1 Introduction

In this paper we consider the question of non-uniquness of weak solutions to the 3D Navier–Stokes equations with fractional viscosity (FVNSE) on \(\mathbb {T}^3\)

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t v + \nabla \cdot (v \otimes v) + \nabla p + \nu (- \Delta )^{\theta } v = 0,\\ \nabla \cdot v = 0, \end{array}\right. } \end{aligned}$$
(1)

where \(\theta \in \mathbb {R}\) is a fixed constant, and for \(u \in C^{\infty }(\mathbb {T}^3)\) with \(\int _{\mathbb {T}^3} u(x) dx =0\), the fractional Laplacian is defined via the Fourier transform as

$$\begin{aligned} \mathcal {F}((- \Delta )^{\theta } u)(\xi ) = |\xi |^{2\theta }\mathcal {F}(u)(\xi ), \quad \xi \in \mathbb {Z}^3. \end{aligned}$$

Definition

(weak solutions) A vector field \(v \in C^0_{weak}(\mathbb {R};L^2(\mathbb {T}^3))\) is called a weak solution to the FVNSE if it solves (1) in the sense of distribution.

When \(\theta = 1\), FVNSE (1) is the standard Navier–Stokes equations. Lions first considered FVNSE (1) in [20], and showed the existence and uniqueness of weak solutions to the initial value problem, which also satisfied the energy equality, for \(\theta \in [5/4,\infty )\) in [21]. Moreover, an analogue of the Caffarelli–Kohn–Nirenberg [6] result was established in [18] for the FVNSE system (1), showing that the Hausdorff dimension of the singular set, in space and time, is bounded by \(5 - 4\theta \) for \(\theta \in (1,5/4)\). The existence, uniqueness, regularity and stability of solutions to the FVNSE have been studied in [17, 26, 28, 29] and references therein. Very recently, using the method of convex integration introduced in [12], Colombo et al. [8] showed the non-uniquenss of Leray weak solutions to FVNSE (1) for \(\theta \in (0,1/5)\) and for \(\theta \in (0,1/3)\) in [13].

In the recent breakthrough work [5], Buckmaster and Vicol obtained non-uniqueness of weak solutions to the three-dimensional Navier–Stokes equations. They developed a new convex integration scheme in Sobolev spaces using intermittent Beltrami flows which combined concentrations and oscillations. Later, the idea of using intermittent flows was used to study non-uniqueness for transport equations in [23,24,25] employing scaled Mikado waves, and for stationary Navier–Stokes equations in [7, 22] employing viscous eddies.

The schemes in [5, 24] are based on the convex integration framework in Hölder spaces for the Euler equations, introduced by De Lellis and Székelyhidi [12], subsequently refined in [2, 3, 10, 15], and culminated in the proof of the second half of the Onsager conjecture by Isett in [16]; also see [4] for a shorter proof. For the first half of the Onsager conjecture, see, e.g., [1, 9], and the references therein.

The main contribution of this note is to show that the results in Buckmaster–Vicol’s paper hold for FVNSE (1) for \(\theta < 5/4\):

Theorem 1

Assume that \(\theta \in [1, 5/4)\). Suppose u is a smooth divergence-free vector field, define on \(\mathbb {R}_+ \times \mathbb {T}^3\), with compact support in time and satisfies the condition

$$\begin{aligned} \int _{\mathbb {T}^3} u(t,x)dx \equiv 0. \end{aligned}$$

Then for any given \(\varepsilon _0 > 0\), there exists a weak solution v to the FVNSE (1), with compact support in time, satisfying

$$\begin{aligned} \Vert v - u\Vert _{L^{\infty }_t W^{2\theta - 1,1}_x} < \varepsilon _0. \end{aligned}$$

As a consequence there are infinitely many weak solutions of the FVNSE (1) which are compactly supported in time; in particular, there are infinitely many weak solutions with initial values zero.

Remark 1

In the above theorem we assume that \(\theta \in [1, 5/4)\). However, using the constructions in [5] with a slightly different choice of parameters, one can actually show that Theorem 1.2 and Theorem 1.3 in [5] hold for the 3D FVNSE, i.e., there exist non-unique weak solutions \(v \in C_t^0 W_x^{\beta ,2}\), with a different \(\beta > 0\), depending on \(\theta \). However, in this paper we choose to prove a weaker result, Theorem 1, in order to simplify the presentation while retaining the main idea.

Remark 2

For the case \(\theta \in (-\infty ,1)\), the same construction also yields weak solutions \(v \in C^0_t L^2_x \cap C^0_t W^{1,1}_x\) with a suitable choice of parameters.

We now make some comments on the analysis in this paper. Using the technique in [5], we adapt a convex integration scheme with intermittent Beltrami flows as the building blocks. The main difficulty in a convex integration scheme for (FVNSE), is the error induced by the frictional viscosity \(\nu (- \Delta )^{\theta } v\), which is greater for a larger exponent \(\theta \). This error is controlled by making full use of the concentration effect of intermittent flows introduced in [5]. As it is shown in the crucial estimate (36), the error is controllable only for \(\theta < 5/4\). Compared with [5], since our goal is to construct weak solutions \(v \in C^0_t L^2_{x,weak} \cap L^{\infty }_t W^{2\theta - 1,1}_x\), we adapt a slightly simpler cut-off function and prove only estimates that are sufficient for this purpose.

2 Outline

2.1 Iteration lemma

Following [5], we consider the approximate system

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t v + \nabla \cdot (v \otimes v) + \nabla p + \nu (- \Delta )^{\theta } v = \nabla \cdot R,\\ \nabla \cdot v = 0, \end{array}\right. } \end{aligned}$$
(2)

where R is a symmetric \(3 \times 3\) matrix.

Lemma 1

(Iteration Lemma for \(L^2\) weak solutions) Let \(\theta \in (-\infty , 5/4)\). Assume \((v_q, R_q)\) is a smooth solution to (2) with

$$\begin{aligned} \Vert R_q\Vert _{L^{\infty }_t L^1_x}&\le \delta _{q+1}, \end{aligned}$$
(3)

for some \(\delta _{q+1} > 0\). Then for any given \(\delta _{q+2} > 0\), there exists a smooth solution \((v_{q+1}, R_{q+1})\) of (2) with

$$\begin{aligned} \Vert R_{q+1}\Vert _{L^{\infty }_t L^1_x}&\le \delta _{q+2}, \end{aligned}$$
(4)
$$\begin{aligned} \text {and}\quad {\text {supp}}_t v_{q+1} \cup {\text {supp}}_t R_{q+1}&\subset N_{\delta _{q+1}}({\text {supp}}_t v_{q} \cup {\text {supp}}_t R_{q}). \end{aligned}$$
(5)

Here for a given set \(A \subset \mathbb {R}\), the \(\delta \)-neighborhood of A is denoted by

$$\begin{aligned} N_{\delta }(A) = \{ y \in \mathbb {R}: \exists y' \in A, |y-y'| < \delta \}. \end{aligned}$$

Furthermore, the increment \(w_{q+1} = v_{q+1} - v_q\) satisfies the estimates

$$\begin{aligned} \Vert w_{q+1}\Vert _{L^{\infty }_t L^2_x}&\le C \delta _{q+1}^{1/2}, \end{aligned}$$
(6)
$$\begin{aligned} \Vert w_{q+1}\Vert _{L^{\infty }_t W^{2 \theta - 1,1}_x}&\le \delta _{q+2}, \end{aligned}$$
(7)

where the positive constant C depends only on \(\theta \).

Proof of Theorem 1

Assume Lemma 1 is valid. Let \(v_0 = u\). Then

$$\begin{aligned} \int _{\mathbb {T}^3} \partial _t v_0(t,x)dx = \frac{d}{dt} \int _{\mathbb {T}^3} v_0(t,x)dx \equiv 0. \end{aligned}$$

Let

$$\begin{aligned} R_0 = \mathcal {R}(\partial _t v_0 + \nu (- \Delta )^{\theta } v_0) + v_0 \otimes v_0 + p_0 I , \quad p_0 = -\frac{1}{3} |v_0|^2, \end{aligned}$$

where \(\mathcal {R}\) is the symmetric anti-divergence operator established in Lemma 5, below. Clearly \((v_0,R_0)\) solves (2). Set

$$\begin{aligned} \delta _1&= \Vert R_0\Vert _{L^{\infty }_t L^1_x}, \\ \delta _{q+1}&= 2^{-q} \varepsilon _0, \quad \text { for } q \ge 1. \end{aligned}$$

Apply Lemma 1 iteratively to obtain smooth solution \((v_q, R_q)\) to (2). It follows from (6) that

$$\begin{aligned} \sum \Vert v_{q+1} - v_{q}\Vert _{L^{\infty }_t L^2_x} = \sum \Vert w_{q+1}\Vert _{L^{\infty }_t L^2_x} \le C \sum \delta _{q+1}^{1/2} < \infty . \end{aligned}$$

Thus \(v_q\) converge strongly to some \(v \in C^0_t L^2_x\). Since \(\Vert R_{q+1}\Vert _{L^{\infty }_t L^1_x} \rightarrow 0\), as \(q \rightarrow \infty \), v is a weak solution to the FVNSE (1). Estimate (7) leads to

$$\begin{aligned} \Vert v - v_0\Vert _{_{L^{\infty }_t W^{2\theta - 1,1}_x}} \le \sum _{q=1}^{\infty } \Vert w_q\Vert _{_{L^{\infty }_t W^{2\theta - 1,1}_x}} \le \sum _{q=1}^{\infty }\delta _{q+1} \le \varepsilon _0. \end{aligned}$$

Furthermore, it follows from (5) that

$$\begin{aligned} {\text {supp}}_t v&\subset \cup _{q \ge 0} {\text {supp}}_t v_q \subset N_{\sum _{q \ge 0} \delta _{q+1}}({\text {supp}}_t u) \subset N_{\delta _1 + \varepsilon _0}({\text {supp}}_t u). \end{aligned}$$

Now we show the existence of infinitely many weak solutions with initial values zero. Let \(u(t,x) = \varphi (t) \sum _{|k| \le N} a_k e^{ik \cdot x}\) with \(a_k \ne 0, a_k \cdot k = 0, a_{-k} = a_k^*\) for all \(|k| \le N\), and \(\varphi \in C_c^{\infty }(\mathbb {R}_+)\). Thus \(\nabla \cdot u = 0\) satisfies the conditions of the theorem. Hence there exists a weak solution v to (1) close enough to u so that \(v \not\equiv 0\). \(\square \)

3 Iteration scheme

3.1 Notations and parameters

For a complex number \(\zeta \in \mathbb {C}\), we denote by \(\zeta ^*\) its complex conjugate. Let us normalize the volume

$$\begin{aligned} |\mathbb {T}^3| = 1. \end{aligned}$$

For smooth functions \(u \in C^{\infty }(\mathbb {T}^3)\) with \(\int _{\mathbb {T}^3} u(x) dx =0\) and \(s \in \mathbb {R}\), we define

$$\begin{aligned} \mathcal {F}(|\nabla |^s u)(\xi ) = |\xi |^{s}\mathcal {F}(u)(\xi ), \quad \xi \in \mathbb {Z}^3. \end{aligned}$$

For \(M, N \in [0,+\infty ]\), denote the Fourier projection of u by

$$\begin{aligned} \mathcal {F} (\mathbb {P}_{[M,N)} u) = {\left\{ \begin{array}{ll} u(\xi ), &{}\quad M \le |\xi | < N, \xi \in \mathbb {Z}^3,\\ 0, &{}\quad \text {otherwise}. \end{array}\right. } \end{aligned}$$

We also denote \(\mathbb {P}_{\le k} = \mathbb {P}_{[0,k)}\) and \(\mathbb {P}_{\ge k} = \mathbb {P}_{[k,+\infty )}\) for \(k > 0\).

Following the notation in [5], we introduce here several parameters \(\sigma , r, \lambda \), with

$$\begin{aligned} 0< \sigma< 1< r< \lambda< \mu< \lambda ^2, \quad \sigma r < 1, \end{aligned}$$
(8)

where \(\lambda = \lambda _{q+1} \in 5\mathbb {N}\) is the ‘frequency’ parameter; \(\sigma \) with \(1/\sigma \in \mathbb {N}\) is a small parameter such that \(\lambda \sigma \in \mathbb {N}\) parameterizes the spacing between frequencies; \(r \in \mathbb {N}\) denotes the number of frequencies along edges of a cube; \(\mu \) measures the amount of temporal oscillation.

Later \(\sigma , r, \mu \) will be chosen to be suitable powers of \(\lambda _{q+1}\). We also fix a constant \(p > 1\) which will be chosen later to be close to 1. The constants implicitly in the notation ‘\(\lesssim \)’ may depend on p but are independent of the parameters \(\sigma , r, \lambda \).

3.2 Intermittent Beltrami flows

We use intermittent Beltrami flows introduced in [5] as the building blocks. Recall some basic facts of Beltrami waves.

Proposition 1

[5, Proposition 3.1] Given \(\overline{\xi } \in \mathbb {S}^2 \cap \mathbb {Q}^3\), let \(A_{\overline{\xi }} \in \mathbb {S}^2 \cap \mathbb {Q}^3\) be such that

$$\begin{aligned} A_{\overline{\xi }} \cdot \overline{\xi } = 0, \quad |A_{\overline{\xi }}| = 1, \quad A_{-\overline{\xi }} = A_{\overline{\xi }}. \end{aligned}$$

Let \(\Lambda \) be a given finite subset of \(\mathbb {S}^2\) such that \(- \Lambda = \Lambda \), and \(\lambda \in \mathbb {Z}\) be such that \(\lambda \Lambda \subset \mathbb {Z}^3\). Then for any choice of coefficients \(a_{\overline{\xi }} \in \mathbb {C}\) with \(a_{\overline{\xi }}^* = a_{-\overline{\xi }}\) the vector field

$$\begin{aligned} W(x) = \sum _{\overline{\xi } \in \Lambda } a_{\overline{\xi }} B_{\overline{\xi }} e^{i \lambda \overline{\xi } \cdot x}, \quad \text { with }\, B_{\overline{\xi }} = \frac{1}{\sqrt{2}}\left( A_{\overline{\xi }} + i \overline{\xi } \times A_{\overline{\xi }}\right) , \end{aligned}$$

is real-valued, divergence-free and satisfies

$$\begin{aligned} \nabla \times W = \lambda W, \quad \nabla \cdot (W \otimes W) = \nabla \frac{|W|^2}{2}. \end{aligned}$$

Furthermore,

$$\begin{aligned} \langle W \otimes W \rangle := \fint _{ \mathbb {T}^3} W \otimes W dx= \sum _{\overline{\xi } \in \Lambda } \frac{1}{2}|a_{(\overline{\xi })}|^2(\mathrm {Id} - \overline{\xi } \otimes \overline{\xi }). \end{aligned}$$

Let \(\Lambda , \Lambda ^+, \Lambda ^- \subset \mathbb {S}^2 \cap \mathbb {Q}^3\) be defined by

$$\begin{aligned} \Lambda ^+&= \left\{ \frac{1}{5}(3e_1 \pm 4e_2), \frac{1}{5}(3e_2 \pm 4e_3), \frac{1}{5}(3e_3 \pm 4e_1) \right\} ,\\ \Lambda ^-&= -\Lambda ^+, \quad \Lambda = \Lambda ^+ \cup \Lambda ^-. \end{aligned}$$

Clearly we have

$$\begin{aligned} 5\Lambda \in \mathbb {Z}^3, \quad \text { and } \quad \min _{\overline{\xi }', \overline{\xi } \in \Lambda , \overline{\xi }'+ \overline{\xi }\ne 0} |\overline{\xi }'+ \overline{\xi }| \ge \frac{1}{5}. \end{aligned}$$
(9)

Also it is direct to check that

$$\begin{aligned} \frac{1}{8}\sum _{\overline{\xi } \in \Lambda }(\mathrm {Id} - \overline{\xi } \otimes \overline{\xi }) = \mathrm {Id}. \end{aligned}$$

In fact, representations of this form exist for symmetric matrices close to the identity. We have the following simple variant of [5, Proposition 3.2].

Proposition 2

Let \(B_{\varepsilon }(\mathrm {Id})\) denote the ball of symmetric matrices, centered at the identity, of radius \(\varepsilon \). Then there exist a constant \(\varepsilon _{\gamma } > 0\) and smooth positive functions \(\gamma _{(\overline{\xi })} \in C^{\infty }(B_{\varepsilon _{\gamma }}(\mathrm {Id}))\), such that

  1. 1.

    \(\gamma _{(\overline{\xi })} = \gamma _{(-\overline{\xi })}\);

  2. 2.

    for each \(R \in B_{\varepsilon _{\gamma }}(\mathrm {Id})\) we have the identity

    $$\begin{aligned} R = \frac{1}{2}\sum _{\overline{\xi } \in \Lambda } \left( \gamma _{(\overline{\xi })}(R)\right) ^2(\mathrm {Id} - \overline{\xi } \otimes \overline{\xi }). \end{aligned}$$

Define the Dirichlet kernel

$$\begin{aligned} D_r(x)&= \frac{1}{(2r+1)^{3/2}} \sum _{\xi \in \Omega _r} e^{i \xi \cdot x}, \quad \Omega _r = \{(j,k,l): j,k,l \in \{-r,\ldots ,r\} \}. \end{aligned}$$

It has the property that, for \(1 < p \le \infty \),

$$\begin{aligned} \Vert D_r\Vert _{L^p} \lesssim r^{3/2 - 3/p}, \quad \Vert D_r\Vert _{L^2} = (2 \pi )^3. \end{aligned}$$

Following [5], for \(\overline{\xi } \in \Lambda ^+\), define a directed and rescaled Dirichlet kernel by

$$\begin{aligned} \eta _{(\overline{\xi })}(t,x) = \eta _{\overline{\xi }, \lambda , \sigma , r, \mu }(t,x) = D_r(\lambda \sigma (\overline{\xi } \cdot x + \mu t, A_{\overline{\xi }} \cdot x, (\overline{\xi } \times A_{\overline{\xi }}) \cdot x)), \end{aligned}$$
(10)

and for \(\overline{\xi } \in \Lambda ^-\), define

$$\begin{aligned} \eta _{(\overline{\xi })}(t,x) = \eta _{-(\overline{\xi })}(t,x). \end{aligned}$$

Note the important identity

$$\begin{aligned} \frac{1}{\mu } \partial _t \eta _{(\overline{\xi })}(t,x) = \pm (\overline{\xi } \cdot \nabla ) \eta _{(\overline{\xi })}(t,x), \quad \overline{\xi } \in \Lambda ^{\pm }. \end{aligned}$$
(11)

Since the map \(x \mapsto \lambda \sigma (\overline{\xi } \cdot x + \mu t, A_{\overline{\xi }} \cdot x, (\overline{\xi } \times A_{\overline{\xi }}) \cdot x)\) is the composition of a rotation by a rational orthogonal matrix mapping \(\{e_1, e_2, e_3\}\) to \(\{ \overline{\xi }, A_{\overline{\xi }}, \overline{\xi } \times A_{\overline{\xi }} \}\), a translation, and a rescaling by integers, for \(1 < p \le \infty \), we have

$$\begin{aligned} \fint _{ \mathbb {T}^3} \eta _{(\overline{\xi })}(t,x)^2(t,x) dx = 1, \quad \Vert \eta _{(\overline{\xi })}\Vert _{L^{\infty }_t L^p_x(\mathbb {T}^3)} \lesssim r^{3/2 - 3/p}. \end{aligned}$$

Let \(W_{(\overline{\xi })}\) be the Beltrami plane wave at frequency \(\lambda \),

$$\begin{aligned} W_{(\overline{\xi })} = W_{\overline{\xi }, \lambda }(x) = B_{\overline{\xi }} e^{i \lambda \overline{\xi } \cdot x}. \end{aligned}$$

Define the intermittent Beltrami wave \(\mathbb {W}_{(\overline{\xi })}\) as

$$\begin{aligned} \mathbb {W}_{(\overline{\xi })}(t,x) := \mathbb {W}_{\overline{\xi },\lambda ,\sigma ,r,\mu }(t,x) = \eta _{(\overline{\xi })}(t,x)W_{(\overline{\xi })}(x). \end{aligned}$$
(12)

It follows from the definitions and (9) that

$$\begin{aligned} \mathbb {P}_{[\frac{\lambda }{2}, 2 \lambda )} \mathbb {W}_{(\overline{\xi })}&= \mathbb {W}_{(\overline{\xi })}, \end{aligned}$$
(13)
$$\begin{aligned} \mathbb {P}_{[\frac{\lambda }{5}, 4 \lambda )} \Big (\mathbb {W}_{(\overline{\xi })} \otimes \mathbb {W}_{(\overline{\xi }')}\Big )&= \mathbb {W}_{(\overline{\xi })} \otimes \mathbb {W}_{(\overline{\xi }')}, \quad \overline{\xi }' \ne -\overline{\xi } . \end{aligned}$$
(14)

The following properties are immediate from the definitions.

Proposition 3

[5, Proposition 3.4] Let \(a_{\overline{\xi }} \in \mathbb {C}\) be constants with \(a_{\overline{\xi }}^* = a_{-\overline{\xi }}\). Let

$$\begin{aligned} W(x) = \sum _{\overline{\xi } \in \Lambda } a_{\overline{\xi }} \mathbb {W}_{(\overline{\xi })}(x). \end{aligned}$$

Then W(x) is real valued. Moreover, for each \(R \in B_{\varepsilon _{\gamma }}(\mathrm {Id})\) we have

$$\begin{aligned} \sum _{\overline{\xi } \in \Lambda } \left( \gamma _{(\overline{\xi })}(R)\right) ^2 \fint _{ \mathbb {T}^3} \mathbb {W}_{(\overline{\xi })} \otimes \mathbb {W}_{(-\overline{\xi })} = \sum _{\overline{\xi } \in \Lambda } \left( \gamma _{(\overline{\xi })}(R)\right) ^2 B_{\overline{\xi }} \otimes B_{-\overline{\xi }} = R. \end{aligned}$$

Proposition 4

[5, Proposition 3.5] For any \(1 < p \le \infty , N \ge 0, K \ge 0\):

$$\begin{aligned} \left\| \nabla ^N \partial _t^K \mathbb {W}_{(\overline{\xi })} \right\| _{L^{\infty }_t L^p_x}&\lesssim \lambda ^N (\lambda \sigma r \mu )^K r^{3/2 - 3/p}, \end{aligned}$$
(15)
$$\begin{aligned} \left\| \nabla ^N \partial _t^K \eta _{(\overline{\xi })} \right\| _{L^{\infty }_t L^p_x}&\lesssim (\lambda \sigma r)^N (\lambda \sigma r \mu )^K r^{3/2 - 3/p}. \end{aligned}$$
(16)

3.3 Perturbations

Let \(\psi (t)\) be a smooth cut-off function such that

$$\begin{aligned} \psi (t) = 1 \text { on } {\text {supp}}_t R_q,\quad {{\,\mathrm{supp}\,}}\psi (t) \subset N_{\delta _{q+1}}({\text {supp}}_t R_q), \quad |\psi '(t)| \le 2 \delta _{q+1}^{-1}. \end{aligned}$$
(17)

Take a smooth increasing function \(\chi \) such that

$$\begin{aligned} \chi (s) = {\left\{ \begin{array}{ll} 1, &{}\quad 0 \le s < 1\\ s, &{}\quad s \ge 2 \end{array}\right. }, \end{aligned}$$

and set

$$\begin{aligned} \rho (t,x) = \varepsilon _{\gamma }^{-1}\delta _{q+1} \chi \left( \delta _{q+1}^{-1}|R_q(t,x)|\right) \psi ^2(t). \end{aligned}$$

where \(\varepsilon _{\gamma }\) is the constant in Proposition 2. Then clearly

$$\begin{aligned} {\text {supp}}_t \rho&\subset N_{\delta _{q+1}}({\text {supp}}_t R_q). \end{aligned}$$
(18)

It follows from the above definition that

$$\begin{aligned}&|R_q|/\rho = \varepsilon _{\gamma }\frac{|R_q|}{\delta _{q+1} \chi \left( \delta _{q+1}^{-1}|R_q(t,x)|\right) \psi ^2} \le \varepsilon _{\gamma } \implies \mathrm {Id} - R_q/\rho \in B_{\varepsilon _{\gamma }}(\mathrm {Id}) \text { on } {{\,\mathrm{supp}\,}}R_q. \end{aligned}$$

Therefore, the amplitude functions

$$\begin{aligned} a_{(\overline{\xi })}(t,x) := \rho ^{1/2}(t,x) \gamma _{(\overline{\xi })}(\mathrm {Id} - \rho (t,x)^{-1}R_q(t,x) ) \end{aligned}$$

are well-defined and smooth. Define the velocity perturbation to be \(w = w_{q+1}\):

$$\begin{aligned} w&= w^{(p)} + w^{(c)} + w^{(t)}, \\ w^{(p)}&= \sum _{\overline{\xi } \in \Lambda } a_{(\overline{\xi })} \mathbb {W}_{(\overline{\xi })} = \sum _{\overline{\xi } \in \Lambda } a_{(\overline{\xi })}(t,x) \eta _{(\overline{\xi })}(t,x) B_{\overline{\xi }} e^{i \lambda \overline{\xi } \cdot x},\\ w^{(c)}&= \frac{1}{\lambda _{q+1}} \sum _{\overline{\xi } \in \Lambda } \nabla \left( a_{(\overline{\xi })}\eta _{(\overline{\xi })} \right) \times W_{(\overline{\xi })},\\ w^{(t)}&= \frac{1}{\mu }\sum _{\overline{\xi } \in \Lambda ^+} \mathbb {P}_{LH} \mathbb {P}_{\ne 0}\left( a_{(\overline{\xi })}^2 \eta _{(\overline{\xi })}^2 \overline{\xi } \right) , \end{aligned}$$

where \(\mathbb {P}_{LH} = \mathrm {Id} - \nabla \Delta ^{-1} {\text {div}}\) is the Leray-Helmholtz projection into divergence-free vector field, and \(\mathbb {P}_{\ne 0} f = f - \fint _{ \mathbb {T}^3} f dx\). It is well-known that \(\mathbb {P}_{LH}\) is bounded on \(L^p, 1< p < \infty \) (see, e.g., [14]). It follows from Proposition 3 that

$$\begin{aligned} \sum _{\overline{\xi } \in \Lambda } a_{(\overline{\xi })}^2 \fint _{ \mathbb {T}^3} \mathbb {W}_{(\overline{\xi })} \otimes \mathbb {W}_{(-\overline{\xi })} dx&= \rho \mathrm {Id} - R_q. \end{aligned}$$
(19)

3.4 Estimates for perturbations

Lemma 2

The following bounds hold:

$$\begin{aligned} \Vert \rho \Vert _{L^{\infty }_t L^1_x}&\le C \delta _{q+1}, \end{aligned}$$
(20)
$$\begin{aligned} \Vert \rho ^{-1}\Vert _{C^0({{\,\mathrm{supp}\,}}R_q)}&\lesssim \delta _{q+1}^{-1}, \end{aligned}$$
(21)
$$\begin{aligned} \Vert \rho \Vert _{C^N_{t,x}}&\le C(\delta _{q+1}, \Vert R_q\Vert _{C^N}), \end{aligned}$$
(22)
$$\begin{aligned} \Vert a_{(\overline{\xi })}\Vert _{L^{\infty }_t L^2_x}&\lesssim \Vert \rho \Vert _{L^{\infty }_t L^1_x}^{1/2} \lesssim \delta _{q+1}^{1/2}, \end{aligned}$$
(23)
$$\begin{aligned} \Vert a_{(\overline{\xi })}\Vert _{C^N_{t,x}}&\le C(\delta _{q+1}, \Vert R_q\Vert _{C^N}). \end{aligned}$$
(24)

Proof

It follows from (3) that

$$\begin{aligned} \Vert \rho (t,\cdot )\Vert _{L^1_x}&=\int _{|R_q| \le \delta _{q+1}} \rho + \int _{|R_q|> \delta _{q+1}} \rho \lesssim \delta _{q+1} + \int _{|R_q| > \delta _{q+1}} |R_q|\\&\le C \delta _{q+1}. \end{aligned}$$

It is direct to verify (21) and (23), while (22) and (24) follow from (17) and (21). \(\square \)

Now we can estimate the time support of \(w_{q+1}\):

$$\begin{aligned} {\text {supp}}_t w_{q+1} \subset {\text {supp}}_t \rho \subset {{\,\mathrm{supp}\,}}\psi \subset N_{\delta _{q+1}}({\text {supp}}_t R_q). \end{aligned}$$
(25)

We need the following Lemma, which is a variant of [5, Lemma 3.6].

Lemma 3

([24, Lemma 2.1]) Let \(f,g \in C^{\infty }(\mathbb {T}^3)\), and g is \((\mathbb {T}/N)^3\) periodic, \(N \in \mathbb {N}\). Then for \(1 \le p \le \infty \),

$$\begin{aligned} \Vert f g\Vert _{L^p} \le \Vert f\Vert _{L^p} \Vert g\Vert _{L^p} + C_p N^{-1/p} \Vert f\Vert _{C^1} \Vert g\Vert _{L^p}. \end{aligned}$$

Let us denote

$$\begin{aligned} \mathcal {C}_N =C\left( \sup _{\overline{\xi } \in \Lambda }\Vert a_{(\overline{\xi })}\Vert _{C^N_{t,x}}\right) \end{aligned}$$
(26)

to be some polynomials depending on \(\sup _{\overline{\xi } \in \Lambda }\Vert a_{(\overline{\xi })}\Vert _{C^N_{t,x}}\).

Lemma 4

Suppose the parameters satisfy (8) and

$$\begin{aligned} r^{3/2} \le \mu . \end{aligned}$$
(27)

Then the following estimates for the perturbations hold:

$$\begin{aligned} \left\| w_{q+1}^{(p)}\right\| _{L^{\infty }_t L^2_x}&\lesssim \delta _{q+1}^{1/2} + (\lambda _{q+1} \sigma )^{-1/2} \mathcal {C}_1, \end{aligned}$$
(28)
$$\begin{aligned} \Vert w_{q+1}\Vert _{L^{\infty }_t L^p_x}&\lesssim r^{3/2-3/p}\mathcal {C}_1, \end{aligned}$$
(29)
$$\begin{aligned} \left\| w_{q+1}^{(c)}\right\| _{L^{\infty }_t L^p_x} + \left\| w_{q+1}^{(t)}\right\| _{L^{\infty }_t L^p_x}&\lesssim (\sigma r + \mu ^{-1}r^{3/2})r^{3/2-3/p}\mathcal {C}_1, \end{aligned}$$
(30)
$$\begin{aligned} \left\| \partial _t w_{q+1}^{(p)}\right\| _{L^{\infty }_t L^p_x} + \left\| \partial _t w_{q+1}^{(c)}\right\| _{L^{\infty }_t L^p_x}&\lesssim \lambda _{q+1} \sigma \mu r^{5/2 - 3/p}\mathcal {C}_2, \end{aligned}$$
(31)
$$\begin{aligned} \Vert |\nabla |^N w_{q+1}\Vert _{L^{\infty }_t L^p_x}&\lesssim r^{3/2-3/p} \lambda _{q+1}^N \mathcal {C}_{N+1}, \end{aligned}$$
(32)

for \(1< p < \infty , N \ge 1\).

Proof

Since \(\mathbb {W}_{(\overline{\xi })}\) is \((\mathbb {T}/\lambda \sigma )^3\) periodic, it follows from (15), (23), and Lemma 3 that

$$\begin{aligned} \left\| w_{q+1}^{(p)}\right\| _{L^{\infty }_t L^2_x}&\lesssim \sum _{\overline{\xi } \in \Lambda } \left( \left\| a_{(\overline{\xi })}\right\| _{L^{\infty }_t L^2_x} + (\lambda _{q+1} \sigma )^{-1/2} \left\| a_{(\overline{\xi })}\right\| _{C^1}\right) \Vert \mathbb {W}_{(\overline{\xi })}\Vert _{L^{\infty }_t L^2_x}\\&\lesssim \delta _{q+1}^{1/2} + (\lambda _{q+1} \sigma )^{-1/2}\mathcal {C}_1. \end{aligned}$$

In view of (8), (15) and (16) yield that

$$\begin{aligned} \left\| w_{q+1}^{(p)}\right\| _{L^{\infty }_t L^p_x}&\lesssim \sum _{\overline{\xi } \in \Lambda } \left\| a_{(\overline{\xi })}\right\| _{C^0}\left\| \mathbb {W}_{(\overline{\xi })}\right\| _{L^{\infty }_t L^p_x}\lesssim r^{3/2-3/p}\mathcal {C}_0,\\ \left\| w_{q+1}^{(c)}\right\| _{L^{\infty }_t L^p_x}&\lesssim \lambda _{q+1}^{-1} \sum _{\overline{\xi } \in \Lambda } \left( \left\| \eta _{(\overline{\xi })} \right\| _{L^{\infty }_t L^p_x} + \left\| \nabla \eta _{(\overline{\xi })} \right\| _{L^{\infty }_t L^p_x}\right) \left\| a_{(\overline{\xi })}\right\| _{C^1}\left\| \mathbb {W}_{(\overline{\xi })}\right\| _{L^{\infty }_t L^p_x}\\&\lesssim (\sigma r) r^{3/2-3/p}\mathcal {C}_1,\\ \left\| w_{q+1}^{(t)}\right\| _{L^{\infty }_t L^p_x}&\lesssim \mu ^{-1}\sum _{\overline{\xi } \in \Lambda ^+} \left\| a_{(\overline{\xi })}^2 \eta _{(\overline{\xi })}^2 \overline{\xi } \right\| _{L^{\infty }_t L^p_x} \lesssim \mu ^{-1} \sum _{\overline{\xi } \in \Lambda ^+}\left\| a_{(\overline{\xi })}^2\right\| _{C^0} \left\| \eta _{(\overline{\xi })} \right\| _{L^{\infty }_t L^{2p}_x}^2 \\&\lesssim \mu ^{-1}r^{3-3/p}\mathcal {C}_0, \end{aligned}$$

where the boundedness of \(\mathbb {P}_{LH}\) and \(\mathbb {P}_{\ne 0}\) on \(L^p\), for \(1< p < \infty \), is used in the first inequality of the estimate for \(\Vert w_{q+1}^{(t)}\Vert _{L^{\infty }_t L^p_x}\). In the same way, we can estimate

$$\begin{aligned} \left\| \partial _t w_{q+1}^{(p)}\right\| _{L^{\infty }_t L^p_x}&\lesssim \sum _{\overline{\xi } \in \Lambda } \left\| \partial _t a_{(\overline{\xi })}\right\| _{C^0} \left\| \mathbb {W}_{(\overline{\xi })}\right\| _{L^{\infty }_t L^p_x} + \left\| a_{(\overline{\xi })}\right\| _{C^0} \left\| \partial _t \mathbb {W}_{(\overline{\xi })}\right\| _{L^{\infty }_t L^p_x}\\&\lesssim \lambda _{q+1} \sigma \mu r^{5/2 - 3/p} \mathcal {C}_1,\\ \left\| \partial _t w_{q+1}^{(c)}\right\| _{L^{\infty }_t L^p_x}&\lesssim \lambda _{q+1}^{-1} \sum _{\overline{\xi } \in \Lambda } \left\| a_{(\overline{\xi })}\right\| _{C^2_{t,x}} \Bigg ( \left\| \eta _{(\overline{\xi })} \right\| _{L^{\infty }_t L^p_x} + \left\| \nabla \eta _{(\overline{\xi })} \right\| _{L^{\infty }_t L^p_x} + \left\| \partial _t \eta _{(\overline{\xi })} \right\| _{L^{\infty }_t L^p_x} \\&\qquad + \left\| \partial _t \nabla \eta _{(\overline{\xi })}\right\| _{L^{\infty }_t L^p_x}\Bigg ) \lesssim \sigma r\lambda _{q+1} \sigma \mu r^{5/2 - 3/p} \mathcal {C}_2 \lesssim \lambda _{q+1} \sigma \mu r^{5/2 - 3/p}\mathcal {C}_2. \end{aligned}$$

For \(N \ge 1\), using (15) and (16), we obtain that

$$\begin{aligned} \left\| \nabla ^N w_{q+1}^{(p)}\right\| _{L^{\infty }_t L^p_x}&\lesssim \sum _{\overline{\xi } \in \Lambda } \sum _{k=0}^N \left\| \nabla ^k a_{(\overline{\xi })}\right\| _{C^0} \left\| \nabla ^{N-k} \mathbb {W}_{(\overline{\xi })}\right\| _{L^{\infty }_t L^p_x} \\&\lesssim \lambda _{q+1}^N r^{3/2 - 3/p}\mathcal {C}_N,\\ \left\| \nabla ^N w_{q+1}^{(c)}\right\| _{L^{\infty }_t L^p_x}&\lesssim \lambda _{q+1}^{-1} \sum _{\overline{\xi } \in \Lambda } \sum _{m=0}^{N}\sum _{k=0}^m \lambda _{q+1}^{N-m} \left\| \nabla ^{k+1} a_{(\overline{\xi })}\right\| _{C^0} \left\| \nabla ^{m-k} \eta _{(\overline{\xi })}\right\| _{L^{\infty }_t L^p_x} \\&\quad + \lambda _{q+1}^{-1} \sum _{\overline{\xi } \in \Lambda } \sum _{m=0}^{N}\sum _{k=0}^m \lambda _{q+1}^{N-m} \left\| \nabla ^{k} a_{(\overline{\xi })}\right\| _{C^0} \left\| \nabla ^{m-k+1} \eta _{(\overline{\xi })}\right\| _{L^{\infty }_t L^p_x}\\&\lesssim \lambda _{q+1}^N r^{3/2 - 3/p} \mathcal {C}_{N+1},\\ \left\| \nabla ^N w_{q+1}^{(t)}\right\| _{L^{\infty }_t L^p_x}&\lesssim \mu ^{-1}\sum _{\overline{\xi } \in \Lambda } \sum _{m=0}^{N} \left\| \nabla ^{N-m}\left( a_{(\overline{\xi })}^2\right) \right\| _{C^0} \sum _{k=0}^m \left\| \nabla ^{k}\eta _{(\overline{\xi })}\right\| _{L^{\infty }_t L^{2p}_x} \left\| \nabla ^{m-k}\eta _{(\overline{\xi })}\right\| _{L^{\infty }_t L^{2p}_x}\\&\lesssim \lambda _{q+1}^N r^{3/2 - 3/p}\frac{(\sigma r)^N r^{3/2}}{\mu } \mathcal {C}_N \lesssim \lambda _{q+1}^N r^{3/2 - 3/p}\mathcal {C}_N, \end{aligned}$$

where we use (8) and (27). \(\square \)

3.5 Estimates for the stress

Let us recall the following operator in [12].

Lemma 5

(symmetric anti-divergence) There exists a linear operator \(\mathcal {R}\), of order \(-1\), mapping vector fields to symmetric matrices such that

$$\begin{aligned} \nabla \cdot \mathcal {R}(u) = u - \fint _{ \mathbb {T}^3} u, \end{aligned}$$
(33)

with standard Calderon–Zygmund estimates, for \(1< p < \infty \),

$$\begin{aligned} \Vert \mathcal {R} \Vert _{L^p \rightarrow W^{1,p}} \lesssim 1, \quad \Vert \mathcal {R} \Vert _{C^0 \rightarrow C^0} \lesssim 1,\quad \Vert \mathcal {R} \mathbb {P}_{\ne 0} u \Vert _{L^p} \lesssim \Vert |\nabla |^{-1}\mathbb {P}_{\ne 0} u\Vert _{L^p}. \end{aligned}$$
(34)

Proof

Suppose \(u \in C^{\infty }(\mathbb {T}^3, \mathbb {R}^3)\) is a smooth vector field. Define

$$\begin{aligned} \mathcal {R}(u) = \frac{1}{4}\left( \nabla \mathbb {P}_{LH} v + (\nabla \mathbb {P}_{LH} v)^T\right) + \frac{3}{4}\left( \nabla v + (\nabla v)^T\right) - \frac{1}{2}(\nabla \cdot v) \mathrm {Id} \end{aligned}$$

where \(v \in C^{\infty }(\mathbb {T}^3, \mathbb {R}^3)\) is the unique solution to \(\Delta v = u - \fint _{ \mathbb {T}^3} u\) with \(\fint _{ \mathbb {T}^3} v = 0\).

It is direct to verify that \(\mathcal {R}(u)\) is a symmetric matrix field depending linearly on u and satisfies (33). Note that \(\mathcal {R}\) is a constant coefficient ellitpic operator of order \(-1\). We refer to [14] for the Calderon-Zygmund estimates \(\Vert \mathcal {R} \Vert _{L^p \rightarrow W^{1,p}} \lesssim 1\) and \(\Vert \mathcal {R} \mathbb {P}_{\ne 0} u \Vert _{L^p} \lesssim \Vert |\nabla |^{-1}\mathbb {P}_{\ne 0} u\Vert _{L^p}\). Combining these with Sobolev embeddings, we have \(\Vert \mathcal {R} u\Vert _{C^{\alpha }} \lesssim \Vert \mathcal {R} u\Vert _{W^{1,4}} \lesssim \Vert u\Vert _{L^4} \lesssim \Vert u\Vert _{C^0}\), with \(\alpha = 1/4\). \(\square \)

We have the following variant of [5, Lemma B.1] in [5].

Lemma 6

Let \(a\in C^2(\mathbb {T}^3)\). For \(1< p < \infty \), and any smooth function \(f \in L^p(\mathbb {T}^3)\), we have

$$\begin{aligned} \Vert |\nabla |^{-1}\mathbb {P}_{\ne 0}(a \mathbb {P}_{\ge k} f)\Vert _{L^p(\mathbb {T}^3)} \lesssim k^{-1} \Vert \nabla ^2 a\Vert _{L^{\infty }(\mathbb {T}^3)} \Vert f\Vert _{L^p(\mathbb {T}^3)}. \end{aligned}$$
(35)

Proof of Lemma 6

We follow the proof in [5]. Note that

$$\begin{aligned} |\nabla |^{-1}\mathbb {P}_{\ne 0}(a \mathbb {P}_{\ge k} f) = |\nabla |^{-1}\mathbb {P}_{\ge k/2}(\mathbb {P}_{\le k/2} a \mathbb {P}_{\ge k} f) + |\nabla |^{-1}\mathbb {P}_{\ne 0}(\mathbb {P}_{\ge k/2} a \mathbb {P}_{\ge k} f). \end{aligned}$$

As direct consequences of the Littlewood–Paley decomposition and Schauder estimates we have the bounds for \(1< p < \infty \) (see, for example, [14])

$$\begin{aligned} \Vert \mathbb {P}_{\le k/2}\Vert _{L^p \rightarrow L^p} \lesssim 1, \quad \Vert |\nabla |^{-1}\mathbb {P}_{\ge k/2} \Vert _{L^p \rightarrow L^p} \lesssim k^{-1}, \quad \Vert |\nabla |^{-1}\mathbb {P}_{\ne 0} \Vert _{L^p \rightarrow L^p} \lesssim 1. \end{aligned}$$

Combining these bounds with Hölder’s inequality and the embedding \(W^{1,4}(\mathbb {T}^3) \subset L^{\infty }(\mathbb {T}^3)\), we obtain

$$\begin{aligned} \Vert |\nabla |^{-1}\mathbb {P}_{\ne 0}(a \mathbb {P}_{\ge k} f)\Vert _{L^p}&\lesssim k^{-1} \Vert \mathbb {P}_{\le k/2} a \mathbb {P}_{\ge k} f \Vert _{L^p} + \Vert \mathbb {P}_{\ge k/2} a \mathbb {P}_{\ge k} f \Vert _{L^p} \\&\lesssim k^{-1} (\Vert \mathbb {P}_{\le k/2} a\Vert _{L^{\infty }} + k\Vert \mathbb {P}_{\ge k/2} a\Vert _{L^{\infty }}) \Vert f\Vert _{L^p} \\&\lesssim k^{-1} (\Vert \nabla \mathbb {P}_{\le k/2} a\Vert _{L^4} + k\Vert \nabla \mathbb {P}_{\ge k/2} a\Vert _{L^4}) \Vert f\Vert _{L^p}\\&\lesssim k^{-1} (\Vert \mathbb {P}_{\le k/2} \nabla a\Vert _{L^4} + k\Vert |\nabla |^{-1} \mathbb {P}_{\ge k/2} |\nabla |\nabla \mathbb {P}_{\ge k/2} a\Vert _{L^4}) \Vert f\Vert _{L^p} \\&\lesssim k^{-1} (\Vert \nabla a\Vert _{L^4} + \Vert \nabla ^2 \mathbb {P}_{\ge k/2} a\Vert _{L^4}) \Vert f\Vert _{L^p} \lesssim k^{-1} \Vert \nabla ^2 a\Vert _{L^4} \Vert f\Vert _{L^p}. \end{aligned}$$

\(\square \)

It follows from the definition of \(w_{q+1}\) that

$$\begin{aligned} \int _{\mathbb {T}^3} w_{q+1} dx&= \int _{\mathbb {T}^3} \frac{1}{\lambda _{q+1}} \sum _{\overline{\xi } \in \Lambda } \nabla \left( a_{(\overline{\xi })}\eta _{(\overline{\xi })} W_{(\overline{\xi })} \right) dx \\&\quad + \int _{\mathbb {T}^3} \frac{1}{\mu }\sum _{\overline{\xi } \in \Lambda ^+}{P}_{LH} \mathbb {P}_{\ne 0}\left( a_{(\overline{\xi })}^2 \eta _{(\overline{\xi })}^2 \overline{\xi } \right) dx = 0. \end{aligned}$$

Hence \(\int _{\mathbb {T}^3} \nu (-\Delta )^{\theta } w_{q+1}dx = 0\) and \(\dfrac{d}{dt} \int _{\mathbb {T}^3} w_{q+1} dx = 0\). We obtain \(R_{q+1}\) by plugging \(v_{q+1} = v_q + w_{q+1}\) in (2), using (33) and the assumption that \((v_q,R_q)\) solves (2):

$$\begin{aligned} \nabla \cdot R_{q+1}&= \nabla \cdot \left[ \mathcal {R}\left( \nu (-\Delta )^{\theta } w_{q+1} + \partial _t w_{q+1}^{(p)} + \partial _t w_{q+1}^{(c)}\right) + v_q \otimes w_{q+1} + w_{q+1} \otimes v_q \right] \\&\quad + \nabla \cdot \left[ \left( w_{q+1}^{(c)} + w_{q+1}^{(t)}\right) \otimes w_{q+1} + w_{q+1}^{(p)} \otimes \left( w_{q+1}^{(c)} + w_{q+1}^{(t)}\right) \right] \\&\quad \times \left[ \nabla \cdot \left( w_{q+1}^{(p)} \otimes w_{q+1}^{(p)} - R_{q}\right) + \partial _t w_{q+1}^{(t)}\right] + \nabla (p_{q+1} - p_q)\\&:= \nabla \cdot (\widetilde{R}_{linear} + \widetilde{R}_{corrector} + \widetilde{R}_{oscillation}) + \nabla (p_{q+1}-p_q). \end{aligned}$$

It follows from Lemma 4 that

$$\begin{aligned} \Vert \widetilde{R}_{corrector} \Vert _{L^{\infty }_t L^p_x}&\lesssim \left( \left\| w_{q+1}^{(c)}\right\| _{L^{\infty }_t L^{2p}_x} + \left\| w_{q+1}^{(t)}\right\| _{L^{\infty }_t L^{2p}_x} \right) \left( \left\| w_{q+1}\right\| _{L^{\infty }_t L^{2p}_x} + \left\| w_{q+1}^{(p)}\right\| _{L^{\infty }_t L^{2p}_x}\right) \\&\lesssim \left( \sigma r + \mu ^{-1}r^{3/2}\right) r^{3-3/p}\mathcal {C}_1. \end{aligned}$$

Noting that \(\nabla \times \dfrac{w_{q+1}^{(p)}}{\lambda _{q+1}} = w_{q+1}^{(p)} + w_{q+1}^{(c)}\), Lemma 4 and (34) yield that

$$\begin{aligned}&\Vert \widetilde{R}_{linear}\Vert _{L^{\infty }_t L^p_x} \nonumber \\&\quad \lesssim \lambda _{q+1}^{-1}\left\| \partial _t \mathcal {R} \nabla \times (w_{q+1}^{(p)})\right\| _{L^{\infty }_t L^p_x} + \left\| \mathcal {R}\left( \nu (-\Delta )^{\theta }w_{q+1}\right) \right\| _{L^{\infty }_t L^p_x} \nonumber \\&\qquad + \Vert v_q \otimes w_{q+1} + w_{q+1} \otimes v_q\Vert _{L^{\infty }_t L^p_x} \nonumber \\&\quad \lesssim \lambda _{q+1}^{-1}\left\| \partial _t w_{q+1}^{(p)}\right\| _{L^{\infty }_t L^p_x} + \left\| |\nabla |^{2\theta - 1} w_{q+1}\right\| _{L^{\infty }_t L^p_x} + \Vert v_q\Vert _{C^0} \Vert w_{q+1}\Vert _{L^{\infty }_t L^p_x}\nonumber \\&\quad \lesssim \sigma \mu r^{5/2-3/p} \mathcal {C}_2 + r^{3/2-3/p}\left( \lambda _{q+1}^{2\theta -1} + \Vert v_q\Vert _{C^0}\right) \mathcal {C}_3. \end{aligned}$$
(36)

This is the crucial estimate to control the fractional viscosity. If we assume that \(p \sim 1, r \sim \lambda _{q+1}^{-1}\), we must have \(\theta < 5/4\) in order that the second term in (36) is small for \(\lambda _{q+1}\) sufficiently large.

It remains to estimate \(\widetilde{R}_{oscillation}\), which can be handled in the same way as in [5]. It follows from (19) that

$$\begin{aligned} \nabla \cdot \left( w_{q+1}^{(p)} \otimes w_{q+1}^{(p)} - R_{q}\right)&= \nabla \cdot \left( \sum _{\overline{\xi }, \overline{\xi }' \in \Lambda } a_{(\overline{\xi })} a_{(\overline{\xi }')} \mathbb {W}_{\overline{\xi }} \otimes \mathbb {W}_{(\overline{\xi }')} - R_q\right) \\&= \nabla \cdot \left( \sum _{\overline{\xi }, \overline{\xi }' \in \Lambda } a_{(\overline{\xi })} a_{(\overline{\xi }')} \mathbb {P}_{\ge \lambda _{q+1} \sigma /2}\mathbb {W}_{(\overline{\xi })} \otimes \mathbb {W}_{(\overline{\xi }')} \right) + \nabla \rho \\&{:=} \sum _{\overline{\xi }, \overline{\xi }' \in \Lambda } E_{(\overline{\xi }, \overline{\xi }')} + \nabla \rho . \end{aligned}$$

Since \(E_{(\overline{\xi }, \overline{\xi }')}\) has zero mean, we can split it as

$$\begin{aligned} E_{(\overline{\xi }, \overline{\xi }')} + E_{(\overline{\xi }', \overline{\xi })}&= \mathbb {P}_{\ne 0} \left( \nabla \left( a_{(\overline{\xi })} a_{(\overline{\xi }')}\right) \cdot \left( \mathbb {P}_{\ge \lambda _{q+1} \sigma /2}\left( \mathbb {W}_{(\overline{\xi })} \otimes \mathbb {W}_{(\overline{\xi }')} + \mathbb {W}_{\overline{\xi '}} \otimes \mathbb {W}_{(\overline{\xi })}\right) \right) \right) \\&\quad + \mathbb {P}_{\ne 0} \left( a_{(\overline{\xi })} a_{(\overline{\xi }')} \nabla \cdot \left( \mathbb {W}_{(\overline{\xi })} \otimes \mathbb {W}_{(\overline{\xi }')} + \mathbb {W}_{\overline{\xi '}} \otimes \mathbb {W}_{(\overline{\xi })}\right) \right) \\&:= E_{(\overline{\xi }, \overline{\xi }',1)} + E_{(\overline{\xi }, \overline{\xi }',2)}. \end{aligned}$$

Using (15), (34) and (35), we obtain

$$\begin{aligned} \left\| \mathcal {R} E_{(\overline{\xi }, \overline{\xi }', 1)}\right\| _{L^{\infty }_t L^p_x}&\lesssim \left\| |\nabla |^{-1} E_{(\overline{\xi }, \overline{\xi }', 1)}\right\| _{L^{\infty }_t L^p_x} \\&\lesssim (\lambda _{q+1} \sigma )^{-1} \left\| a_{(\overline{\xi })} a_{(\overline{\xi }')}\right\| _{C^3} \left\| \mathbb {W}_{(\overline{\xi })} \otimes \mathbb {W}_{(\overline{\xi }')}\right\| _{L^{\infty }_t L^p_x}\\&\lesssim (\lambda _{q+1} \sigma )^{-1} \left\| a_{(\overline{\xi })} a_{(\overline{\xi }')}\right\| _{C^3} \left\| \mathbb {W}_{(\overline{\xi })}\right\| _{L^{\infty }_t L^{2p}_x} \left\| \mathbb {W}_{(\overline{\xi }')}\right\| _{L^{\infty }_t L^{2p}_x}\\&\lesssim (\lambda _{q+1} \sigma )^{-1} r^{3-3/p} \mathcal {C}_3. \end{aligned}$$

Recall the vector identity \(A \cdot \nabla B + B \cdot \nabla A = \nabla (A \cdot B) - A \times (\nabla \times B) - B \times (\nabla \times A)\). For \(\overline{\xi }, \overline{\xi }' \in \Lambda \), using the anti-symmetry of the cross product, we can write

$$\begin{aligned}&\nabla \cdot \left( \mathbb {W}_{(\overline{\xi })} \otimes \mathbb {W}_{(\overline{\xi }')} + \mathbb {W}_{(\overline{\xi }')} \otimes \mathbb {W}_{(\overline{\xi })} \right) \\&\quad = \left( W_{(\overline{\xi })} \otimes W_{(\overline{\xi }')} + W_{(\overline{\xi }')} \otimes W_{(\overline{\xi })} \right) \nabla \left( \eta _{(\overline{\xi })} \eta _{(\overline{\xi }')} \right) \\&\qquad + \eta _{(\overline{\xi })} \eta _{(\overline{\xi }')} \left( W_{(\overline{\xi })} \cdot \nabla W_{(\overline{\xi }')} + W_{(\overline{\xi }')} \cdot \nabla W_{(\overline{\xi })} \right) \\&\quad = \left( W_{(\overline{\xi }')} \cdot \nabla \left( \eta _{(\overline{\xi })} \eta _{(\overline{\xi }')} \right) \right) W_{(\overline{\xi })} + \left( W_{(\overline{\xi })} \cdot \nabla \left( \eta _{(\overline{\xi })} \eta _{(\overline{\xi }')} \right) \right) W_{\overline{\xi }' } \\&\qquad + \eta _{(\overline{\xi })} \eta _{(\overline{\xi }')} \nabla \left( W_{(\overline{\xi })} \cdot W_{(\overline{\xi }')} \right) . \end{aligned}$$

For the term \(E_{(\overline{\xi }, \overline{\xi }',2)}\), first consider the case \(\overline{\xi } + \overline{\xi '} \ne 0\). It follows from the above identity and (14) that

$$\begin{aligned}&a_{(\overline{\xi })} a_{(\overline{\xi }')} \nabla \cdot \left( \mathbb {W}_{(\overline{\xi })} \otimes \mathbb {W}_{(\overline{\xi }')} + \mathbb {W}_{(\overline{\xi }')} \otimes \mathbb {W}_{(\overline{\xi })}\right) \\&\quad = a_{(\overline{\xi })} a_{(\overline{\xi }')} \nabla \cdot \mathbb {P}_{\ge \lambda _{q+1}/10} \left( \eta _{(\overline{\xi })} \eta _{(\overline{\xi }')} \left( W_{(\overline{\xi })} \otimes W_{(\overline{\xi }')} + W_{(\overline{\xi }')} \otimes W_{(\overline{\xi })} \right) \right) \\&\quad = a_{(\overline{\xi })} a_{(\overline{\xi }')} \mathbb {P}_{\ge \lambda _{q+1}/10} \left( \nabla \left( \eta _{(\overline{\xi })} \eta _{(\overline{\xi }')} \right) \cdot \left( W_{(\overline{\xi })} \otimes W_{(\overline{\xi }')} + W_{(\overline{\xi }')} \otimes W_{(\overline{\xi })} \right) \right) \\&\qquad + a_{(\overline{\xi })} a_{(\overline{\xi }')} \mathbb {P}_{\ge \lambda _{q+1}/10} \left( \eta _{(\overline{\xi })} \eta _{(\overline{\xi }')} \nabla \left( W_{(\overline{\xi })} \cdot W_{(\overline{\xi }')}\right) \right) \\&\quad = a_{(\overline{\xi })} a_{(\overline{\xi }')} \mathbb {P}_{\ge \lambda _{q+1}/10} \left( \nabla \left( \eta _{(\overline{\xi })} \eta _{(\overline{\xi }')} \right) \cdot \left( W_{(\overline{\xi })} \otimes W_{(\overline{\xi }')} + W_{(\overline{\xi }')} \otimes W_{(\overline{\xi })} \right) \right) \\&\qquad + \nabla \left( a_{(\overline{\xi })} a_{(\overline{\xi }')} \mathbb {W}_{(\overline{\xi })} \cdot \mathbb {W}_{(\overline{\xi }')} \right) - \nabla \left( a_{(\overline{\xi })} a_{(\overline{\xi }')}\right) \mathbb {P}_{\ge \lambda _{q+1}/10}\left( \mathbb {W}_{(\overline{\xi })} \cdot \mathbb {W}_{(\overline{\xi }')} \right) \\&\qquad - a_{(\overline{\xi })} a_{(\overline{\xi }')} \mathbb {P}_{\ge \lambda _{q+1}/10} \left( \left( W_{(\overline{\xi })} \cdot W_{(\overline{\xi }')}\right) \nabla \left( \eta _{(\overline{\xi })} \eta _{(\overline{\xi }')}\right) \right) , \end{aligned}$$

where the second term is a pressure, the third can be estimated analogously to \(E_{(\overline{\xi }, \overline{\xi }', 1)}\). Also note that the first and fourth term can estimated analogously. Using (16), (34) and (35), we obtain

$$\begin{aligned}&\left\| \mathcal {R} \left( a_{(\overline{\xi })} a_{(\overline{\xi }')} \mathbb {P}_{\ge \lambda _{q+1}/10} \left( \nabla \left( \eta _{(\overline{\xi })} \eta _{(\overline{\xi }')} \right) \cdot \left( W_{(\overline{\xi })} \otimes W_{(\overline{\xi }')} + W_{(\overline{\xi }')} \otimes W_{(\overline{\xi })} \right) \right) \right) \right\| _{L^{\infty }_t L^p_x}\\&\quad \lesssim \lambda _{q+1}^{-1} \left\| a_{(\overline{\xi })} a_{(\overline{\xi }')}\right\| _{C^3}\left\| \nabla \left( \eta _{(\overline{\xi })} \eta _{(\overline{\xi }')} \right) \right\| _{L^{\infty }_t L^p_x}\\&\quad \lesssim \sigma r^{4-3/p} \mathcal {C}_3. \end{aligned}$$

Now consider \(E_{(\overline{\xi }, -\overline{\xi },2)}\). We can write

$$\begin{aligned} \nabla \cdot \left( \mathbb {W}_{(\overline{\xi })} \otimes \mathbb {W}_{(-\overline{\xi })} + \mathbb {W}_{(-\overline{\xi })} \otimes \mathbb {W}_{(\overline{\xi })} \right)&= \left( W_{(-\overline{\xi })} \cdot \nabla \eta _{(\overline{\xi })}^2 \right) W_{(\overline{\xi })} + \left( W_{(\overline{\xi })} \cdot \nabla \eta _{(\overline{\xi })}^2 \right) W_{(-\overline{\xi })}\\&= \left( A_{\overline{\xi }} \cdot \nabla \eta _{(\overline{\xi })}^2\right) A_{\overline{\xi }} + \left( (\overline{\xi } \times A_{\overline{\xi }}) \cdot \nabla \eta _{(\overline{\xi })}^2\right) \left( \overline{\xi } \times A_{\overline{\xi }}\right) \\&= \nabla \xi _{(\overline{\xi })}^2 - \left( \overline{\xi } \cdot \nabla \eta _{(\overline{\xi })}^2\right) \overline{\xi } \\&= \nabla \eta _{(\overline{\xi })}^2 - \frac{\overline{\xi }}{\mu }\partial _t \eta _{(\overline{\xi })}^2, \end{aligned}$$

where we use (11) and the fact that \(\{ \overline{\xi }, A_{\overline{\xi }}, \overline{\xi } \times A_{\overline{\xi }} \}\) forms an orthonormal basis of \(\mathbb {R}^3\). Therefore, we can write

$$\begin{aligned} E_{(\overline{\xi }, -\overline{\xi },2)}&= \mathbb {P}_{\ne 0}\left( a_{(\overline{\xi })}^2 \nabla \mathbb {P}_{\ge \lambda _{q+1} \sigma /2} \eta _{(\overline{\xi })}^2 - a_{(\overline{\xi })}^2 \frac{\overline{\xi }}{\mu }\partial _t \eta _{(\overline{\xi })}^2 \right) \\&= \nabla \left( a_{(\overline{\xi })}^2 \mathbb {P}_{\ge \lambda _{q+1} \sigma /2} \eta _{(\overline{\xi })}^2 \right) - \mathbb {P}_{\ne 0}\left( \mathbb {P}_{\ge \lambda _{q+1} \sigma /2}(\eta _{(\overline{\xi })}^2) \nabla a_{(\overline{\xi })}^2 \right) \\&\quad - \mu ^{-1} \partial _t \mathbb {P}_{\ne 0}\left( a_{(\overline{\xi })}^2 \eta _{(\overline{\xi })}^2 \overline{\xi } \right) + \mu ^{-1} \mathbb {P}_{\ne 0} \left( \partial _t\left( a_{(\overline{\xi })}^2\right) \eta _{(\overline{\xi })}^2 \overline{\xi } \right) . \end{aligned}$$

Using the identity \(\mathrm {Id} - \mathbb {P}_{LH} = \nabla \Delta ^{-1} {\text {div}}\) , we obtain

$$\begin{aligned} \sum _{\overline{\xi }} E_{(\overline{\xi }, -\overline{\xi },2)} + \partial _t w_{q+1}^{(t)}&= \nabla \sum _{\overline{\xi }} \left( a_{(\overline{\xi })}^2 \mathbb {P}_{\ge \lambda _{q+1} \sigma /2} \eta _{(\overline{\xi })}^2 \right) - \nabla \sum _{\overline{\xi }} \mu ^{-1} \Delta ^{-1} \nabla \cdot \partial _t \left( a_{(\overline{\xi })}^2 \eta _{(\overline{\xi })}^2 \overline{\xi }\right) \\&\quad - \sum _{\overline{\xi }} \mathbb {P}_{\ne 0}\left( \mathbb {P}_{\ge \lambda _{q+1} \sigma /2}(\eta _{(\overline{\xi })}^2) \nabla a_{(\overline{\xi })}^2 \right) + \mu ^{-1} \sum _{\overline{\xi }} \mathbb {P}_{\ne 0} \left( \partial _t\left( a_{(\overline{\xi })}^2\right) \eta _{(\overline{\xi })}^2 \overline{\xi } \right) , \end{aligned}$$

where the first and second terms are pressure terms. Using (16), (34) and (35), we obtain

$$\begin{aligned} \Vert \mathcal {R} \mathbb {P}_{\ne 0}\left( \mathbb {P}_{\ge \lambda _{q+1} \sigma /2}(\eta _{(\overline{\xi })}^2) \nabla a_{(\overline{\xi })}^2 \right) \Vert _{L^{\infty }_t L^p_x}&\lesssim (\lambda _{q+1} \sigma )^{-1} \Vert \eta _{(\overline{\xi })} \Vert _{L^{\infty }_t L^{2p}_x}^2\mathcal {C}_3 \\&\lesssim (\lambda _{q+1} \sigma )^{-1} r^{3-3/p} \mathcal {C}_3. \end{aligned}$$

It follows from (16) and (34) that

$$\begin{aligned} \mu ^{-1}\Vert \mathcal {R} \mathbb {P}_{\ne 0} \left( \partial _t\left( a_{(\overline{\xi })}^2\right) \eta _{(\overline{\xi })}^2 \overline{\xi } \right) \Vert _{L^{\infty }_t L^p_x}&\lesssim \mu ^{-1}\Vert \partial _t\left( a_{(\overline{\xi })}^2\right) \eta _{(\overline{\xi })}^2 \overline{\xi } \Vert _{L^{\infty }_t L^p_x} \\&\lesssim \mu ^{-1} r^{3-3/p} \mathcal {C}_1. \end{aligned}$$

Let us now give the explicit definition of \(\widetilde{R}_{oscillation}\):

$$\begin{aligned} \widetilde{R}_{oscillation}&= \sum _{\overline{\xi }, \overline{\xi }' \in \Lambda } \mathbb {P}_{\ne 0} \left( \nabla (a_{(\overline{\xi })} a_{(\overline{\xi }')}) \cdot (\mathbb {P}_{\ge \lambda _{q+1} \sigma /2}(\mathbb {W}_{(\overline{\xi })} \otimes \mathbb {W}_{(\overline{\xi }')} + \mathbb {W}_{\overline{\xi '}} \otimes \mathbb {W}_{(\overline{\xi })}) ) \right) \\&\quad + \sum _{\overline{\xi }, \overline{\xi }' \in \Lambda , \overline{\xi } \ne \overline{\xi }' } a_{(\overline{\xi })} a_{(\overline{\xi }')} \mathbb {P}_{\ge \lambda _{q+1}/10} \left( \nabla \left( \eta _{(\overline{\xi })} \eta _{(\overline{\xi }')} \right) \cdot \left( W_{(\overline{\xi })} \otimes W_{(\overline{\xi }')} + W_{(\overline{\xi }')} \otimes W_{(\overline{\xi })} \right) \right) \\&\quad - \sum _{\overline{\xi }, \overline{\xi }' \in \Lambda , \overline{\xi } \ne \overline{\xi }' } \nabla \left( a_{(\overline{\xi })} a_{(\overline{\xi }')}\right) \mathbb {P}_{\ge \lambda _{q+1}/10}\left( \mathbb {W}_{(\overline{\xi })} \cdot \mathbb {W}_{(\overline{\xi }')} \right) \\&\quad - \sum _{\overline{\xi }, \overline{\xi }' \in \Lambda , \overline{\xi } \ne \overline{\xi }' } a_{(\overline{\xi })} a_{(\overline{\xi }')} \mathbb {P}_{\ge \lambda _{q+1}/10} \left( \left( W_{(\overline{\xi })} \cdot W_{(\overline{\xi }')}\right) \nabla \left( \eta _{(\overline{\xi })} \eta _{(\overline{\xi }')}\right) \right) \\&\quad - \sum _{\overline{\xi } \in \Lambda } \mathbb {P}_{\ne 0}\left( \mathbb {P}_{\ge \lambda _{q+1} \sigma /2}(\eta _{(\overline{\xi })}^2) \nabla a_{(\overline{\xi })}^2 \right) + \mu ^{-1} \sum _{\overline{\xi } \in \Lambda } \mathbb {P}_{\ne 0} \left( \partial _t\left( a_{(\overline{\xi })}^2\right) \eta _{(\overline{\xi })}^2 \overline{\xi } \right) . \end{aligned}$$

Finally, we estimate the time support of \(R_{q+1}\). Using (25) we obtain

$$\begin{aligned} {\text {supp}}_t R_{q+1} \subset {\text {supp}}_t w_{q+1} \cup {\text {supp}}_t R_{q} \subset N_{\delta _{q+1}}({\text {supp}}_t R_q). \end{aligned}$$

Now we choose the parameters \(r, \sigma , \mu \). Fix \(\alpha \) so that

$$\begin{aligned} \max \left\{ 0,\frac{2}{3}(2\theta - 1)\right\}< \alpha < 1, \end{aligned}$$

which is possible since \(\theta \in (-\infty ,5/4)\). Fix

$$\begin{aligned} r = \lambda ^{\alpha }_{q+1}, \quad \sigma = \lambda ^{-(\alpha + 1)/2}_{q+1}, \quad \mu = \lambda ^{(5\alpha + 1)/4}_{q+1}. \end{aligned}$$
(37)

Clearly (27) is satisfied. Choose \(p > 1\) sufficiently close to 1 so that

$$\begin{aligned}&-\frac{\alpha +1}{2} + \frac{5\alpha + 1}{4} +\left( \frac{5}{2} - \frac{3}{p}\right) \alpha< 0 , \quad \left( \frac{3}{2} - \frac{3}{p}\right) \alpha + \max (0,2\theta - 1)< 0, \\&\quad -\frac{5\alpha + 1}{4} + \left( \frac{9}{2} - \frac{3}{p}\right) \alpha< 0, \quad -\frac{1-\alpha }{2} + \left( 3 - \frac{3}{p}\right) \alpha < 0. \end{aligned}$$

Note that \(\mathcal {C}_N\) is independent of \(\lambda _{q+1}\), due to (24). Combining the above estimates with Lemma 4, it is easy to check that, by taking \(\lambda _{q+1}\) sufficiently large, we arrive at (4), (6) and (7). This completes the proof of Lemma 1.