\mathfrak {t}}_L$ and $\tau \le \lambda {\mathfrak {t}}_L$

$$\begin{aligned} \Vert \partial _\tau \Psi + i(k \cdot {\tilde{v}})\Psi \Vert _{C^{r_3}_x}&\le C \sum _{j=1}^{r_3} \mu ^{j-1} \Vert {\tilde{v}}\Vert _{C_x}^{j-1} \Vert {\tilde{v}}\Vert _{C^{r_3}_x} \\&\le C \mu ^{{r_3}-1} L_n^{({r_3}+1)/2} \varsigma _{n+1}^{({r_3}-1)(\alpha -1)}(D_n \ell ^{1-{r_3}} + \varsigma _{n+1}^{\alpha -1}). \end{aligned}$$

Using (B.6), the conclusion is immediate. $\square$

Proof of (4.2) and (4.6). By (3.5) the bounds for $\partial _\tau \Psi$ are exactly the same as for $\Psi$, up to a multiplicative factor $\Vert {\tilde{v}}\Vert _{C_x} \le CL_n^{1/2}\varsigma _{n+1}^{\alpha -1}$ in front. Since the largest power of L comes from $\Gamma$, by (B.3) and (B.4) the desired estimates are obtained by (4.1) and (4.5) by multiplying with an additional factor $\varsigma _{n+1}^{\alpha -1}$. $\square$

Proof of (4.4) and (4.8). Let us now prove the estimates for $\partial _s a_k$. We have

$$\begin{aligned} \partial _s a_k&= {\textbf{1}}_{\{k \in \Lambda _j\}} \partial _s\sqrt{\rho _\ell } \, \Gamma \, \Psi + {\textbf{1}}_{\{k \in \Lambda _j\}} \sqrt{\rho _\ell } \, \partial _s\Gamma \, \Psi + {\textbf{1}}_{\{k \in \Lambda _j\}} \sqrt{\rho _\ell } \, \Gamma \, \partial _s\Psi , \end{aligned}$$

so that

$$\begin{aligned} \Vert \partial _s a_k \Vert _{C_x}&\le \Vert \partial _s\sqrt{\rho _\ell } \Vert _{C_x} \Vert \Gamma \Vert _{C_x} \Vert \Psi \Vert _{C_x} \\&\quad + \nonumber \Vert \sqrt{\rho _\ell } \Vert _{C_x} \Vert \partial _s \Gamma \Vert _{C_x} \Vert \Psi \Vert _{C_x} \\&\quad + \nonumber \Vert \sqrt{\rho _\ell } \Vert _{C_x} \Vert \Gamma \Vert _{C_x} \Vert \partial _s \Psi \Vert _{C_x}, \end{aligned}$$

(B.8)

and for $r \in \mathbb {N}$, $r \ge 1$

$$\begin{aligned} \Vert \partial _s a_k \Vert _{C^r_x}&\le C\sum _{r_1+r_2+r_3=r} \Vert \partial _s\sqrt{\rho _\ell } \Vert _{C^{r_1}_x} \Vert \Gamma \Vert _{C^{r_2}_x} \Vert \Psi \Vert _{C^{r_3}_x} \\&\quad + \nonumber C\sum _{r_1+r_2+r_3=r} \Vert \sqrt{\rho _\ell } \Vert _{C^{r_1}_x} \Vert \partial _s \Gamma \Vert _{C^{r_2}_x} \Vert \Psi \Vert _{C^{r_3}_x} \\&\quad + \nonumber C\sum _{r_1+r_2+r_3=r} \Vert \sqrt{\rho _\ell } \Vert _{C^{r_1}_x} \Vert \Gamma \Vert _{C^{r_2}_x} \Vert \partial _s \Psi \Vert _{C^{r_3}_x}. \end{aligned}$$

(B.9)

The quantities $\Vert \sqrt{\rho _\ell } \Vert _{C^{r_1}_x}$, $\Vert \Gamma \Vert _{C^{r_2}_x}$ and $\Vert \Psi \Vert _{C^{r_3}_x}$ have been bounded during previous steps of the proof; we are left to provide analogous bounds for $\partial _s\sqrt{\rho _\ell }$, $\partial _s \Gamma$ and $\partial _s\Psi$.

Bounds on $\partial _s \sqrt{\rho _\ell }$. Rewrite, for fixed $s \le {\mathfrak {t}}_L$:

$$\begin{aligned} \Vert \partial _s \sqrt{\rho _\ell } \Vert _{C_x} \le C \delta _n^{-1/2} \Vert \partial _s \rho _\ell \Vert _{C_x}, \end{aligned}$$

(B.10)

and for every ${r_1} \ge 1$

$$\begin{aligned} \Vert \partial _s \sqrt{\rho _\ell } \Vert _{C^{r_1}_x}&\le C \sum _{h_1+h_2=r_1} \Vert \partial _s \rho _\ell \Vert _{C^{h_1}_x} \Vert {\rho _\ell }^{-1/2}\Vert _{C^{h_2}_x} . \end{aligned}$$

(B.11)

It holds

$$\begin{aligned} \partial _s \rho _\ell&= \frac{\partial _s |\mathring{R}_\ell |^2}{r_0 \sqrt{\eta ^2 \delta _{n+1}^2 + |\mathring{R}_\ell |^2}} + \partial _s \gamma _n = \frac{\sum _{i,j}\mathring{R}_\ell ^{i,j} \partial _s \mathring{R}_\ell ^{i,j}}{r_0 \sqrt{\eta ^2 \delta _{n+1}^2 + |\mathring{R}_\ell |^2}} + \partial _s \gamma _n. \end{aligned}$$

Now, notice that by (3.9) we have $|\partial _s \gamma _n | \le C\Vert {\tilde{e}} \Vert _{C^1_{\le {\mathfrak {t}}}} \le C{\overline{e}}+C{\overline{e}}^{1/2}D_n$, where the second inequality is justified by our iterative assumptions (2.5) and (2.9), and (recall $\chi _\ell \ge 0$ to apply Cauchy-Schwartz inequality to $v_n \chi _\ell ^{1/2}$ and $\chi _\ell ^{1/2}$)

$$\begin{aligned} \int _{\mathbb {T}^3} |v_\ell (x,t)|^2 dx&= \int _{\mathbb {T}^3} \left| \int _{\mathbb {T}^3 \times \mathbb {R}} v_n(x-y,t-s) \chi _\ell (y,s) dy ds\right| ^2 dx \\&\le \int _{\mathbb {T}^3} \int _{\mathbb {T}^3 \times \mathbb {R}} |v_n(x-y,t-s)|^2 \chi _\ell (y,s) dy ds dx \\&\le \int _{\mathbb {T}^3 \times \mathbb {R}} e(t-s) \chi _\ell (y,s) dy ds \le {\overline{e}}. \end{aligned}$$

Moreover, $\Vert \partial _s \mathring{R}_\ell \Vert _{C^h_x} \le CL_n \delta _{n+1} \ell ^{-1-h}$ for every $h \in \mathbb {N}$, by standard mollification estimates. Hence, using [16, Proposition 4.2] to bound the Hölder norm of the term under square root and recalling (3.1), we get

$$\begin{aligned} \Vert \partial _s \rho _\ell \Vert _{C_x}&\le C L_n^2 \delta _{n+1} \ell ^{-1}, \\ \Vert \partial _s \rho _\ell \Vert _{C^{h_1}_x}&\le C L_n^{2h_1+2} \delta _{n+1} \ell ^{-1-h_1}. \end{aligned}$$

Plugging back into (B.10) and (B.11) we obtain

$$\begin{aligned} \Vert \partial _s \sqrt{\rho _\ell } \Vert _{C_x}&\le C L_n^2 \delta _n^{-1/2} \delta _{n+1} \ell ^{-1}, \\ \Vert \partial _s \sqrt{\rho _\ell } \Vert _{C^{r_1}_x}&\le CL_n^{3{r_1}+1}\delta _n^{-1/2}\delta _{n+1} \ell ^{-1-r_1}. \end{aligned}$$

Bounds on $\partial _s \Gamma$. Since $\gamma _k^{(j)}$ is smooth on $B_{r_0(Id)}$ and $R_\ell /\rho _\ell$ takes values in $B_{\frac{r_0}{2}(Id)}$, we can bound

$$\begin{aligned} \Vert \partial _s \Gamma \Vert _{C_x}&\le C \left\| \,\partial _s \frac{\mathring{R}_\ell }{\rho _\ell } \right\| _{C_x} , \quad \Vert \partial _s \Gamma \Vert _{C^{r_2}_x} \le C \left\| \,\partial _s \frac{\mathring{R}_\ell }{\rho _\ell }\right\| _{C^{r_2}_x}. \end{aligned}$$

On the other hand,

$$\begin{aligned} \partial _s \frac{\mathring{R}_\ell }{\rho _\ell }&= \frac{\partial _s \mathring{R}_\ell }{\rho _\ell } - \frac{\mathring{R}_\ell \partial _s \rho _\ell }{\rho _\ell ^2}, \end{aligned}$$

therefore, taking also into account the previous estimates for $\rho _\ell$

$$\begin{aligned} \Vert \partial _s \Gamma \Vert _{C_x}&\le C L_n^3 \delta _n^{-1} \delta _{n+1} \ell ^{-1}, \\ \Vert \partial _s \Gamma \Vert _{C^{r_2}_x}&\le C L_n^{3{r_2}+2} \delta _n^{-2} \delta _{n+1}^2 \ell ^{-1-{r_2}}. \end{aligned}$$

Bounds on $\partial _s \Psi$. Finally, since $\Psi$ depends on s only through ${\tilde{v}}(y,s)$:

$$\begin{aligned} \partial _s \Psi&= D_v \psi _k^{(j)} ({\tilde{v}},\tau ) \cdot \partial _s {\tilde{v}}, \end{aligned}$$

and recalling (3.4) and [16, Proposition 4.1] for the Hölder seminorm of compositions, we get

$$\begin{aligned} \Vert \partial _s \Psi \Vert _{C_x}&\le C \mu \Vert \partial _s {\tilde{v}} \Vert _{C_x}, \\ \Vert \partial _s \Psi \Vert _{C^{r_3}_x}&\le C \mu \Vert \partial _s {\tilde{v}} \Vert _{C_x} + C|D_v \psi _k^{(j)}| \Vert \partial _s {\tilde{v}}\Vert _{C^{r_3}_x} \\&\quad + C\sum _{i=1}^{r_3} |D_v^{1+i} \psi _k^{(j)}| \Vert {\tilde{v}}\Vert _{C_x}^{i-1} \Vert {\tilde{v}}\Vert _{C^{r_3}_x} \Vert \partial _s {\tilde{v}}\Vert _{C_x} \\&\le C \mu \Vert \partial _s {\tilde{v}} \Vert _{C^{r_3}_x} + C \mu ^{{r_3}+1} L_n^{{r_3}} \varsigma _{n+1}^{({r_3}-1)(\alpha -1)} (D_n \ell ^{1-{r_3}} + \varsigma _{n+1}^{\alpha -1}) \Vert \partial _s {\tilde{v}} \Vert _{C_x}. \end{aligned}$$

The partial derivative of ${\tilde{v}}$ with respect to s is given by $\partial _s {v}_\ell + \ddot{\phi }_{n+1}$, therefore

$$\begin{aligned} \Vert \partial _s {\tilde{v}} \Vert _{C_x}&\le {C L_n D_n + C L \varsigma _{n+1}^{\alpha -2} \le C L_n \varsigma _{n+1}^{\alpha -2}}; \\ \Vert \partial _s {\tilde{v}} \Vert _{C^{r_3}_x}&\le {C L_n D_n \ell ^{-{r_3}} + C L \varsigma _{n+1}^{\alpha -2} \le C L_n (D_n \ell ^{-r_3} + \varsigma _{n+1}^{\alpha -2} ).} \end{aligned}$$

Therefore

$$\begin{aligned} \Vert \partial _s \Psi \Vert _{C_x}&\le C L_n \mu \varsigma _{n+1}^{\alpha -2}, \\ \Vert \partial _s \Psi \Vert _{C^{r_3}_x}&\le C L_n^{r_3+1} \mu ^{r_3+1} \varsigma _{n+1}^{r_3(\alpha -1)-1} (D_n \ell ^{1-{r_3}} +\varsigma _{n+1}^{\alpha -1}). \end{aligned}$$

Collecting everything and plugging into (B.8) and (B.9), we finally arrive to:

$$\begin{aligned} \Vert \partial _s a_k \Vert _{C_x}&\le CL_n^{7/2} \delta _n^{1/2} \mu \varsigma _{n+1}^{\alpha -2}, \end{aligned}$$

$$\begin{aligned} \Vert \partial _s a_k \Vert _{C^r_x}&\le C L_n^{3r+5/2} \delta _n^{1/2} \mu ^{r+1} \varsigma _{n+1}^{r(\alpha -1)-1} (D_n \ell ^{1-r}+\varsigma _{n+1}^{\alpha -1}). \end{aligned}$$

$\square$

Appendix C: Hölder and Besov estimates

Lemma C.1

Let $\phi =\phi _n$, $n \in \mathbb {N}$. For any $\delta \in [0,1)$ and $r \in \mathbb {N}$, $r \le \kappa$ there exists a constant $C=C(\delta ,r)$ such that the following holds almost surely for every $L \in \mathbb {N}$, $L \ge 1$. For every f on $\mathbb {T}^3 \times (\infty ,{\mathfrak {t}}_L]$ of class $C_{\le {\mathfrak {t}}_L}C^{r+\delta }_x$ and every fixed $t \le {\mathfrak {t}}_L$

$$\begin{aligned} \Vert f \circ \phi \Vert _{C^{r+\delta }_x}&\le C L^{r+\delta }\Vert f\Vert _{C^{r+\delta }_x}, \\ \Vert f \circ \phi ^{-1}\Vert _{C^{r+\delta }_x}&\le C L^{r+\delta }\Vert f\Vert _{C^{r+\delta }_x}, \end{aligned}$$

where $f \circ \phi$ denotes the map $\mathbb {T}^3 \times (\infty ,{\mathfrak {t}}_L] \ni (x,t) \mapsto f(\phi (x,t),t)$, and similarly for $f \circ \phi ^{-1}$.

Proof

Fix $t \le {\mathfrak {t}}_L$. For $r=0$, we have

$$\begin{aligned}{}[ f \circ \phi ]_{C^{\delta }_x}&= \sup _{x \ne y \in \mathbb {T}^3} \frac{|f(\phi (x,t),t)-f(\phi (y,t),t)|}{|x-y|^\delta } \\&= \sup _{x \ne y \in \mathbb {T}^3} \frac{|f(\phi (x,t),t)-f(\phi (y,t),t)|}{|\phi (x,t)-\phi (y,t)|^\delta } \frac{|\phi (x,t)-\phi (y,t)|^\delta }{|x-y|^\delta } \le C^\delta L^\delta \Vert f\Vert _{C^{\delta }_x}. \end{aligned}$$

In the previous line we have used $|\phi (x,t)-\phi (y,t)| \le \Vert \phi (\cdot ,t)\Vert _{C^1_x} |x-y| \le CL |x-y|$ since $t \le {\mathfrak {t}}_L$, by the very definition of the stopping time ${\mathfrak {t}}_L$.

For $r>0$, we argue as follows. Let ${\textbf{k}}$ be any multi-index of order $k\le r$; we use Faà di Bruno formula and the previous decomposition to deduce

$$\begin{aligned}{}[ D^{\textbf{k}} (f \circ \phi ) ]_{C^{\delta }_x}&\le \sum _{\begin{array}{c} r_1,\dots ,r_k\\ |{\textbf{r}}|=r_1+\dots +r_k \end{array}} C_{r_i,k} \Vert (D^{\textbf{r}}f) \circ \phi \Vert _{C^{\delta }_x} \prod _{j=1}^{k} \Vert \phi \Vert _{C^j_x}^{r_j} \\&\le \sum _{\begin{array}{c} r_1,\dots ,r_k\\ |{\textbf{r}}|=r_1+\dots +r_k \end{array}} C_{r_i,k} \Vert f\Vert _{C^{|{\textbf{r}}|+\delta }_x} C^\delta L^\delta \prod _{j=1}^{k} \Vert \phi \Vert _{C^j_x}^{r_j} \\&\le \sum _{\begin{array}{c} r_1,\dots ,r_k\\ |{\textbf{r}}|=r_1+\dots +r_k \end{array}} C_{r_i,k} \Vert f\Vert _{C^{|{\textbf{r}}|+\delta }_x} C^{|{\textbf{r}}|+\delta }L^{|{\textbf{r}}|+\delta } , \end{aligned}$$

where the sum is taken over all $r_i$ such that $r_1+2r_2+\dots +kr_k = k$ and multi-indices ${\textbf{r}}$ with $|{\textbf{r}}|=r_1+\dots +r_m$, and $C_{r_i,k}$ are constants depending only on k and $r_1,\dots ,r_k$. In the last inequality we have used $\Vert \phi \Vert _{C^j_x} \le CL$ since $j \le \kappa$ and $t \le {\mathfrak {t}}_L$ is fixed. We conclude using $|{\textbf{r}}| \le k \le r$ and taking the sum over all multi-indices of order $\le r$. The second inequality is proved in a similar fashion. $\square$

Lemma C.2

Let $\phi =\phi _n$, $n \in \mathbb {N}$. For every $\alpha \in (0,2)$, $\alpha \ne 1$ there exists $C=C(\alpha )$ such that for every $L \in \mathbb {N}$, $L\ge 1$ and every continuous function f on $\mathbb {T}^3 \times (-\infty ,{\mathfrak {t}}_L]$ it holds

$$\begin{aligned} \Vert f \circ \phi \Vert _{B^{-\alpha }_{\infty ,\infty }}&\le CL^{4-\alpha }\Vert f \Vert _{B^{-\alpha }_{\infty ,\infty }}, \\ \Vert f \circ \phi ^{-1} \Vert _{B^{-\alpha }_{\infty ,\infty }}&\le CL^{4-\alpha }\Vert f \Vert _{B^{-\alpha }_{\infty ,\infty }}. \end{aligned}$$

Proof

We only prove the second claim, the other being similar. Let us preliminarily prove that for every smooth g and $L\ge 1$ it holds

$$\begin{aligned} \Vert g \circ \phi \Vert _{B^{\alpha }_{1,1}}&\le CL^{4-\alpha }\Vert g \Vert _{B^{\alpha }_{1,1}}. \end{aligned}$$

(C.1)

By continuity of the operator $(Id-\Delta )^{-1}:B^{\alpha -2}_{1,1}\rightarrow B^{\alpha }_{1,1}$ [45, Proposition A.6] it holds $\Vert g \circ \phi \Vert _{B^{\alpha }_{1,1}} \le C \Vert (Id-\Delta ) (g \circ \phi ) \Vert _{B^{\alpha -2}_{1,1}}$. Since g is smooth we have

$$\begin{aligned} (Id-\Delta ) (g \circ \phi ) = g \circ \phi - \sum _{j,k = 1}^d \partial _{x_j} g \circ \phi \,\partial _{x_k}^2 \phi ^j - \sum _{j,k,\ell = 1}^d \partial _{x_j} \partial _{x_\ell } g \circ \phi \, \partial _{x_k}\phi ^j \partial _{x_k} \phi ^\ell \end{aligned}$$

By paraproduct estimates for Besov functions (see [32, Lemma 2.2] or [45, Proposition A.7]) and Lemma 2.2

$$\begin{aligned} \Vert g \circ \phi \Vert _{B^{\alpha }_{1,1}}&\le C \Vert (Id-\Delta ) (g \circ \phi ) \Vert _{B^{\alpha -2}_{1,1}} \\&\le C \Vert g \circ \phi \Vert _{B^{\alpha -2}_{1,1}} + CL \sum _{j=1}^d \Vert \partial _{x_j}g \circ \phi \Vert _{B^{\alpha -2}_{1,1}}\\ {}&\quad + CL^2 \sum _{j,\ell =1}^d \Vert \partial _{x_j}\partial _{x_\ell }g \circ \phi \Vert _{B^{\alpha -2}_{1,1}} \\&\le CL^{2-\alpha } \Vert g \Vert _{B^{\alpha -2}_{1,1}} + CL^{3-\alpha } \sum _{j=1}^d \Vert \partial _{x_j}g \Vert _{B^{\alpha -2}_{1,1}}\\ {}&\quad + CL^{4-\alpha } \sum _{j,\ell =1}^d \Vert \partial _{x_j}\partial _{x_\ell }g\Vert _{B^{\alpha -2}_{1,1}}, \end{aligned}$$

where in the last line we have used the duality $B^{\alpha -2}_{1,1} = (B^{2-\alpha }_{\infty ,\infty })^* = (C^\alpha _x)^*$, with equivalence of norms, the measure preserving property of $\phi ^{-1}$, and previous Lemma C.1. By continuity of derivatives (Proposition A.5 in [45]) we conclude the proof of (C.1).

Let us now come back to the estimate for $f \circ \phi ^{-1}$. By duality, and since any function $g \in B^{\alpha }_{1,1}$ can be approximated by smooth functions in the $B^{\alpha }_{1,1}$ topology, there exists a constant C such that

$$\begin{aligned} \Vert f \circ \phi ^{-1} \Vert _{B^{-\alpha }_{\infty ,\infty }}&= \sup _{\begin{array}{c} g \in B^{\alpha }_{1,1},\\ \Vert g\Vert _{B^{\alpha }_{1,1}}=1 \end{array}} \int _{\mathbb {T}^d} (f \circ \phi ^{-1}) g dx \le \sup _{\begin{array}{c} g \in C^\infty ,\\ \Vert g\Vert _{B^{\alpha }_{1,1}}\le 2 \end{array}} \int _{\mathbb {T}^d} (f \circ \phi ^{-1}) g dx. \end{aligned}$$

Thus, since f is continuous, $\phi$ is measure preserving and by (C.1)

$$\begin{aligned} \sup _{\begin{array}{c} g \in C^\infty ,\\ \Vert g\Vert _{B^{\alpha }_{1,1}}\le 2 \end{array}} \int _{\mathbb {T}^d} (f \circ \phi ^{-1}) g dx&= \sup _{\begin{array}{c} g \in C^\infty ,\\ \Vert g\Vert _{B^{\alpha }_{1,1}}\le 2 \end{array}} \int _{\mathbb {T}^d} f (g\circ \phi ) dx \\&\le \sup _{\begin{array}{c} h \in C^\infty ,\\ \Vert h\Vert _{B^{\alpha }_{1,1}}\le CL^{4-\alpha } \end{array}} \int _{\mathbb {T}^d} f h dx \le CL^{4-\alpha } \Vert f\Vert _{B^{-\alpha }_{\infty ,\infty }}. \end{aligned}$$

$\square$

Lemma C.3

Let $\phi =\phi _n$, $n \in \mathbb {N}$. For any $\beta \in (0,\alpha ]$ there exists a constant $C=C(\beta ,\alpha )$ such that almost surely for every $L \in \mathbb {N}$, $L \ge 1$, for every smooth function f on $\mathbb {T}^3 \times (\infty ,{\mathfrak {t}}_L]$ and given $t \le {\mathfrak {t}}_L$ it holds

$$\begin{aligned} \Vert f \circ \phi \Vert _{C^\beta _t C_x}&\le CL^{\beta /\alpha } \Vert f\Vert _{C_t C^{\beta /\alpha }_x} + \Vert f\Vert _{C^\beta _t C_x}, \\ \Vert f \circ \phi ^{-1} \Vert _{C^\beta _t C_x}&\le CL^{\beta /\alpha } \Vert f\Vert _{C_t C^{\beta /\alpha }_x} + \Vert f\Vert _{C^\beta _t C_x}. \end{aligned}$$

Proof

Let us only prove the claim for $f \circ \phi$. The $C_t C_x$ norm of $f \circ \phi$ is already controlled by the right-hand-side thanks to previous Lemma C.1, thus we only need a bound for the Hölder seminorm $[ f \circ \phi ]_{C^\beta _t C_x}$. We have

$$\begin{aligned}{}[ f \circ \phi ]_{C^\beta _t C_x}&= \sup _{t \ne s \le {\mathfrak {t}}_L} \sup _{x \in \mathbb {T}^3} \frac{|f(\phi (x,t),t)-f(\phi (x,s),s)|}{|t-s|^\beta } \\&\le \sup _{t \ne s \le {\mathfrak {t}}_L} \sup _{x \in \mathbb {T}^3} \frac{|f(\phi (x,t),t)-f(\phi (x,s),t)|}{|\phi (x,t)-\phi (x,s)|^{\beta /\alpha }} \frac{{|\phi (x,t)-\phi (x,s)|^{\beta /\alpha }}}{|t-s|^\beta } \\&\quad + \sup _{t \ne s \le {\mathfrak {t}}_L} \sup _{x \in \mathbb {T}^3} \frac{|f(\phi (x,s),t)-f(\phi (x,s),s)|}{|t-s|^{\alpha }} \\&\le C^{\beta /\alpha } L^{\beta /\alpha } \Vert f\Vert _{C_t C^{\beta /\alpha }_x} + \Vert f\Vert _{C^\beta _t C_x}. \end{aligned}$$

$\square$

Lemma C.4

(Schauder estimates) Let $\phi =\phi _n$, $n \in \mathbb {N}$. For any $\delta \in (0,1)$ and any $r\in \mathbb {N}$, $r + 2 \le \kappa$, there exists a constant $C=C(\delta ,r)$ with the following properties holding almost surely for every $L \in \mathbb {N}$, $L \ge 1$. If $\psi , \Psi : \mathbb {T}^3 \times (\infty ,{\mathfrak {t}}_L] \rightarrow \mathbb {R}$ are the unique zero-average solutions of

$$\begin{aligned} \Delta ^\phi \psi = f, \quad \Delta ^\phi \Psi = \mathord {\textrm{div}}^\phi F, \qquad t\in (\infty ,{\mathfrak {t}}_L], \end{aligned}$$

then for every fixed $t \le {\mathfrak {t}}_L$ we have

$$\begin{aligned} \Vert \psi \Vert _{C^{r+2+\delta }_x}&\le C L^{2r+2+2\delta } \Vert f\Vert _{C^{r+\delta }_x}, \\ \Vert \Psi \Vert _{C^{r+1+\delta }_x}&\le C L^{2r+1+2\delta }\Vert F\Vert _{C^{r+\delta }_x}. \end{aligned}$$

Moreover we have the almost sure estimates for every $v:\mathbb {T}^3 \times (\infty ,{\mathfrak {t}}_L] \rightarrow \mathbb {R}$, $A:\mathbb {T}^3 \times (\infty ,{\mathfrak {t}}_L] \rightarrow \mathbb {R}^{3\times 3}$ and $t \le {\mathfrak {t}}_L$:

$$\begin{aligned} \Vert {\mathcal {Q}}^\phi v\Vert _{C^{r+\delta }_x}&\le C L^{2r+2\delta }\Vert v\Vert _{C^{r+\delta }_x}, \end{aligned}$$

(C.2)

$$\begin{aligned} \Vert {\mathcal {P}}^\phi v\Vert _{C^{r+\delta }_x}&\le C L^{2r+2\delta }\Vert v\Vert _{C^{r+\delta }_x}, \end{aligned}$$

(C.3)

$$\begin{aligned} \Vert {\mathcal {R}}^\phi v\Vert _{C^{r+1+\delta }_x}&\le C L^{2r+1+2\delta }\Vert v\Vert _{C^{r+\delta }_x}, \end{aligned}$$

(C.4)

$$\begin{aligned} \Vert {\mathcal {R}}^\phi (\mathord {\textrm{div}}^\phi A)\Vert _{C^{r+\delta }_x}&\le C L^{2r+2\delta } \Vert A\Vert _{C^{r+\delta }_x}, \end{aligned}$$

(C.5)

$$\begin{aligned} \Vert {\mathcal {R}}^\phi {\mathcal {Q}}^\phi (\mathord {\textrm{div}}^\phi A)\Vert _{C^{r+\delta }_x}&\le C L^{2r+2\delta } \Vert A\Vert _{C^{r+\delta }_x}. \end{aligned}$$

(C.6)

Finally, for $\phi ,v,L$ as above it holds

$$\begin{aligned} \Vert {\mathcal {R}}^\phi v\Vert _{C^\delta _x}&\le C L^{5+4\delta } \Vert v\Vert _{B^{\delta -1}_{\infty ,\infty }}. \end{aligned}$$

(C.7)

Proof

By the very definition of the operators $\Delta ^\phi$ and $\mathord {\textrm{div}}^\phi$ we have $\Delta (\psi \circ \phi ^{-1}) = f \circ \phi ^{-1}$ and $\Delta (\Psi \circ \phi ^{-1}) = \mathord {\textrm{div}}\,(F \circ \phi ^{-1})$ for every $t \le {\mathfrak {t}}_L$; since $\phi$ and $\phi ^{-1}$ are smooth in space, by Lemma C.1 and the usual Schauder estimates for the Laplace problem we get

$$\begin{aligned} \Vert \psi \Vert _{C^{r+2+\delta }_x} \lesssim L^{r+2+\delta }\Vert \psi \circ \phi ^{-1}\Vert _{C^{r+2+\delta }_x}&\lesssim L^{r+2+\delta }\Vert f \circ \phi ^{-1}\Vert _{C^{r+\delta }_x}\\ {}&\lesssim L^{2r+2+2\delta }\Vert f\Vert _{C^{r+\delta }_x}, \\ \Vert \Psi \Vert _{C^{r+1+\delta }_x} \lesssim L^{r+1+\delta }\Vert \Psi \circ \phi ^{-1}\Vert _{C^{r+1+\delta }_x}&\lesssim L^{r+1+\delta }\Vert F \circ \phi ^{-1}\Vert _{C^{r+\delta }_x}\\ {}&\lesssim L^{2r+1+2\delta }\Vert F\Vert _{C^{r+\delta }_x}, \end{aligned}$$

where the implicit constant in the inequalities above depends on r and $\delta$. Estimates (C.2)–(C.6) can be deduced from the analogous esimates for ${\mathcal {Q}},{\mathcal {P}}$ and ${\mathcal {R}}$ contained in [16, Proposition 4.3] with a similar argument. Finally, (C.7) can be derived by the explicit expression of ${\mathcal {P}}^\phi ,{\mathcal {R}}^\phi$, Schauder estimates in Besov spaces with negative exponent and Lemma C.2; for instance (recall footnote 6)

$$\begin{aligned} \Vert \nabla ^{\phi } {\mathcal {P}}^\phi u \Vert _{C^\delta _x}&\lesssim L^{1+2\delta } \Vert u\Vert _{C^{1+\delta }_x} \lesssim L^{2+3\delta } \Vert u \circ \phi ^{-1}\Vert _{C^{1+\delta }_x}\\ {}&\lesssim L^{2+3\delta } \left\| \left( v-\int _{\mathbb {T}^3}v\right) \circ \phi ^{-1}\right\| _{C^{\delta -1}_x} \\&\lesssim L^{5+4\delta } \left\| v-\int _{\mathbb {T}^3}v\right\| _{C^{\delta -1}_x} \lesssim L^{5+4\delta } \left\| v\right\| _{C^{\delta -1}_x}. \end{aligned}$$

$\square$

The next proposition is analogous to [16, Proposition 4.4].

Proposition C.5

(Stationary phase Lemma) Let $\phi =\phi _n$, $n \in \mathbb {N}$. Take a smooth function $a\in C^{\infty }(\mathbb {T}^3)$ and let $k\in \mathbb {Z}^3{\setminus }\{0\}$ and $\lambda \ge 1$ be fixed. Define $f(x) :=a(x)e^{i\lambda k \cdot x}$, $x \in \mathbb {T}^3$.

(i) For any $r\in \mathbb {N}$, we have almost surely for every $t\in \mathbb {R}$

$$\begin{aligned} \left| \int _{\mathbb {T}^3}f(\phi (x,t))\,dx\right| \le \frac{[a]_{C^r_x}}{\lambda ^r}. \end{aligned}$$

(C.8)

(ii) For any $\delta \in (0,1)$ and $r\in \mathbb {N}$, $r + 1 \le \kappa$, we have almost surely for every $L \in \mathbb {N}$, $L \ge 1$ and $t \in (\infty ,{\mathfrak {t}}_L]$

$$\begin{aligned} \Vert {\mathcal {R}}^\phi (f \circ \phi )\Vert _{C^\delta _x} \le CL^\delta \left( \lambda ^{\delta -1}\Vert a\Vert _{C_x} + \lambda ^{\delta -r}[a]_{C^r_x} + \lambda ^{-r}[a]_{C^{r+\delta }_x} \right) , \end{aligned}$$

(C.9)

$$\begin{aligned} \Vert {\mathcal {R}}^\phi {\mathcal {Q}}^\phi (f \circ \phi )\Vert _{C^\delta _x} \le CL^\delta \left( \lambda ^{\delta -1}\Vert a\Vert _{C_x} + \lambda ^{\delta -r}[a]_{C^r_x} + \lambda ^{-r}[a]_{C^{r+\delta }_x} \right) , \end{aligned}$$

(C.10)

where $C=C(\delta ,r)$.

Proof

The proof follows immediately after reduction to [16, Proposition 4.4]: for the first part of the proposition, we use $\int _{\mathbb {T}^3}f(\phi (x,t))dx= \int _{\mathbb {T}^3}f(x)dx$ almost surely for every t since $\phi$ is measure preserving; for the second part, we use Lemma C.1 and the estimates for $\Vert {\mathcal {R}} f\Vert _{C^\delta _x}$, $\Vert {\mathcal {R}}{\mathcal {Q}} f\Vert _{C^\delta _x}$ proved in [16]. $\square$

Lemma C.6

Let $\alpha \in \mathbb {R}$, $p_1,p_2 \in (1,\infty )$ such that $1/p_1+1/p_2 = 1$, then for every $\delta >0$ there exists C such that for every $f \in B^{\alpha +\delta }_{p_1,\infty }(\mathbb {T}^3)$ with zero mean and $g \in L^{p_2}(\mathbb {T}^3)$ the convolution on the torus $f *_{\mathbb {T}^3} g$ belongs to $B^{\alpha }_{\infty ,\infty }(\mathbb {T}^3)$ and

$$\begin{aligned} \Vert f *_{\mathbb {T}^3} g \Vert _{B^{\alpha }_{\infty ,\infty }} \le C \Vert f \Vert _{B^{\alpha +\delta }_{p_1,\infty }} \Vert g\Vert _{L^{p_2}}. \end{aligned}$$

Proof

First of all, notice that the convolution $f *_{\mathbb {T}^3} g$ is well defined also for $\alpha <0$ by duality, and has zero mean because f does. We recall (see Lemma 2.1 in [32]) that for every $1 \le p_1 \le p_2 \le \infty$ and $1 \le q_1 \le q_2 \le \infty$ it holds $B^{\alpha }_{p_1,q_1} \subset B^{\alpha -3(1/p_1-1/p_2)}_{p_2,q_2}$ and for every $p \in (1,\infty )$ it holds $B^\alpha _{p,1} \subset W^{\alpha ,p} \subset B^\alpha _{p,\infty } \subset B^{\alpha -\delta }_{p,1}$, with continuity of the inclusions.

Let $p \in (1,\infty )$. Then, $\Vert f *_{\mathbb {T}^3} g \Vert _{B^{\alpha }_{\infty ,\infty }} \lesssim \Vert f *_{\mathbb {T}^3} g \Vert _{B^{\alpha +3/p}_{p,\infty }}$, and since $f *g$ is zero-mean

$$\begin{aligned} \Vert f *_{\mathbb {T}^3} g \Vert _{B^{\alpha +3/p}_{p,\infty }}&\lesssim \Vert (-\Delta )^{\alpha /2+3/(2p)}f *_{\mathbb {T}^3} g \Vert _{B^0_{p,\infty }} \lesssim \Vert (-\Delta )^{\alpha /2+3/(2p)}f *_{\mathbb {T}^3} g \Vert _{L^p} \\&\lesssim \Vert (-\Delta )^{\alpha /2+3/(2p)}f *_{\mathbb {T}^3} g \Vert _{L^\infty } \lesssim \Vert (-\Delta )^{\alpha /2+3/(2p)}f\Vert _{L^{p_1}} \Vert g \Vert _{L^{p_2}}, \end{aligned}$$

where in the last line we have used Hölder inequality on $\mathbb {T}^3$. Thus since $p_1 \in (1,\infty )$ and f is zero mean

$$\begin{aligned} \Vert (-\Delta )^{\alpha /2+3/(2p)}f\Vert _{L^{p_1}} \lesssim \Vert f\Vert _{W^{\alpha +3/p,p_1}} \lesssim \Vert f\Vert _{B^{\alpha +3/p}_{p_1,1}} \lesssim \Vert f\Vert _{B^{\alpha +4/p}_{p_1,\infty }}. \end{aligned}$$

We conclude taking $p > 4/\delta$. $\square$

Lemma C.7

Let $\ell =2^{-N}$ for some positive $N \in \mathbb {N}$. Let f be a smooth function on $\mathbb {R}^d$ with $\text{ supp }(f) \subset (0,2\pi )^d$, and denote

$$\begin{aligned} f^0&:=f - (2\pi )^{-d}\int _{[0,2\pi ]^d} f, \\ f_\ell&:=\ell ^{-d}f(\cdot /\ell ), \\ f^0_\ell&:=f_\ell - (2\pi )^{-d}\int _{[0,2\pi ]^d} f_\ell = f_\ell - (2\pi )^{-d}\int _{[0,2\pi ]^d} f. \end{aligned}$$

Extend f, $f^0$, $f_\ell$ and $f^0_\ell$ periodically on $\mathbb {T}^d$. Then for every $\alpha \in \mathbb {R}$, $p \in [1,\infty ]$ it holds

$$\begin{aligned} \Vert f^0_\ell \Vert _{B^{\alpha }_{p,\infty }} = \ell ^{d/p-d-\alpha } \Vert f^0 \Vert _{B^\alpha _{p,\infty }} \le \ell ^{d/p-d-\alpha } \Vert f \Vert _{B^\alpha _{p,\infty }}. \end{aligned}$$

Proof

For every $k \in \mathbb {Z}^d$ one can compute the Fourier coefficients (renormalizing constants omitted)

$$\begin{aligned} \hat{f^0_\ell }(k) :=\int _{[0,2\pi ]^d}{f^0_\ell }(x)e^{-ik\cdot x} dx = {\textbf{1}}_{\{k \ne 0\}} \hat{f_\ell }(k) = {\textbf{1}}_{\{k \ne 0\}} {\textbf{1}}_{\{k \in 2^N\mathbb {Z}^d\}} 2^{Nd} {\hat{f}}(2^{-N}k), \end{aligned}$$

(C.11)

where the last identity comes from

$$\begin{aligned} \int _{[0,2\pi ]^d} f_\ell (x) e^{-ik\cdot x} dx&= 2^{Nd} \int _{[0,2\pi ]^d} f(2^N x) e^{-ik\cdot x} dx \\ {}&= 2^{Nd} \int _{[0,2\pi ]^d} \sum _{h \in \mathbb {Z}^d} {\hat{f}}(h) e^{i (2^N h-k) \cdot x} dx. \end{aligned}$$

Let $\rho _j$, $j\ge -1$ a Littlewood–Paley partition of unity as that of Proposition 2.10 in [2]. We can choose the radial function $\varphi$ in that proposition with support in [7/8, 31/16]. Denote $K_j :={\mathscr {F}}^{-1} \rho _j$ the inverse Fourier transform of $\rho _j$. For $j \ge 0$ it holds $\int _{\mathbb {R}^d}{K_j}(x) dx = \rho _j(0)=0$ and

$$\begin{aligned}&{\mathscr {F}}^{-1} \rho _j(x) = \int _{\mathbb {R}^d} \rho _j(\xi ) e^{i\xi \cdot x} d\xi \\ {}&\quad = \int _{\mathbb {R}^d} \rho (2^{-j}\xi ) e^{i\xi \cdot x} d\xi = 2^{jd} \int _{\mathbb {R}^d} \rho _j(\zeta ) e^{i 2^j \zeta \cdot x} d\zeta = 2^{jd} {\mathscr {F}}^{-1} \rho (2^j x), \end{aligned}$$

and therefore for every $j \ge N$ we have

$$\begin{aligned} \Delta _j f^0_\ell&= K_j *f^0_\ell = \int _{\mathbb {R}^d} {\mathscr {F}}^{-1} \rho _j(y)f^0_\ell (\cdot -y) dy \\&= \int _{\mathbb {R}^d} {\mathscr {F}}^{-1} \rho _j(y)f_\ell (\cdot -y) dy \\&= \int _{\mathbb {R}^d} 2^{jd} {\mathscr {F}}^{-1} \rho (2^j y) 2^{Nd} f(2^N \cdot - 2^N y) dy \\&= 2^{Nd} \int _{\mathbb {R}^d} 2^{(j-N)d} {\mathscr {F}}^{-1} \rho (2^{j-N} z) f(2^N \cdot - z) dz =2^{Nd} \Delta _{j-N} f(2^N \cdot ). \end{aligned}$$

We have used that $K_j$ has zero average in the third line. On the other hand, for $-1 \le j \le N-1$ we have $\Delta _j f^0_\ell =0$ since $\Delta _j f^0_\ell = {\mathscr {F}}^{-1} (\rho (2^{-j} \cdot ) \hat{f^0_\ell })) = {\mathscr {F}}^{-1} (\rho _j \hat{f^0_\ell }))$, and $\rho _j, \hat{f^0_\ell }$ have disjoint supports ($\rho _j$ has support in the annulus $2^{N-1}7/8 \le |\xi | \le 2^{N-1}31/16 < 2^N$ and $\hat{f^0_\ell }$ is non-zero only on $2^N \mathbb {Z}^d {\setminus } \{0\}$ by (C.11)). Therefore

$$\begin{aligned} \Vert f^0_\ell \Vert _{B^\alpha _{p,\infty }}&= \sup _{j \ge -1} 2^{\alpha j} \Vert \Delta _j f^0_\ell \Vert _{L^p(\mathbb {T}^d)} \\&= \sup _{j \ge N} 2^{\alpha j} \Vert \Delta _j f^0_\ell \Vert _{L^p(\mathbb {T}^d)} \\&= 2^{Nd} \sup _{j \ge N} 2^{\alpha j} \Vert \Delta _{j-N} f(2^N \cdot )\Vert _{L^p(\mathbb {T}^d)} \\&= 2^{Nd(1-1/p)} \sup _{j \ge N} 2^{\alpha j} \Vert \Delta _{j-N} f\Vert _{L^p(\mathbb {T}^d)} \\&= 2^{N\alpha +Nd(1-1/p)} \sup _{j \ge N} 2^{\alpha (j-N)} \Vert \Delta _{j-N} f\Vert _{L^p(\mathbb {T}^d)} \\&= 2^{N\alpha +Nd(1-1/p)} \Vert f^0 \Vert _{B^\alpha _{p,\infty }} \le 2^{N\alpha +Nd(1-1/p)} \Vert f \Vert _{B^\alpha _{p,\infty }}. \end{aligned}$$

$\square$