Global existence and non-uniqueness for the Cauchy problem associated to 3D Navier–Stokes equations perturbed by transport noise

Abstract

We show global existence and non-uniqueness of probabilistically strong, analytically weak solutions of the three-dimensional Navier–Stokes equations perturbed by Stratonovich transport noise. We can prescribe either: (i) any divergence-free, square integrable intial condition; or (ii) the kinetic energy of solutions up to a stopping time, which can be chosen arbitrarily large with high probability. Solutions enjoy some Sobolev regularity in space but are not Leray–Hopf.

1 Introduction

In the present note we consider the Navier–Stokes equations on the three-dimensional torus $\mathbb {T}^3=(\mathbb {R}/2\pi )^3$ perturbed by Stratonovich transport noise:

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

(1.1)

The noise is characterized by a finite collections of smooth divergence-free vector fields $\{\sigma _k\}_{k \in I}$ and i.i.d. standard Brownian motions $\{B^k\}_{k \in I}$ on a given filtered probability space $(\Omega ,\mathcal {F},\{\mathcal {F}_t\}_{t \ge 0}, \mathbb {P})$ satisfying the usual conditions.

A probabilistically strong, analytically weak solution to (1.1) is a stochastic process u, progressively measurable with respect to $\{\mathcal {F}_t\}_{t\ge 0}$, such that (1.1) holds true when integrated in time and space against a smooth, divergence free, compactly supported test function. Notice that the pressure p is uniquely determined by u and the relations (assuming each term makes sense at least as a distribution)

$$\begin{aligned} \mathord {\textrm{div}}\,(u \cdot \nabla ) u \, dt + \sum _{k \in I} \mathord {\textrm{div}}\,(\sigma _k \cdot \nabla ) u \bullet dB^k + \Delta p \, dt = 0, \quad \int _{\mathbb {T}^3} p = 0. \end{aligned}$$

Hereafter, for $p \in [1,\infty ]$ we denote $L^p_x:= L^p(\mathbb {T}^3,\mathbb {R}^n)$, $n \in \mathbb {N}$ depending on the context. A similar convention will be in force for other spaces of functions on $\mathbb {T}^3$ considered in the following, as the Sobolev spaces $W^{s,p}_x$, $s \in \mathbb {R}$, and $H^s_x:= W^{s,2}_x$. Our goal is to prove global existence and non-uniqueness of probabilistically strong, analytically weak solutions to (1.1) emanating from any prescribed divergence-free initial condition $u_0 \in L^2_x$. In particular, we can prove the following:

Theorem 1.1

Given any divergence-free $u_0 \in L^2_x$ almost surely and independent of the Brownian motions $\{B^k\}_{k \in I}$, there exist infinitely many probabilistically strong, analytically weak solutions of (1.1) of class $u \in L^p_{loc}([0,\infty ), L^2_x) \cap L^2_{loc}([0,\infty ), W^{1,1}_x) \cap C_{loc}([0,\infty ), H^{-1}_x)$ for every $p \in [1,4)$ almost surely with initial condition $u|_{t=0} = u_0$.

If we don’t require $u|_{t=0}$ to attain a particular initial datum $u_0$, then we can produce slightly more regular solutions and even prescribe the kinetic energy thereof up to an arbitrary large stopping time.

Theorem 1.2

There exists $\gamma >0$ with the following property. For every $\varkappa \in (0,1)$ and $T \in (0,\infty )$ there exists a stopping time $\mathfrak {t}$ such that $\mathbb {P} \{ \mathfrak {t} \ge T \} \ge \varkappa$ and, for every energy profile e on $[0,\infty )$ satisfying ${\underline{e}}:= \inf _{t \ge 0} e(t) > 0$ and ${\overline{e}}:= \Vert e \Vert _{C^1_t} < \infty$, there exists a probabilistically strong, analytically weak solution of (1.1) of class $u \in C_{loc}([0,\infty ),H^{\gamma }_x \cap W^{1+\gamma ,1}_x)$ almost surely with kinetic energy

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

Moreover, if $u_1$, $u_2$ are the solutions associated with kinetic energies $e_1$, $e_2$ and $e_1(t)=e_2(t)$ for every $t \in [0,T/2]$, then $u_1(x,t)=u_2(x,t)$ for every $x \in \mathbb {T}^3$ and $t \in [0,T/2]$.

Our proofs strongly rely on convex integration techniques developed in previous works, both deterministic and stochastic. For deterministic Navier–Stokes, solutions with prescribed energy profile were constructed in [3, 4] using intermittent Beltrami flows. Then, using intemittent jets as building blocks (first introduced in [2]), [5] proved non-uniqueness results for (deterministic) power-law fluids with prescribed initial condition. In [18] these results are extended to Navier–Stokes equations perturbed with additive noise. The main difficulty overcome by [18] consists in controlling the interplay between the noise and the convex integration scheme, via the introduction of a sequence of stopping times and gluing solutions on different time intervals. More recently, [17] applied for the first time a convex integration scheme to an equation perturbed by noise of transport type, thanks to a flow transformation: roughly speaking, the stochastic PDE perturbed by transport noise is reduced to a random PDE (that is, a partial differential equation with random coefficients but without stochastic integrals) and the latter is solved by a suitably designed convex integration scheme. In particular, in [17] the authors considered Euler equations. All these works have significantly influenced the present one.

The main novelties contained in this note, apart from the results here presented, are mainly of technical nature. Particularly non-trivial is the fact that, contrary to previous convex integration schemes for Navier–Stokes equations, here we need to include iterative estimates in $L^1_x$ for the pressure. Those estimates are not deduced directly from the expression of the perturbation, but rather require ad hoc arguments based on the introduction of an auxiliary iterative scheme, see Sects. 3.3 and 3.5 for details. Also, worth of mention is the bound $p<4$ in Theorem 1.1, due to the lack of uniform-in-time control over the $L^1_x$ norm of the Reynolds stress.

To close this introduction, let us mention the main motivations behind our work. In recent years, Stratonovich transport noise has shown regularizing properties (in the sense of improved well-posedness) in transport equation [12], point vortices dynamics [13], Vlasov-Poisson equations [10], Navier–Stokes equations in vorticity from [14], and related models [1, 11, 19, 20]. Furthermore, its appearance in fluid dynamics equations is motivated by multiscale arguments [9, 15, 16].

Our results, however, rule out the possibility that Navier–Stokes equations in velocity form be regularized by transport noise (at least in the sense of restoring uniqueness; delay or prevention of blow-up of regular solutions remain plausible).

Moreover, we would like to point out that the mere global existence of probabilistically strong solutions to the Cauchy problem associated to Navier–Stokes equations in dimension three was unknown before this work. However, our solutions do not enjoy the desirable $L^\infty ([0,\infty ),L^2_x) \cap L^2([0,\infty ),H^1_x)$ regularity of Leray-Hopf weak solutions. Whether probabilistically strong solutions – either Leray-Hopf or not – can be also obtained without the use of convex integration techniques remains an important open problem.

2 Preliminaries

2.1 Flow transformation and reduction to a random PDE

As already discussed in [17], applying the flow transformation

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

where we denote by $\phi :\mathbb {T}^3 \times [0,\infty ) \rightarrow \mathbb {T}^3$ the flow of measure preserving diffeomorphisms of the torus given by

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

(2.1)

it is possible to rewrite the SPDE (1.1) as a random PDE:

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t v + \mathord {\textrm{div}}^\phi (v \otimes v) + \nabla ^{\phi }q = \Delta ^{\hspace{-0.05cm}\phi }v, \\ \mathord {\textrm{div}}^\phi v = 0 . \end{array}\right. } \end{aligned}$$

(2.2)

Here $v \otimes v$ denotes the tensor product and the symbols $\mathord {\textrm{div}}^\phi$, $\nabla ^{\phi }$, $\Delta ^{\hspace{-0.05cm}\phi }$ are abbreviations for the space-time dependent differential operators

$$\begin{aligned}&\mathord {\textrm{div}}^\phi v := [\mathord {\textrm{div}}\,(v \circ \phi ^{-1})] \circ \phi , \quad \nabla ^{\phi }q := [\nabla (q \circ \phi ^{-1})] \circ \phi , \quad \\&\Delta ^{\hspace{-0.05cm}\phi }v := [\Delta (v \circ \phi ^{-1})] \circ \phi . \end{aligned}$$

By the same arguments of [17, Proposition A.1] one can see that systems (1.1) and (2.2) are equivalent, namely u is a probabilistically strong, analytically weak solution of (1.1) if and only if v is a probabilistically strong, analytically weak solution of (2.2). We point out that in (1.1) and (2.2) the pressure terms p and q are always implicitly reconstructed from the velocity fields (assuming for instance null space average) and do not appear in the distributional formulation of the equations, when integrating against smooth test functions h satisfying $\mathord {\textrm{div}}\,h = 0$ and $\mathord {\textrm{div}}^\phi h = 0$, respectively. Thus, noticing also that $\int _{\mathbb {T}^3} |u(x,t)|^2 dx = \int _{\mathbb {T}^3} |v(x,t)|^2 dx$ for every t since $\phi$ is measure preserving, it is sufficient to prove Theorem 1.1 and Theorem 1.2 for solutions v of (2.2).

2.2 Mollification of the noise

Without loss of generality we assume that every realisation of the $\mathbb {R}^{|I|}$-valued driving noise $B=(B^k)_{k \in I}$ has Hölder $C^{1/2-}_{loc}$ time regularity. Since we need smoothness of the flow during the construction, we can argue as in [17] and introduce a smooth mollifier $\vartheta :\mathbb {R}\rightarrow \mathbb {R}$ with support contained in (0, 1), and define for $t \in \mathbb {R}$ and some parameter $\varsigma _n>0$, $n \in \mathbb {N}$ to be properly chosen below:

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

with the convention that $B(t-s)=B(0)=0$ whenever $t-s$ is negative. Notice that $B_n$ is smooth at every time $t \in \mathbb {R}$ and it is identically zero for negatives times, being the support of $\vartheta$ contained in (0, 1). Finally, define $\phi _n$ as the unique solution of the integral equation

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

(2.3)

Notice that $\phi _n(x,t)= \phi _n(x,0) = x$ for $t<0$, and for every fixed $t \in \mathbb {R}$ the map $\phi _n(\cdot ,t)$ is measure preserving. We extend the flow $\phi$ defined by (2.1) to $t<0$ similarly.

For any $r \ge 0$, stopping time $\mathfrak {t}$ and Banach space E, we denote $C^r_\mathfrak {t} E:= C^r((-\infty ,\mathfrak {t}],E)$. The following is a consequence of [17, Lemma 2.2].

Lemma 2.1

Fix $T \in (0,\infty )$, $\varkappa \in (0,1)$, $\beta \in [0,1/4]$ and $\kappa ,r \in \mathbb {N}$. Then there exist a constant C, a stopping time $\mathfrak {t}$ such that $\mathbb {P} \{ \mathfrak {t} \ge T \} \ge \varkappa$, and a sequence of stopping times $\mathfrak {t}=\mathfrak {t}_1 \le ... \le \mathfrak {t}_L \le ...$ such that $\mathfrak {t}_L \rightarrow \infty$ almost surely as $L \rightarrow \infty$ and for every $L,n \in \mathbb {N}$, $L \ge 1$ the following hold

$$\begin{aligned} \Vert \phi _{n+1} - \phi _n \Vert _{C^\beta _{\mathfrak {t}_L} C^\kappa _x} \le CL (n+1) \varsigma _n^{1/4-\beta }, \quad \Vert \phi _n \Vert _{C^{1/4}_{\mathfrak {t}_L} C^\kappa _x} \le CL, \\ \Vert \phi _{n+1}^{-1} - \phi _n^{-1} \Vert _{C^\beta _{\mathfrak {t}_L} C^\kappa _x} \le CL (n+1) \varsigma _n^{1/4-\beta }, \quad \Vert \phi _n^{-1} \Vert _{C^{1/4}_{\mathfrak {t}_L} C^\kappa _x} \le CL, \end{aligned}$$

as well as

$$\begin{aligned} \Vert \phi _n \Vert _{C^r_{\mathfrak {t}_L} C^\kappa _x} \le CL \varsigma _n^{{1/4}-r}, \quad \Vert \phi _n^{-1} \Vert _{C^r_{\mathfrak {t}_L} C^\kappa _x} \le CL \varsigma _n^{{1/4}-r}. \end{aligned}$$

Remark 2.2

In the previous lemma we are in fact only assuming that B has H-Hölder trajectories for some $H>1/4$. This is coherent with the fact that we can replace the noise B in (1.1) with a fractional Brownian motion $B^H$ with Hurst parameter $H>1/4$, see also [17, Remark 2.3].

Remark 2.3

(On globality-in-time of solutions) Both Theorem 1.1 and Theorem 1.2 state the existence of global-in-time solutions to (1.1), which we obtain by construction. Indeed, our convex integration scheme relies on perturbations defined for every time $t>0$. Then we use the stopping times $\mathfrak {t}_L$ just to give good bounds on suitable norms of the solutions up to time $\mathfrak {t}_L$, depending in L as $L^{\mathfrak {m}}$, $\mathfrak {m}:=m^{m(1+\log \log L)}$ for some integer m independent of L. This approach was previously pursued in [17] for Euler equations. However, for the sake of simplicity we prefer to work only on the time interval $[0,\mathfrak {t}]$, omitting the verifications on $[0,\mathfrak {t}_L]$ for $L>1$.

2.3 Intermittent jets

The main building blocks of our convex integration scheme are the so-called intermittent jets, first introduced in [2], cf. also [3, Section 7.4].

Recall [3, Lemma 6.6] (see also [6, Lemma 2.4]), according to which there exists a finite set $\Lambda \subset \mathbb {S}^2\cap \mathbb {Q}^3$ such that for each $\xi \in \Lambda$ there exists a $C^\infty$-function $\gamma _\xi :\overline{B_{1/2}}(\textrm{Id})\rightarrow \mathbb {R}$ such that

$$\begin{aligned} R=\sum _{\xi \in \Lambda }\gamma _\xi ^2(R)\, \xi \otimes \xi , \quad \forall R \in \overline{B_{1/2}}(\textrm{Id}). \end{aligned}$$

Here $\mathbb {S}^2\cap \mathbb {Q}^3$ is the set of points of the unit sphere in $\mathbb {R}^3$ with rational coordinates, and $\overline{B_{1/2}}(\textrm{Id}) \subset \mathcal {S}^{3\times 3}$ denotes the closed ball of radius 1/2 around the identity matrix $\textrm{Id}$, in the space of $3 \times 3$ symmetric matrices.

For each $\xi \in \Lambda$ let $A_\xi \in \mathbb {S}^2\cap \mathbb {Q}^3$ be an orthogonal vector to $\xi$. Then for each $\xi \in \Lambda$ we have that $\{\xi , A_\xi , \xi \times A_\xi \}\subset \mathbb {S}^2\cap \mathbb {Q}^3$ is an orthonormal basis of $\mathbb {R}^3$. We label by $n_*$ the smallest positive natural such that $\{n_*\xi , n_*A_\xi , n_*\xi \times A_\xi \}\subset \mathbb {Z}^3$ for every $\xi \in \Lambda$.

Next, let $\Theta :\mathbb {R}^2 \rightarrow \mathbb {R}$ and $\psi :\mathbb {R}\rightarrow \mathbb {R}$ be smooth functions with support in the ball of radius 1 (of the respective domains), renormalized such that $\theta :=-\Delta \Theta$ and $\psi$ obey $\Vert \theta \Vert _{L^2} = 2\pi$ and $\Vert \psi \Vert _{L^2} = (2\pi )^{1/2}$. For parameters $r_\perp , r_{//}>0$ such that $r_\perp \ll r_{//}\ll 1$, define the rescaled cut-off functions

$$\begin{aligned}{} & {} \theta _{r_\perp }(x_1,x_2) = \frac{1}{r_\perp } \theta \left( \frac{x_1}{r_\perp },\frac{x_2}{r_\perp }\right) , \quad \Theta _{r_\perp }(x_1,x_2) = \frac{1}{r_\perp } \Theta \left( \frac{x_1}{r_\perp },\frac{x_2}{r_\perp }\right) , \quad \\{} & {} \psi _{r_{//}}(x_3) = \frac{1}{r_{//}^{1/2}} \psi \left( \frac{x_3}{r_{//}}\right) . \end{aligned}$$

We periodize $\theta _{r_\perp }, \Theta _{r_\perp }$ and $\psi _{r_{//}}$ so that they are viewed as periodic functions on $\mathbb {T}^2, \mathbb {T}^2$ and $\mathbb {T}$ respectively. Consider a large real number $\lambda$ such that $\lambda r_\perp \in \mathbb {N}$, and a large time oscillation parameter $\mu >0$. For every $\xi \in \Lambda$ we introduce

$$\begin{aligned} \begin{aligned} \psi _{\xi }(x,t)&:= \psi _{r_{//}}(n_*r_\perp \lambda (x\cdot \xi +\mu t)), \\ \Theta _{\xi }(x)&:= \Theta _{r_{\perp }}(n_*r_\perp \lambda (x-\alpha _\xi )\cdot A_\xi , n_*r_\perp \lambda (x-\alpha _\xi )\cdot (\xi \times A_\xi )), \\ \theta _{\xi }(x)&:= \theta _{r_{\perp }}(n_*r_\perp \lambda (x-\alpha _\xi )\cdot A_\xi , n_*r_\perp \lambda (x-\alpha _\xi )\cdot (\xi \times A_\xi )), \end{aligned} \end{aligned}$$

where $x \in \mathbb {T}^3$, $t \in \mathbb {R}$ and $\alpha _\xi \in \mathbb {R}^3$ are shifts to ensure that $\{\Theta _{\xi }\}_{\xi \in \Lambda }$ have mutually disjoint support.

The intermittent jets $W_{\xi }:\mathbb {T}^3\times \mathbb {R}\rightarrow \mathbb {R}^3$ are then defined for every $\xi \in \Lambda$ as

$$\begin{aligned} W_{\xi }(x,t):= \xi \,\psi _{\xi }(x,t) \, \theta _{\xi }(x). \end{aligned}$$

Each $W_{\xi }$ has zero space average and is $(\mathbb {T}/r_\perp \lambda )^3$-periodic in its space variable. Moreover, by the choice of $\alpha _\xi$ we have that

$$\begin{aligned} W_{\xi }\otimes W_{\xi '}\equiv 0, \quad \text {for } \xi \ne \xi '\in \Lambda , \end{aligned}$$

and for every $t \in \mathbb {R}$

$$\begin{aligned} \frac{1}{(2\pi )^3}\int _{\mathbb {T}^3} W_{\xi }(x,t)\otimes W_{\xi }(x,t)dx=\xi \otimes \xi . \end{aligned}$$

These facts, combined with [3, Lemma 6.6], imply that

$$\begin{aligned} \frac{1}{(2\pi )^3} \sum _{\xi \in \Lambda }\gamma _\xi ^2(R) \int _{\mathbb {T}^3}W_{\xi }(x,t)\otimes W_{\xi }(x,t)dx=R, \quad \forall R \in \overline{B_{1/2}}(\textrm{Id}), \quad \forall t \in \mathbb {R}.\nonumber \\ \end{aligned}$$

(2.4)

Since $W_{\xi }$ is not divergence free, we introduce the compressibility corrector term

$$\begin{aligned} W_{\xi }^{(c)}:= \frac{1}{n_*^2\lambda ^2} \nabla \psi _{\xi }\times \text {curl}(\Theta _{\xi }\xi ) = \mathord {\textrm{curl}}\,\mathord {\textrm{curl}}\,V_{\xi }-W_{\xi }, \quad V_{\xi }:= \frac{1}{n_*^2\lambda ^2}\,\xi \,\psi _\xi \,\Theta _\xi , \end{aligned}$$

so that

$$\begin{aligned} \mathord {\textrm{div}}\,\left( W_{\xi }+W_{\xi }^{(c)}\right) = 0. \end{aligned}$$

Next, we recall the key bounds from [3, Section 7.4]. For integers $0 \le N, M\le 10$ and $p\in [1,\infty ]$ the following hold provided $r_{//}^{-1}\ll r_{\perp }^{-1}\ll \lambda$

$$\begin{aligned} \Vert \nabla ^N \partial _t^M \psi _{\xi }\Vert _{C_t L^p_x}&\lesssim r_{//}^{1/p-1/2} \left( \frac{r_\perp \lambda }{r_{//}}\right) ^N \left( \frac{r_\perp \lambda \mu }{r_{//}}\right) ^M, \\ \Vert \nabla ^N \theta _{\xi }\Vert _{ L^p_x}+\Vert \nabla ^N\Theta _{\xi }\Vert _{ L^p_x}&\lesssim r^{2/p-1}_\perp \lambda ^N, \\ \Vert \nabla ^N \partial _t^M W_{\xi }\Vert _{C_t L^p_x}&\lesssim r^{2/p-1}_\perp r_{//}^{1/p-1/2}\lambda ^N\left( \frac{r_\perp \lambda \mu }{r_{//}}\right) ^M, \\ \Vert \nabla ^N \partial _t^MW_{\xi }^{(c)}\Vert _{C_t L^p_x}&\lesssim r_\perp ^{2/p} r_{//}^{1/p-3/2}\lambda ^N\left( \frac{r_\perp \lambda \mu }{r_{//}}\right) ^M, \\ \Vert \nabla ^N \partial _t^M V_{\xi } \Vert _{C_t L^p_x}&\lesssim r^{2/p-1}_\perp r_{//}^{1/p-1/2}\lambda ^{N-2}\left( \frac{r_\perp \lambda \mu }{r_{//}}\right) ^M, \end{aligned}$$

where $\lesssim$ is an abbreviation for inequality $\le$ up to an unimportant multiplicative constant, that we keep implicit for notational simplicity. In the lines above, the implicit constants may depend on p, but are independent of N, M (because we restricted to $N,M \le 10$) and $\lambda , r_\perp , r_{//}, \mu$.

As a notational convention, hereafter we denote with the bold symbols $\varvec{\psi }_\xi$, $\varvec{\theta }_\xi$ etc. the respective quantities precomposed with the flow $\phi _{n+1}$, namely $\varvec{\psi }_\xi := \psi _\xi \circ \phi _{n+1}$, $\varvec{\theta }_\xi := \theta _\xi \circ \phi _{n+1}$ etc. When composing with the flow $\phi _{n+1}$ these bounds become

$$\begin{aligned} \Vert \nabla ^N \partial _t^M \varvec{\psi }_{\xi }\Vert _{C_t L^p_x}&\lesssim r_{//}^{1/p-1/2} \left( \frac{r_\perp \lambda }{r_{//}}\right) ^N \left( \frac{r_\perp \lambda \mu }{\varsigma _{n+1}r_{//}}\right) ^M, \\ \Vert \nabla ^N \partial _t^M \varvec{\theta }_{\xi }\Vert _{C_t L^p_x}+\Vert \nabla ^N \partial _t^M \varvec{\Theta }_{\xi }\Vert _{C_t L^p_x}&\lesssim r^{2/p-1}_\perp \lambda ^N \left( \frac{\lambda }{\varsigma _{n+1}} \right) ^M, \\ \Vert \nabla ^N \partial _t^M \textbf{W}_{\xi }\Vert _{C_t L^p_x}&\lesssim r^{2/p-1}_\perp r_{//}^{1/p-1/2}\lambda ^N\left( \frac{r_\perp \lambda \mu }{\varsigma _{n+1}r_{//}}\right) ^M, \\ \Vert \nabla ^N \partial _t^M \textbf{W}_{\xi }^{(c)}\Vert _{C_t L^p_x}&\lesssim r_\perp ^{2/p} r_{//}^{1/p-3/2}\lambda ^N\left( \frac{r_\perp \lambda \mu }{\varsigma _{n+1}r_{//}}\right) ^M, \\ \Vert \nabla ^N \partial _t^M \textbf{V}_{\xi } \Vert _{C_t L^p_x}&\lesssim r^{2/p-1}_\perp r_{//}^{1/p-1/2}\lambda ^{N-2}\left( \frac{r_\perp \lambda \mu }{\varsigma _{n+1}r_{//}}\right) ^M. \end{aligned}$$

3 Construction of solutions with prescribed energy

As usual in convex integration schemes, for every $n \in \mathbb {N}$ we will consider the Navier–Stokes-Reynolds system for $t \in \mathbb {R}$

$$\begin{aligned} \partial _t v_n&+ \mathord {\textrm{div}}^{\phi _n}(v_n\otimes v_n) + \nabla ^{\phi _n}q_n - \Delta ^{\hspace{-0.05cm}\phi _n}v_n = \mathord {\textrm{div}}^{\phi _n}\mathring{R}_n, \end{aligned}$$

(3.1)

where the Reynolds stress $\mathring{R}_n$ takes values in the space of $3 \times 3$ symmetric traceless matrices. Similarly to what done in [17] we do not impose the divergence-free condition $\mathord {\textrm{div}}\,^{\phi _n} v_n = 0$ at every $n \in \mathbb {N}$, but rather we only require a decay of certain norms of $\mathord {\textrm{div}}\,^{\phi _n} v_n$ along the iteration. For technical reasons we also require $\int _{\mathbb {T}^3} v_n = 0$ for every $n \in \mathbb {N}$. Then, given a sequence $(v_n,q_n,\phi _n,\mathring{R}_n)$ of progressively measurable solutions to the random Navier–Stokes-Reynolds system (3.1), we construct a solution v of (2.2) showing the convergences, with respect to suitable topologies:

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

Notice that $\phi _n \rightarrow \phi$ by construction, and we do not need to prove the convergence of $q_n$ since the limit q can be recovered by v, $\phi$, and the condition $\int _{\mathbb {T}^3} q = 0$.

3.1 Iterative assumptions and main proposition

We shall define

$$\begin{aligned} \lambda _n := a^{b^n}, \qquad \delta _n := \lambda _1^{2\beta } \lambda _n^{-2\beta }, \end{aligned}$$

for some parameters $a \in 2\mathbb {N}$, $b \in 7\mathbb {N}$ sufficiently large and $\beta \in (0,1)$ sufficiently small.

Without loss of generality we can suppose that the energy profile e in the statement of Theorem 1.2 is defined for negative times also, and $\inf _{t \in \mathbb {R}} e(t)> {\underline{e}} > 0$ and $\Vert e \Vert _{C^1_t}< {\overline{e}} < \infty$ (possibly changing the values of ${\underline{e}}$ and ${\overline{e}}$).

We require the following iterative estimates on the energy error

$$\begin{aligned} \frac{3}{4} \delta _{n+1} e(t) \le e(t) - \int _{\mathbb {T}^3} |v_n(x,t)|^2 dx \le \frac{5}{4} \delta _{n+1} e(t), \quad \forall t \le \mathfrak {t}, \end{aligned}$$

(3.2)

as well as the integrability bounds

$$\begin{aligned} \Vert q_n \Vert _{C_{\mathfrak {t}} L^1_x} \le C_q \lambda _n^{1/1000}, \qquad \Vert \mathring{R}_n \Vert _{C_{\mathfrak {t}} L^1_x} \le C_R \delta _{n+2}, \end{aligned}$$

(3.3)

the derivative controls

$$\begin{aligned}{} & {} \Vert v_n \Vert _{C_{\mathfrak {t}}W_x^{1,1}} \le C_v \sum _{k=0}^n \delta _{k+2}^2, \qquad \Vert v_n \Vert _{C^1_{\mathfrak {t},x}} \le C_v \lambda _n^{6}, \qquad \Vert v_n \Vert _{C^1_{\mathfrak {t}}C_x^3} \le C_v \lambda _n^{12},\qquad \end{aligned}$$

(3.4)

$$\begin{aligned}{} & {} \Vert q_n \Vert _{C_{\mathfrak {t}} C_x^2} \le C_q \lambda _n^{12}, \qquad \Vert \mathring{R}_n \Vert _{C_{\mathfrak {t}} C_x^1} \le C_R \lambda _n^{20}, \end{aligned}$$

(3.5)

and the estimates on the divergence

$$\begin{aligned} \Vert \mathord {\textrm{div}}^{\phi _n}v_n \Vert _{C_{\mathfrak {t}} H_x^{-1}} \le C_v \delta _{n+3}^3, \qquad \Vert \mathord {\textrm{div}}^{\phi _n}v_n \Vert _{C_{\mathfrak {t}} L^1_x} \le C_v \delta _{n+3}^3. \end{aligned}$$

(3.6)

for some universal constants $C_v,C_q \in (0,\infty )$ and $C_R \in (0,{\overline{e}}/48)$.

Theorem 1.2 is then a consequence of the following:

Proposition 3.1

There exists a choice of parameters $a,b,\beta$ with the following property. Let $(v_n,q_n,\phi _n,\mathring{R}_n)$, $n \in \mathbb {N}$ be a solution of (2.2) with $\phi _n$ given by (2.3) and satisfying the inductive estimates (3.2), (3.3), (3.4) and (3.6). Then there exists a quadruple $(v_{n+1},q_{n+1},\phi _{n+1},\mathring{R}_{n+1})$, solution of (2.2) with $\phi _{n+1}$ given by (2.3), satisfying the same inductive estimates with n replaced by $n+1$ and such that

$$\begin{aligned} \Vert v_{n+1} - v_{n} \Vert _{C_{\mathfrak {t}}L_x^2} \le C_v \delta _{n+1}^{1/2}, \qquad \Vert v_{n+1} - v_{n} \Vert _{C_{\mathfrak {t}}W_x^{1,1}} \le C_v \delta _{n+3}^2. \end{aligned}$$

(3.7)

Moreover, $(v_{n+1},q_{n+1},\mathring{R}_{n+1})$ at any time t depends only on the values of $(v_n,q_n,\mathring{R}_n,\phi _{n+1},e)$ at times $s \le t$.

Proof of Theorem 1.2

Starting the iteration from $(v_0,q_0,\mathring{R}_0)=(0,0,0)$ it is easy to check that $\{v_n\}_{n \in \mathbb {N}}$ is a Cauchy sequence in $C_\mathfrak {t} H_x^\gamma \cap C_\mathfrak {t} W_x^{1+\gamma ,1}$. Moreover, $\mathring{R}_n$ and $\mathord {\textrm{div}}^{\phi _n}v_n$ converge to zero in $C_\mathfrak {t} L^1_x$ and thus the limit v is indeed a solution of (2.2) on $[0,\mathfrak {t}]$. Then (3.2) guarantees the solution has the desired energy profile. Moreover, v is progressively measurable as the limit of progressive processes. Convergences on an arbitrary time interval $[0,\mathfrak {t}_L]$, $L>1$ has been previously discussed, and are omitted. We just mention the fact that on $[0,\mathfrak {t}_L]$ the iterative estimates gain a factor $L_n:= L^{m^n}$ for some m (see also [17, Proposition 2.4]) and all the convergences (except for the energy profile) hold true as soon as $m<b$. Since b will be taken large in what follows, simply taking $m:=b-1$ gives the desired result. Finally, if $e_1$, $e_2$ are two energy profiles and $e_1(t)=e_2(t)$ for every $t\in [0,T/2]$, then the two associated solutions coincide up to time T/2 by the last part of the proposition. $\square$

The remainder of this section is devoted to the proof of Proposition 3.1.

3.2 Perturbation of the velocity

The building blocks of the perturbation are the intermittent jets presented in Sect. 2.3. First of all, we fix a choice for the parameters:

$$\begin{aligned} r_\perp := \lambda _{n+1}^{-6/7}, \quad r_{//}:= \lambda _{n+1}^{-4/7}, \quad \mu := \lambda _{n+1}^{9/7}, \end{aligned}$$

as well as the mollification parameters

$$\begin{aligned} \ell := \lambda _{n+1}^{-\frac{3}{2} \alpha } \lambda _n^{-100}, \qquad \varsigma _n = \frac{\lambda _n^{-1/100}}{(n+1)^4}. \end{aligned}$$

In the expression above $\alpha \in (0,1)$ is a sufficiently small parameter to be chosen later. We require that $\ell ^{-1}$ is an integer power of 2, which is true for instance if $\alpha b \in 2\mathbb {N}$.

Next, let $\chi _1 \in C^\infty _c([-1,1]^3 \times [0,1))$ be a standard mollifier, and denote $\chi _\ell$ the rescaled kernel $\chi _\ell (x,t):= \ell ^{-4}\chi _1(x/\ell ,t/\ell )$, and define

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

The new velocity field $v_{n+1}$ will be constructed as a perturbation of $v_\ell$:

$$\begin{aligned} v_{n+1} := v_\ell + w_{n+1}, \end{aligned}$$

where $w_{n+1}:= w^{(p)}_{n+1} + w^{(c)}_{n+1} + w^{(t)}_{n+1}$ is split in three different contributions described below.

3.2.1 The principal perturbation $w^{(p)}_{n+1}$

Following [18], we define the energy pumping term.¹

$$\begin{aligned} \gamma _n (t)&:= \frac{1}{3(2\pi )^3} \left( e(t)(1-\delta _{n+2}) - \int _{\mathbb {T}^3} |v_n (x,t)|^2 dx \right) , \quad t \in (-\infty ,\mathfrak {t} ], \end{aligned}$$

and denote $\gamma _\ell := \gamma _n *\chi _\ell$ and $\rho := 2 \sqrt{\ell ^2 + |\mathring{R}_\ell |^2} + \gamma _\ell$. For every $\xi \in \Lambda$ let us introduce the amplitude functions

$$\begin{aligned} \textbf{a}_\xi := \rho ^{1/2} \gamma _{\xi } \left( \textrm{Id} - \frac{\mathring{R}_\ell }{\rho } \right) , \qquad a_\xi := \textbf{a}_\xi \circ \phi _{n+1}^{-1}, \end{aligned}$$

(3.8)

and the principal part of the perturbation

$$\begin{aligned} w^{(p)}_{n+1} := \sum _{\xi \in \Lambda } \textbf{a}_{\xi } \textbf{W}_{\xi } := \left( \sum _{\xi \in \Lambda } a_{\xi } W_{\xi } \right) \circ \phi _{n+1}, \end{aligned}$$

(3.9)

where we use the bold symbols to indicate precomposition with the flow $\phi _{n+1}$. By (2.4) and the fact that $\phi _{n+1}$ is measure preserving it holds

$$\begin{aligned} w^{(p)}_{n+1} \otimes w^{(p)}_{n+1} + \mathring{R}_\ell - \rho Id&= \sum _{\xi \in \Lambda } \textbf{a}_{\xi }^2\, \Pi _{\ne 0} (\textbf{W}_{\xi } \otimes \textbf{W}_{\xi }) \nonumber \\&= \left( \sum _{\xi \in \Lambda } a_{\xi }^2\, \Pi _{\ne 0} (W_{\xi } \otimes W_{\xi }) \right) \circ \phi _{n+1}. \end{aligned}$$

(3.10)

3.2.2 The compressibility corrector $w^{(c)}_{n+1}$

Denote $\mathcal {P}$ the classical Leray projector on zero-average, divergence free velocity fields and $\mathcal {Q} = Id - \mathcal {P}$ its orthogonal. Recall from [17] the operators $\mathcal {P}^{\phi _n}, \mathcal {Q}^{\phi _n}$, $n \in \mathbb {N}$, acting on a given $v \in C^\infty (\mathbb {T}^3,\mathbb {R}^3)$ as

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

The compressibility corrector is made of two different contributions, namely

$$\begin{aligned} w^{(c)}_{n+1} := w^{(c,1)}_{n+1}+w^{(c,2)}_{n+1}, \end{aligned}$$

where

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

(3.11)

is as in [17] and is needed to reduce the size of $\mathord {\textrm{div}}^{\phi _{n+1}}v_\ell$, whereas

$$\begin{aligned} w^{(c,2)}_{n+1}&:= \sum _{\xi \in \Lambda } \mathord {\textrm{curl}}^{\phi _{n+1}}(\nabla ^{\phi _{n+1}}\textbf{a}_{\xi } \times \textbf{V}_{\xi }) + \nabla ^{\phi _{n+1}}\textbf{a}_{\xi } \times \mathord {\textrm{curl}}^{\phi _{n+1}}\textbf{V}_{\xi } + \textbf{a}_{\xi } \textbf{W}^{(c)}_{\xi } \nonumber \\&:= \left( \sum _{\xi \in \Lambda } \mathord {\textrm{curl}}\,(\nabla a_{\xi } \times V_{\xi }) + \nabla a_{\xi } \times \mathord {\textrm{curl}}\,V_{\xi } + a_{\xi } W^{(c)}_{\xi } \right) \circ \phi _{n+1}, \end{aligned}$$

(3.12)

serves to compensate for the divergence of the principal corrector since

$$\begin{aligned} w^{(p)}_{n+1}+w^{(c,2)}_{n+1}&= \sum _{\xi \in \Lambda } \mathord {\textrm{curl}}^{\phi _{n+1}}\mathord {\textrm{curl}}^{\phi _{n+1}}(\textbf{a}_{\xi } \textbf{V}_{\xi } ) \\&= \left( \sum _{\xi \in \Lambda } \mathord {\textrm{curl}}\,\mathord {\textrm{curl}}\,(a_{\xi } V_{\xi } ) \right) \circ \phi _{n+1} \end{aligned}$$

and thus $\mathord {\textrm{div}}^{\phi _{n+1}}(w^{(p)}_{n+1}+w^{(c,2)}_{n+1}) = 0$.

3.2.3 The temporal corrector $w^{(t)}$

Finally, the temporal corrector $w^{(t)}_{n+1}$ is defined as

$$\begin{aligned} w^{(t)}_{n+1}&:= - \frac{1}{\mu } \mathcal {P}^{\phi _{n+1}} \Pi _{\ne 0} \sum _{\xi \in \Lambda } \textbf{a}_{\xi }^2\, \varvec{\theta }_{\xi }^2\, \varvec{\psi }_{\xi }^2\, \xi \nonumber \\&:= - \frac{1}{\mu } \left( \mathcal {P} \Pi _{\ne 0} \sum _{\xi \in \Lambda } a_{\xi }^2 \theta _{\xi }^2 \psi _{\xi }^2 \xi \right) \circ \phi _{n+1}. \end{aligned}$$

(3.13)

By definition it holds

$$\begin{aligned} \partial _t w^{(t)}_{n+1}&= - \frac{1}{\mu } \left( \mathcal {P} \Pi _{\ne 0} \sum _{\xi \in \Lambda } \partial _t \left( a_{\xi }^2 \theta _{\xi }^2 \psi _{\xi }^2 \xi \right) \right) \circ \phi _{n+1} \\ \nonumber&\quad - \frac{1}{\mu } {{\dot{\phi }}}_{n+1} \cdot \left[ \nabla \left( \mathcal {P} \Pi _{\ne 0} \sum _{\xi \in \Lambda } a_{\xi }^2 \theta _{\xi }^2 \psi _{\xi }^2 \xi \right) \circ \phi _{n+1} \right] \\ \nonumber&= - \frac{1}{\mu } \mathcal {P}^{\phi _{n+1}} \left( \Pi _{\ne 0} \left( \sum _{\xi \in \Lambda } \partial _t \left( a_{\xi }^2 \theta _{\xi }^2 \psi _{\xi }^2 \xi \right) \right) \circ \phi _{n+1} \right) \\ \nonumber&\quad - \frac{1}{\mu } {{\dot{\phi }}}_{n+1} \cdot \left[ \nabla \left( \mathcal {P} \Pi _{\ne 0} \sum _{\xi \in \Lambda } a_{\xi }^2 \theta _{\xi }^2 \psi _{\xi }^2 \xi \right) \circ \phi _{n+1} \right] \\ \nonumber&= - \frac{1}{\mu } \mathcal {P}^{\phi _{n+1}} \Pi _{\ne 0} \sum _{\xi \in \Lambda } \partial _t \left( \textbf{a}_{\xi }^2 \varvec{\theta }_{\xi }^2 \varvec{\psi }_{\xi }^2 \xi \right) \\ \nonumber&\quad + \frac{1}{\mu } \mathcal {P}^{\phi _{n+1}} \Pi _{\ne 0} \left( {{\dot{\phi }}}_{n+1} \cdot \left[ \sum _{\xi \in \Lambda } \nabla \left( a_{\xi }^2 \theta _{\xi }^2 \psi _{\xi }^2 \xi \right) \circ \phi _{n+1} \right] \right) \\ \nonumber&\quad - \frac{1}{\mu } {{\dot{\phi }}}_{n+1} \cdot \left[ \nabla \left( \mathcal {P} \Pi _{\ne 0} \sum _{\xi \in \Lambda } a_{\xi }^2 \theta _{\xi }^2 \psi _{\xi }^2 \xi \right) \circ \phi _{n+1} \right] . \end{aligned}$$

(3.14)

3.2.4 Estimates on the velocity

The iterative estimates (3.4) on the velocity $v_{n+1}$ and (3.7) on the increment $w_{n+1} = v_{n+1}-v_n$ (actually, with $W_x^{1,1}$ replaced by $W_x^{1,p}$ for some $p>1$ sufficiently close to one) are easily obtained from the corresponding bounds in [18, Section 3] and the fact that the flow $\phi _{n+1}$ is measure preserving.

First, by (3.23), (3.24) and (3.28) in [18] it holds $\Vert \rho \Vert _{C_{\mathfrak {t}}L^1_x} \le C \delta _{n+1}$ for some universal constant C, as well as

$$\begin{aligned} \Vert a_\xi \Vert _{C_{\mathfrak {t},x}}&= \Vert \textbf{a}_\xi \Vert _{C_{\mathfrak {t},x}} \lesssim \Vert \rho \Vert _{C_{\mathfrak {t},x}}^{1/2} \lesssim \ell ^{-2} \delta _{n+1}^{1/2}, \\ \Vert a_\xi \Vert _{C_{\mathfrak {t}}L_x^2}&= \Vert \textbf{a}_\xi \Vert _{C_{\mathfrak {t}}L_x^2} \lesssim \Vert \rho \Vert _{C_{\mathfrak {t}}L^1_x}^{1/2} \lesssim \delta _{n+1}^{1/2}, \end{aligned}$$

whereas by (3.25) and (3.34) therein we have for every $N = 1,2,...10$ and $M=0,1$

$$\begin{aligned} \Vert \rho \Vert _{C^N_{\mathfrak {t},x}}&\lesssim \ell ^{2-7N} \delta _{n+1}, \\ \Vert a_\xi \Vert _{C^M_{\mathfrak {t}}C^N_x}&\lesssim \varsigma _{n+1}^{-M} \Vert \textbf{a}_\xi \Vert _{C^{M+N}_{\mathfrak {t},x}} \lesssim \varsigma _{n+1}^{-M} \ell ^{-8-7(M+N)} \delta _{n+1}^{1/2}. \end{aligned}$$

By the previous inequalities, together with [3, Lemma 7.4], the fact that $\phi _{n+1}$ is measure preserving, and $W_\xi = \textbf{W}_\xi \circ \phi _{n+1}^{-1}$ is $1/r_\perp \lambda$ periodic in its space variable, one has

$$\begin{aligned} \Vert w^{(p)}_{n+1} \Vert _{C_{\mathfrak {t}}L_x^2} = \Vert w^{(p)}_{n+1} \circ \phi _{n+1}^{-1}\Vert _{C_{\mathfrak {t}}L_x^2}&\le \frac{C_v}{2} \delta _{n+1}^{1/2} , \end{aligned}$$

where the constant $C_v$ is universal.

For $p \in (1,\infty )$, the bounds (3.43), (3.44), (3.45) read as

$$\begin{aligned} \Vert w^{(p)}_{n+1} \Vert _{C_{\mathfrak {t}}L_x^p}&\lesssim \delta _{n+1}^{1/2} \ell ^{-8} r_\perp ^{2/p-1} r_{//}^{1/p-1/2}, \\ \Vert w^{(c,2)}_{n+1} \Vert _{C_{\mathfrak {t}}L_x^p}&\lesssim \delta _{n+1}^{1/2} \ell ^{-22} r_\perp ^{2/p} r_{//}^{1/p-3/2}, \\ \Vert w^{(t)}_{n+1} \Vert _{C_{\mathfrak {t}}L_x^p}&\lesssim \delta _{n+1} \ell ^{-16} r_\perp ^{2/p-1} r_{//}^{1/p-2} \lambda _{n+1}^{-1}, \end{aligned}$$

Moreover, arguing as in [17, Lemma 4.9] and invoking iterative assumption (3.6) we get

$$\begin{aligned} \Vert w^{(c,1)}_{n+1} \Vert _{C_{\mathfrak {t}}L_x^2}&\lesssim \Vert \mathord {\textrm{div}}^{\phi _n}v_n \Vert _{C_{\mathfrak {t}}H_x^{-1}} \lesssim \delta _{n+3}^3. \end{aligned}$$

In particular, all the bounds above with $p=2$ guarantee that the iterative assumption on $\Vert v_{n+1} - v_n \Vert _{C_\mathfrak {t}L_x^2}$ holds true, up to choosing the parameter a large enough so to absorb all the implicit constants in the previous inequalities.

We also need to bound the $W_x^{1,p}$ norm of the increment for some $p>1$. It holds (see (3.52) and (3.51) in [18])

$$\begin{aligned} \Vert w^{(p)}_{n+1} + w^{(c,2)}_{n+1} \Vert _{C_{\mathfrak {t}}W_x^{1,p}}&\lesssim \sum _{\xi \in \Lambda } \Vert \mathord {\textrm{curl}}^{\phi _{n+1}}\mathord {\textrm{curl}}^{\phi _{n+1}}(\textbf{a}_\xi \textbf{V}_\xi ) \Vert _{C_{\mathfrak {t}}W_x^{1,p}} \\&\lesssim r_\perp ^{2/p-1} r_{//}^{1/p-1/2} \ell ^{-8} \lambda _{n+1}, \\ \Vert w^{(t)}_{n+1} \Vert _{C_{\mathfrak {t}}W_x^{1,p}}&\lesssim \frac{1}{\mu } \sum _{\xi \in \Lambda } \Vert \textbf{a}_\xi ^2 \,\varvec{\theta }_\xi ^2 \,\varvec{\psi }_\xi ^2 \Vert _{C_{\mathfrak {t}}W_x^{1,p}} \\&\lesssim r_\perp ^{2/p-2} r_{//}^{1/p-1} \ell ^{-16} \lambda _{n+1}^{-2/7}. \end{aligned}$$

In addition, by interpolation and assuming $p \in (1,2)$ it holds

$$\begin{aligned} \Vert w^{(c,1)}_{n+1} \Vert _{C_{\mathfrak {t}}W_x^{1,p}}&\lesssim \Vert \mathord {\textrm{div}}^{\phi _n}v_n \Vert _{C_{\mathfrak {t}}L_x^p} \\&\lesssim \Vert \mathord {\textrm{div}}^{\phi _n}v_n \Vert _{C_{\mathfrak {t}}L^1_x}^{2-p} \Vert v_n \Vert _{C^1_{\mathfrak {t},x}}^{p-1} \\&\lesssim \delta _{n+3}^{3(2-p)} \lambda _{n}^{6(p-1)}, \end{aligned}$$

and thus $\Vert v_{n+1}-v_n\Vert _{C_{\mathfrak {t}}W_x^{1,p}} \lesssim \delta _{n+3}^2$ at least when $\beta$ is sufficiently small, $p < \min \{ 101/100, 1+ \beta b^3 /100 \}$, and a is taken large enough. The iterative bound (3.4) on $\Vert v_{n+1}\Vert _{C_{\mathfrak {t}}W_x^{1,1}}$ descends immediately.

Finally, for the $C^1_{\mathfrak {t},x}$ and $C^1_{\mathfrak {t}}C_x^3$ norms of the incremental velocity we have the very loose estimates (cf. (3.48), (3.49) and (3.50) in [18])

$$\begin{aligned} \Vert w^{(p)}_{n+1} + w^{(c,2)}_{n+1} \Vert _{C^1_{\mathfrak {t},x}}&\lesssim \sum _{\xi \in \Lambda } \Vert \mathord {\textrm{curl}}^{\phi _{n+1}}\mathord {\textrm{curl}}^{\phi _{n+1}}(\textbf{a}_\xi \textbf{V}_\xi ) \Vert _{C^1_{\mathfrak {t},x}} \\&\lesssim \sum _{\xi \in \Lambda } \varsigma _{n+1}^{-2} \Vert \textbf{a}_\xi \Vert _{C^1_{\mathfrak {t}}C^2_x} (\Vert \textbf{V}_\xi \Vert _{C^1_{\mathfrak {t}}C_x^2} + \Vert \textbf{V}_\xi \Vert _{C_{\mathfrak {t}}C_x^3}) \\&\lesssim \sum _{\xi \in \Lambda } \varsigma _{n+1}^{-3} \ell ^{-29} (\Vert \textbf{V}_\xi \Vert _{C^1_{\mathfrak {t}}W_x^{4,2}} + \Vert \textbf{V}_\xi \Vert _{C_{\mathfrak {t}}W_x^{5,2}}) \\&\lesssim \varsigma _{n+1}^{-4} \ell ^{-29} r_\perp r_{//}^{-1}\lambda _{n+1}^3 \mu \lesssim \lambda _{n+1}^5, \\ \Vert w^{(p)}_{n+1} + w^{(c,2)}_{n+1} \Vert _{C^1_{\mathfrak {t}}C_x^3}&\lesssim \varsigma _{n+1}^{-3} \lambda _{n+1}^2 \sum _{\xi \in \Lambda } \Vert \mathord {\textrm{curl}}^{\phi _{n+1}}\mathord {\textrm{curl}}^{\phi _{n+1}}(\textbf{a}_\xi \textbf{V}_\xi ) \Vert _{C_{\mathfrak {t}}C_x^3} \\&\lesssim \varsigma _{n+1}^{-3} \lambda _{n+1}^2 \sum _{\xi \in \Lambda } \Vert \textbf{a}_\xi \Vert _{C_{\mathfrak {t}}C^5_x} \Vert \textbf{V}_\xi \Vert _{C_{\mathfrak {t}}C_x^5} \\&\lesssim \varsigma _{n+1}^{-3} \lambda _{n+1}^2 \sum _{\xi \in \Lambda } \ell ^{-43} \Vert \textbf{V}_\xi \Vert _{C_{\mathfrak {t}}W_x^{7,2}} \\&\lesssim \varsigma _{n+1}^{-3} \ell ^{-43} \lambda _{n+1}^7 \lesssim \lambda _{n+1}^{8}, \end{aligned}$$

and

$$\begin{aligned} \Vert w^{(t)}_{n+1} \Vert _{C^1_{\mathfrak {t},x}}&\lesssim \frac{\lambda _{n+1}^2}{\varsigma _{n+1}\mu } \sum _{\xi \in \Lambda } \Vert \textbf{a}_\xi ^2 \,\varvec{\theta }_\xi ^2 \,\varvec{\psi }_\xi ^2 \Vert _{C_{\mathfrak {t},x}} + \frac{1}{\mu } \sum _{\xi \in \Lambda } \Vert \textbf{a}_\xi ^2 \,\varvec{\theta }_\xi ^2 \,\varvec{\psi }_\xi ^2 \Vert _{C_{\mathfrak {t}}C_x^1} \\&\lesssim \frac{\lambda _{n+1}^2}{\varsigma _{n+1}\mu } \sum _{\xi \in \Lambda } \Vert \textbf{a}_\xi \Vert _{C_{\mathfrak {t},x}}^2 \Vert \varvec{\theta }_\xi ^2 \,\varvec{\psi }_\xi ^2 \Vert _{C_{\mathfrak {t},W_x^{2,2}}} + \frac{1}{\mu } \sum _{\xi \in \Lambda } \Vert \textbf{a}_\xi \Vert _{C_{\mathfrak {t}}C_x^1}^2 \Vert \varvec{\theta }_\xi ^2 \,\varvec{\psi }_\xi ^2 \Vert _{C_{\mathfrak {t}}W_x^{3,2}} \\&\lesssim \varsigma _{n+1}^{-1} \mu ^{-1} \ell ^{-30} r_\perp ^{-1} r_{//}^{-1/2}\lambda _{n+1}^6 \lesssim \lambda _{n+1}^6, \\ \Vert w^{(t)}_{n+1} \Vert _{C^1_{\mathfrak {t}}C_x^3}&\lesssim \frac{\lambda _{n+1}^2}{\varsigma _{n+1}\mu } \sum _{\xi \in \Lambda } \Vert \textbf{a}_\xi ^2 \,\varvec{\theta }_\xi ^2 \,\varvec{\psi }_\xi ^2 \Vert _{C_{\mathfrak {t}}C_x^3} \\&\lesssim \frac{\lambda _{n+1}^2}{\varsigma _{n+1}\mu } \sum _{\xi \in \Lambda } \Vert \textbf{a}_\xi ^2 \Vert _{C_{\mathfrak {t}}C_x^3} \Vert \varvec{\theta }_\xi ^2 \,\varvec{\psi }_\xi ^2 \Vert _{C_{\mathfrak {t}}W_x^{5,2}} \\&\lesssim \varsigma _{n+1}^{-1}\mu ^{-1} \ell ^{-58} r_\perp ^{-1} r_{//}^{-1/2} \lambda _{n+1}^{12} \lesssim \lambda _{n+1}^{12}. \end{aligned}$$

For $w^{(c,1)}_{n+1}$ we have instead

$$\begin{aligned} \Vert w^{(c,1)}_{n+1}\Vert _{C^3_{\mathfrak {t},x}}&\lesssim \ell ^{-5} \Vert \mathcal {Q}^{\phi _n}v_n\Vert _{C_{\mathfrak {t}}L_x^2} \lesssim \ell ^{-5} \Vert \mathord {\textrm{div}}^{\phi _n}v_n \Vert _{C_{\mathfrak {t}}H_x^{-1}} \lesssim \ell ^{-5} \delta _{n+3}^3. \end{aligned}$$

Notice that these estimates guarantee that (3.4) are satisfied provided $\beta$ (resp. a) is taken once more sufficiently small (resp. large).

3.2.5 Estimates on the divergence

To verify the iterative bounds (3.6) it is necessary to reformulate Lemma 4.7 of [17] so to estimate the difference $(\mathord {\textrm{div}}^{\phi _{n+1}}- \mathord {\textrm{div}}^{\phi _n})\,v$ in the $L_x^p$ and $L_x^2$ scales. The product map $(f,g) \mapsto fg$ is continuous from $L_x^p \times C_x \rightarrow L_x^p$ and $H_x^{-1} \times C_x^2 \rightarrow H_x^{-1}$, therefore we have:

Lemma 3.2

For every $p \in [1,\infty ]$ 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 it holds for every $t \le \mathfrak {t}$

$$\begin{aligned} \Vert G(t)\Vert _{H_x^{-1}}&\le C (n+1)\varsigma _n^{1/4} \Vert v \Vert _{L_x^2}, \\ \Vert G(t)\Vert _{L_x^p}&\le C (n+1)\varsigma _n^{1/4} \Vert v \Vert _{W_x^{1,p}}. \end{aligned}$$

Then, arguing as in [17], rewrite

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

(3.15)

By [17, Lemma 4.4] and iterative assumptions it holds

$$\begin{aligned} \left\| \mathord {\textrm{div}}^{\phi _{n+1}}v_\ell - \left( \mathord {\textrm{div}}^{\phi _{n+1}}v_n \right) *\chi _\ell \right\| _{C_{\mathfrak {t},x}}&\lesssim \ell ^{1/4} \Vert v_n\Vert _{C^1_{\mathfrak {t},x}} \lesssim \lambda _{n}^{-10}, \\ \left\| \left( \mathord {\textrm{div}}^{\phi _n}Q^{\phi _n}v_n\right) *\chi _\ell - \mathord {\textrm{div}}^{\phi _n}\left( (Q^{\phi _n}v_n) *\chi _\ell \right) \right\| _{C_{\mathfrak {t},x}}&\lesssim \ell ^{1/4} \Vert Q^{\phi _n} v_n\Vert _{C_{\mathfrak {t}}C^1_x} \\&\lesssim \ell ^{1/4} \Vert v_n\Vert _{C^1_{\mathfrak {t},x}} \lesssim \lambda _{n}^{-10}. \end{aligned}$$

A fortiori, these quantities satisfy the same bounds in $C_{\mathfrak {t}}H_x^{-1}$ and $C_{\mathfrak {t}}L_x^p$.

The other two terms on the right-hand-side of (3.15) are controlled with Lemma 3.2 and the bounds on $\Vert v_n\Vert _{C_{\mathfrak {t}}L_x^2},\Vert v_n\Vert _{C_{\mathfrak {t}}W_x^{1,p}}$, and using $(n+1) \varsigma _{n}^{1/4} = \lambda _n^{-1/400} \le \frac{C_v}{4} \delta _{n+3}^3$ for $\beta$ sufficiently small.

3.2.6 Estimate on the energy

The control on the energy is quite standard. Rewrite

$$\begin{aligned} |v_{n+1}|^2&= |v_\ell |^2 + |w^{(p)}_{n+1}|^2 + |w^{(c)}_{n+1}+w^{(t)}_{n+1}|^2 + 2 v_\ell \cdot w^{(p)}_{n+1} \\&\quad + 2 (v_\ell + w^{(p)}_{n+1})\cdot (w^{(c)}_{n+1}+w^{(t)}_{n+1}). \end{aligned}$$

Since $\Vert w^{(c)}_{n+1}+w^{(t)}_{n+1} \Vert _{C_{\mathfrak {t}}L_x^2} \lesssim \delta _{n+3}^2$ it holds

$$\begin{aligned}&\left| \int _{\mathbb {T}^3} (v_\ell + w^{(p)}_{n+1})\cdot (w^{(c)}_{n+1}+w^{(t)}_{n+1}) dx \right|&\lesssim \Vert v_\ell + w^{(p)}_{n+1} \Vert _{C_{\mathfrak {t}}L_x^2} \Vert w^{(c)}_{n+1}\\&\quad +w^{(t)}_{n+1} \Vert _{C_{\mathfrak {t}}L_x^2} \lesssim \delta _{n+3}^2. \end{aligned}$$

The term $v_\ell \cdot w^{(p)}_{n+1}$ is controlled in the following way:

$$\begin{aligned} \int _{\mathbb {T}^3} |v_\ell \cdot w^{(p)}_{n+1}| dx&\lesssim \Vert v_\ell \Vert _{C_t L_x^\infty } \Vert w^{(p)}_{n+1}\Vert _{C_tL^1_x} \lesssim \ell ^{-10} r_\perp r_{//}^{1/2} \lesssim \delta _{n+3}^2. \end{aligned}$$

By (3.10) and using that $\textrm{tr}(\mathring{R}_\ell )=0$ and $\phi _{n+1}$ is measure preserving, we have

$$\begin{aligned} \int _{\mathbb {T}^3} \left( |w^{(p)}_{n+1}|^2 - 3\rho \right) dx&= \sum _{\xi \in \Lambda } \int _{\mathbb {T}^3} \textbf{a}_\xi ^2 \,\Pi _{\ne 0}|\textbf{W}_\xi |^2 dx = \sum _{\xi \in \Lambda } \int _{\mathbb {T}^3} {a}_\xi ^2 \,\Pi _{\ne 0}|{W}_\xi |^2 dx \\&= \sum _{\xi \in \Lambda } \int _{\mathbb {T}^3} {a}_\xi ^2 \,\Pi _{\ge r_\perp \lambda _{n+1}/2}|{W}_\xi |^2 dx \\&\lesssim \Vert a_\xi ^2 \Vert _{C_\mathfrak {t}C_x^9} (\lambda _{n+1}r_\perp )^{-9} \Vert |W_\xi |^2 \Vert _{C_\mathfrak {t}L_x^2} \\&\lesssim \ell ^{7-16\cdot 9} (\lambda _{n+1}r_\perp )^{-9} r_\perp ^{-1} r_{//}^{-1/2} \lesssim \lambda _{n+1}^{-1/10}. \end{aligned}$$

The last line is justified by the fact that ${W}_\xi$ is $r_\perp \lambda _{n+1}$ periodic. On the other hand,

$$\begin{aligned} \int _{\mathbb {T}^3} 3\rho dx = \gamma _\ell - \gamma _n + 6 \int _{\mathbb {T}^3} \sqrt{\ell ^2 + |\mathring{R}_\ell |^2} dx + e(t)(1-\delta _{n+2}) - \int _{\mathbb {T}^3} |v_n(x,t)|^2 dx. \end{aligned}$$

Putting all together, we can rewrite the energy error at level $n+1$ as

$$\begin{aligned} e(t)(1-\delta _{n+2}) - \int _{\mathbb {T}^3} |v_{n+1}|^2 dx&= \gamma _n-\gamma _\ell + 6\int _{\mathbb {T}^3} \sqrt{\ell ^2 + |\mathring{R}_\ell |^2} dx\\&\quad + \int _{\mathbb {T}^3} |v_{\ell }|^2 - |v_{n}|^2 dx + r_{n+1}, \end{aligned}$$

where by the previous lines $r_{n+1}$ is a reminder satisfying $|r_{n+1}| \le \frac{{\overline{e}}}{8} \delta _{n+2}$ (up to choosing a sufficiently large). Then (3.2) is recovered noticing that all the other terms on the right-hand-side of the equation above are smaller than ${\overline{e}}\ell + 4C_v^2\lambda _n^{6} \ell + 6 C_R \delta _{n+2} \le \frac{{\overline{e}}}{8} \delta _{n+2}$.

3.3 The oscillatory term and the new pressure $q_{n+1}$

The temporal corrector serves to reduce the oscillatory term

$$\begin{aligned} \partial _t w^{(t)}_{n+1} + \Pi _{\ne 0}\sum _{\xi \in \Lambda } \textbf{a}_{\xi }^2 \mathord {\textrm{div}}^{\phi _{n+1}}(\textbf{W}_{\xi } \otimes \textbf{W}_{\xi }). \end{aligned}$$

For classical intermittent jets the key identity was

$$\begin{aligned} \mathord {\textrm{div}}\,(W_{\xi } \otimes W_{\xi })&= 2 (W_{\xi } \cdot \nabla \psi _{\xi }) \theta _{\xi } \xi = \frac{1}{\mu } \theta _{\xi }^2 \partial _t \psi _{\xi }^2 \xi = \frac{1}{\mu }\partial _t \left( \theta _{\xi }^2 \psi _{\xi }^2 \xi \right) , \end{aligned}$$

which holds true because $\xi \cdot \nabla \psi = \mu ^{-1} \partial _t \psi$. However, for us it holds instead

$$\begin{aligned} \mathord {\textrm{div}}^{\phi _{n+1}}(\textbf{W}_{\xi } \otimes \textbf{W}_{\xi })&= \mathord {\textrm{div}}\,(W_{\xi } \otimes W_{\xi }) \circ \, \phi _{n+1} \nonumber \\&= \frac{1}{\mu } \partial _t \left( \theta _{\xi }^2 \psi _{\xi }^2 \xi \right) \circ \phi _{n+1} \nonumber \\&= \frac{1}{\mu } \partial _t \left( \varvec{\theta }_{\xi }^2 \varvec{\psi }_{\xi }^2 \xi \right) - \frac{1}{\mu } {\dot{\phi }}_{n+1} \cdot \left[ \nabla \left( \theta _{\xi }^2 \psi _{\xi }^2 \xi \right) \circ \phi _{n+1} \right] .\qquad \end{aligned}$$

(3.16)

Therefore by (3.14) and (3.16) we have

$$\begin{aligned} \partial _t&w^{(t)}_{n+1} + \Pi _{\ne 0}\sum _{\xi \in \Lambda } \textbf{a}_{\xi }^2 \, \mathord {\textrm{div}}^{\phi _{n+1}}(\textbf{W}_{\xi } \otimes \textbf{W}_{\xi }) \nonumber \\&= \frac{1}{\mu } \mathcal {Q}^{\phi _{n+1}} \Pi _{\ne 0} \sum _{\xi \in \Lambda } \partial _t \left( \textbf{a}_{\xi }^2 \varvec{\theta }_{\xi }^2 \varvec{\psi }_{\xi }^2 \xi \right) \nonumber \\&\quad + \frac{1}{\mu } \mathcal {P}^{\phi _{n+1}} \Pi _{\ne 0} \,{{\dot{\phi }}}_{n+1} \cdot \left[ \left( \sum _{\xi \in \Lambda } \nabla \left( a_{\xi }^2 \theta _{\xi }^2 \psi _{\xi }^2 \xi \right) \right) \circ \phi _{n+1} \right] \nonumber \\&\quad - \frac{1}{\mu } {{\dot{\phi }}}_{n+1} \cdot \left[ \nabla \left( \mathcal {P} \Pi _{\ne 0} \sum _{\xi \in \Lambda } a_{\xi }^2 \theta _{\xi }^2 \psi _{\xi }^2 \xi \right) \circ \phi _{n+1} \right] \nonumber \\&\quad - \frac{1}{\mu }\Pi _{\ne 0} \sum _{\xi \in \Lambda } (\partial _t \textbf{a}_{\xi }^2) \varvec{\theta }_{\xi }^2 \varvec{\psi }_{\xi }^2 \xi \nonumber \\&\quad - \frac{1}{\mu } \Pi _{\ne 0} \sum _{\xi \in \Lambda } \textbf{a}_{\xi }^2\, {\dot{\phi }}_{n+1} \cdot \left[ \nabla \left( \theta _{\xi }^2 \psi _{\xi }^2 \xi \right) \circ \phi _{n+1} \right] . \end{aligned}$$

(3.17)

The lines second to five on the right-hand-side will be shown to be small in $L_x^p$ in Sect. 3.4. The key fact is that in these terms there is no time derivative acting on $\psi _\xi$ or $\varvec{\psi }_\xi$, and thus the factor $\mu ^{-1}$ in front dominates by our choice of parameters.

On the other hand, the first term on the right-hand-side is the gradient of a pressure (in the sense that equals $\nabla ^{\phi _{n+1}} {\tilde{P}}$ for some ${\tilde{P}}$) but it needs to be manipulated further, nonetheless. Indeed, we can not prove our iterative estimate on the $\Vert q_n \Vert _{C_\mathfrak {t}L^1_x}$ by estimating the increment ${\tilde{P}}$ in $C_\mathfrak {t}L^1_x$.

We rewrite

$$\begin{aligned} \frac{1}{\mu } \mathcal {Q}^{\phi _{n+1}} \Pi _{\ne 0} \sum _{\xi \in \Lambda } \partial _t \left( \textbf{a}_{\xi }^2 \varvec{\theta }_{\xi }^2 \varvec{\psi }_{\xi }^2 \xi \right)&= \frac{1}{\mu } \mathcal {Q} \left( \Pi _{\ne 0} \sum _{\xi \in \Lambda } [\partial _t \left( \textbf{a}_{\xi }^2 \varvec{\theta }_{\xi }^2 \varvec{\psi }_{\xi }^2 \xi \right) ] \circ \phi _{n+1}^{-1} \right) \circ \phi _{n+1} \nonumber \\&= \frac{1}{\mu } \mathcal {Q} \left( \Pi _{\ne 0} \sum _{\xi \in \Lambda } \partial _t \left( {a}_{\xi }^2 \theta _{\xi }^2 \psi _{\xi }^2 \xi \right) \right) \circ \phi _{n+1} \nonumber \\&\quad + \frac{1}{\mu } \mathcal {Q} \left( \Pi _{\ne 0} ({{\dot{\phi }}}_{n+1} \circ \phi _{n+1}^{-1}) \cdot \left[ \sum _{\xi \in \Lambda } \nabla \left( {a}_{\xi }^2 \theta _{\xi }^2 \psi _{\xi }^2 \xi \right) \right] \right) \circ \phi _{n+1} \nonumber \\&=: (\nabla P) \circ \phi _{n+1} + \frac{1}{\mu } \mathcal {Q}^{\phi _{n+1}} \Pi _{\ne 0} \nonumber \\&\left( {{\dot{\phi }}}_{n+1} \cdot \left[ \sum _{\xi \in \Lambda }\nabla \left( {a}_{\xi }^2 \theta _{\xi }^2 \psi _{\xi }^2 \xi \right) \circ \phi _{n+1} \right] \right) \nonumber \\&=\nabla ^{\phi _{n+1}}(P \circ \phi _{n+1}) + \frac{1}{\mu } \mathcal {Q}^{\phi _{n+1}} \Pi _{\ne 0} \nonumber \\&\left( {{\dot{\phi }}}_{n+1} \cdot \left[ \sum _{\xi \in \Lambda } \nabla \left( {a}_{\xi }^2 \theta _{\xi }^2 \psi _{\xi }^2 \xi \right) \circ \phi _{n+1} \right] \right) , \end{aligned}$$

(3.18)

where the pressure increment P is implicitly defined in the second-to-last line. The other term does not have any time derivative and therefore can be easily absorbed into the Reynolds stress $\mathring{R}_{n+1}$, in particular it does not need to be incorporated in the new pressure $q_{n+1}$.

As a consequence, recalling the bounds $\Vert \rho \Vert _{C^2_{\mathfrak {t},x}} \lesssim \ell ^{-12}$ and $\Vert \rho \Vert _{C_{\mathfrak {t}}L^1_x} \lesssim \delta _{n+1}$ and defining the new pressure as

$$\begin{aligned} q_{n+1} := q_\ell - \rho - P \circ \phi _{n+1}, \end{aligned}$$

(3.19)

the iterative estimate (3.4) on the derivative of the pressure is satisfied since

$$\begin{aligned} \Vert P \circ \phi _{n+1} \Vert _{C_\mathfrak {t}C_x^2}&\lesssim \left\| \frac{1}{\mu } \Pi _{\ne 0} \sum _{\xi \in \Lambda } \partial _t ({a}_{\xi }^2 \theta _{\xi }^2 \psi _{\xi }^2 \xi ) \right\| _{C_\mathfrak {t}C_x^2} \\&\lesssim \frac{\lambda _{n+1}^2}{\varsigma _{n+1}\mu } \sum _{\xi \in \Lambda } \left\| {a}_{\xi }^2 \theta _{\xi }^2 \psi _{\xi }^2 \right\| _{C_\mathfrak {t}C_x^2} \\&\lesssim \frac{\lambda _{n+1}^2}{\varsigma _{n+1}\mu } \sum _{\xi \in \Lambda } \Vert {a}_{\xi }\Vert _{C_\mathfrak {t}C_x^2}^2 \Vert \theta _{\xi }^2 \psi _{\xi }^2 \Vert _{C_\mathfrak {t}W_x^{4,2}} \\&\lesssim \varsigma _{n+1}^{-1} \mu ^{-1} \ell ^{-44} r_\perp ^{-1} r_{//}^{-1/2} \lambda _{n+1}^{10} \lesssim \lambda _{n+1}^{10}, \end{aligned}$$

whereas the integrability bound (3.3) is satisfied as long as we can prove $\Vert P \circ \phi _{n+1}\Vert _{L^1_x} = \Vert P\Vert _{L^1_x} \le \lambda _{n+1}^{1/1000}$ for every $n \in \mathbb {N}$, assuming at least $C_q \ge 2$ and $\beta$ sufficiently small. We postpone the verification of the latter inequality to Sect. 3.5.

3.4 The Reynolds stress $\mathring{R}_{n+1}$

Let us recall from [7] the operator $\mathcal {R}$ that acts as left inverse of the operator $\text{ div }$. Namely, for every $v \in C^\infty (\mathbb {T}^3,\mathbb {R}^3)$ let $\mathcal {R}v$ be the matrix-valued function defined in [7, 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 we have

$$\begin{aligned} \mathcal {R}^{\phi _{n+1}} v :=[\mathcal {R}(v \circ \phi _{n+1}^{-1})] \circ \phi _{n+1}, \quad \mathord {\textrm{div}}^{\phi _{n+1}}(\mathcal {R}^{\phi _{n+1}} v) = v - \frac{1}{(2\pi )^3}\int _{\mathbb {T}^3}v. \end{aligned}$$

Then, we shall choose the new Reynolds stress $\mathring{R}_{n+1}$ such that

$$\begin{aligned} \mathring{R}_{n+1} :=\mathcal {R}^{\phi _{n+1}} \left( \partial _t v_{n+1} + \mathord {\textrm{div}}^{\phi _{n+1}}(v_{n+1} \otimes v_{n+1}) + \nabla ^{\phi _{n+1}}q_{n+1} - \Delta ^{\hspace{-0.05cm}\phi _{n+1}}v_{n+1}\right) .\nonumber \\ \end{aligned}$$

(3.20)

It is easy to check that the term inside the parentheses has zero space average by construction, and thus (3.20) gives a solution to the Navier–Stokes-Reynolds system at level $n+1$. In addition, it can be conveniently decomposed as

$$\begin{aligned} \partial _t v_{n+1}&+ \mathord {\textrm{div}}^{\phi _{n+1}}(v_{n+1}\otimes v_{n+1}) + \nabla ^{\phi _{n+1}}q_{n+1} - \Delta ^{\hspace{-0.05cm}\phi _{n+1}}v_{n+1} \\&= \nonumber \underbrace{\left[ \partial _t w^{(p)}_{n+1} + \partial _t w^{(c)}_{n+1} + \mathord {\textrm{div}}^{\phi _{n+1}}(w_{n+1} \otimes v_\ell + v_\ell \otimes w_{n+1} ) - \Delta ^{\hspace{-0.05cm}\phi _{n+1}}w_{n+1} \right] }_{=\,linear\, error} \\&\quad + \nonumber \underbrace{\left[ \mathord {\textrm{div}}^{\phi _{n+1}}\left( w^{(p)}_{n+1} \otimes w^{(p)}_{n+1} +\mathring{R}_\ell \right) + \partial _t w^{(t)}_{n+1} + \nabla ^{\phi _{n+1}}(q_{n+1}-q_\ell )\right] }_{=\,oscillation\, error} \\&\quad + \nonumber \underbrace{\left[ \mathord {\textrm{div}}^{\phi _{n+1}}(v_\ell \otimes v_\ell - (v_n \otimes v_n) *\chi _\ell )\right] }_{=\,mollification\, error\,I} \\&\quad + \nonumber \underbrace{\left[ \mathord {\textrm{div}}^{\phi _n}\left( \left( v_n \otimes v_n\right) *\chi _\ell + q_\ell Id - \mathring{R}_\ell \right) - \left( \mathord {\textrm{div}}^{\phi _n}\left( v_n \otimes v_n + q_n Id - \mathring{R}_n \right) \right) *\chi _\ell \right] }_{=\,mollification\, error\, II} \\&\quad + \nonumber \underbrace{\left[ \chi _\ell *\Delta ^{\hspace{-0.05cm}\phi _n}v_n - \Delta ^{\hspace{-0.05cm}\phi _n}v_\ell \right] }_{=\,mollification\, error \,III} \\&\quad + \nonumber \underbrace{\left[ (\Delta ^{\hspace{-0.05cm}\phi _n}- \Delta ^{\hspace{-0.05cm}\phi _{n+1}})\, v_\ell +\left( \mathord {\textrm{div}}^{\phi _{n+1}}-\mathord {\textrm{div}}^{\phi _n}\right) ((v_n \otimes v_n) *\chi _\ell - \mathring{R}_\ell + q_\ell Id) \right] }_{=\,flow\, error} \\&\quad + \nonumber \underbrace{\left[ \mathord {\textrm{div}}^{\phi _{n+1}}( w^{(p)}_{n+1} \otimes (w^{(c)}_{n+1}+w^{(t)}_{n+1}) + (w^{(c)}_{n+1}+w^{(t)}_{n+1}) \otimes w_{n+1}) \right] }_{=\,corrector\, error}. \end{aligned}$$

In this way, since the operator $\mathcal {R}^{\phi _{n+1}}$ is linear, we are able to separately control the different contributions to the new Reynolds stress $\mathring{R}_{n+1}$. Thus, the estimate (3.3) on the Reynolds stress in $L^1_x$ (actually in $L_x^p$ for some $p>1$) is obtained in a standard way, making use of the antidivergence $\mathcal {R}^{\phi _{n+1}}$. The “new" error terms coming from the composition with the flow are easily controlled assuming $\varsigma _{n+1}^{-1} \ll \lambda _{n+1} \ll \mu$. The key is that in those errors we don’t have time derivatives of the intermittent jets.

3.4.1 Linear error

The terms

$$\begin{aligned} \mathcal {R}^{\phi _{n+1}} \mathord {\textrm{div}}^{\phi _{n+1}}(w_{n+1} \otimes v_\ell + v_\ell \otimes w_{n+1}) \end{aligned}$$

and

$$\begin{aligned} \mathcal {R}^{\phi _{n+1}} \Delta ^{\hspace{-0.05cm}\phi _{n+1}}w_{n+1} = \mathcal {R}^{\phi _{n+1}} \mathord {\textrm{div}}^{\phi _{n+1}}\nabla ^{\phi _{n+1}}w_{n+1} \end{aligned}$$

are controlled using that $\mathcal {R}^{\phi _{n+1}} \mathord {\textrm{div}}^{\phi _{n+1}}:L_x^p \rightarrow L_x^p$ is bounded, Young convolution inequality and

$$\begin{aligned} \Vert v_\ell \otimes w_{n+1}\Vert _{L_x^p} + \Vert \nabla ^{\phi _{n+1}}w_{n+1}\Vert _{L_x^p}&\lesssim \Vert v_\ell \Vert _{L_x^\infty }\Vert w_{n+1}-w^{(c,1)}_{n+1}\Vert _{L_x^p} \\&\quad + \Vert v_\ell \Vert _{L_x^{2p'}}\Vert w^{(c,1)}_{n+1}\Vert _{L_x^{2}} + \Vert w_{n+1}\Vert _{W_x^{1,p}} \\&\lesssim \Vert v_\ell \Vert _{L_x^\infty }\Vert w_{n+1}-w^{(c,1)}_{n+1}\Vert _{L_x^p} \\&\quad + \ell ^{-4 \frac{p'-1}{p'}} \Vert v_n\Vert _{L_x^2}\Vert w^{(c,1)}_{n+1}\Vert _{L_x^2}\\ {}&\quad + \Vert w_{n+1}\Vert _{W_x^{1,p}} \lesssim \delta _{n+3}^2. \end{aligned}$$

Here $p'$ is such that $1/2p' + 1/2 = 1/p$, in particular $p' \rightarrow 1^+$ as $p \rightarrow 1^+$, so that the factor $\ell ^{-4 \frac{p'-1}{p'}}\delta _{n+3} \lesssim 1$ for p sufficiently small (depending only on $\alpha$, $\beta$ and b but not on a; in particular we can always increase the value of a to absorb any implicit constant).

As for the other term, recall

$$\begin{aligned} w^{(p)}_{n+1} + w^{(c,2)}_{n+1}&= \mathord {\textrm{curl}}^{\phi _{n+1}}\mathord {\textrm{curl}}^{\phi _{n+1}}\textbf{V} = \left( \mathord {\textrm{curl}}\,\mathord {\textrm{curl}}\,V \right) \circ \phi _{n+1}, \\ \partial _t w^{(p)}_{n+1} + \partial _t w^{(c,2)}_{n+1}&= \left( \mathord {\textrm{curl}}\,\mathord {\textrm{curl}}\,\partial _t V \right) \circ \phi _{n+1} \\&\quad + {\dot{\phi }}_{n+1} \cdot \left[ \left( \nabla \mathord {\textrm{curl}}\,\mathord {\textrm{curl}}\,V \right) \circ \phi _{n+1} \right] \\&= \mathord {\textrm{curl}}^{\phi _{n+1}}\left( (\mathord {\textrm{curl}}\,\partial _t V) \circ \phi _{n+1}\right) \\&\quad + {\dot{\phi }}_{n+1} \cdot \left[ \left( \nabla \mathord {\textrm{curl}}\,\mathord {\textrm{curl}}\,V \right) \circ \phi _{n+1} \right] , \end{aligned}$$

and use that $\mathcal {R}^{\phi _{n+1}}\mathord {\textrm{curl}}^{\phi _{n+1}}$ and $\mathcal {R}^{\phi _{n+1}}$ are bounded on $L_x^p$ plus the estimates (take p close to one)

$$\begin{aligned} \Vert (\mathord {\textrm{curl}}\,\partial _t V) \circ \phi _{n+1} \Vert _{L_x^p}&\lesssim \Vert V \Vert _{C^1_{\mathfrak {t}} W_x^{1,p}} \\&\lesssim r_\perp ^{2/p} r_{//}^{1/p-3/2} \mu \lesssim \lambda _{n+1}^{-1/10}, \\ \Vert {\dot{\phi }}_{n+1} \cdot \left[ \left( \nabla \mathord {\textrm{curl}}\,\mathord {\textrm{curl}}\,V \right) \circ \phi _{n+1} \right] \Vert _{L_x^p}&\lesssim \varsigma _{n+1}^{-1} \Vert V \Vert _{C_{\mathfrak {t}} W_x^{3,p}} \\&\lesssim \varsigma _{n+1}^{-1} \lambda _{n+1} r_\perp ^{2/p-1} r_{//}^{1/p-1/2} \lesssim \lambda _{n+1}^{-1/10}. \end{aligned}$$

It only remains to control $\mathcal {R}^{\phi _{n+1}} \partial _t w^{(c,1)}_{n+1}$, which can be done as in [17]. More precisely, one writes $\partial _t w^{(c,1)}_{n+1}$ as $- (\mathcal {Q}^{\phi _n} v_n) *\partial _t \chi ^0_\ell$ (we use the zero-mean version of $\partial _t \chi _\ell$) and bounds (using Lemma C.7 and adapting Lemma C.6 therein with any $\delta >0$, $1+1/p=1/p_1 + 1/p_2$ and $p_2$ sufficiently close to 1)

$$\begin{aligned} \Vert \mathcal {R}^{\phi _{n+1}} \partial _t w^{(c,1)}_{n+1} \Vert _{C_{\mathfrak {t}}L_x^p}&\lesssim \Vert (\mathcal {Q}^{\phi _n} v_n) *\partial _t \chi ^0_\ell \Vert _{C_{\mathfrak {t}}W_x^{-1,p}} \\&\lesssim \ell \sup _{s \le t} \Vert (\mathcal {Q}^{\phi _n} v_n)(\cdot ,t-s) *_{\mathbb {T}^3} \partial _t \chi ^0_\ell (\cdot ,s)\Vert _{C_{\mathfrak {t}}W_x^{-1,p}} \\&\lesssim \ell \sup _{s \le t} \Vert (\mathcal {Q}^{\phi _n} v_n)(\cdot ,t-s) *_{\mathbb {T}^3} \partial _t \chi ^0_\ell (\cdot ,s)\Vert _{C_{\mathfrak {t}}B^{\delta -1}_{p,\infty }} \\&\lesssim \ell \sup _{s \le t} \Vert \mathcal {Q}^{\phi _n} v_n\Vert _{C_{\mathfrak {t}}L_x^{p_1}} \Vert \partial _t \chi ^0_\ell \Vert _{C_{\mathfrak {t}}B^{2\delta -1}_{p_2,\infty }} \\&\lesssim \ell ^{-3\delta } \delta _{n+3}^3 \lesssim \delta _{n+3}^2. \end{aligned}$$

In the lines above $B^{\alpha }_{p,q}:= B^{\alpha }_{p,q}(\mathbb {T}^3)$, $\alpha \in \mathbb {R}$ and $p,q \in [1,\infty ]$, denotes the Besov space on the three dimensional torus, cf. [17].

3.4.2 Oscillation error

Recalling (3.10), it holds

$$\begin{aligned} \partial _t w^{(t)}_{n+1}&+ \mathord {\textrm{div}}^{\phi _{n+1}}\left( w^{(p)}_{n+1} \otimes w^{(p)}_{n+1} + \mathring{R}_\ell - \rho Id \right) \\&= \partial _t w^{(t)}_{n+1} + \mathord {\textrm{div}}^{\phi _{n+1}}\left( \sum _{\xi \in \Lambda } \textbf{a}_{\xi }^2\, \Pi _{\ne 0} (\textbf{W}_{\xi } \otimes \textbf{W}_{\xi }) \right) \\&= \partial _t w^{(t)}_{n+1} + \Pi _{\ne 0}\sum _{\xi \in \Lambda } \textbf{a}_{\xi }^2 \mathord {\textrm{div}}^{\phi _{n+1}}(\textbf{W}_{\xi } \otimes \textbf{W}_{\xi }) \\&\quad + \Pi _{\ne 0} \sum _{\xi \in \Lambda } \nabla ^{\phi _{n+1}}\textbf{a}_{\xi }^2 \cdot \Pi _{\ne 0}(\textbf{W}_{\xi } \otimes \textbf{W}_{\xi }). \end{aligned}$$

According to the same estimates as for $R_{osc}^{(x)}$ in [18, page 22] we can bound for p sufficiently small (recall that $\textbf{a}_\xi$ and $a_\xi$ enjoy the same bounds on space derivatives up to unimportant multiplicative constants)

$$\begin{aligned}&\left\| \mathcal {R}^{\phi _{n+1}} \left( \Pi _{\ne 0} \sum _{\xi \in \Lambda } \nabla ^{\phi _{n+1}}\textbf{a}_{\xi }^2 \cdot \Pi _{\ne 0}(\textbf{W}_{\xi } \otimes \textbf{W}_{\xi }) \right) \right\| _{C_\mathfrak {t}L_x^p} \\&= \left\| \mathcal {R}\left( \sum _{\xi \in \Lambda } \nabla a_{\xi }^2 \cdot \Pi _{\ne 0} (W_{\xi } \otimes W_{\xi }) \right) \right\| _{C_\mathfrak {t}L_x^p} \\&= \left\| \mathcal {R}\left( \sum _{\xi \in \Lambda } \nabla a_{\xi }^2 \cdot \Pi _{\ge r_\perp \lambda _{n+1}/2} (W_{\xi } \otimes W_{\xi }) \right) \right\| _{C_\mathfrak {t}L_x^p} \\&\lesssim \ell ^{-23} r_\perp ^{2/p-3} r_{//}^{1/p-1} \lambda _{n+1}^{-1} \lesssim \lambda _{n+1}^{-1/10}. \end{aligned}$$

As for the other terms, we have already seen by (3.17) and (3.18) the decomposition

$$\begin{aligned} \partial _t&w^{(t)}_{n+1} + \Pi _{\ne 0}\sum _{\xi \in \Lambda } \textbf{a}_{\xi }^2 \, \mathord {\textrm{div}}^{\phi _{n+1}}(\textbf{W}_{\xi } \otimes \textbf{W}_{\xi }) \\&= \nabla ^{\phi _{n+1}}(P \circ \phi _{n+1}) \\&\quad + \frac{1}{\mu } \Pi _{\ne 0} \, {{\dot{\phi }}}_{n+1} \cdot \left[ \sum _{\xi \in \Lambda } \nabla \left( {a}_{\xi }^2 \theta _{\xi }^2 \psi _{\xi }^2 \xi \right) \circ \phi _{n+1} \right] \\&\quad - \frac{1}{\mu } {{\dot{\phi }}}_{n+1} \cdot \left[ \nabla \left( \mathcal {P} \Pi _{\ne 0} \sum _{\xi \in \Lambda } a_{\xi }^2 \theta _{\xi }^2 \psi _{\xi }^2 \xi \right) \circ \phi _{n+1} \right] \\&\quad - \frac{1}{\mu }\Pi _{\ne 0} \sum _{\xi \in \Lambda } (\partial _t \textbf{a}_{\xi }^2) \varvec{\theta }_{\xi }^2 \varvec{\psi }_{\xi }^2 \xi \\&\quad - \frac{1}{\mu } \Pi _{\ne 0} \sum _{\xi \in \Lambda } \textbf{a}_{\xi }^2\, {\dot{\phi }}_{n+1} \cdot \left[ \nabla \left( \theta _{\xi }^2 \psi _{\xi }^2 \xi \right) \circ \phi _{n+1} \right] , \end{aligned}$$

where we can bound

$$\begin{aligned} \frac{1}{\mu } \left\| \Pi _{\ne 0} \sum _{\xi \in \Lambda } (\partial _t \textbf{a}_{\xi }^2) \varvec{\theta }_{\xi }^2 \varvec{\psi }_{\xi }^2 \xi \right\| _{C_\mathfrak {t}L_x^p}&\lesssim \frac{1}{\mu } \sum _{\xi \in \Lambda } \Vert a_{\xi } \Vert _{C_{\mathfrak {t},x}} \Vert a_{\xi } \Vert _{C^1_{\mathfrak {t},x}} \Vert \theta _{\xi }^2 \psi _{\xi }^2 \Vert _{C_\mathfrak {t}L_x^p} \\&\lesssim \mu ^{-1} \varsigma _{n+1}^{-1} \ell ^{-19} r_{//}^{1/p-1} r_\perp ^{2/p-2} \lesssim \lambda _{n+1}^{-1}, \end{aligned}$$

and all the other terms except $\nabla ^{\phi _{n+1}}(P \circ \phi _{n+1})$ with

$$\begin{aligned} \frac{\varsigma _{n+1}^{-1}}{\mu } \sum _{\xi \in \Lambda } \Vert a_{\xi }^2 \theta _{\xi }^2 \psi _{\xi }^2 \Vert _{C_\mathfrak {t}W_x^{1,p}}&\lesssim \frac{\varsigma _{n+1}^{-1}}{\mu } \Vert a_{\xi } \Vert _{C_{t,x}} \Vert a_{\xi } \Vert _{C^1_{\mathfrak {t},x}} \Vert \theta _{\xi } \Vert _{L_x^{2p}}^2 \Vert \psi _{\xi } \Vert _{L_x^{2p}}^2 \\&\quad + \frac{\varsigma _{n+1}^{-1}}{\mu } \Vert a_{\xi } \Vert _{C_{t,x}}^2 \Vert \theta _{\xi } \Vert _{L_x^{2p}} \Vert \nabla \theta _{\xi } \Vert _{L_x^{2p}} \Vert \psi _{\xi }\Vert _{L_x^{2p}}^2 \\&\quad + \frac{\varsigma _{n+1}^{-1}}{\mu } \Vert a_{\xi } \Vert _{C_{t,x}}^2 \Vert \theta _{\xi } \Vert _{L_x^{2p}}^2 \Vert \nabla \psi _{\xi } \Vert _{L_x^{2p}} \Vert \psi _{\xi }\Vert _{L_x^{2p}} \\&\lesssim \varsigma _{n+1}^{-1} \mu ^{-1} \ell ^{-16} r_\perp ^{2/p-2} r_{//}^{1/p-1} ( \ell ^{-7} + \lambda _{n+1} + r_\perp r_{//}^{-1} \lambda _{n+1} ) \\&\lesssim \lambda _{n+1}^{-1/10}. \end{aligned}$$

3.4.3 Mollification error

In order to control the mollification error, use

$$\begin{aligned} \Vert \mathcal {R}^{\phi _{n+1}} \mathord {\textrm{div}}^{\phi _{n+1}}(v_\ell \otimes v_\ell - (v_n \otimes v_n) *\chi _\ell ) \Vert _{C_{\mathfrak {t}}L_x^p}&\lesssim \Vert v_\ell \otimes v_\ell - (v_n \otimes v_n) *\chi _\ell \Vert _{C_{\mathfrak {t}}L_x^p} \\&\lesssim \ell \lambda _{n}^{6} \lesssim \delta _{n+3}^2, \end{aligned}$$

and by [17, Lemma 4.4] with $G = v_n \otimes v_n + q_n Id - \mathring{R}_n$

$$\begin{aligned} \left\| \mathord {\textrm{div}}^{\phi _n}\left( G *\chi _\ell \right) - \left( \mathord {\textrm{div}}^{\phi _n}G \right) *\chi _\ell \right\| _{C_{\mathfrak {t},x}} \lesssim \ell ^{1/4} \Vert G\Vert _{C_{\mathfrak {t}}C_x^1} \lesssim \ell ^{1/4} \lambda _n^{20} \lesssim \delta _{n+3}^2. \end{aligned}$$

As for the term $\chi _\ell *\Delta ^{\hspace{-0.05cm}\phi _n}v_n - \Delta ^{\hspace{-0.05cm}\phi _n}v_\ell$, rewrite

$$\begin{aligned} \chi _\ell *\Delta ^{\hspace{-0.05cm}\phi _n}v_n - \Delta ^{\hspace{-0.05cm}\phi _n}v_\ell&= \chi _\ell *\mathord {\textrm{div}}^{\phi _n}\nabla ^{\phi _n}v_n - \mathord {\textrm{div}}^{\phi _n}(\chi _\ell *\nabla ^{\phi _n}v_n ) \\&\quad + \mathord {\textrm{div}}^{\phi _n}(\chi _\ell *\nabla ^{\phi _n}v_n ) - \mathord {\textrm{div}}^{\phi _n}\nabla ^{\phi _n}v_\ell , \end{aligned}$$

and use the same lemma with $G=\nabla ^{\phi _n}v_n$ (or $G=v_n$ and replacing $\mathord {\textrm{div}}^{\phi _n}$ with $\nabla ^{\phi _n}$)

$$\begin{aligned} \left\| \chi _\ell *\mathord {\textrm{div}}^{\phi _n}\nabla ^{\phi _n}v_n - \mathord {\textrm{div}}^{\phi _n}(\chi _\ell *\nabla ^{\phi _n}v_n ) \right\| _{C_{\mathfrak {t},x}}&\lesssim \ell ^{1/4} \Vert v_n\Vert _{C_{\mathfrak {t}}C^2_x} \\&\lesssim \ell ^{1/4} \lambda _n^{12} \lesssim \delta _{n+3}^2, \\ \left\| \mathcal {R}^{\phi _{n+1}} \left( \mathord {\textrm{div}}^{\phi _n}(\chi _\ell *\nabla ^{\phi _n}v_n ) - \mathord {\textrm{div}}^{\phi _n}\nabla ^{\phi _n}v_\ell \right) \right\| _{C_{\mathfrak {t}}L_x^p}&\lesssim \Vert \chi _\ell *\nabla ^{\phi _n}v_n - \nabla ^{\phi _n}v_\ell \Vert _{C_{\mathfrak {t}}L_x^p} \\&\lesssim \ell ^{1/4} \lambda _n^{12} \lesssim \delta _{n+3}^2. \end{aligned}$$

3.4.4 Flow error

Use Lemma 3.2 with $G = \nabla ^{\phi _n}v_\ell$ or $G=(v_n \otimes v_n) *\chi _\ell - \mathring{R}_\ell + q_\ell Id$ (or again with $G=v_\ell$ and replacing $(\mathord {\textrm{div}}^{\phi _{n+1}}-\mathord {\textrm{div}}^{\phi _n})$ with $(\nabla ^{\phi _{n+1}}-\nabla ^{\phi _n})$ ) to get

$$\begin{aligned} \left\| (\Delta ^{\hspace{-0.05cm}\phi _n}- \Delta ^{\hspace{-0.05cm}\phi _{n+1}}) v_\ell \right\| _{C_{\mathfrak {t}}W_x^{-1,p}}&\le \left\| (\mathord {\textrm{div}}^{\phi _n}- \mathord {\textrm{div}}\,^{\phi _{n+1}}) \nabla ^{\phi _n}v_\ell \right\| _{C_{\mathfrak {t}}W_x^{-1,p}} \\&+ \left\| \mathord {\textrm{div}}\,^{\phi _{n+1}}(\nabla ^{\phi _n}- \nabla ^{\phi _{n+1}}) v_\ell \right\| _{C_{\mathfrak {t}}W_x^{-1,p}} \\&\lesssim (n+1) \varsigma _n^{1/4} \Vert v_\ell \Vert _{C_{\mathfrak {t}}W_x^{1,p}} \lesssim \delta _{n+3}^{2}, \end{aligned}$$

and for some $\delta >0$ depending only on p and such that $\delta \rightarrow 0^+$ as $p\rightarrow 1^+$

$$\begin{aligned}&\left\| \left( \mathord {\textrm{div}}^{\phi _{n+1}}-\mathord {\textrm{div}}^{\phi _n}\right) ((v_n \otimes v_n) *\chi _\ell - \mathring{R}_\ell + q_\ell Id) \right\| _{C_{\mathfrak {t}}W_x^{-1,p}} \\&\qquad \lesssim (n+1) \varsigma _n^{1/4} \Vert (v_n \otimes v_n) *\chi _\ell - \mathring{R}_\ell + q_\ell Id \Vert _{C_{\mathfrak {t}}L_x^p} \\&\qquad \lesssim (n+1) \varsigma _n^{1/4} \ell ^{-\delta } \left( \Vert v_n \Vert _{C_{\mathfrak {t}}L_x^2} + \Vert \mathring{R}_n\Vert _{C_{\mathfrak {t}}L^1_x} + \Vert q_n \Vert _{C_{\mathfrak {t}}L^1_x} \right) \\&\qquad \lesssim (n+1) \varsigma _n^{1/4} \ell ^{-\delta } \lambda _n^{1/1000} \lesssim \delta _{n+3}^2. \end{aligned}$$

We point out that the previous inequalities hold true at least choosing p sufficiently close to 1, but again not depending on a, so that implicit constants can be absorbed taking a large enough.

3.4.5 Correction error

It is sufficient to control, recalling previous bounds

$$\begin{aligned} \Vert w^{(p)}_{n+1} \otimes (w^{(c)}_{n+1}+w^{(t)}_{n+1}) \Vert _{C_{\mathfrak {t}}L_x^p}&\lesssim \Vert w^{(p)}_{n+1} \Vert _{C_{\mathfrak {t}}L_x^{2p}} \Vert w^{(c)}_{n+1}+w^{(t)}_{n+1} \Vert _{C_{\mathfrak {t}}L_x^{2p}} \\&\lesssim \ell ^{-p'} (\delta _{n+3}^3 + \ell ^{-22}\lambda _{n+1}^{-1/10} ) \lesssim \delta _{n+3}^2, \\ \Vert (w^{(c)}_{n+1}+w^{(t)}_{n+1}) \otimes w_{n+1} \Vert _{C_{\mathfrak {t}}L_x^p}&\lesssim \Vert w_{n+1} \Vert _{C_{\mathfrak {t}}L_x^{2p}} \Vert w^{(c)}_{n+1}+w^{(t)}_{n+1} \Vert _{C_{\mathfrak {t}}L_x^{2p}} \\&\lesssim \ell ^{-p'} (\delta _{n+3}^3 + \ell ^{-22}\lambda _{n+1}^{-1/10} ) \lesssim \delta _{n+3}^2. \end{aligned}$$

Putting all together, the proof of the bound (3.3) on $\Vert \mathring{R}_{n+1}\Vert _{C_\mathfrak {t}L_x^1}$ is proved up to noticing

$$\begin{aligned} \Vert \mathring{R}_{n+1}\Vert _{C_\mathfrak {t}L_x^1} \lesssim \Vert \mathring{R}_{n+1}\Vert _{C_\mathfrak {t}L_x^p} \lesssim \delta _{n+3}^2 + \lambda _{n+1}^{-1/10} \le C_R \delta _{n+3} \end{aligned}$$

up to suitable choice of parameters.

3.4.6 Estimate on $\Vert \mathring{R}_{n+1} \Vert _{C_\mathfrak {t}C_x^1}$

This comes easily from the Navier–Stokes-Reynolds equation itself and the bounds on the derivatives of $v_{n+1}$, $q_{n+1}$ already proved in Sects. 3.2 and 3.3. Indeed, by (3.20)

$$\begin{aligned} \mathring{R}_{n+1} :=\mathcal {R}^{\phi _{n+1}} \left( \partial _t v_{n+1} + \mathord {\textrm{div}}^{\phi _{n+1}}(v_{n+1} \otimes v_{n+1}) + \nabla ^{\phi _{n+1}}q_{n+1} - \Delta ^{\hspace{-0.05cm}\phi _{n+1}}v_{n+1}\right) , \end{aligned}$$

and thus for every $\delta >0$ it holds

$$\begin{aligned} \Vert \mathring{R}_{n+1} \Vert _{C_\mathfrak {t}C^{1+\delta }_x}&\lesssim \Vert \partial _t v_{n+1} + \mathord {\textrm{div}}^{\phi _{n+1}}(v_{n+1} \otimes v_{n+1}) + \nabla ^{\phi _{n+1}}q_{n+1} - \Delta ^{\hspace{-0.05cm}\phi _{n+1}}v_{n+1} \Vert _{C_\mathfrak {t}C^{\delta }_x} \\&\lesssim \Vert v_{n+1} \Vert _{C^1_{\mathfrak {t},x}} \Vert v_{n+1} \Vert _{C^1_\mathfrak {t}C^3_x} + \Vert q_{n+1} \Vert _{C_\mathfrak {t}C^2_x} \lesssim \lambda _{n+1}^{18}. \end{aligned}$$

3.5 Estimate on the pressure

As already explained in Sect. 3.3, in order to prove the $L_x^1$ estimate on the pressure we only need to bound P.

Let us therefore consider the auxiliary Navier–Stokes-Reynolds system

$$\begin{aligned} \partial _t {\tilde{u}}_n + \mathord {\textrm{div}}\,({\tilde{u}}_n \otimes {\tilde{u}}_n) + \nabla {\tilde{p}}_n - \Delta {\tilde{u}}_n = \mathord {\textrm{div}}\,\mathring{{\tilde{R}}}_n, \end{aligned}$$

with $\mathord {\textrm{div}}\,{\tilde{u}}_n = 0$ and $\int _{\mathbb {T}^3} {\tilde{u}}_n = 0$. Notice that the differential operators above are not composed with the flow $\phi _{n}$. Then, starting from the triple $({\tilde{u}}_0,{\tilde{p}}_0,\mathring{{\tilde{R}}}_0) = 0$ and iterating the system $({\tilde{u}}_n,{\tilde{p}}_n,\mathring{{\tilde{R}}}_n)$ simultaneously with the iteration for $({v}_n,{p}_n,\mathring{{R}}_n)$ according to

$$\begin{aligned} {\tilde{u}}_{n+1} := {\tilde{u}}_\ell + {\tilde{w}}_{n+1} := {\tilde{u}}_\ell + {\tilde{w}}^{(p)}_{n+1} + {\tilde{w}}^{(c)}_{n+1}+ {\tilde{w}}^{(t)}_{n+1}, \end{aligned}$$

where for every $n \in \mathbb {N}$ we define

$$\begin{aligned} {\tilde{w}}^{(p)}_{n+1}&:= \sum _{\xi \in \Lambda } a_{\xi } W_{\xi }, \\ {\tilde{w}}^{(c)}_{n+1}&:= \sum _{\xi \in \Lambda } \mathord {\textrm{curl}}\,(\nabla a_{\xi } \times V_{\xi }) + \nabla a_{\xi } \times \mathord {\textrm{curl}}\,V_{\xi } + a_{\xi } W^{(c)}_{\xi }, \\ {\tilde{w}}^{(t)}_{n+1}&:= - \frac{1}{\mu } \mathcal {P} \Pi _{\ne 0} \sum _{\xi \in \Lambda } a_{\xi }^2 \theta _{\xi }^2 \psi _{\xi }^2 \xi , \end{aligned}$$

and

$$\begin{aligned} {\tilde{p}}_{n+1}&:= {\tilde{p}}_\ell - P, \\ \mathring{{\tilde{R}}}_{n+1}&:= \mathcal {R} \left( \partial _t {\tilde{v}}_{n+1} + \mathord {\textrm{div}}\,({\tilde{u}}_{n+1} \otimes {\tilde{u}}_{n+1}) + \nabla {\tilde{p}}_{n+1} - \Delta {\tilde{u}}_{n+1} \right) , \end{aligned}$$

one can prove the bounds

$$\begin{aligned} \Vert {\tilde{u}}_{n+1}-{\tilde{u}}_n \Vert _{C_\mathfrak {t}L_x^2} \le C_u \delta _{n+1}^{1/2}, \quad \Vert {\tilde{u}}_n \Vert _{C^1_{\mathfrak {t},x}} \le C_u \lambda _n^{6}, \\ \Vert \mathring{{\tilde{R}}}_n \Vert _{C_\mathfrak {t}L_x^1} \le C_R 3^n, \quad \Vert \mathring{{\tilde{R}}}_n \Vert _{C_\mathfrak {t}C^1_x} \le C_R \lambda _n^{20}, \end{aligned}$$

for every $n \in \mathbb {N}$. Indeed, we have used $\rho$ and $a_\xi$ exactly as before, and so they are constructed from $\mathring{{R}}_\ell$ and not $\mathring{{\tilde{R}}}_\ell$; in particular, the bounds on ${\tilde{u}}_n$ and $\mathring{{\tilde{R}}}_n$ in $C^1$ as well as the bound on the increment ${\tilde{u}}_{n+1}-{\tilde{u}}_n$ are readily proved.

The only slight difference is the estimate on $\mathring{{\tilde{R}}}_n$ in $L^1_x$, for which we argue as follows. First, one has the following decomposition of $\mathord {\textrm{div}}\,\mathring{{\tilde{R}}}_{n+1}$

$$\begin{aligned} \partial _t {\tilde{u}}_{n+1}&+ \mathord {\textrm{div}}\,({\tilde{u}}_{n+1}\otimes {\tilde{u}}_{n+1}) + \nabla {\tilde{p}}_{n+1} - \Delta {\tilde{u}}_{n+1} \\&= \nonumber \underbrace{\left[ \partial _t {\tilde{w}}^{(p)}_{n+1} + \partial _t {\tilde{w}}^{(c)}_{n+1} + \mathord {\textrm{div}}\,({\tilde{w}}_{n+1} \otimes {\tilde{u}}_\ell + {\tilde{u}}_\ell \otimes {\tilde{w}}_{n+1} ) - \Delta {\tilde{w}}_{n+1} \right] }_{=\,linear\, error} \\&\quad + \nonumber \underbrace{\left[ \mathord {\textrm{div}}\,\left( {\tilde{w}}^{(p)}_{n+1} \otimes {\tilde{w}}^{(p)}_{n+1} + \mathring{R}_\ell \circ \phi _{n+1}^{-1} - \rho \circ \phi _{n+1}^{-1} Id \right) + \partial _t {\tilde{w}}^{(t)}_{n+1} + \nabla P \right] }_{=\,oscillation\, error} \\&\quad + \nonumber \underbrace{\left[ \mathord {\textrm{div}}\,({\tilde{u}}_\ell \otimes {\tilde{u}}_\ell - ({\tilde{u}}_n \otimes {\tilde{u}}_n) *\chi _\ell )\right] }_{=\,mollification\, error} \\&\quad + \nonumber \underbrace{\left[ \mathord {\textrm{div}}\,( {\tilde{w}}^{(p)}_{n+1} \otimes ({\tilde{w}}^{(c)}_{n+1}+{\tilde{w}}^{(t)}_{n+1}) + ({\tilde{w}}^{(c)}_{n+1}+{\tilde{w}}^{(t)}_{n+1}) \otimes {\tilde{w}}_{n+1}) \right] }_{=\,corrector\, error} \\&\quad + \nonumber \underbrace{\left[ \mathord {\textrm{div}}\,\left( \rho \circ \phi _{n+1}^{-1} Id - \mathring{R}_\ell \circ \phi _{n+1}^{-1} + \mathring{{\tilde{R}}}_\ell \right) \right] }_{=\,remainder\, error}. \end{aligned}$$

Notice that

$$\begin{aligned} w^{(p)}_{n+1} \otimes w^{(p)}_{n+1}&= \sum _{\xi \in \Lambda } a_{\xi }^2\, \Pi _{\ne 0} (W_{\xi } \otimes W_{\xi }) + \rho \circ \phi _{n+1}^{-1} Id - \mathring{R}_\ell \circ \phi _{n+1}^{-1}, \end{aligned}$$

and hence the fast-fast interaction of the perturbation is not compensated by $\mathring{{\tilde{R}}}_\ell$ but rather by the “old" terms $\rho \circ \phi _{n+1}^{-1} Id + \mathring{{R}}_\ell \circ \phi _{n+1}^{-1}$, for which however we have bounds in $L_x^p$. All the other terms are controlled as usual, and therefore a remainder error proportional to

$$\begin{aligned} \Vert \rho \circ \phi _{n+1}^{-1} Id - \mathring{R}_\ell \circ \phi _{n+1}^{-1} + \mathring{{\tilde{R}}}_\ell \Vert _{L_x^p} \le 2 C_R 3^n \end{aligned}$$

in the $C_\mathfrak {t}L^1_x$ norm of $\mathring{{\tilde{R}}}_{n+1}$ accumulates along the iteration. Notice that the auxiliary Reynolds stress $\mathring{{\tilde{R}}}_{n+1}$ is not small, but this is not important for our purposes since we only use it to recover estimates on P. Indeed, we know

$$\begin{aligned} \partial _t {\tilde{u}}_n + \mathord {\textrm{div}}\,({\tilde{u}}_n \otimes {\tilde{u}}_n) + \nabla {\tilde{p}}_n - \Delta {\tilde{u}}_n&= \mathord {\textrm{div}}\,\mathring{{\tilde{R}}}_n, \\ \partial _t {\tilde{u}}_{n+1} + \mathord {\textrm{div}}\,({\tilde{u}}_{n+1} \otimes {\tilde{u}}_{n+1}) + \nabla {\tilde{p}}_{n+1} - \Delta {\tilde{u}}_{n+1}&= \mathord {\textrm{div}}\,\mathring{{\tilde{R}}}_{n+1}, \end{aligned}$$

and $\mathord {\textrm{div}}\,{\tilde{u}}_n = \mathord {\textrm{div}}\,{\tilde{u}}_{n+1} = 0$ (the perturbation ${\tilde{w}}_{n+1}$ is divergence free). Thus

$$\begin{aligned} \Delta {\tilde{p}}_n&= \mathord {\textrm{div}}\,\mathord {\textrm{div}}\,(\mathring{{\tilde{R}}}_n- {\tilde{u}}_n \otimes {\tilde{u}}_n), \end{aligned}$$

(3.21)

$$\begin{aligned} \Delta {\tilde{p}}_{n+1}&= \mathord {\textrm{div}}\,\mathord {\textrm{div}}\,(\mathring{{\tilde{R}}}_{n+1}- {\tilde{u}}_{n+1} \otimes {\tilde{u}}_{n+1}), \end{aligned}$$

(3.22)

and Schauder estimates imply for every $p>1$ sufficiently small (independent of n)

$$\begin{aligned} \Vert {\tilde{p}}_n \Vert _{L_x^1} \lesssim \Vert {\tilde{p}}_n \Vert _{L_x^p}&\lesssim \Vert \mathring{{\tilde{R}}}_n \Vert _{L_x^p} + \Vert {\tilde{u}}_n \Vert _{L_x^{2p}}^2 \\&\lesssim \Vert \mathring{{\tilde{R}}}_n \Vert _{L_x^p} + \sum _{k=1}^{n} \Vert {\tilde{u}}_{k} - {\tilde{u}}_{k-1} \Vert _{L_x^{2p}}^2 \le \tfrac{1}{2} \lambda _n^{1/1000}, \end{aligned}$$

and similarly for $n+1$. In addition, we know that P is related to ${\tilde{p}}_n,{\tilde{p}}_{n+1}$ via the formula

$$\begin{aligned} {\tilde{p}}_{n+1} = {\tilde{p}}_\ell - P, \end{aligned}$$

so that we have

$$\begin{aligned} \Vert P \Vert _{L_x^1} \le \Vert {\tilde{p}}_{n+1} \Vert _{L_x^1} + \Vert {\tilde{p}}_n \Vert _{L_x^1} \le \lambda _{n+1}^{1/1000}, \end{aligned}$$

concluding the proof.

Remark 3.3

We critically need (3.21) and (3.22) for the “deterministic" pressures ${\tilde{p}}_n$, ${\tilde{p}}_{n+1}$. Indeed, it does not seem possible to recover directly good estimates on P, and thus any estimate on the pressure $q_{n+1}$ must be obtained from the equation itself. However, in the stochastic case this is not feasible: indeed

$$\begin{aligned} \Delta ^{\hspace{-0.05cm}\phi _{n+1}}q_{n+1} = \mathord {\textrm{div}}^{\phi _{n+1}}\mathord {\textrm{div}}^{\phi _{n+1}}(\mathring{R}_{n+1} - v_{n+1} \otimes v_{n+1} + \nabla ^{\phi _{n+1}}v_{n+1} ) - \mathord {\textrm{div}}^{\phi _{n+1}}(\partial _t v_{n+1}), \end{aligned}$$

where the additional terms come from the fact that $\mathord {\textrm{div}}^{\phi _{n+1}}v_{n+1} \ne 0$ in general, and moreover $\mathord {\textrm{div}}^{\phi _{n+1}}$ and $\partial _t$ do not commute. But the last term in the previous inequality prevents us from obtaining any good bound on $\Vert q_{n+1} \Vert _{L_x^1}$, since we can only estimate $\Vert \partial _t v_{n+1} \Vert _{W_x^{-1,1}}$ with $\Vert v_{n+1}\Vert _{C^1_{\mathfrak {t},x}} \lesssim \lambda _{n+1}^{6}$.

4 Construction of solutions with prescribed initial condition

In this second part we want to modify the previous construction so to prescribe the initial value of the solution to (1.1). Let $u_0 \in L_x^2$ with $\mathord {\textrm{div}}\,u_0 = 0$ in the sense of distributions. We shall assume $u_0$ independent of the Brownian motion B and replace the filtration $\{\mathcal {F}_t\}_{t \ge 0}$ with the augmented canonical filtration generated by $(u_0,B)$. Moreover, without loss of generality we can suppose that

$$\begin{aligned} \Vert u_0 \Vert _{L_x^2} \le M, \quad \mathbb {P}-\text{ almost } \text{ surely } \end{aligned}$$

for some finite constant M. Indeed, for a general initial condition $u_0 \in L_x^2$ almost surely one defines $\Omega _M:= \{ M-1 \le \Vert u_0 \Vert _{L_x^2} < M\}$. Then, given the existence of infinitely many solutions $u_M$ on each $\Omega _M$ one can define $u:= \sum _{M \in \mathbb {N}, M \ge 1} u_M \textbf{1}_{\Omega _M}$, solving the equation with initial condition $u_0$.

In order to prescribe the initial value of solutions we follow the approach of [5] and [18]. However, differently from what done in Sect. 3 here we need to use the so-called Da Prato-Debussche trick to produce a good initial triple $(v_0,q_0,\mathring{R}_0)$ to start the iteration with.

Formally, let Z be the solution of the Stokes system with transport noise emanating from $u_0$:

$$\begin{aligned} {\left\{ \begin{array}{ll} d Z + \sum _{k \in I} (\sigma _k \cdot \nabla ) Z \bullet dB^k + \nabla p_Z dt = \Delta Z dt, \\ \mathord {\textrm{div}}\,Z = 0, \\ Z|_{t=0} = u_0, \end{array}\right. }\nonumber \\ \end{aligned}$$

(4.1)

and let u be any solution of (1.1) with the same initial condition. Then the difference $V:= u-Z$ solves

$$\begin{aligned} {\left\{ \begin{array}{ll} d V + \mathord {\textrm{div}}\,\left( (V+Z) \otimes (V+Z) \right) dt + \sum _{k \in I} (\sigma _k \cdot \nabla ) V \bullet dB^k + \nabla p_V \,dt = \Delta V dt, \\ \mathord {\textrm{div}}\,V = 0, \\ V|_{t=0} = 0, \end{array}\right. }\nonumber \\ \end{aligned}$$

(4.2)

which is equivalent, by composition with the flow $(v,q,z):= (V,p_V,Z) \circ \phi$, to the system

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t v + \mathord {\textrm{div}}^\phi \left( (v+z) \otimes (v+z) \right) + \nabla ^{\phi }q = \Delta ^{\hspace{-0.05cm}\phi }v, \\ \mathord {\textrm{div}}^\phi v = 0, \\ v|_{t=0} = 0. \end{array}\right. }\nonumber \\ \end{aligned}$$

(4.3)

Viceversa, via the inverse transformation $u=(v+z) \circ \phi ^{-1}$, any solution of (4.3) yields a solution to (1.1) satisfying $u|_{t=0}=u_0$. Therefore, in order to construct (multiple) solutions of (4.3) one can introduce for every $n\in \mathbb {N}$ a smooth approximation $z_n$ of z and solve the Navier-Stokes-Reynolds system on $t \in \mathbb {R}$

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t v_n + \mathord {\textrm{div}}^{\phi _n}\left( (v_n+z_n) \otimes (v_n+z_n) \right) + \nabla ^{\phi _n}q_n - \Delta ^{\hspace{-0.05cm}\phi _n}v_n = \mathord {\textrm{div}}^{\phi _n}\mathring{R}_n, \\ v_n|_{t = 0} = 0. \end{array}\right. }\nonumber \\ \end{aligned}$$

(4.4)

Then, a solution to (4.3) is obtained showing the convergences $v_n \rightarrow v$, $z_n \rightarrow z$ and $\mathring{R}_n$, $\mathord {\textrm{div}}^{\phi _n}v_n \rightarrow 0$ in suitable spaces as $n \rightarrow \infty$.

Remark 4.1

Before moving on, let us first comment on the Da Prato-Debussche trick. Usually, it is used in semilinear stochastic equations with additive noise to “remove" the stochastic integral; this is obtained taking the difference between any hypothetical solution of the original equation and the stochastic convolution, and observing that the difference solves a random PDE. In our case, however, since the noise is multiplicative we still have a stochastic integral in (4.2), and we need the flow transformation anyway to reformulate the problem as a random PDE. The point is that here we use the Da Prato-Debussche trick only to produce a good initial triple $(v_0,q_0,\mathring{R}_0)$. More specifically, the iteration will be started at $v_0 = q_0 = 0$ and $\mathring{R}_0 = z_0 \otimes z_0$, extended to negative times with their value at time $t=0$; it is in fact this extension to negative times that we are not able to perform otherwise.

4.1 Approximation of z

Let us first study the Stokes system (4.1). Via Galerkin approximation and compactness, it is easy to obtain a probabilistically weak, analytically weak, progressively measurable solution Z in $L^\infty (\Omega , L^\infty _t L_x^2 \cap L^2_t H_x^1)$ satisfying almost surely

$$\begin{aligned} \Vert Z(t) \Vert _{L_x^2}^2 + 2 \int _0^t \Vert Z(s) \Vert _{H_x^1}^2 ds \le \Vert u_0 \Vert _{L_x^2}^2 \le M^2. \end{aligned}$$

Moreover, pathwise uniqueness for the equation in the space $L^2 (\Omega , L^\infty _t L_x^2 \cap L^2_t H_x^1)$ descends for instance from application of [8, Proposition A.1] with $\psi = 1$ and $\varphi (u) = u^2$. Thus Z is a probabilistically strong solution by Yamada-Watanabe Theorem.

Moreover, by Itō isometry one can prove for every $p<\infty$

$$\begin{aligned} \mathbb {E} \left\| Z(t)-Z(s)\right\| _{H_x^{-1}}^p \lesssim M^p |t-s|^{p/2}, \end{aligned}$$

where the implicit constant depends only on p, and thus by Kolmogorov continuity criterion we can assume without loss of generality that Z is almost surely $\gamma$-Hölder continuous for every $\gamma <1/2$, as a process taking values in $H^{-1}_x$. Thus, modifying the definition of our stopping times $\mathfrak {t}_L$ from Lemma 2.1 we can assume

$$\begin{aligned} \Vert Z\Vert _{C^{1/4}_{\mathfrak {t}_L} H_x^{-1}} \le C_Z L M \end{aligned}$$

almost surely for some $C_Z \in (0,\infty )$. Similarly to what previously done in Sect. 3, for simplicity we only work up to time $\mathfrak {t}=\mathfrak {t}_1$ hereafter. Furthermore, in this section we shall suppose $\mathfrak {t} \ge 1$ with probability larger than 1/2 (this is always possible modifying the value of $C_Z$) and also $\mathfrak {t} \le 2$ almost surely.

We approximate z via Fourier projection in space, $z_n:= \Pi _{\le \kappa _n} z$ for some frequency cut-off $\kappa _n:= \lambda _{n+1}^{\alpha /16} \rightarrow \infty$. By Sobolev embedding we have the estimates (also use arguments as in [17, Lemma C.2] to deduce $\Vert z\Vert _{C^{1/4}_{\mathfrak {t}} H_x^{-1}} \lesssim \Vert Z\Vert _{C^{1/4}_{\mathfrak {t}} H_x^{-1}}$)

$$\begin{aligned} \Vert z_n \Vert _{L^\infty _{\mathfrak {t},x}} \lesssim M\kappa _n^2, \qquad \Vert z_n \Vert _{L^\infty _\mathfrak {t}C_x^2}&\lesssim M\kappa _n^4, \qquad \Vert z_n\Vert _{C^{1/4}_{\mathfrak {t},x}} \lesssim M\kappa _n^3. \end{aligned}$$

We shall also need a control on $z_{n+1}-z_n$ in $L^2_x$. Using $\Vert Z\Vert _{L^2_\mathfrak {t}H^1_x} \lesssim M$ we have

$$\begin{aligned} \Vert z_{n+1} - z_n \Vert _{L^2_tL^2_x}&\lesssim M \kappa _n^{-1}. \end{aligned}$$

Notice that the previous bound is not uniform with respect to time. As a consequence, in this section we will iteratively control the Reynolds stress $\mathring{R}_n$ in $L^2_\mathfrak {t} L^1_x$ and the velocity $v_n$ in $L^4_\mathfrak {t} L^2_x$, in contrast with uniform-in-time controls of Sect. 3.

4.2 Iterative assumptions

Let the parameters $\lambda _n$, $\delta _n$, $r_\perp$, $r_{//}$, $\mu$, $\ell$ and $\varsigma _n$ be given by the same formulas as in Sect. 3, but with possibly different values of the parameters a, b, $\alpha$, $\beta$.

Let $\gamma _n:= 2^{-n}$ for $n \in \mathbb {N}{\setminus } \{n_0\}$ and $\gamma _{n_0} \in \{0,1\}$ be given, for some $n_0 \in \mathbb {N}$. Also, introduce a sequence of times $\tau _n:= 2^{-n}$ and assume in addition $\ell \le \varsigma _n \le \tau _{n+1}$. In this section, the following iterative assumptions shall be in force for some $C_v, C_q, C_R \in (0,\infty )$.

First, an assumption on the $L_x^2$ and $W_x^{1,1}$ norms of $v_n$

$$\begin{aligned} v_n&= 0, \quad \forall t \in [-\tau _n,\tau _n \wedge \mathfrak {t}], \end{aligned}$$

(4.5)

$$\begin{aligned} { \int _{-\tau _n}^\mathfrak {t} \Vert v_n(t) \Vert ^4_{L_x^2} dt}&\le 6 C_v^4 (n+1)^2, \end{aligned}$$

(4.6)

$$\begin{aligned} \Vert v_n(t) \Vert _{W_x^{1,1}}&\le C_v \sum _{k=0}^n \delta _{k+1}^2, \quad \forall t \in (\tau _n \wedge \mathfrak {t} , \mathfrak {t} ]. \end{aligned}$$

(4.7)

Also, a control on the derivatives

$$\begin{aligned} \Vert v_n \Vert _{C^1_{\mathfrak {t},x}} \le C_v \lambda _n^{6}, \qquad \Vert v_n \Vert _{C^1_{\mathfrak {t}}C_x^3} \le C_v \lambda _n^{12}, \end{aligned}$$

(4.8)

$$\begin{aligned} \Vert q_n \Vert _{L^\infty _\mathfrak {t} C^2_x} \le C_q \lambda _n^{12}, \qquad \Vert \mathring{R}_n \Vert _{L^\infty _\mathfrak {t} C^1_x} \le C_R \lambda _n^{20} \lambda _{n+1}^{\alpha /4}. \end{aligned}$$

(4.9)

The Reynolds stress and the pressure need to be controlled in $L^2_t L^1_x$. We have

$$\begin{aligned} \int _{-\tau _n}^{\tau _{n-1} \wedge \mathfrak {t}} \Vert \mathring{R}_n(t) \Vert _{L_x^1}^2 dt&\le C_R^2 (n+1)^2 \end{aligned}$$

(4.10)

$$\begin{aligned} \int _{\tau _{n-1} \wedge \mathfrak {t}}^{\mathfrak {t}} \Vert \mathring{R}_n(t) \Vert _{L_x^1}^2 dt&\le C_R^2 \delta _{n+1}^2, \end{aligned}$$

(4.11)

$$\begin{aligned} \int _{-\tau _n}^{\mathfrak {t}} \Vert q_n(t) \Vert _{L_x^1}^2 dt&\le C_q^2 \lambda _n^{1/500}. \end{aligned}$$

(4.12)

Finally, the estimates on the divergence

$$\begin{aligned} {\Vert \mathord {\textrm{div}}^{\phi _n}v_n \Vert _{L^4_\mathfrak {t} H_x^{-1}}} \le C_v \delta _{n+2}^3, \qquad \Vert \mathord {\textrm{div}}^{\phi _n}v_n \Vert _{L^\infty _\mathfrak {t} L_x^1} \le C_v \delta _{n+2}^3. \end{aligned}$$

(4.13)

Proposition 4.2

There exist a choice of parameters $a,b,\alpha ,\beta$ and a constant $C_e \in (0,\infty )$ with the following property. Let $(v_n,z_n,q_n,\phi _n,\mathring{R}_n)$, $n \in \mathbb {N}$ be a solution of (4.4) with $z_n = \Pi _{\le \kappa _n} z$ and $\phi _n$ given by (2.3), and satisfying the inductive estimates (4.5) to (4.13). Then there exists a quintuple $(v_{n+1},z_{n+1},q_{n+1},\phi _{n+1},\mathring{R}_{n+1})$, solution of (4.4) with $z_{n+1} = \Pi _{\le \kappa _{n+1}} z$ and $\phi _{n+1}$ given by (2.3), satisfying the same inductive estimates with n replaced by $n+1$ and such that

$$\begin{aligned} \int _{\tau _{n-2} \wedge \mathfrak {t}}^\mathfrak {t} \Vert v_{n+1}(t)-v_n(t) \Vert _{L^2_x}^4 dt&\le C_v^4 \gamma _{n+1}^2, \end{aligned}$$

(4.14)

$$\begin{aligned} \int _{\tau _{n+1} \wedge \mathfrak {t}}^{\tau _{n-2} \wedge \mathfrak {t}} \Vert v_{n+1}(t)-v_n(t) \Vert _{L^2_x}^4 dt&\le C_v^4 (n+1)^2, \end{aligned}$$

(4.15)

$$\begin{aligned} \Vert v_{n+1}(t)-v_n(t) \Vert _{L_x^2}&= 0, \quad \forall t \in [-\tau _{n+1}, \tau _{n+1} \wedge \mathfrak {t}], \end{aligned}$$

(4.16)

$$\begin{aligned} \Vert v_{n+1}(t)-v_n(t) \Vert _{W_x^{1,1}}&\le C_v \delta _{n+2}^2, \quad \forall t \in [-\tau _{n+1}, \mathfrak {t}], \end{aligned}$$

(4.17)

$$\begin{aligned} \int _{\tau _{n-2} \wedge \mathfrak {t}}^\mathfrak {t} | \Vert v_{n+1} (t)\Vert _{L_x^2}^2 -\Vert v_n (t)\Vert _{L_x^2}^2 - 3 \gamma _{n+1} |\, dt&\le C_e \delta _{n+1}. \end{aligned}$$

(4.18)

Proof of Theorem 1.1

We apply Proposition 4.2 iteratively, starting the iteration from $v_0 = q_0 = 0$, $\mathring{R}_0 = z_0 \otimes z_0$. Recall that by definition $z_0 = \Pi _{\le 1} z$, so that $\Vert \mathring{R}_0 \Vert _{L^\infty _\mathfrak {t}C^1_x} \lesssim \Vert u_0\Vert _{L_x^2}^2 \le M^2$ and the iterative assumptions hold true. Assumptions (4.14)-(4.15)-(4.16) (resp. (4.17)) guarantee that $\{ v_n\}_{n \in \mathbb {N}}$ is a Cauchy sequence in $L^p_\mathfrak {t} L^2_x$ for every $p < 4$ (resp. in $C_\mathfrak {t}W^{1,1}_x$). Denote $v:= \lim _{n \rightarrow \infty } v_n$, which satisfies $v|_{t=0} = 0$ by (4.5) and is progressively measurable. By (4.10) and (4.11) we deduce $\mathring{R}_n \rightarrow 0$ in $L^1_{\mathfrak {t},x}$. Taking into account also (4.13) and $z_n \rightarrow z$ in $L^2_{\mathfrak {t},x}$ we deduce that v is a solution of (4.3) on the time interval $[0,\mathfrak {t}]$. The regularity claimed in the statement of the theorem holds true for $u = (v+z) \circ \phi ^{-1}$ since with probability one $v \in L^p_\mathfrak {t} L^2_x \cap C_\mathfrak {t}W^{1,1}_x$ for every $p<4$ and $z \in L^\infty _\mathfrak {t} L^2_x \cap L^2_\mathfrak {t} H^1_x \cap C^{1/4}_\mathfrak {t} H^{-1}_x$. General time intervals $[0,\mathfrak {t}_L]$ are dealt with including a factor $L_n=L^{m^{n+1}}$ in the iterative assumptions, $m<b$.

Next, we prove non-uniqueness of solutions. Choose $n_0 \ge 4$ such that

$$\begin{aligned} \int _{\tau _{n_0-3} \wedge \mathfrak {t}}^\mathfrak {t} | \Vert v_{n_0}(t)\Vert ^2_{L_x^2} - \Vert v_{n_0-1}(t)\Vert ^2_{L_x^2} - 3\gamma _{n_0} | dt \le 1/3. \end{aligned}$$

This distinguishes the two solutions $v^1,v^2$ obtained imposing $\gamma _{n_0}=1$ and $\gamma _{n_0}=0$ respectively. Indeed, $\mathfrak {t} > 1$ with probability at least 1/2, and thus the length of the time interval $[\tau _{n_0-3} \wedge \mathfrak {t}, \mathfrak {t}]$ is at least 1/2 (since $n_0 \ge 4$) with probability at least 1/2. Thus it must be

$$\begin{aligned} \left| \int _{\tau _{n_0-3} \wedge \mathfrak {t}}^{\mathfrak {t}} \left( \Vert v^1(t)\Vert ^2_{L_x^2} - \Vert v^1_{n_0-1}(t)\Vert ^2_{L_x^2} - 3 \right) dt \right|&\le 1/3 + |\mathfrak {t}| \sum _{n> n_0} \gamma _n \le 2/3, \\ \left| \int _{\tau _{n_0-3} \wedge \mathfrak {t}}^{\mathfrak {t}} \left( \Vert v^2(t)\Vert ^2_{L_x^2} - \Vert v^2_{n_0-1}(t)\Vert ^2_{L_x^2} \right) dt \right|&\le 1/3 + |\mathfrak {t}| \sum _{n > n_0} \gamma _n \le 2/3, \end{aligned}$$

where we used $\mathfrak {t} \le 2$ almost surely. Since $v^1_{n_0-1}=v^2_{n_0-1}$ we have

$$\begin{aligned}&\int _{\tau _{n_0-3} \wedge \mathfrak {t}}^{\mathfrak {t}} \Vert v^1(t)\Vert ^2_{L_x^2} dt \ge -2/3 + 3/2 + \int _{\tau _{n_0-3} \wedge \mathfrak {t}}^{\mathfrak {t}} \Vert v^1_{n_0-1}(t)\Vert ^2_{L_x^2} dt \ge -4/3 + 3/2 \\&+ \int _{\tau _{n_0-3} \wedge \mathfrak {t}}^{\mathfrak {t}} \Vert v^2(t)\Vert ^2_{L_x^2} dt \end{aligned}$$

with probability at least 1/2. $\square$

4.3 Convex integration scheme

In the remainder of the section we prove Proposition4.2. Since the construction is similar to that of Sect. 3, many estimates will be similar to those already seen before, and they will be omitted when possible.

4.3.1 Perturbation of the velocity

Following [18], define

$$\begin{aligned} \rho&:= 2 \sqrt{\ell ^2 + |\mathring{R}_\ell |^2} + \frac{\gamma _{n+1}}{(2\pi )^3}. \end{aligned}$$

The amplitude functions $\textbf{a}_\xi$ and $a_\xi$ are defined accordingly via the formula (3.8), as well as the perturbations $w^{(p)}_{n+1}$, $w^{(c)}_{n+1}:= w^{(c,1)}_{n+1} + w^{(c,2)}_{n+1}$ and $w^{(t)}_{n+1}$, formulas (3.9), (3.11), (3.12), and (3.13) respectively. We point out that, although we use the same symbols as in Sect. 3, here the objects are different because $\rho$ and $\mathring{R}_n$ are different. Then, given a smooth non-decreasing cut-off $\chi :\mathbb {R}\rightarrow [0,1]$ identically equal to 0 on $(-\infty ,\tau _{n+1}]$ and to 1 on $[\tau _n, \infty )$, define

$$\begin{aligned} {\tilde{w}}^{(p)}_{n+1} := {w}^{(p)}_{n+1} \chi , \quad {\tilde{w}}^{(c)}_{n+1} := {w}^{(c)}_{n+1} \chi , \quad {\tilde{w}}^{(t)}_{n+1} := {w}^{(t)}_{n+1} \chi ^2. \end{aligned}$$

4.3.2 Estimates on the velocity

By definition we have for every $t \le \mathfrak {t}$

$$\begin{aligned} \Vert \rho (t) \Vert _{L_x^1}&\lesssim \ell + \Vert \mathring{R}_\ell (t) \Vert _{L_x^1} + \gamma _{n+1}, \end{aligned}$$

(4.19)

$$\begin{aligned} \Vert a_\xi (t)\Vert _{L_x^2}&= \Vert \textbf{a}_\xi (t)\Vert _{L_x^2} \lesssim \Vert \rho (t)\Vert _{L_x^1}^{1/2}, \end{aligned}$$

(4.20)

with universal implicit constant. By Sobolev embedding $W^{5,1}_{\mathfrak {t},x} \subset C_{\mathfrak {t},x}$ and the bounds on $\mathring{R}_n$ in $L^2_\mathfrak {t} L^1_x$ given by (4.10) and (4.11) then we have

$$\begin{aligned} \Vert a_\xi \Vert _{C_\mathfrak {t} C_x}^2&= \Vert \textbf{a}_\xi \Vert _{C_\mathfrak {t}C_x}^2 \lesssim \Vert \rho \Vert _{C_\mathfrak {t}C_x} \lesssim \ell ^{-5} (n+1), \end{aligned}$$

(4.21)

whereas by (3.25) and (3.34) therein we have for every $N = 1,2,...10$ and $M=0,1$

$$\begin{aligned} \Vert \rho \Vert _{C^N_{\mathfrak {t},x}}&\lesssim \nonumber \ell ^{2-8N} (n+1), \\ \Vert a_\xi \Vert _{C^M_{\mathfrak {t}}C^N_x}&\lesssim \varsigma _{n+1}^{-M} \Vert \textbf{a}_\xi \Vert _{C^{M+N}_{\mathfrak {t},x}} \lesssim \varsigma _{n+1}^{-M} \ell ^{-10-8(M+N)} (n+1)^{1/2}. \end{aligned}$$

(4.22)

The different exponents of $\ell$ in the previous lines with respect to those in Sect. 3 are due to the fact that our iterative assumption on $\mathring{R}_n$ is a bound in $L^2_\mathfrak {t}L^1_x$ instead of $C_\mathfrak {t}L^1_x$.

Let us now give estimates for the velocity increments. For $t \in [-\tau _{n+1},\tau _{n+1} \wedge \mathfrak {t}]$ we have ${\tilde{w}}_{n+1} (t) \equiv 0$ because of the cut-off, and therefore (4.16) holds.

The principal part of the perturbation ${\tilde{w}}^{(p)}_{n+1}$ is controlled in $L^2_x$ as follows. By (4.19), (4.20), (4.22) and [3, Lemma 7.4] one has for every $t \le \mathfrak {t}$

$$\begin{aligned} \Vert {\tilde{w}}_{n+1}^{(p)} (t)\Vert _{L_x^2} = \Vert ({\tilde{w}}_{n+1}^{(p)}\circ \phi _{n+1}^{-1}) (t)\Vert _{L_x^2} \lesssim \Vert \mathring{R}_{\ell }(t) \Vert _{L^1_x}^{1/2} + \gamma _{n+1}^{1/2}. \end{aligned}$$

In particular, by assumptions (4.10) and (4.11) it holds for some universal $C_v \in (0,\infty )$

$$\begin{aligned} \int _{\tau _{n-2} \wedge \mathfrak {t}}^\mathfrak {t} \Vert {\tilde{w}}_{n+1}^{(p)}(t) \Vert _{L^2_x}^4 dt&\le \frac{C_v^4}{2} \gamma _{n+1}^2, \\ \int _{\tau _{n+1} \wedge \mathfrak {t}}^{\tau _{n-2} \wedge \mathfrak {t}} \Vert {\tilde{w}}_{n+1}^{(p)}(t) \Vert _{L^2_x}^4 dt&\le \frac{C_v^4}{2} (n+1)^2. \end{aligned}$$

On the other hand, using (4.21) we have for $p \in (1,\infty )$ and $t \le \mathfrak {t}$ (recall that $\delta _{n+1} \lesssim \gamma _{n+1} \lesssim n+1$ for every n)

$$\begin{aligned} \Vert {\tilde{w}}^{(p)}_{n+1}(t) \Vert _{L_x^p}&\lesssim (n+1)^{1/2} \ell ^{-10} r_\perp ^{2/p-1} r_{//}^{1/p-1/2}, \\ \Vert {\tilde{w}}^{(c,2)}_{n+1}(t) \Vert _{L_x^p}&\lesssim (n+1)^{1/2} \ell ^{-26} r_\perp ^{2/p} r_{//}^{1/p-3/2}, \\ \Vert {\tilde{w}}^{(t)}_{n+1}(t) \Vert _{L_x^p}&\lesssim (n+1)\, \ell ^{-20} r_\perp ^{2/p-1} r_{//}^{1/p-2} \lambda _{n+1}^{-1}. \end{aligned}$$

Finally, $\Vert w^{(c,1)}_{n+1}\Vert _{L^4_\mathfrak {t}L_x^2} \lesssim \Vert \mathord {\textrm{div}}^{\phi _n}v_n \Vert _{L^4_\mathfrak {t}H_x^{-1}} \lesssim \delta _{n+2}^3$. Thus (4.14) and (4.15) hold true taking $p=2$ above and a sufficiently large.

Also, arguing as in Sect. 3 it is easy to check (4.17) and (4.8) (all the additional factors $\ell ^{-1}$, as well as the time derivative of the cut-off $|\chi '|\lesssim 2^n$ can be absorbed into some positive power of $\lambda _{n+1}$). Moreover, the estimates on the divergence (4.13) descend from the bounds on $\Vert v_n\Vert _{L^4_\mathfrak {t}L_x^2}$, $\Vert v_n\Vert _{L^\infty _\mathfrak {t}W_x^{1,1}}$ as in Sect. 3, and are omitted.

4.3.3 Estimate on the energy

The energy estimate (4.18), contrary to the corresponding assumption (3.2) of Sect. 3, does not provide a prescribed energy profile for the velocity field; instead, it serves to quantify how much energy is pumped into the system with the perturbation ${\tilde{w}}_{n+1}$. Incidentally, tuning the parameter $\gamma _{n_0}$ as in the proof of Theorem 1.1, this gives non-uniqueness of solutions.

Let $t \in [0,\mathfrak {t}]$ be given. By the same computations as in Sect. 3, we have

$$\begin{aligned} | \Vert v_{n+1}(t)\Vert ^2_{L_x^2} - \Vert v_n(t)\Vert ^2_{L_x^2} - 3\gamma _{n+1} | \lesssim \delta _{n+1} + \Vert \mathring{R}_\ell (t)\Vert _{L_x^1}. \end{aligned}$$

Therefore, since we are assuming $\mathfrak {t} \le 2$ almost surely and using (4.11), we obtain

$$\begin{aligned} \int _{\tau _{n-2} \wedge \mathfrak {t}}^\mathfrak {t} | \Vert v_{n+1}(t)\Vert ^2_{L_x^2} - \Vert v_n(t)\Vert ^2_{L_x^2} - 3\gamma _{n+1} | dt \lesssim \delta _{n+1}, \end{aligned}$$

where the implicit constant is universal (denoted $C_e$ in (4.18)).

4.3.4 The pressure $q_{n+1}$ and Reynolds stress $\mathring{R}_{n+1}$

We shall define the new pressure as

$$\begin{aligned} q_{n+1} = q_\ell - ( \rho + P \circ \phi _{n+1} )\chi ^2 , \end{aligned}$$

where P has zero space average and is given implicitly by

$$\begin{aligned} \nabla P = \frac{1}{\mu } \mathcal {Q} \left( \Pi _{\ne 0} \sum _{\xi \in \Lambda } \partial _t \left( {a}_{\xi }^2 \theta _{\xi }^2 \psi _{\xi }^2 \xi \right) \right) . \end{aligned}$$

Let us recover the expression for the Reynolds stress at level $n+1$. Denote for simplicity $G_n:= (v_n+z_n) \otimes (v_n+z_n) + q_n Id - \mathring{R}_n$ and $G_\ell := G_n *\chi _\ell$. It holds

$$\begin{aligned} \partial _t v_{n+1}&+ \mathord {\textrm{div}}^{\phi _{n+1}}\left( (v_{n+1}+z_{n+1}) \otimes (v_{n+1}+z_{n+1}) \right) + \nabla ^{\phi _{n+1}}q_{n+1} - \Delta ^{\hspace{-0.05cm}\phi _{n+1}}v_{n+1} \\&= \nonumber \underbrace{\left[ \chi \partial _t w^{(p+c)}_{n+1} + \mathord {\textrm{div}}^{\phi _{n+1}}({\tilde{w}}_{n+1} \otimes (v_\ell +z_{n+1}) + (v_\ell +z_{n+1}) \otimes {\tilde{w}}_{n+1} ) - \Delta ^{\hspace{-0.05cm}\phi _{n+1}}{\tilde{w}}_{n+1} \right] }_{=\,linear\, error} \\&\quad + \nonumber \underbrace{\left[ \chi ^2 \mathord {\textrm{div}}^{\phi _{n+1}}\left( w^{(p)}_{n+1} \otimes w^{(p)}_{n+1} +\mathring{R}_\ell \right) + \chi ^2 \partial _t w^{(t)}_{n+1} + {\nabla ^{\phi _{n+1}}(q_{n+1}-q_\ell )}\right] }_{=\,oscillation\, error} \\&\quad + \nonumber \underbrace{\left[ w^{(p+c)}_{n+1}\chi ' + 2 w^{(t)}_{n+1} \chi \chi ' + (1-\chi ^2) \mathord {\textrm{div}}^{\phi _{n+1}}\mathring{R}_\ell \right] }_{=\,cut-off\, error} \\&\quad + \nonumber \underbrace{\left[ \mathord {\textrm{div}}^{\phi _{n+1}}((v_\ell +z_\ell ) \otimes (v_\ell +z_\ell ) - ((v_n+z_n) \otimes (v_n+z_n)) *\chi _\ell )\right] }_{=\,mollification\, error\,I} \\&\quad + \nonumber \underbrace{\left[ \chi _\ell *\Delta ^{\hspace{-0.05cm}\phi _n}v_n - \Delta ^{\hspace{-0.05cm}\phi _n}v_\ell + \mathord {\textrm{div}}^{\phi _n}G_\ell - \left( \mathord {\textrm{div}}^{\phi _n}G_n \right) *\chi _\ell \right] }_{=\,mollification\, error\, II} \\&\quad + \nonumber \underbrace{\left[ (\Delta ^{\hspace{-0.05cm}\phi _n}- \Delta ^{\hspace{-0.05cm}\phi _{n+1}})\, v_\ell +\left( \mathord {\textrm{div}}^{\phi _{n+1}}-\mathord {\textrm{div}}^{\phi _n}\right) G_\ell \right] }_{=\,flow\, error} \\&\quad + \nonumber \underbrace{\left[ \mathord {\textrm{div}}^{\phi _{n+1}}( {\tilde{w}}^{(p)}_{n+1} \otimes {\tilde{w}}^{(c+t)}_{n+1} + {\tilde{w}}^{(c+t)}_{n+1} \otimes {\tilde{w}}_{n+1}) \right] }_{=\,corrector\, error} \\&\quad + \nonumber \underbrace{\left[ \mathord {\textrm{div}}^{\phi _{n+1}}( (z_{n+1}-z_\ell ) \otimes (v_\ell +z_{n+1}) + (v_\ell +z_\ell ) \otimes (z_{n+1}-z_\ell )) \right] }_{=\,corrector \, error\,II}. \end{aligned}$$

Then, the estimates for $\mathring{R}_{n+1}$ and $q_{n+1}$ descend as in Sect. 3. More precisely, first one proves (4.9) using the construction of $q_{n+1}$ and the equation satisfied by $\mathring{R}_{n+1}$, plus the estimates on the velocity and $\Vert z_{n+1}\Vert _{L^\infty _\mathfrak {t} C_x^2} \lesssim M \kappa _{n+1}^{-4} = M \lambda _{n+2}^{\alpha /4}$. Notice that the presence of $z_{n+1}$ is the reason why we have the additional factor $\lambda _{n+2}^{\alpha /4}$ in the $L^\infty _\mathfrak {t} C_x^1$ norm of $\mathring{R}_{n+1}$.

Then, the bounds (4.10) and (4.11) are recovered with the usual approach, with only minor differences:

• The many factors $(n+1)$ in the iterative estimates clearly play no role, since $\delta _n$, $\lambda _n^{-1}$ decay exponentially fast. Moreover, the additional $\lambda _{n+2}^{\alpha /4}$ in the $L^\infty _\mathfrak {t} C_x^1$ norm of $\mathring{R}_{n+1}$ does not affect the estimate in $L^2_\mathfrak {t} L^1_x$, since the former only enters in the mollification error and still $\ell ^{1/4} \lambda _n^{20} \lambda _{n+1}^{\alpha /4} = \ell ^{-\alpha /8} \lambda _n^{-5} \lesssim \delta _{n+2}^2$ can be made arbitrary small.

• The term involving the increment $z_{n+1}-z_\ell$ can be bound in $L^2_{\mathfrak {t},x}$ observing that

  $$\begin{aligned} \Vert z_{n+1} - z_n \Vert _{L^2_\mathfrak {t}L^2_x}&\lesssim M \kappa _n^{-1} = M \lambda _{n+1}^{-\alpha /16} \lesssim \delta _{n+2}^2, \\ \Vert z_n - z_\ell \Vert _{L^2_\mathfrak {t}L^2_x}&\lesssim \ell ^{1/4} \Vert z_n\Vert _{C^{1/4}_{\mathfrak {t},x}} \lesssim M \ell ^{1/4} \kappa _n^{3} \lesssim \delta _{n+2}^2. \end{aligned}$$

• The new cut-off error

  $$\begin{aligned} \mathcal {R}^{\phi _{n+1}} \left( w^{(p+c)}_{n+1}\chi ' + 2 w^{(t)}_{n+1} \chi \chi ' + (1-\chi ^2) \mathord {\textrm{div}}^{\phi _{n+1}}\mathring{R}_\ell \right) \end{aligned}$$

  is such that the perturbations $w^{(p+c)}_{n+1}$ and $w^{(t)}_{n+1}$ are sufficiently small in $L^2_\mathfrak {t} L_x^p$, $p>1$ close to one, to compensate for the additional factor $2^n$ coming from $|\chi '|$. Finally, the remaining $\mathcal {R}^{\phi _{n+1}} (1-\chi ^2) \mathord {\textrm{div}}^{\phi _{n+1}}\mathring{R}_\ell = (1-\chi ^2) \mathring{R}_\ell$ gives the factor $n+1$ for $t \le \tau _{n}$.

To conclude, (4.12) is deduced by arguments similar to those of Sect. 3.