Second Derivatives Estimate of Suitable Solutions to the 3D Navier–Stokes Equations

Abstract

We study the second spatial derivatives of suitable weak solutions to the incompressible Navier–Stokes equations in dimension three. We show that it is locally $L ^{\frac{4}{3}, q}$ for any $q > \frac{4}{3}$, which improves from the current result of $L ^{\frac{4}{3}, \infty }$. Similar improvements in Lorentz space are also obtained for higher derivatives of the vorticity for smooth solutions. We use a blow-up technique to obtain nonlinear bounds compatible with the scaling. The local study works on the vorticity equation and uses De Giorgi iteration. In this local study, we can obtain any regularity of the vorticity without any a priori knowledge of the pressure. The local-to-global step uses a recently constructed maximal function for transport equations.

1 Introduction

We study the three dimensional incompressible Navier–Stokes equations

$$\begin{aligned} \partial _tu + u \cdot \nabla u + \nabla P = \varDelta u, \qquad {\text {div}}u = 0. \end{aligned}$$

(1)

Here $u: (0, T) \times {\mathbb {R}}^3 \rightarrow {\mathbb {R}}^3$ and $P: (0, T) \times {\mathbb {R}}^3 \rightarrow {\mathbb {R}}$ represent the velocity field and the pressure field of a fluid in ${\mathbb {R}}^3$, within a finite or infinite timespan of length T. Initial condition

$$\begin{aligned} u (0, \cdot ) = u _0 \in L ^2 ({\mathbb {R}}^3) \end{aligned}$$

is given by a divergence-free velocity profile $u _0$ of finite energy.

Leray [11] and Hopf [8] proved the existence of weak solutions for all time. They constructed solutions $u \in C _w (0, \infty ; L ^2 ({\mathbb {R}}^3)) \cap L ^2 (0, \infty ; \dot{H} ^1 ({\mathbb {R}}^3))$ corresponding to each aforementioned initial value, and satisfying (1) in the sense of distribution. A weak solution is called a Leray-Hopf solution if it satisfies energy inequality

$$\begin{aligned} \frac{1}{2} \Vert u (t) \Vert _{L ^2 ({\mathbb {R}}^3)} ^2 + \Vert \nabla u \Vert _{L ^2 ((0, t) \times {\mathbb {R}}^3)} ^2 \le \frac{1}{2} \Vert u _0 \Vert _{L ^2 ({\mathbb {R}}^3)} ^2 \end{aligned}$$

for every $t > 0$. Since Leray and Hopf much work has been developed in regard to the uniqueness and regularity of weak solutions. Nonuniqueness of weak solutions was proven very recently by Buckmaster and Vicol using convex integration scheme [1]. However, the question of the uniqueness of Leray-Hopf solutions still remains open. The uniqueness is related with the regularity of solutions by the Ladyženskaya-Prodi-Serrin criteria [7, 10, 14, 20, 21]: if the velocity belongs to any space interpolating $L ^2 _t L ^\infty _x$ and $L ^\infty _t L ^3 _x$ then it is actually smooth, and hence unique. The endpoint case $L ^\infty _t L ^3 _x$ came much later by Iskauriaza, Serëgin and Shverak [9]. These spaces require $\frac{1}{6}$ higher spatial integrability than the energy space provides, which is ${\mathcal {E}} = L ^\infty _t L ^2 _x \cap L ^2 _t \dot{H} ^1 _x$.

At the level of energy space, Scheffer began to study the partial regularity for a class of Leray-Hopf solutions, called suitable weak solutions [16,17,18,19]. These solutions exist globally and satisfy the following local energy inequality:

$$\begin{aligned} \partial _t\frac{|u|^2}{2} + {\text {div}}\left( u\left( \frac{|u|^2}{2} + P \right) \right) + |\nabla u| ^2 \le \varDelta \frac{|u|^2}{2}. \end{aligned}$$

Scheffer showed the singular set, at which the solution is unbounded nearby, has time-space Hausdorff dimension at most $\frac{5}{3}$. This result was later improved by Caffarelli, Kohn and Nirenberg in [2] (see also [12, 23]), where they showed the 1-dimensional Hausdorff measure of the singular set is zero. We will investigate the regularity of suitable weak solutions. In the periodic setting, Constantin constructed suitable weak solutions whose second derivatives have space-time integrability $L ^{\frac{4}{3} - \varepsilon }$ for any $\varepsilon > 0$, provided the initial vorticities are bounded measures [6]. This was improved by Lions to a slightly better space $L ^{\frac{4}{3}, \infty }$, a Lorentz space which corresponds to weak $L ^\frac{4}{3}$ space [13]. These estimates are extended to higher derivatives of smooth solutions by one of the authors and Choi using blow-up arguments: $L ^{p, \infty } _{\mathrm {loc}}$ space-time boundedness for $(-\varDelta ) ^\frac{\alpha }{2} \nabla ^n u$, where $p = \frac{4}{n + \alpha + 1}$, $n \ge 1$, $0 \le \alpha < 2$ [5, 22]. They also constructed suitable weak solutions satisfying these bounds for $n + \alpha < 3$.

The aim of this paper is to improve these regularity results in Lorentz space. The main result is the following. Note that the estimate does not rely on the size of the pressure.

Theorem 1

Suppose we have a smooth solution u to the Navier–Stokes equations in $(0, T) \times {\mathbb {R}}^3$ for some $0 < T \le \infty$ with smooth divergence free initial data $u _0 \in L ^2$. Then for any integer $n \ge 0$, for any real number $q > 1$, the vorticity $\omega = {\text {curl}}u$ satisfies

$$\begin{aligned} \left\| |\nabla ^n \omega | ^\frac{4}{n+2} \mathbf{1} _{\{|\nabla ^n \omega | ^\frac{4}{n+2} > C _n t^{-2}\}} \right\| _{L ^{1, q}((0, T) \times {\mathbb {R}}^3)} \le C _{q, n} \Vert u _0 \Vert _{L ^2} ^2 \end{aligned}$$

(2)

for some constant $C _n$ depending on n and $C _{q, n}$ depending only on q and n, uniform in T. The above estimate (2) also holds for suitable weak solutions with only $L ^2$ divergence free initial data in the case $n = 1$.

This theorem gives the following improvement on the second derivatives:

Corollary 1

Let u be a suitable weak solution in $(0, \infty ) \times {\mathbb {R}}^3$ with initial data $u _0 \in L ^2$. Then for any $q > \frac{4}{3}$, $K \subset \subset (0, \infty ) \times {\mathbb {R}}^3$, there exists a constant $C _{q, K}$ depending on q and K such that the following holds

$$\begin{aligned} \Vert \nabla ^2 u \Vert _{L ^{\frac{4}{3}, q} (K)} \le C _{q, K} \left( \Vert u _0 \Vert _{L ^2} ^\frac{3}{2} + 1 \right) . \end{aligned}$$

Let us explain the main ideas of the proof. Similar as previous work on higher derivatives, the proof is also based on blow-up techniques. In particular, we blow up the equation along a trajectory, using the scaling symmetry and the Galilean invariance of the Navier–Stokes equations. That is, if we fix an initial time $t _0$ and move the frame of reference along some X(t), and zoom in into $\varepsilon$ scale, then it is easy to verify that ${{\tilde{u}}} (s, y)$ and ${{\tilde{P}}} (s, y)$, defined by

$$\begin{aligned} \frac{1}{\varepsilon }{{\tilde{u}}} \left( \frac{t - t _0}{\varepsilon ^2}, \frac{x - X (t)}{\varepsilon } \right)&:= u (t, x) - \dot{X}(t) \nonumber \\ \frac{1}{\varepsilon ^2} {{\tilde{P}}} \left( \frac{t - t _0}{\varepsilon ^2}, \frac{x - X (t)}{\varepsilon } \right)&:= P(t, x) + x \cdot {\ddot{X}}(t), \end{aligned}$$

(3)

also satisfy the Navier–Stokes equation

$$\begin{aligned} \partial _s {{\tilde{u}}} + {{\tilde{u}}} \cdot \nabla {{\tilde{u}}} + \nabla {{\tilde{P}}} = \varDelta {{\tilde{u}}}, \qquad {\text {div}}{{\tilde{u}}} = 0. \end{aligned}$$

We develop the following local theorem for ${{\tilde{u}}}$ and ${{\tilde{P}}}$. Note that it needs nothing from the pressure. Denote $B _r \subset {\mathbb {R}}^3$ to be a ball centered at the origin with radius r, and $Q _r = (-r ^2, 0) \times B _r \subset {\mathbb {R}}^4$ to be a space-time cylinder.

Theorem 2

(Local Theorem) There exists a universal constant $\eta _1 > 0$, such that for any suitable weak solution u to the Navier–Stokes equations in $(-4, 0) \times {\mathbb {R}}^3$ satisfying

$$\begin{aligned}&\int _{B _1} u (t, x) \phi (x) {\mathrm {d}}x = 0 \qquad {\mathrm{a.e. }} t \in (-4, 0), \end{aligned}$$

(4)

$$\begin{aligned}&\Vert \nabla u \Vert _{L ^{p _1} _t L ^{q _1} _x (Q _2)} + \Vert \omega \Vert _{L ^{p _2} _t L ^{q _2} _x (Q _2)} \le \eta _1, \end{aligned}$$

(5)

where $\phi \in C _c ^\infty (B _1)$ is a non-negative function with $\int \phi = 1$, $\omega = {\text {curl}}u$ is the vorticity, $\frac{4}{3} \le p _1 \le \infty$, $1 \le p _2 \le \infty$, $1 \le q _1, q _2 < 3$ satisfying

$$\begin{aligned} \frac{1}{p _1} + \frac{1}{p _2} < 1, \qquad \frac{1}{q _1} + \frac{1}{q _2} \le \frac{7}{6}, \end{aligned}$$

then for any integer $n \ge 0$, we have

$$\begin{aligned} \Vert \nabla ^n \omega \Vert _{L ^\infty (Q _{8 ^{-n-2}})} \le C _n \end{aligned}$$

for some constant $C _n$ depending only on n.

Let us illustrate the ideas of how to go from this local theorem towards the main result. We want to choose a “pivot quantity”, blow up near a point, and use this quantity to control $\nabla ^n \omega$. When we patch the local results together, we will obtain a nonlinear bound with the same scaling as the pivot quantity, so we want the pivot quantity to have the best possible scaling. The ideal pivot quantities would be $\int |\nabla u| ^2 {\mathrm {d}}x {\mathrm {d}}t$ and $\int |\nabla ^2 P| {\mathrm {d}}x {\mathrm {d}}t$. $\int |u| ^\frac{10}{3} {\mathrm {d}}x {\mathrm {d}}t$ has a worse scaling and should not be used. However, we still need to control the flux in the local theorem, so we want to take out the mean velocity and control u by $\nabla u$ using Poincaré’s inequality.

In order to take out the mean velocity, we choose X(t) to be the trajectory of the mollified flow so that (4) can be realized. Notice that a cylinder $Q _r$ in the local (s, y) coordinate will be transformed into a “skewed cylinder” growing along X(t) in the global (t, x) coordinate. One of the authors recently constructed a maximal function ${\mathcal {M}}_{\mathcal {Q}}$ associated with these cylinders [24], which serves as a bridge between the local theorem and the global result, and is one of the main reasons for the improvement in this paper. The idea is, if locally the vorticity gradient can be controlled in $L ^\infty$ by the integral of something in the skewed cylinder, and the integral in a skewed cylinder can be controlled by the maximal function ${\mathcal {M}}_{\mathcal {Q}}$, then vorticity gradient is pointwise bounded by the maximal function.

If one uses $\int |\nabla u| ^2 {\mathrm {d}}x {\mathrm {d}}t$ and $\int |\nabla ^2 P| {\mathrm {d}}x {\mathrm {d}}t$ as the pivot quantity, then unfortunately the best possible outcome would just be an $L ^{1, \infty }$ bound, as obtained in [24]. The reason is, the maximal function is bounded on $L ^p$ for $p > 1$, but for $p = 1$ it is only bounded from $L ^1$ to $L ^{1, \infty }$. Unfortunately $|\nabla u| ^2$ and $|\nabla ^2 P|$ are both $L ^1$ quantities, so ${\mathcal {M}}_{\mathcal {Q}}\left( |\nabla u| ^2 + |\nabla ^2 P|\right)$ is only $L ^{1, \infty }$. We need two things to improve from $L ^{1, \infty }$: replace $\int |\nabla u| ^2$ by $\int |\nabla u| ^p$, and drop the pressure $\nabla ^2 P$.

Suppose we could use $(\int |\nabla u| ^p {\mathrm {d}}x {\mathrm {d}}t) ^\frac{2}{p}$ as the pivot quantity for some $p < 2$, then we can majorize it by ${\mathcal {M}}_{\mathcal {Q}}\left( |\nabla u| ^p\right) ^\frac{2}{p} \in L ^1$, since $\frac{2}{p}>1$ and ${\mathcal {M}}_{\mathcal {Q}}$ is bounded in $L ^\frac{2}{p}$. However, this poses significant difficulties in the local theorem. The nonlinear term $u \cdot \nabla u$ is quadratic, and if we only have a subquadratic integrability to begin with, we cannot treat this quadratic transport term as a source term because it is not integrable. Observe that what we lack is the temporal integrability rather than the spatial one: if p is slightly smaller than two, then $u \cdot \nabla u$ is still $L ^{\frac{3}{2}-}$ in space, but $L ^{1-}$ in time. To overcome this difficulty, we write $u \cdot \nabla u$ as $\omega \times u$ up to a gradient term, and put $L ^{2-} _t L^{6-} _x$ on u and $L ^{2+} _t L ^{2-} _x$ on $\omega$. We compensate the lower integrability term by pairing with a higher integrability term to make $\omega \times u$ integrable. $L ^{2+} _t L ^{2-} _x$ of $\omega$ can be interpolated between $L ^{2-} _t L^{2-} _x$ and $L ^\infty _t L ^1 _x$, while the latter is controlled by $L ^2 _{t, x}$ of $\nabla u$. Since $L ^{2+} _t L ^{2-} _x$ is closer to $L ^{2-} _t L^{2-} _x$ than to $L ^\infty _t L ^1 _x$, the pivot quantity that we use is actually $\delta ^{-\nu } \Vert \nabla u \Vert ^2 _{L ^p} + \delta \Vert \nabla u \Vert ^2 _{L ^2}$ for $\nu$ close to 0. By using more subquadratic integrability and a tiny bit of the quadratic one, we can complete the task by interpolation. That is why we obtain $L ^{1, q}$ in the end: it interpolates $L ^1$ bound from $\Vert \nabla u\Vert _{L ^p}$ and $L ^{1, \infty }$ bound from $\Vert \nabla u \Vert _{L ^2}$. Unfortunately we still miss the endpoint $L ^1$.

The second task is more subtle and technical. Without any information on the pressure, we don’t have any control on the nonlocal effect. However, the role of the pressure is not important at the vorticity level; if we take the curl of the Navier–Stokes equation, the pressure will disappear and we are left with the vorticity equation involving only local quantities:

$$\begin{aligned} \partial _t\omega + u \cdot \nabla \omega - \omega \cdot \nabla u = \varDelta \omega . \end{aligned}$$

(6)

Inspired by Chamorro, Lemarié-Rieusset and Mayoufi [4], we introduce a new velocity variable $v = -{\text {curl}}\varphi ^\sharp \varDelta ^{-1}\varphi \omega$ using only local information of vorticity ($\varphi$ and $\varphi ^\sharp$ are spatial cut-off functions), and this helps us to prove the local theorem. This is another main reason for the improvement in this paper. Consequently, the bounds we obtain in the end are on the vorticity $\omega$ rather than on the velocity u.

This paper is organized as follow: in the preliminary Section 2 we introduce the analysis tools to the reader. We show how to rigorously derive the main results from the local theorem in Section 3, and then deal with technicalities of the local theorem in the later sections. The proof of the local theorem consists of three parts. Section 4 introduces the new variables v, and shows the smallness of v in the energy space. Then we use De Giorgi iteration argument in Section 5 to prove boundedness of v. Finally, we inductively bound $\omega$ and all its higher derivatives in Section 6.

2 Preliminary

In this section, we introduce a few tools that we are going to use in the paper, including the maximal function, Lorentz space, and Helmholtz decomposition.

2.1 Maximal Function Associated with Skewed Cylinders

This is recently developed for incompressible flows in [24]. We quote useful results here without proof.

Suppose $u \in L ^p (0, T; \dot{W} ^{1, p} ({\mathbb {R}}^3; {\mathbb {R}}^3))$ is a vector field in ${\mathbb {R}}^3$. Fix $\phi \in C _c ^\infty (B _1)$ to be a nonnegative function with $\int \phi = 1$ through out the paper. For $\varepsilon > 0$ define $\phi _\varepsilon (x) = \varepsilon ^{-3} \phi (-x/\varepsilon )$, and let $u _\varepsilon (t, \cdot ) = u (t, \cdot ) * \phi _\varepsilon$ be the mollified velocity. For a fixed (t, x) we let X(s) solve the following initial value problem:

$$\begin{aligned} {\left\{ \begin{array}{ll} \dot{X} (s) = u _\varepsilon (s, X (s)), \\ X (t) = x. \end{array}\right. } \end{aligned}$$

The skewed parabolic cylinder $Q _\varepsilon (t, x)$ is then defined to be

$$\begin{aligned} Q _\varepsilon (t, x) :{=} \left\{ (t + \varepsilon ^2 s, X (t) + \varepsilon y) : -9 \le s \le 0, y \in B _3 \right\} . \end{aligned}$$

(7)

We use ${\mathcal {M}}$ to denote the spatial Hardy-Littlewood maximal function, which is defined by

$$\begin{aligned} {\mathcal {M}}(f)(t, x) = \sup _{r > 0} \fint _{B _r(x)} |f(t, y)| {\mathrm {d}}y. \end{aligned}$$

Then we construct the space-time maximal function adapted to the flow.

Theorem 3

(${\mathcal {Q}}$-Maximal Function) There exists a universal constant $\eta _0$ such that the following is true. We say $Q _\varepsilon (t, x)$ is admissible if $Q _\varepsilon (t, x) \subset (0, T) \times {\mathbb {R}}^3$ and

$$\begin{aligned} \varepsilon ^2 \fint _{Q _\varepsilon (t, x)} {\mathcal {M}}(|\nabla u|) {\mathrm {d}}x {\mathrm {d}}t \le \eta _0. \end{aligned}$$

(8)

Define the maximal function

$$\begin{aligned} {\mathcal {M}}_{\mathcal {Q}}(f) (t, x) :{=} \sup _{\varepsilon > 0} \left\{ \fint _{Q _\varepsilon (t, x)} |f(s, y)| {\mathrm {d}}s {\mathrm {d}}y: Q _\varepsilon (t, x) \text { is admissible} \right\} . \end{aligned}$$

If u is divergence free and ${\mathcal {M}}(|\nabla u|) \in L ^q$ for some $1 \le q \le \infty$, then ${\mathcal {M}}_{\mathcal {Q}}$ is bounded from $L ^1 ((0, T) \times {\mathbb {R}}^3)$ to $L ^{1, \infty } ((0, T) \times {\mathbb {R}}^3)$ and from $L ^p ((0, T) \times {\mathbb {R}}^3)$ to itself for any $p > 1$ with norm depending on p.

An important consequence of the weak type (1, 1) bound of the Hardy-Littlewood maximal function is the Lebesgue differentiation theorem in ${\mathbb {R}}^n$. Similarly, we can use the ${\mathcal {Q}}$-maximal function to prove the ${\mathcal {Q}}$-Lebesgue differentiation theorem.

Theorem 4

(${\mathcal {Q}}$-Lebesgue Differentiation Theorem) Let $f \in L ^1 _{\mathrm {loc}} ((0, T) \times {\mathbb {R}}^3)$. Then for almost every $(t, x) \in (0, T) \times {\mathbb {R}}^3$,

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0} \fint _{Q _\varepsilon (t, x)} |f (s, y) - f (t, x)| {\mathrm {d}}s {\mathrm {d}}y = 0. \end{aligned}$$

In this case we say (t, x) is a ${\mathcal {Q}}$-Lebesgue point of f.

2.2 Lorentz Space

Let $(X, \mu )$ be a measure space. Recall that for a measurable function f, its decreasing rearrangement is defined as

$$\begin{aligned} f ^* (\lambda ) :{=} \inf \left\{ \alpha> 0: \mu (\{|f| > \alpha \}) < \lambda \right\} , \qquad \lambda \ge 0. \end{aligned}$$

For $0< p < \infty$, $0 < q \le \infty$, Lorentz space $L ^{p,q} (X)$ is defined as the set of functions f for which

$$\begin{aligned} \Vert f \Vert _{L ^{p,q} (X)} :{=} \Vert \lambda ^\frac{1}{p} f ^* \Vert _{L ^q (\frac{{\mathrm {d}}\lambda }{\lambda })} = \Vert \lambda ^{\frac{1}{p} - \frac{1}{q}} f ^* (\lambda ) \Vert _{L ^q} < \infty . \end{aligned}$$

Now we introduct the interpolation lemma for Lorentz spaces.

Lemma 1

(Interpolation of Lorentz Spaces) Let $\nu > 0$ be a fixed positive number. Assume $f _0 \in L ^{p _0, q _0}$, $f _1 \in L ^{p _1, q _1}$, where $0< p _0, p _1< \infty , 0 < q _0, q _1 \le \infty$. If f is a measurable function satisfying

$$\begin{aligned} 2|f| \le \delta f _0 + \delta ^{-\nu } f _1 \qquad \text { for all } \delta > 0, \end{aligned}$$

(9)

then $f \in L ^{p, q}$, where

$$\begin{aligned} \frac{1}{p} = \frac{\nu }{1+\nu } \frac{1}{p _0} + \frac{1}{1+\nu } \frac{1}{p _1}, \qquad \frac{1}{q} = \frac{\nu }{1+\nu } \frac{1}{q _0} + \frac{1}{1+\nu } \frac{1}{q _1}. \end{aligned}$$

Proof

It is easy to check from the definition of decreasing rearrangement that if $h \le f + g$, then $h ^* (2 \lambda ) \le (f + g) ^* (2 \lambda ) \le f ^* (\lambda ) + g ^* (\lambda )$. Thus (9) implies

$$\begin{aligned} 2 |f ^* (2\lambda )| \le \delta f ^* _0 (\lambda )+ \delta ^{-\nu } f ^* _1 (\lambda ), \qquad \text { for all}\, \lambda \ge 0, \delta > 0. \end{aligned}$$

Set $\theta = \frac{1}{1+\nu }$, $\delta = f ^* _0 (\lambda )^{-\theta } f ^* _1 (\lambda )^{\theta }$, then

$$\begin{aligned} 2|f ^* (2\lambda )|&\le f ^* _0 (\lambda )^{-\theta } f ^* _1 (\lambda )^{\theta } f ^* _0 (\lambda )+ f ^* _0 (\lambda )^{\nu \theta } f ^* _1 (\lambda )^{-\nu \theta } f ^* _1 (\lambda )\\&= 2 f ^* _0 (\lambda )^{1 - \theta } f ^* _1 (\lambda )^\theta . \end{aligned}$$

Therefore,

$$\begin{aligned} \Vert f \Vert _{L ^{p, q}}&= \Vert \lambda ^{\frac{1}{p} - \frac{1}{q}} f ^* (\lambda ) \Vert _{L ^q} = C \Vert \lambda ^{\frac{1}{p} - \frac{1}{q}} f ^* (2\lambda )\Vert _{L ^q} \\&\le C \Vert \lambda ^{\frac{1-\theta }{p _0}-\frac{1-\theta }{q _0}} f ^* _0 (\lambda )^{1-\theta }\cdot \lambda ^{\frac{\theta }{p _1}-\frac{\theta }{q _1}} f ^* _1 (\lambda )^\theta \Vert _{L ^q} \\&\le C \Vert \lambda ^{\frac{1-\theta }{p _0}-\frac{1-\theta }{q _0}} f ^* _0 (\lambda )^{1-\theta } \Vert _{L ^{\frac{q _0}{1 - \theta }}} \Vert \lambda ^{\frac{\theta }{p _1} - \frac{\theta }{q _1}} f ^* _1 (\lambda )^\theta \Vert _{L ^\frac{q _1}{\theta }} \\&= C \Vert \lambda ^{\frac{1}{p_0}-\frac{1}{q _0}} f ^* _0 (\lambda )\Vert _{L ^{q _0}} ^{1 - \theta } \Vert \lambda ^{\frac{1}{p _1}-\frac{1}{q _1}} f ^* _1 (\lambda )\Vert _{L ^{q _1}} ^\theta \\&= C \Vert f _0\Vert _{L ^{p _0, q _0}} ^{1 - \theta } \Vert f _1\Vert _{L ^{p _1, q _1}} ^{\theta }, \end{aligned}$$

where $C = 2 ^\frac{1}{p}$. $\square$

We would also like to mention that Riesz transform is bounded on Lorentz space. The proof can be found in [3]. See [15] for general Lorentz spaces.

Lemma 2

For $1< p < \infty$, $1 \le q \le \infty$, $R _{ij} = \partial _i \partial _j \varDelta ^{-1}$ is a bounded linear operator from $L ^{p,q} ({\mathbb {R}}^n)$ to itself. As a spatial operator, it is also bounded in time-space from $L ^{p,q} ((0, T) \times {\mathbb {R}}^n)$ to itself.

2.3 Helmholtz Decomposition

First, recall two vector calculus identities:

$$\begin{aligned} \nabla (u \cdot v)&= (u \cdot \nabla ) v + (v \cdot \nabla ) u + u \times {\text {curl}}v + v \times {\text {curl}}u, \end{aligned}$$

(10)

$$\begin{aligned} {\text {curl}}(u \times v)&= u {\text {div}}v - v {\text {div}}u + (v \cdot \nabla ) u - (u \cdot \nabla ) v. \end{aligned}$$

(11)

For operators A and B, denote $[A, B] = AB - BA$ to be their commutator. Define ${\mathbb {P}} _{{\text {curl}}}= -{\text {curl}}{\text {curl}}\varDelta ^{-1}$ and ${\mathbb {P}} _{\nabla }= \nabla \varDelta ^{-1}{\text {div}}= {\text {Id}}- {\mathbb {P}} _{{\text {curl}}}$ to be the Helmholtz decomposition. Then we compute the following commutators:

$$\begin{aligned} {[}\varphi , {\text {curl}}] u&= -\nabla \varphi \times u, \end{aligned}$$

(12)

$$\begin{aligned} {[}\varphi , \varDelta ] u&= -2 \nabla \varphi \cdot \nabla u - (\varDelta \varphi ) u = -2 {\text {div}}(\nabla \varphi \otimes u) + (\varDelta \varphi ) u, \end{aligned}$$

(13)

$$\begin{aligned} {[}\varphi , \varDelta ^{-1}] u&= \varDelta ^{-1}\left\{ 2 \nabla \varphi \cdot \nabla \varDelta ^{-1}u + (\varDelta \varphi ) \varDelta ^{-1}u\right\} , \end{aligned}$$

(14)

$$\begin{aligned} {[}\varphi , {\mathbb {P}} _{{\text {curl}}}] u&= \nabla \varphi \times {\text {curl}}\varDelta ^{-1}u + \nabla \varphi {\text {div}}\varDelta ^{-1}u - \varDelta ^{-1}u \varDelta \varphi \nonumber \\&\quad + (\varDelta ^{-1}u \cdot \nabla ) \nabla \varphi - (\nabla \varphi \cdot \nabla ) \varDelta ^{-1}u \nonumber \\&\quad + {\mathbb {P}} _{{\text {curl}}}\left\{ 2 \nabla \varphi \cdot \nabla \varDelta ^{-1}u + (\varDelta \varphi ) \varDelta ^{-1}u\right\} . \end{aligned}$$

(15)

The first two are straightforward. The third uses

$$\begin{aligned}{}[\varphi , \varDelta ^{-1}] = -\varDelta ^{-1}[\varphi , \varDelta ] \varDelta ^{-1}, \end{aligned}$$

and the last one is because

$$\begin{aligned} {[}\varphi , {\mathbb {P}} _{{\text {curl}}}]&= [\varphi , -{\text {curl}}{\text {curl}}\varDelta ^{-1}] \\&= -[\varphi , {\text {curl}}] {\text {curl}}\varDelta ^{-1}- {\text {curl}}[\varphi , {\text {curl}}] \varDelta ^{-1}- {\text {curl}}{\text {curl}}[\varphi , \varDelta ^{-1}], \\ [\varphi , {\mathbb {P}} _{{\text {curl}}}] u&= \nabla \varphi \times {\text {curl}}\varDelta ^{-1}u + {\text {curl}}(\nabla \varphi \times \varDelta ^{-1}u) \\&\quad - {\text {curl}}{\text {curl}}\varDelta ^{-1}\left\{ 2 \nabla \varphi \cdot \nabla \varDelta ^{-1}u + (\varDelta \varphi ) \varDelta ^{-1}u\right\} , \end{aligned}$$

so we can expand ${\text {curl}}(\nabla \varphi \times \varDelta ^{-1}u)$ by (11).

Lemma 3

$\partial _i [\varphi , {\mathbb {P}} _{{\text {curl}}}]$ and $[\varphi , {\mathbb {P}} _{{\text {curl}}}] \partial _i$ are both bounded linear operator from $L ^p$ to $L ^p$ for any $1< p < \infty$, i.e.

$$\begin{aligned} \Vert \partial _i [\varphi , {\mathbb {P}} _{{\text {curl}}}] u \Vert _{L ^p} + \Vert [\varphi , {\mathbb {P}} _{{\text {curl}}}] \partial _i u \Vert _{L ^p} \le C _{p, \varphi } \Vert u \Vert _{L ^p}. \end{aligned}$$

Proof

First, we observe that by Jacobi identity $[\varphi , {\mathbb {P}} _{{\text {curl}}}] \partial _i$ and $\partial _i [\varphi , {\mathbb {P}} _{{\text {curl}}}]$ differ by

$$\begin{aligned}{}[[\varphi , {\mathbb {P}} _{{\text {curl}}}], \partial _i] = [\varphi , [{\mathbb {P}} _{{\text {curl}}}, \partial _i]] - [{\mathbb {P}} _{{\text {curl}}}, [\varphi , \partial _i]] = 0 - [{\mathbb {P}} _{{\text {curl}}}, \partial _i \varphi ], \end{aligned}$$

which is bounded from $L ^p$ to $L ^p$ for any p, because both ${\mathbb {P}} _{{\text {curl}}}$ and multiplication by $\partial _i \varphi$ are bounded from $L ^p$ to $L ^p$, so we can complete the proof by duality. For $1< p < 3$, set $\frac{1}{p ^*} = \frac{1}{p} - \frac{1}{3}$, from (15) we can see that

$$\begin{aligned} \Vert [\varphi , {\mathbb {P}} _{{\text {curl}}}] \partial _i u \Vert _{L ^p}&\lesssim \Vert \nabla \varDelta ^{-1}\partial _i u \Vert _{L ^{p} ({\mathbb {R}}^3)} + C _{p, \varphi } \Vert \varDelta ^{-1}\partial _i u \Vert _{L ^{p} ({\text {supp}}\varphi )} \\&\lesssim \Vert u \Vert _{L ^{p} ({\mathbb {R}}^3)} + C _{p, \varphi } \Vert \partial _i \varDelta ^{-1}u \Vert _{L ^{p ^*} ({\text {supp}}\varphi )} \\&\le C \Vert u \Vert _{L ^p ({\mathbb {R}}^3)}. \end{aligned}$$

For $\frac{3}{2}< p < \infty$, set $1 - \frac{1}{p} = \frac{1}{q} = \frac{1}{q ^*} + \frac{1}{3}$, then $1< p, q, q^* < \infty$. Take any $u \in L ^p ({\mathbb {R}}^3)$ and any vector field $v \in L ^q ({\mathbb {R}}^3)$ to get

$$\begin{aligned} \int \partial _i [\varphi , {\mathbb {P}} _{{\text {curl}}}] u \cdot v {\mathrm {d}}x&= -\int [\varphi , {\mathbb {P}} _{{\text {curl}}}] u \cdot \partial _i v {\mathrm {d}}x \\&= \int u \cdot [\varphi , {\mathbb {P}} _{{\text {curl}}}] \partial _i v {\mathrm {d}}x \\&\le \Vert u \Vert _{L ^p ({\mathbb {R}}^3)} (\Vert v \Vert _{L ^{q} ({\mathbb {R}}^3)} + \Vert \partial _i \varDelta ^{-1}v\Vert _{L ^{q} ({\text {supp}}\varphi )}) \\&\le \Vert u \Vert _{L ^p ({\mathbb {R}}^3)} (\Vert v \Vert _{L ^{q} ({\mathbb {R}}^3)} + C _{q, \varphi } \Vert \partial _i \varDelta ^{-1}v\Vert _{L ^{q ^*} ({\text {supp}}\varphi )}) \\&\le C \Vert u \Vert _{L ^p ({\mathbb {R}}^3)} \Vert v \Vert _{L ^{q} ({\mathbb {R}}^3)}. \end{aligned}$$

$\square$

Corollary 2

$\partial _i [\varphi , {\mathbb {P}} _{\nabla }]$ and $[\varphi , {\mathbb {P}} _{\nabla }] \partial _i$ are both bounded linear operator from $L ^p$ to $L ^p$ for any $1< p < \infty$:

Proof

${\text {Id}}= {\mathbb {P}} _{\nabla }+ {\mathbb {P}} _{{\text {curl}}}$ commutes with $\varphi$, so $[\varphi , {\mathbb {P}} _{\nabla }] = -[\varphi , {\mathbb {P}} _{{\text {curl}}}]$. $\square$

Because of the smoothing effect of the Laplace potential, we have the following.

Lemma 4

Let $\varphi \in C _c ^\infty ({\mathbb {R}}^3)$ be supported away from some openset ${\varOmega } \subset {\mathbb {R}}^3$, that is, ${\text {dist}} ({\text {supp}}\varphi , {\varOmega }) = d > 0$. Then for any $f \in L ^1 _{\mathrm {loc}} ({\mathbb {R}}^3)$, $k > 0$,

$$\begin{aligned} \Vert \varDelta ^{-1}(\varphi f) \Vert _{C ^k ({\varOmega })} \lesssim _{k, d} \Vert f \Vert _{L ^1 ({\text {supp}}\varphi )}. \end{aligned}$$

We also have

$$\begin{aligned} \Vert {\mathbb {P}} _{\nabla }(\varphi f) \Vert _{C ^k ({\varOmega })}, \Vert {\mathbb {P}} _{{\text {curl}}}(\varphi f) \Vert _{C ^k ({\varOmega })} \lesssim _{k, d} \Vert f \Vert _{L ^1 ({\text {supp}}\varphi )}. \end{aligned}$$

3 Proof of the Main Results

In this section, we show that the Local Theorem 2 leads to the main results. First, we show the pivot quantity is indeed enough to bound $\nabla ^n \omega$.

Lemma 5

There exists $\eta _2 > 0$ such that the following holds. Let $\frac{11}{6}< p < 2$, $\frac{2-p}{p-1} < \nu \le \frac{7p-12}{6-p}$. If u is a suitable solution to the Navier–Stokes equations in $(-9, 0) \times {\mathbb {R}}^3$ satisfying the condition

$$\begin{aligned}&\int _{B _1} u (t, x) \phi (x) {\mathrm {d}}x = 0, \qquad {\mathrm{a.e. }}\, t \in (-9, 0), \end{aligned}$$

(16)

$$\begin{aligned}&\delta ^{-\nu } \left( \int _{Q _3} |\nabla u| ^p {\mathrm {d}}x {\mathrm {d}}t\right) ^\frac{1}{p} \le \eta _2, \end{aligned}$$

(17)

$$\begin{aligned}&\delta \int _{Q _3} |\nabla u| ^{2} {\mathrm {d}}x {\mathrm {d}}t \le \eta _2, \end{aligned}$$

(18)

for some $\delta \le \eta _2$,

then we have for any $n \ge 0$,

$$\begin{aligned} \Vert \nabla ^n \omega \Vert _{L ^\infty _{t, x} (Q _{8 ^{-n-2}})} \le C _n. \end{aligned}$$

Here $C _n$ is the same constant in Theorem 2.

Proof

First, we claim that

$$\begin{aligned} \delta \Vert \omega \Vert _{L ^\infty (-4, 0; L ^1 (B _2))} \le C \eta _2. \end{aligned}$$

(19)

Formally, we can take the dot product of both sides of the vorticity equation (1) with $\omega ^0 :{=} \frac{\omega }{|\omega |}$, and recalling the convexity inequality $\omega ^0 \cdot \varDelta \omega \le \varDelta |\omega |$, we have

$$\begin{aligned} (\partial _t+ u \cdot \nabla - \varDelta ) |\omega | - \omega \cdot \nabla u \cdot \omega ^ 0 \le 0. \end{aligned}$$

(20)

Let $\psi \in C _c ^\infty ((-9, 0] \times {\mathbb {R}}^3)$ be a cut-off function such that $\mathbf{1} _{Q _2} \le \psi \le \mathbf{1} _{Q _3}$. Multiply (20) by $\psi$ and then integrate in space to get

$$\begin{aligned} \frac{{\mathrm {d}}}{{\mathrm {d}}t} \int \psi |\omega | {\mathrm {d}}x&\le \int \left[ (\partial _t+ u \cdot \nabla + \varDelta ) \psi \right] |\omega | {\mathrm {d}}x + \int \psi \omega \cdot \nabla u \cdot \omega ^0 {\mathrm {d}}x \\&\le C \int _{B _3} 1 + |u| ^2 + |\nabla u| ^2 {\mathrm {d}}x \le C \left( 1 + \int _{B _3} |\nabla u| ^2 {\mathrm {d}}x\right) \end{aligned}$$

for some large universal constant $C > 1$. The last step uses Poincaré’s inequaliting and (16). Integrate in time, we obtain

$$\begin{aligned} \Vert \omega \Vert _{L ^\infty (-4, 0; L ^1 (B _2))} \le C \left( 1 + \frac{\eta _2}{\delta }\right) \le 2 C \frac{\eta _2}{\delta }. \end{aligned}$$

This proves the claim. A more rigorous proof can be obtained by difference quotient same as in Constantin [6] or Lions [13] Theorem 3.6, so we omit the details.

Now we interpolate between (17) and (19). Let $\theta = \frac{1}{1+\nu }$,

$$\begin{aligned} \Vert \omega \Vert _{L ^{p _2} _t L ^{q _2} _x (Q _2)} \le \Vert \omega \Vert _{L ^p (Q _2)} ^\theta \Vert \omega \Vert _{L ^\infty _t L ^1 _x (Q _2)} ^{1 - \theta } \le (2 C) ^{1-\theta } \eta _2 \delta ^{\theta \nu + \theta - 1} \le \frac{1}{2} \eta _1, \end{aligned}$$

where we choose $\eta _2 = \frac{\eta _1}{4C + 1} \le \frac{1}{2} \eta _1$ from Theorem 2, and $p _2, q _2$ are determined by

$$\begin{aligned} \frac{1}{p _2} = \frac{\theta }{p}, \qquad \frac{1}{q _2} = \frac{\theta }{p} + 1 - \theta . \end{aligned}$$

Combining the above with (17) we have

$$\begin{aligned} \Vert \nabla u \Vert _{L ^p _t L ^p _x (Q _2)} + \Vert \omega \Vert _{L ^{p _2} _t L ^{q _2} _x (Q _2)} \le \frac{1}{2} \eta _1 + \frac{1}{2} \eta _1 = \eta _1. \end{aligned}$$

(21)

By the choice of $\theta$ and the range of $\nu$,

$$\begin{aligned} \frac{1}{p} + \frac{1}{p _2}&= \frac{1}{p} + \frac{1}{p (1 + \nu )} = \frac{2 + \nu }{p (1 + \nu )} < 1, \\ \frac{1}{p} + \frac{1}{q _2}&= \frac{1}{p} + \frac{1 + \nu p}{p (1 + \nu )} = \frac{2 + \nu + \nu p}{p (1 + \nu )} \le \frac{7}{6}. \end{aligned}$$

One can also easily check that $p < 2$ implies $q _2 < 2$, and thus by (16) and (21) the requirements of the Local Theorem 2 are satisfied with $p _1 = q _1 = p$, and it completes the proof of the lemma. $\square$

Now we transform this lemma into the global coordinate. Recall that $Q _\varepsilon (t, x)$ is defined by (7).

Corollary 3

There exists $\eta _3 > 0$ such that, if for some $\delta \le \eta _2$,

$$\begin{aligned} \delta ^{-2\nu } \left( \fint _{Q _\varepsilon (t, x)} |\nabla u| ^p {\mathrm {d}}x {\mathrm {d}}t \right) ^\frac{2}{p} + \delta \fint _{Q _\varepsilon (t, x)} |\nabla u| ^2 {\mathrm {d}}x {\mathrm {d}}t \le \eta _3 \varepsilon ^{-4}, \end{aligned}$$

(22)

then

$$\begin{aligned} |\nabla ^n \omega (t, x)| \le C _n \varepsilon ^{-n-2}. \end{aligned}$$

Proof

Define ${{\tilde{u}}}$ by (3). Then (22) implies

$$\begin{aligned}&\delta ^{-2\nu } \left( \fint _{Q _3} |\nabla {{\tilde{u}}}| ^p {\mathrm {d}}x {\mathrm {d}}t \right) ^\frac{2}{p} \le \eta _3, \qquad \delta \fint _{Q _3} |\nabla {{\tilde{u}}}| ^2 {\mathrm {d}}x {\mathrm {d}}t \le \eta _3 \\&\quad \Rightarrow \delta ^{-\nu } \left( \int _{Q _3} |\nabla {{\tilde{u}}}| ^p {\mathrm {d}}x {\mathrm {d}}t \right) ^\frac{1}{p} \le \eta _3 ^\frac{1}{2} |Q _3| ^\frac{1}{p}, \qquad \delta \int _{Q _3} |\nabla {{\tilde{u}}}| ^2 {\mathrm {d}}x {\mathrm {d}}t \le \eta _3 |Q _3|. \end{aligned}$$

Moreover, (16) is satisfied by ${{\tilde{u}}}$. Therefore, if we choose $\eta _3$ such that

$$\begin{aligned} \max \left\{ \eta _3 ^\frac{1}{2} |Q _3| ^\frac{1}{p}, \eta _3 |Q _3| \right\} = \eta _2, \end{aligned}$$

then by Lemma 5, ${{\tilde{\omega }}} :{=} {\text {curl}}{{\tilde{u}}}$ has bounded derivatives at (0, 0), and thus finish the proof of the corollary by scaling. $\square$

Then we use the maximal function to go from the local bound to a global bound.

Proof of Theorem 1

First, we fix $\frac{11}{6}< p < 2$, $\frac{2-p}{p-1} < \nu \le \frac{7p-12}{6-p}$. Let $\eta<< 1$ be a small constant to be specified later. Finally we fix a $0< \delta < \infty$. For $(t, x) \in (0, T) \times {\mathbb {R}}^3$, define

$$\begin{aligned} I (\varepsilon ) = \varepsilon ^4 \left[ \delta ^{-2\nu } \left( \fint _{Q _{\varepsilon } (t, x)} |{\mathcal {M}}(\nabla u)| ^p \right) ^\frac{2}{p} + \delta \fint _{Q _\varepsilon (t, x)} |{\mathcal {M}}(\nabla u)| ^2 \right] . \end{aligned}$$

If (t, x) is both a ${\mathcal {Q}}$-Lebesgue point of $|{\mathcal {M}}(\nabla u)| ^p$ and of $|{\mathcal {M}}(\nabla u)| ^2$, then we claim that there exists a positive $\varepsilon = \varepsilon _{(t, x)}$ such that one of two cases is true:

• Case 1. ${3 \varepsilon _{(t, x)} < t ^\frac{1}{2}}$, and ${I (\varepsilon _{(t, x)}) = \eta }$.

• Case 2. ${3 \varepsilon _{(t, x)} = t ^\frac{1}{2}}$, and ${I (\varepsilon _{(t, x)}) \le \eta }$.

This is because, by Theorem 4,

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0} I (\varepsilon ) = 0 ^4 \left[ \delta ^{-2 \nu } (|{\mathcal {M}}\left( |\nabla u|\right) (t, x)| ^p) ^\frac{2}{p} + \delta |{\mathcal {M}}\left( |\nabla u|\right) (t, x)| ^2 \right] = 0, \end{aligned}$$

and $I (\varepsilon )$ is a continuous function of $\varepsilon$.

On the one hand, in both cases we have $I (\varepsilon ) \le \eta$, which implies that

$$\begin{aligned} \delta ^{-\nu } \varepsilon ^2 \left( \fint _{Q _{\varepsilon } (t, x)} |{\mathcal {M}}(\nabla u)| ^p \right) ^\frac{1}{p} \le \sqrt{\eta }, \qquad \delta ^\frac{1}{2} \varepsilon ^2 \left( \fint _{Q _\varepsilon (t, x)} |{\mathcal {M}}(\nabla u)| ^2 \right) ^\frac{1}{2} \le \sqrt{\eta }. \end{aligned}$$

If we set $\eta < \eta _0 ^2$, then depending on $\delta \ge 1$ or $\delta \le 1$, one of the two would imply admissibility condition (8) by Jensen’s inequality. Therefore $Q _\varepsilon (t, x)$ is admissible and

$$\begin{aligned} I (\varepsilon ) \le \varepsilon ^4 \left[ \delta ^{-2\nu } {\mathcal {M}}_{\mathcal {Q}}({\mathcal {M}}(\nabla u) ^p) ^\frac{2}{p} + \delta {\mathcal {M}}_{\mathcal {Q}}({\mathcal {M}}(\nabla u) ^2) \right] , \end{aligned}$$

so we can combine two cases and conclude

$$\begin{aligned} \varepsilon ^{-4} _{(t, x)} \le \max \left\{ \frac{1}{\eta }\left[ \delta ^{-2\nu } {\mathcal {M}}_{\mathcal {Q}}({\mathcal {M}}(\nabla u) ^p) ^\frac{2}{p} + \delta {\mathcal {M}}_{\mathcal {Q}}({\mathcal {M}}(\nabla u) ^2) \right] , 81 t ^{-2} \right\} . \end{aligned}$$

(23)

On the other hand, if we set $\eta < \eta _3$, then in both cases $I (\varepsilon ) \le \eta _3$. If $\delta \le \eta _2$ one would have

$$\begin{aligned} |\nabla ^n \omega (t, x)| \le C _n \varepsilon ^{-n-2} \end{aligned}$$

(24)

by Corollary 3. If $\delta > \eta _2$, notice that by Jensen’s inequality,

$$\begin{aligned} \left( \fint _{Q _{\varepsilon } (t, x)} |{\mathcal {M}}(\nabla u)| ^p \right) ^\frac{2}{p} \le \fint _{Q _\varepsilon (t, x)} |{\mathcal {M}}(\nabla u)| ^2, \end{aligned}$$

so

$$\begin{aligned} I (\varepsilon )&\ge \varepsilon ^4 \left[ (\delta ^{-2\nu } + \delta - \eta _2) \left( \fint _{Q _{\varepsilon } (t, x)} |{\mathcal {M}}(\nabla u)| ^p \right) ^\frac{2}{p} + \eta _2 \fint _{Q _\varepsilon (t, x)} |{\mathcal {M}}(\nabla u)| ^2 \right] \\&\ge \varepsilon ^4 \left[ (1 - \eta _2) \left( \fint _{Q _{\varepsilon } (t, x)} |{\mathcal {M}}(\nabla u)| ^p \right) ^\frac{2}{p} + \eta _2 \fint _{Q _\varepsilon (t, x)} |{\mathcal {M}}(\nabla u)| ^2 \right] \\&\ge (1 - \eta _2) \eta _2 ^{2 \nu } \varepsilon ^4 \left[ \eta _2 ^{-2\nu } \left( \fint _{Q _{\varepsilon } (t, x)} |{\mathcal {M}}(\nabla u)| ^p \right) ^\frac{2}{p} + \eta _2 \fint _{Q _\varepsilon (t, x)} |{\mathcal {M}}(\nabla u)| ^2 \right] . \end{aligned}$$

If we require $\eta < (1 - \eta _2) \eta _2 ^{2 \nu } \eta _3$, then

$$\begin{aligned} \varepsilon ^4 \left[ \eta _2 ^{-2\nu } \left( \fint _{Q _{\varepsilon } (t, x)} |{\mathcal {M}}(\nabla u)| ^p \right) ^\frac{2}{p} + \eta _2 \fint _{Q _\varepsilon (t, x)} |{\mathcal {M}}(\nabla u)| ^2 \right] \le \eta _3. \end{aligned}$$

Again by Corollary 3, we would still have (24). In conclusion, we choose

$$\begin{aligned} \eta = \min \left\{ \eta _0 ^2, (1 - \eta _2) \eta _2 ^{2 \nu } \eta _3 \right\} . \end{aligned}$$

Then for any $0< \delta < \infty$ one would have

$$\begin{aligned} |\nabla ^n \omega (t, x)| ^\frac{4}{n+2} \le C _n ^\frac{4}{n+2} \max \left\{ \frac{1}{\eta }\left[ \delta ^{-2\nu } {\mathcal {M}}_{\mathcal {Q}}({\mathcal {M}}(\nabla u) ^p) ^\frac{2}{p} + \delta {\mathcal {M}}_{\mathcal {Q}}({\mathcal {M}}(\nabla u) ^2) \right] , 81 t ^{-2} \right\} , \end{aligned}$$

by putting (24) and (23) together. Denote $f = |\nabla ^n \omega | ^\frac{4}{n+2}$, and we denote $f _1 = {\mathcal {M}}_{\mathcal {Q}}({\mathcal {M}}(|\nabla u|) ^p) ^\frac{2}{p}$, $f _2 = {\mathcal {M}}_{\mathcal {Q}}({\mathcal {M}}(|\nabla u|) ^2)$. Then we have almost everywhere

$$\begin{aligned} f \mathbf{1} _{\{f > C _n t ^{-2}\}} \lesssim _n \delta ^{-2 \nu } f _1 + \delta f _2. \end{aligned}$$

By Theorem 3,

$$\begin{aligned} \Vert f _1 \Vert _{L ^1}&\le C _p \Vert {\mathcal {M}}(\nabla u) ^p \Vert _{L ^\frac{2}{p}} ^\frac{2}{p} \lesssim C _p \Vert {\mathcal {M}}(\nabla u) ^2 \Vert _{L ^1} \le C _p \Vert \nabla u\Vert _{L ^2} ^2, \\ \Vert f _2 \Vert _{L ^{1, \infty }}&\le C _1 \Vert {\mathcal {M}}(\nabla u) ^2 \Vert _{L ^1} \le C _1 \Vert \nabla u\Vert _{L ^2} ^2. \end{aligned}$$

Finally, by the interpolation between Lorentz spaces Lemma 1,

$$\begin{aligned} \Vert f \mathbf{1} _{\{f > C _n t ^{-2}\}} \Vert _{L ^{1, 1 + 2 \nu }} \lesssim _{p, n} \Vert \nabla u \Vert _{L ^2 ((0, T) \times {\mathbb {R}}^3)} ^2 \le \Vert u _0 \Vert _{L ^2 ({\mathbb {R}}^3)} ^2. \end{aligned}$$

This proves the theorem for $q \ge 1 + 2 \nu$. Recall that p can be arbitrarily chosen between $\frac{11}{6}$ and 2, and $\nu$ can be chosen between $\frac{2-p}{p-1}$ and $\frac{7p-12}{6-p}$, so $\nu$ can be arbitrarily small, therefore we prove the theorem for any $q > 1$. $\square$

Estimates on $\nabla ^2 u$ can be obtained by a Riesz transform of $\varDelta u = -{\text {curl}}\omega$.

Proof of Corollary 1

We can put $K \subset (t _0, T) \times B _R$ for some $t _0, T, R > 0$. Denote $Q = (t _0, T) \times B _{2R}$. Let $\rho \in C _c ^\infty ({\mathbb {R}}^3)$ be a smooth spatial cut-off function between $\mathbf{1} _{B _R} \le \rho \le \mathbf{1} _{B _{2R}}$. Then

$$\begin{aligned}&\Vert \varDelta (\rho u) \Vert _{L ^{\frac{4}{3},q} ((t _0, T) \times {\mathbb {R}}^3)} \lesssim _\rho \Vert \varDelta u \Vert _{L ^{\frac{4}{3},q} (Q)} + \Vert \nabla u \Vert _{L ^{\frac{4}{3},q} (Q)} + \Vert u \Vert _{L ^{\frac{4}{3},q} (Q)}. \end{aligned}$$

Since $\varDelta u = -{\text {curl}}\omega$, the case $n = 1$ of Theorem 1 gives

$$\begin{aligned} \Vert \varDelta u \mathbf{1} _{\{|\varDelta u| > C _1 t ^{-\frac{3}{2}}\}} \Vert _{L ^{\frac{4}{3}, q} ((0, T) \times {\mathbb {R}}^3)} \le C _q \Vert u _0 \Vert _{L ^2 ({\mathbb {R}}^3)} ^\frac{3}{2}, \end{aligned}$$

so

$$\begin{aligned} \Vert \varDelta u \Vert _{L ^{\frac{4}{3},q} (Q)} \le C _q \Vert u _0 \Vert _{L ^2 ({\mathbb {R}}^3)} ^\frac{3}{2} + C _1 \Vert t ^{-\frac{3}{2}} \Vert _{L ^{\frac{4}{3}} (Q)} \lesssim C _q \Vert u _0 \Vert _{L ^2 ({\mathbb {R}}^3)} ^\frac{3}{2} + C _1 \left( \frac{R ^3}{t _0}\right) ^\frac{3}{4}. \end{aligned}$$

As for lower order terms,

$$\begin{aligned} \Vert \nabla u \Vert _{L ^\frac{4}{3} (Q)}&\lesssim \Vert \nabla u \Vert _{L ^2 (Q)}, \\ \Vert u \Vert _{L ^\frac{4}{3} (Q)}&\le \Vert u \Vert _{L ^\infty _t L ^2 _x (Q)}. \end{aligned}$$

For Leray-Hopf solution, $\Vert \nabla u\Vert _{L ^\infty _t L ^2 _x \cap L ^2 _t \dot{H} ^1 _x ((0, T) \times {\mathbb {R}}^3)} \le \Vert u _0\Vert _{L ^2}$, so

$$\begin{aligned} \Vert \varDelta (\rho u) \Vert _{L ^{\frac{4}{3},q} ((t _0, T) \times {\mathbb {R}}^3)}&\lesssim _{q, K} \Vert u _0 \Vert _{L ^2 ({\mathbb {R}}^3)} ^\frac{3}{2} + 1 + \Vert u _0 \Vert _{L ^2 ({\mathbb {R}}^3)} \lesssim \Vert u _0 \Vert _{L ^2 ({\mathbb {R}}^3)} ^\frac{3}{2} + 1. \end{aligned}$$

Because Riesz transform is bounded from $L ^{\frac{4}{3}, q} ((t _0, T) \times {\mathbb {R}}^3)$ to itself by Lemma 2,

$$\begin{aligned} \Vert \nabla ^2 u \Vert _{L ^{\frac{4}{3},q} (K)} \le \Vert \nabla ^2 (\rho u) \Vert _{L ^{\frac{4}{3},q} (Q)} \lesssim _{q, K} \Vert u _0 \Vert _{L ^2 ({\mathbb {R}}^3)} ^\frac{3}{2} + 1. \end{aligned}$$

$\square$

Remark 1

For smooth solutions to the Navier–Stokes equation, we have $L ^{1, q}$ estimate for the third derivatives for any $q > 1$,

$$\begin{aligned} \left\| \nabla ^2 \omega \mathbf{1} _{\{|\nabla ^2 \omega | > C t ^{-2}\}} \right\| _{L ^{1, q} ((0, T) \times {\mathbb {R}}^3)} \le C _q \Vert u _0 \Vert _{L ^2} ^2. \end{aligned}$$

4 Local Study: Part One, Initial Energy

The next three sections are dedicated to the proof of the Local Theorem 2. In [22], the proof of the local theorem consists of the following three parts:

• Step 1. Show the velocity u is locally small in the energy space ${\mathcal {E}} = L ^{\infty } _t L ^2 _x \cap L ^2 _t H ^1 _x$.

• Step 2. Use De Giorgi iteration and the truncation method developed in [23] to show u is locally bounded in $L ^\infty$.

• Step 3. Bootstrap to higher regularity by differentiating the original equation.

In our case, directly working with u is difficult due to the lack of control on the pressure, which is nonlocal. Therefore, we would like to work on vorticity, whose evolution is governed by (6) and only involves local quantities. Since $\omega$ is one derivative of u, we have less integrability to do any parabolic regularization, and we don’t have the local energy inequality to perform De Giorgi iteration. This motivates us to work on minus one derivative of $\omega$, but instead of $\omega$ we use a localization of $\omega$. Similar as [4], we introduce a new local quantity

$$\begin{aligned} \boxed { v :{=} -{\text {curl}}\varphi ^\sharp \varDelta ^{-1}\varphi {\text {curl}}u = -{\text {curl}}\varphi ^\sharp \varDelta ^{-1}\varphi \omega . }, \end{aligned}$$

where $\varphi$ and $\varphi ^\sharp$ are a pair of fixed smooth spatial cut-off functions, which are defined between $\mathbf{1} _{B _\frac{6}{5}} \le \varphi \le \mathbf{1} _{B _\frac{5}{4}}$, $\mathbf{1} _{B _\frac{4}{3}} \le \varphi ^\sharp \le \mathbf{1} _{B _\frac{3}{2}}$. This v is divergence free and compactly supported. It will help us get rid of the pressure P, while staying in the same space as u: it scales the same as u, has the same regularity, inherit a local energy inequality from u, and its evolution only depends on local information. We will follow the same three steps above, but we will work on v instead of u.

For convenience, from now on we will use $\eta$ to denote a small universal constant depending only on the smallness of $\eta _1$, such that $\lim _{\eta _1 \rightarrow 0} \eta = 0$. Similar as the constant C, the value of $\eta$ may change from line to line. The purpose of this section is to obtain the smallness of v in the energy space ${\mathcal {E}}$, which is the following proposition:

Proposition 1

Under the same assumptions of the Local Theorem 2, we have

$$\begin{aligned} \Vert v \Vert _{{\mathcal {E}} (Q _1)} ^2 = \sup _{t \in (-1, 0)} \int _{B _1} |v (t)| ^2 {\mathrm {d}}x + \int _{Q _1} |\nabla v| ^2 {\mathrm {d}}x \le \eta . \end{aligned}$$

(25)

For convenience, define $q _3$, $q _4$, $q _5$ by

$$\begin{aligned} \frac{1}{q _3} = \frac{1}{q _1} - \frac{1}{3}, \qquad \frac{1}{q _4} = \frac{1}{q _2} - \frac{1}{3}, \qquad \frac{1}{q _5} = \left( \frac{1}{q _3} - \frac{1}{3}\right) _+. \end{aligned}$$

4.1 Equations of v

We use (10) in (1) to rewrite the equation of u, then take the curl to rewrite the equation of $\omega$, finally apply $-{\text {curl}}\varphi ^\sharp \varDelta ^{-1}\varphi$ on the vorticity equation to obtain the equation of v:

$$\begin{aligned} \partial _tu + {\mathbb {P}} _{{\text {curl}}}(\omega \times u)&= \varDelta u, \nonumber \\ \partial _t\omega + {\text {curl}}(\omega \times u)&= \varDelta \omega , \nonumber \\ \partial _tv - {\text {curl}}\varphi ^\sharp \varDelta ^{-1}\varphi {\text {curl}}(\omega \times u)&= - {\text {curl}}\varphi ^\sharp \varDelta ^{-1}\varphi \varDelta \omega . \end{aligned}$$

(26)

The second term of (26) is

$$\begin{aligned} {\text {curl}}\varphi ^\sharp \varDelta ^{-1}\varphi {\text {curl}}(\omega \times u)&= {\mathbf {B}}- {\mathbb {P}} _{{\text {curl}}}(\varphi \omega \times u), \end{aligned}$$

where ${\mathbf {B}}$ denotes the quadratic commutator

$$\begin{aligned} {\mathbf {B}}&:{=} - {\text {curl}}(1 - \varphi ^\sharp ) \varDelta ^{-1}\varphi {\text {curl}}(\omega \times u) + {\text {curl}}\varDelta ^{-1}[\varphi , {\text {curl}}] (\omega \times u) \\&= - {\text {curl}}(1 - \varphi ^\sharp ) \varDelta ^{-1}\varphi {\text {curl}}(\omega \times u) + {\text {curl}}\varDelta ^{-1}(-\nabla \varphi \times (\omega \times u)). \end{aligned}$$

Here we used (12). The right hand side of (26) is

$$\begin{aligned} -{\text {curl}}\varphi ^\sharp \varDelta ^{-1}\varphi \varDelta \omega&= \varDelta v + {\mathbf {L}}, \end{aligned}$$

where ${\mathbf {L}}$ denotes the linear commutator

$$\begin{aligned} {\mathbf {L}}&:{=} [-{\text {curl}}\varphi ^\sharp \varDelta ^{-1}\varphi , \varDelta ] \omega \\&= -{\text {curl}}[\varphi ^\sharp \varDelta ^{-1}\varphi , \varDelta ] \omega \\&= -{\text {curl}}[\varphi ^\sharp , \varDelta ] \varDelta ^{-1}\varphi \omega -{\text {curl}}\varphi ^\sharp \varDelta ^{-1}[\varphi , \varDelta ] \omega \\&= -{\text {curl}}[\varphi ^\sharp , \varDelta ] \varDelta ^{-1}\varphi \omega + {\text {curl}}\varphi ^\sharp \varDelta ^{-1}\left( 2 {\text {div}}(\nabla \varphi \otimes \omega ) - (\varDelta \varphi ) \omega \right) . \end{aligned}$$

Here we used (13). Therefore we have the equation for v as the following:

$$\begin{aligned} \partial _tv + {\mathbb {P}} _{{\text {curl}}}(\varphi \omega \times u) = {\mathbf {B}}+ {\mathbf {L}}+ \varDelta v. \end{aligned}$$

(27)

We observe the following localization decomposition:

Lemma 6

We can decompose

$$\begin{aligned} \varphi u = v + w, \qquad \varphi \omega = {\text {curl}}v + \varpi , \end{aligned}$$

where w and $\varpi$ are harmonic inside $B _1$.

Proof

We can compute v by

$$\begin{aligned} v&= -{\text {curl}}\varphi ^\sharp \varDelta ^{-1}\varphi {\text {curl}}u \\&= {\text {curl}}(1 - \varphi ^\sharp ) \varDelta ^{-1}\varphi {\text {curl}}u - {\text {curl}}\varDelta ^{-1}\varphi {\text {curl}}u \\&= {\text {curl}}(1 - \varphi ^\sharp ) \varDelta ^{-1}\varphi \omega -{\text {curl}}\varDelta ^{-1}[\varphi , {\text {curl}}] u + {\mathbb {P}} _{{\text {curl}}}(\varphi u) \\&= {\text {curl}}(1 - \varphi ^\sharp ) \varDelta ^{-1}\varphi \omega + {\text {curl}}\varDelta ^{-1}(\nabla \varphi \times u) - {\mathbb {P}} _{\nabla }(\varphi u) + \varphi u \\&= {\text {curl}}(1 - \varphi ^\sharp ) \varDelta ^{-1}\varphi \omega + {\text {curl}}\varDelta ^{-1}(\nabla \varphi \times u) - \nabla \varDelta ^{-1}(\nabla \varphi \cdot u) + \varphi u, \end{aligned}$$

using ${\text {div}}u = 0$. We denote

$$\begin{aligned} w :{=} -{\text {curl}}(1 - \varphi ^\sharp ) \varDelta ^{-1}\varphi \omega -{\text {curl}}\varDelta ^{-1}(\nabla \varphi \times u) + \nabla \varDelta ^{-1}(\nabla \varphi \cdot u), \end{aligned}$$

which implies the first decomposition $\varphi u = v + w$. By taking the curl,

$$\begin{aligned} {\text {curl}}(\varphi u)&= {\text {curl}}v + {\text {curl}}w, \\ \nabla \varphi \times u + \varphi \omega&= {\text {curl}}v - {\text {curl}}{\text {curl}}(1 - \varphi ^\sharp ) \varDelta ^{-1}\varphi \omega - {\text {curl}}{\text {curl}}\varDelta ^{-1}(\nabla \varphi \times u) \\&= {\text {curl}}v - {\text {curl}}{\text {curl}}(1 - \varphi ^\sharp ) \varDelta ^{-1}\varphi \omega + {\mathbb {P}} _{{\text {curl}}}(\nabla \varphi \times u). \end{aligned}$$

We denote

$$\begin{aligned} \varpi&:{=} - {\text {curl}}{\text {curl}}(1 - \varphi ^\sharp ) \varDelta ^{-1}\varphi \omega - {\mathbb {P}} _{\nabla }(\nabla \varphi \times u) \\&= - {\text {curl}}{\text {curl}}(1 - \varphi ^\sharp ) \varDelta ^{-1}\varphi \omega - \nabla \varDelta ^{-1}{\text {div}}(\nabla \varphi \times u) \\&= - {\text {curl}}{\text {curl}}(1 - \varphi ^\sharp ) \varDelta ^{-1}\varphi \omega + \nabla \varDelta ^{-1}(\nabla \varphi \cdot \omega ), \end{aligned}$$

which implies the second decomposition $\varphi \omega = {\text {curl}}v + \varpi$. We can easily see that $\varDelta w$ and $\varDelta \varpi$ are both the sum of a smooth function supported outside $B _\frac{3}{2}$ and the Newtonian potential of something supported inside ${\text {supp}} (\nabla \varphi ) \subset B _{\frac{5}{4}} \setminus B _{\frac{6}{5}}$, so they are harmonic inside $B _1$. $\square$

Using this decomposition, we can continue to expand

$$\begin{aligned} {\mathbb {P}} _{{\text {curl}}}(\varphi \omega \times u)&= \varphi \omega \times u - {\mathbb {P}} _{\nabla }(\varphi \omega \times u) \\&= \omega \times v + \omega \times w - \frac{1}{2} {\mathbb {P}} _{\nabla }\left( ({\text {curl}}v + \varpi ) \times u + \omega \times (v + w) \right) \\&= \omega \times v - \frac{1}{2} {\mathbb {P}} _{\nabla }\left( {\text {curl}}v \times u + \omega \times v \right) - {\mathbf {W}}, \end{aligned}$$

where ${\mathbf {W}}$ denotes the remainders involving w and $\varpi$,

$$\begin{aligned} {\mathbf {W}}:{=} -\omega \times w + \frac{1}{2} {\mathbb {P}} _{\nabla }\left( \varpi \times u + \omega \times w \right) . \end{aligned}$$

By subtracting (11) from (10), for divergence free u, v we have

$$\begin{aligned} {\text {curl}}v \times u + {\text {curl}}u \times v = -\nabla (u \cdot v) + 2 u \cdot \nabla v + {\text {curl}}(u \times v), \end{aligned}$$

so

$$\begin{aligned} {\mathbb {P}} _{{\text {curl}}}(\varphi \omega \times u)&= \omega \times v + \frac{1}{2} \nabla (u \cdot v) - {\mathbb {P}} _{\nabla }{\text {div}}(u \otimes v) - {\mathbf {W}}\\&= \omega \times v + \nabla \left( \frac{1}{2} u \cdot v - \varDelta ^{-1}{\text {div}}{\text {div}}(u \otimes v) \right) - {\mathbf {W}}. \end{aligned}$$

For convenience, denote the Riesz operator

$$\begin{aligned} {\mathbf {R}}= \frac{1}{2} {\text {tr}}- \varDelta ^{-1}{\text {div}}{\text {div}}\end{aligned}$$

Finally, we have the equation of v as

$$\begin{aligned} \partial _tv + \omega \times v + \nabla {\mathbf {R}}(u \otimes v) = {\mathbf {B}}+ {\mathbf {L}}+ {\mathbf {W}}+ \varDelta v, \qquad {\text {div}}v = 0. \end{aligned}$$

(28)

We now check the spatial integrability of these new terms.

Lemma 7

For any $1< p < \infty$,

$$\begin{aligned}&\Vert v \Vert _{L ^p}, \Vert \nabla w \Vert _{L ^p}, \Vert \varpi \Vert _{L ^p} \lesssim \Vert \omega \Vert _{L ^1 (B _2)} + \Vert u \Vert _{L ^p (B _2)}, \\&\Vert \nabla v \Vert _{L ^p}, \Vert \nabla \varpi \Vert _{L ^p} \lesssim \Vert \omega \Vert _{L ^p (B _2)}, \\&\Vert \nabla ^2 w \Vert _{L ^p} \lesssim \Vert u \Vert _{W ^{1,p}} (B _2). \end{aligned}$$

If we denote $q = (\frac{1}{p} - \frac{1}{3}) _+ ^{-1}$, then

$$\begin{aligned} \Vert {\mathbf {B}}\Vert _{L ^q (B _2)}&\lesssim \Vert \omega \times u \Vert _{L ^p (B _2)}, \\ \Vert {\mathbf {L}}\Vert _{L ^p (B _2)}&\lesssim \Vert \omega \Vert _{L ^p (B _2)} , \\ \Vert {\mathbf {W}}\Vert _{L ^p (B _2)}&\lesssim \Vert \omega \times w \Vert _{L ^p (B _2)} + \Vert \varpi \times u \Vert _{L ^p (B _2)} . \end{aligned}$$

Proof

$v, w, \varpi$ are all supported inside $B _2$, so

$$\begin{aligned} \Vert v \Vert _{L ^p}&\le \Vert \varphi u \Vert _{L ^p} + \Vert w \Vert _{L ^p} \lesssim \Vert u \Vert _{L ^p (B _2)} + \Vert \nabla w \Vert _{L ^p}, \\ \Vert \nabla w \Vert _{L ^p}&\le \Vert \nabla {\text {curl}}(1 - \varphi ^\sharp ) \varDelta ^{-1}\varphi \omega \Vert _{L ^p (B _2)} + \Vert \nabla {\text {curl}}\varDelta ^{-1}(\nabla \varphi \times u) \Vert _{L ^p} \\&\quad + \Vert \nabla ^2 \varDelta ^{-1}(\nabla \varphi \cdot u) \Vert _{L ^p} \\&\lesssim \Vert (1 - \varphi ^\sharp ) \varDelta ^{-1}\varphi \omega \Vert _{C ^2} + \Vert \nabla \varphi \times u \Vert _{L ^p} + \Vert \nabla \varphi \cdot u \Vert _{L ^p} \\&\lesssim \Vert \omega \Vert _{L ^1 (B _2)} + \Vert u \Vert _{L ^p (B _2)}, \\ \Vert \varpi \Vert _{L ^p}&\le \Vert {\text {curl}}{\text {curl}}(1 - \varphi ^\sharp ) \varDelta ^{-1}\varphi \omega \Vert _{L ^p (B _2)} + \Vert \nabla \varDelta ^{-1}(\nabla \varphi \cdot \omega ) \Vert _{L ^p} \\&\le \Vert (1 - \varphi ^\sharp ) \varDelta ^{-1}\varphi \omega \Vert _{C ^2} + \Vert \nabla \varDelta ^{-1}{\text {div}}(\nabla \varphi \times u) \Vert _{L ^p} \\&\le \Vert \omega \Vert _{L ^1 (B _2)} + \Vert \nabla \varphi \times u \Vert _{L ^p} \\&\le \Vert \omega \Vert _{L ^1 (B _2)} + \Vert u \Vert _{L ^p (B _2)}. \end{aligned}$$

Here we used Lemma 4 since $\varphi$ and $1 - \varphi ^\sharp$ are supported away from each other, and we also used the boundedness of Riesz transform by Lemma 2. Their derivatives are bounded by

$$\begin{aligned} \Vert \nabla v \Vert _{L ^p}&= \Vert \nabla {\text {curl}}\varphi ^\sharp \varDelta ^{-1}\varphi \omega \Vert _{L ^p} \\&\le \Vert \nabla {\text {curl}}\varDelta ^{-1}\varphi \omega \Vert _{L ^p} + \Vert \nabla {\text {curl}}(1 - \varphi ^\sharp ) \varDelta ^{-1}\varphi \omega \Vert _{L ^p (B _2)} \\&\lesssim \Vert \omega \Vert _{L ^p (B _2)} + \Vert \omega \Vert _{L ^1 (B _2)} \lesssim \Vert \omega \Vert _{L ^p (B _2)}, \\ \Vert \nabla ^2 w \Vert _{L ^p}&\le \Vert \nabla ^2 {\text {curl}}(1 - \varphi ^\sharp ) \varDelta ^{-1}\varphi \omega \Vert _{L ^p (B _2)} + \Vert \nabla ^2 {\text {curl}}\varDelta ^{-1}(\nabla \varphi \times u) \Vert _{L ^p} \\&\qquad + \Vert \nabla ^3 \varDelta ^{-1}(\nabla \varphi \cdot u) \Vert _{L ^p} \\&\lesssim \Vert \omega \Vert _{L ^1 (B _2)} + \Vert u \Vert _{W ^{1, p} (B _2)} \lesssim \Vert u \Vert _{W ^{1, p} (B _2)}, \\ \Vert \nabla \varpi \Vert _{L ^p}&\le \Vert \nabla {\text {curl}}{\text {curl}}(1 - \varphi ^\sharp ) \varDelta ^{-1}\varphi \omega \Vert _{L ^p (B _2)} + \Vert \nabla ^2 \varDelta ^{-1}(\nabla \varphi \cdot \omega ) \Vert _{L ^p} \\&\lesssim \Vert \omega \Vert _{L ^1 (B _2)} + \Vert \omega \Vert _{L ^p (B _2)} \lesssim \Vert \omega \Vert _{L ^p (B _2)}. \end{aligned}$$

The proof for ${\mathbf {B}}, {\mathbf {L}}, {\mathbf {W}}$ are similar, so we omit it here. $\square$

Since $u \in {\mathcal {E}}$ and $\omega \in L ^\infty _t L ^1 _x$, it can be seen from the above lemma that $v, \nabla w, \varpi \in {\mathcal {E}}$, thus

$$\begin{aligned} \Vert {\mathbf {B}}\Vert _{L ^3 (B _2)}&\lesssim \Vert \omega \times u \Vert _{L ^\frac{3}{2} (B _2)} \in L ^1 _t, \\ \Vert {\mathbf {L}}\Vert _{L ^2 (B _2)}&\lesssim \Vert \omega \Vert _{L ^2 (B _2)} \in L ^2 _t, \\ \Vert {\mathbf {W}}\Vert _{L ^\frac{3}{2} (B _2)}&\lesssim \Vert \omega \times w \Vert _{L ^\frac{3}{2} (B _2)} + \Vert \varpi \times u \Vert _{L ^\frac{3}{2} (B _2)} \in L ^2 _t, \end{aligned}$$

therefore ${\mathbf {B}}, {\mathbf {L}}, {\mathbf {W}}\in L ^1 _t L ^3 _{\mathrm {loc}, x} + L ^2 _x L ^\frac{3}{2} _{\mathrm {loc},x}$. In the appendix we prove the suitability for v: it satisfies the following local energy inequality:

$$\begin{aligned} \partial _t\frac{|v| ^2}{2} + |\nabla v| ^2 + {\text {div}}\left[ v {\mathbf {R}}(u \otimes v) \right]&\le \varDelta \frac{|v| ^2}{2} + v \cdot ({\mathbf {B}}+ {\mathbf {L}}+ {\mathbf {W}}). \end{aligned}$$

(29)

4.2 Energy Estimate

Multipling (29) by $\varphi ^4$ then integrating over ${\mathbb {R}}^3$ yields

$$\begin{aligned}&\frac{{\mathrm {d}}}{{\mathrm {d}}t} \int \varphi ^4 \frac{|v| ^2}{2} {\mathrm {d}}x + \int \varphi ^4 |\nabla v| ^2 {\mathrm {d}}x \\&\quad \le \int \frac{|v| ^2}{2} \varDelta \varphi ^4 {\mathrm {d}}x + \int (v \cdot \nabla \varphi ^4) {\mathbf {R}}(u \otimes v) {\mathrm {d}}x \\&\qquad + \int \varphi ^4 v \cdot {\mathbf {B}}{\mathrm {d}}x + \int \varphi ^4 v \cdot {\mathbf {L}}{\mathrm {d}}x + \int \varphi ^4 v \cdot {\mathbf {W}}{\mathrm {d}}x . \end{aligned}$$

Let us discuss these terms. For the first four terms on the right hand side,

$$\begin{aligned} I _\varDelta&:{=} \int \frac{|v| ^2}{2} \varDelta \varphi ^4 {\mathrm {d}}x \le C \Vert \varphi ^2 v\Vert _{L ^2} \Vert v\Vert _{L ^2}, \end{aligned}$$

(30)

$$\begin{aligned} I _{\mathbf {R}}&:{=} \int (v \cdot \nabla \varphi ^4) {\mathbf {R}}(u \otimes v) {\mathrm {d}}x \le C \Vert \varphi ^2 v\Vert _{L ^2} \Vert {\mathbf {R}}(u \otimes v)\Vert _{L ^2} \nonumber \\&\le C \Vert \varphi ^2 v\Vert _{L ^2} \Vert u \otimes v\Vert _{L ^2} , \end{aligned}$$

(31)

$$\begin{aligned} I _{\mathbf {B}}&:{=} \int \varphi ^4 v \cdot {\mathbf {B}}{\mathrm {d}}x \le \Vert \varphi ^2 v \Vert _{L ^2} \Vert \varphi ^2 {\mathbf {B}}\Vert _{L ^2} \nonumber \\&\le C \Vert \varphi ^2 v \Vert _{L ^2} \Vert \omega \times u \Vert _{L ^\frac{6}{5} (B _2)}, \end{aligned}$$

(32)

$$\begin{aligned} I _{\mathbf {L}}&:{=} \int \varphi ^4 v \cdot {\mathbf {L}}{\mathrm {d}}x \le \Vert \varphi ^\frac{2}{3} |v| ^\frac{1}{3} \Vert _{L ^6} \Vert |v| ^\frac{2}{3} \Vert _{L ^{q _3}} \Vert \varphi ^2 {\mathbf {L}}\Vert _{L ^{q _2}} \nonumber \\&\le \Vert \varphi ^2 v \Vert _{L ^2} ^\frac{1}{3} \Vert v \Vert _{L ^{q _3}} ^\frac{2}{3} \Vert \omega \Vert _{L ^{q _2} (B _2)}. \end{aligned}$$

(33)

Here we use Hölder’s inequality, and $\varphi$ is compactly supported in $B _2$ and $\frac{1}{q_2} + \frac{1}{q _3} + \frac{1}{6} \le 1$. For the ${\mathbf {W}}$ term,

$$\begin{aligned} I _{\mathbf {W}}&:{=} \int \varphi ^4 v \cdot {\mathbf {W}}{\mathrm {d}}x \\&= -\int \varphi ^4 v \cdot \omega \times w {\mathrm {d}}x + \frac{1}{2} \int \varphi ^4 v \cdot {\mathbb {P}} _{\nabla }\left( \varpi \times u + \omega \times w \right) {\mathrm {d}}x \\&= -I _{{\mathbf {W}}1} + \frac{1}{2} I _{{\mathbf {W}}2}. \end{aligned}$$

For the first one, we break it as

$$\begin{aligned} I _{{\mathbf {W}}1} = \int \varphi ^4 v \cdot \omega \times w {\mathrm {d}}x&= \int \varphi ^3 v \times {\text {curl}}v \cdot w {\mathrm {d}}x + \int \varphi ^3 v \cdot \varpi \times w {\mathrm {d}}x. \end{aligned}$$

Using (10),

$$\begin{aligned} v \times {\text {curl}}v = \frac{1}{2} \nabla |v| ^2 - (v \cdot \nabla ) v, \end{aligned}$$

we have

$$\begin{aligned} \int \varphi ^3 v \times {\text {curl}}v \cdot w {\mathrm {d}}x&= -\frac{1}{2} \int |v| ^2 {\text {div}}(\varphi ^3 w) {\mathrm {d}}x + \int v \cdot \nabla (\varphi ^3 w) \cdot v {\mathrm {d}}x \\&\le C \Vert \varphi ^2 v\Vert _{L ^2} \left( \Vert \nabla w \otimes v\Vert _{L ^2} + \Vert w \otimes v\Vert _{L ^2} \right) . \end{aligned}$$

The remaining is of lower order:

$$\begin{aligned} \int \varphi ^3 v \cdot \varpi \times w {\mathrm {d}}x \le C \Vert \varphi ^2 v\Vert _{L ^2} \Vert \varpi \times w\Vert _{L ^2}. \end{aligned}$$

For the second one,

$$\begin{aligned} I _{{\mathbf {W}}2}&= \int {\mathbb {P}} _{\nabla }(\varphi ^4 v) \cdot \left( \varpi \times u + \omega \times w \right) {\mathrm {d}}x\\&\le \Vert {\mathbb {P}} _{\nabla }(\varphi ^4 v) \Vert _{L ^6} \Vert \varpi \times u + \omega \times w \Vert _{L ^\frac{6}{5}}, \end{aligned}$$

where

$$\begin{aligned} \Vert {\mathbb {P}} _{\nabla }(\varphi ^4 v) \Vert _{L ^6} = \Vert \nabla \varDelta ^{-1}{\text {div}}(\varphi ^4 v) \Vert _{L ^6} = \Vert \nabla \varDelta ^{-1}(v \cdot \nabla \varphi ^4) \Vert _{L ^6}&\le C \Vert \varphi ^2 v\Vert _{L ^2}. \end{aligned}$$

Thus $I _{\mathbf {W}}$ can be bounded by

$$\begin{aligned} I _{\mathbf {W}}\le C \Vert \varphi ^2 v\Vert _{L ^2} \left( \Vert \nabla w \otimes v\Vert _{L ^2} + \Vert \varpi \times w\Vert _{L ^2} + \Vert \varpi \times u + \omega \times w \Vert _{L ^\frac{6}{5}} \right) . \end{aligned}$$

(34)

In summary, we conclude that, for $-4 \le t \le 0$,

$$\begin{aligned} \frac{{\mathrm {d}}}{{\mathrm {d}}t} \int \varphi ^4 \frac{|v| ^2}{2} {\mathrm {d}}x + \int \varphi ^4 |\nabla v| ^2 {\mathrm {d}}x&\le I _\varDelta + I _{\mathbf {R}}+ I _{\mathbf {B}}+ I _{\mathbf {L}}+ I _{\mathbf {W}}, \end{aligned}$$

(35)

with good estimates on each of the term on the right.

4.3 Proof of Proposition 1

First we check the integrability of each terms.

Lemma 8

(Integrability) Given conditions (4) and (5), we have

$$\begin{aligned} \Vert u \Vert _{L ^{p _1} _t L ^{q _3} _x (Q _2)}&\le \eta , \nonumber \\ \Vert \varphi \omega \Vert _{L ^{p _1} _t L ^{q _1} _x ((-4, 0) \times {\mathbb {R}}^3)}&\le \eta , \qquad \Vert \varphi \omega \Vert _{L ^{p _2} _t L ^{q _2} _x ((-4, 0) \times {\mathbb {R}}^3)} \le \eta , \nonumber \\ \Vert \nabla v\Vert _{L ^{p _1} _t L ^{q _1} _x ((-4, 0) \times {\mathbb {R}}^3)}&\le \eta , \qquad \Vert \nabla v\Vert _{L ^{p _2} _t L ^{q _2} _x ((-4, 0) \times {\mathbb {R}}^3)} \le \eta , \nonumber \\ \Vert v \Vert _{L ^{p _1} _t L ^{q _3} _x ((-4, 0) \times {\mathbb {R}}^3)}&\le \eta , \qquad \Vert v \Vert _{L ^{p _2} _t L ^{q _4} _x ((-4, 0) \times {\mathbb {R}}^3)} \le \eta , \nonumber \\ \Vert \nabla w \Vert _{L ^{p _1} _t L ^{q _3} _x ((-4, 0) \times {\mathbb {R}}^3)}&\le \eta , \nonumber \\ \Vert w \Vert _{L ^{p _1} _t L ^{q _5} _x ((-4, 0) \times {\mathbb {R}}^3)}&\le \eta , \end{aligned}$$

(36)

$$\begin{aligned} \Vert \varpi \Vert _{L ^{p _1} _t L ^{q _3} _x ((-4, 0) \times {\mathbb {R}}^3)}&\le \eta , \qquad \Vert \varpi \Vert _{L ^{p _2} _t L ^{q _4} _x ((-4, 0) \times {\mathbb {R}}^3)} \le \eta . \end{aligned}$$

(37)

Proof

Integrability of u is obtained by Sobolev embedding and that $\varphi u$ has average 0. Integrability of $\varphi \omega$ is given. The remaining are consequences of Lemma 7 and Sobolev embedding. $\square$

Proof of Proposition 1

We prove Proposition 1 using a Grönwall argument. Multiply (35) by an increasing smooth function $\psi _1 (t)$ with $\psi _1 (t) = 0$ for $t \le -2$, $\psi _1 (t) = 1$ for $t \ge -1$, we have

$$\begin{aligned}&\frac{{\mathrm {d}}}{{\mathrm {d}}t} \left( \psi _1 (t) \int \varphi ^4 \frac{|v| ^2}{2} {\mathrm {d}}x \right) + \psi _1 (t) \int \varphi ^4 |\nabla v| ^2 {\mathrm {d}}x \\&\quad = \psi ' _1 (t) \int \varphi ^4 \frac{|v| ^2}{2} {\mathrm {d}}x + \psi _1 (t) \left( I _\varDelta + I _{\mathbf {R}}+ I _{\mathbf {B}}+ I _{\mathbf {L}}+ I _{\mathbf {W}}\right) . \end{aligned}$$

Formally, we can integrate from $-4$ to $t < 0$ and to get

$$\begin{aligned}&\psi _1 (t) \int \varphi ^4 \frac{|v| ^2}{2} {\mathrm {d}}x + \int _{-2} ^t \psi _1 (s) \int \varphi ^4 |\nabla v| ^2 {\mathrm {d}}x \\&\quad = \int _{-2} ^t \psi ' _1 (s) \int \varphi ^4 \frac{|v| ^2}{2} {\mathrm {d}}x {\mathrm {d}}t + \int _{-2} ^t \psi _1 (s) \left( I _{{\varDelta },{\mathbf {R}},{\mathbf {B}},{\mathbf {L}},{\mathbf {W}}} \right) {\mathrm {d}}t. \end{aligned}$$

This integration is justified since v satisfies the local energy inequality (29) in distribution, and $\psi _1 (t) \varphi ^4 (x) \in C _c ^\infty ((-4, 0] \times B _2)$. Because of (30), (31), (32), (33), (34), and

$$\begin{aligned} \Vert \varphi ^2 v \Vert _{L ^2 (B _2)}, \Vert \varphi ^2 v \Vert _{L ^2 (B _2)} ^\frac{1}{3} \le C \left( 1 + \int \varphi ^4 |v| ^2 {\mathrm {d}}x \right) , \end{aligned}$$

we can conclude that

$$\begin{aligned}&\frac{{\mathrm {d}}}{{\mathrm {d}}t} \left( \psi _1 (t) \int \varphi ^4 \frac{|v| ^2}{2} {\mathrm {d}}x \right) + \psi _1 (t) \int \varphi ^4 |\nabla v| ^2 {\mathrm {d}}x \\&\quad \le C {\varPhi } (t) \left( 1 + \psi _1 (t) \int \varphi ^4 \frac{|v| ^2}{2} {\mathrm {d}}x \right) , \end{aligned}$$

where

$$\begin{aligned} {\varPhi } (t)&= \psi ' _1 (t) \int \varphi ^4 \frac{|v| ^2}{2} {\mathrm {d}}x \\&\quad + \Vert v \Vert _{L ^2} + \Vert u \otimes v\Vert _{L ^2} + \Vert \omega \times u \Vert _{L ^\frac{6}{5} (B _2)} \\&\quad + \Vert v \Vert _{L ^{q _3} _x} ^\frac{2}{3} \Vert \omega \Vert _{L ^{q _2} _x (B _2)} + \Vert \nabla w \otimes v\Vert _{L ^2} \\&\quad + \Vert \varpi \times w\Vert _{L ^2} + \Vert \varpi \times u + \omega \times w \Vert _{L ^\frac{6}{5}} \\&\le \Vert v \Vert _{L ^2} ^2 + \Vert v \Vert _{L ^2} + \Vert v \Vert _{L ^{q _4} _x} \Vert u \Vert _{L ^{q _3} _x (B _2)} + \Vert \omega \Vert _{L ^{q _2} _x (B _2)} \Vert u\Vert _{L ^{q _3} _x (B _2)} \\&\quad + \Vert v \Vert _{L ^{q _3} _x} ^\frac{2}{3} \Vert \omega \Vert _{L ^{q _2} _x (B _2)} + \Vert \nabla w \Vert _{L ^{q _3} _x} \Vert v \Vert _{L ^{q _4} _x} \\&\quad + \Vert \varpi \Vert _{L ^{q _4} _x} \Vert w \Vert _{L ^{q _3} _x} \\&\quad + \Vert \varpi \Vert _{L ^{q _4} _x} \Vert u\Vert _{L ^{q _3} _x} + \Vert \omega \Vert _{L ^{q _2} _x} \Vert w\Vert _{L ^{q _3} _x} \\&\le \left( \Vert v \Vert _{L ^{q _3} _x} + \Vert v \Vert _{L ^{q _3} _x} ^\frac{1}{2} + \Vert u \Vert _{L ^{q _3} _x (B _2)} + \Vert \nabla w \Vert _{L ^{q _3} _x} + \Vert w\Vert _{L ^{q _3} _x} \right) \\&\quad \times \left( \Vert v \Vert _{L ^{q _4} _x} + \Vert v \Vert _{L ^{q _4} _x} ^\frac{1}{2} + \Vert \omega \Vert _{L ^{q _2} _x} + \Vert \varpi \Vert _{L ^{q _4} _x} \right) \end{aligned}$$

Here we used interpolation for $\Vert v \Vert _{L ^2} ^2 \le \Vert v \Vert _{L ^{q _3} _x} \Vert v \Vert _{L ^{q _4} _x}.$ Therefore

$$\begin{aligned} \Vert {\varPhi } \Vert _{L ^1 _t}&\lesssim \left\| \left( \Vert v \Vert _{L ^{q _3} _x} + \Vert v \Vert _{L ^{q _3} _x} ^\frac{1}{2} + \Vert u \Vert _{L ^{q _3} _x (B _2)} + \Vert \nabla w \Vert _{L ^{q _3} _x} + \Vert w\Vert _{L ^{q _3} _x} \right) \right\| _{L ^{p _1} _t} \\&\qquad \times \left\| \left( \Vert v \Vert _{L ^{q _4} _x} + \Vert v \Vert _{L ^{q _4} _x} ^\frac{1}{2} + \Vert \omega \Vert _{L ^{q _2} _x} + \Vert \varpi \Vert _{L ^{q _4} _x} \right) \right\| _{L ^{p _2} _t} \le \eta . \end{aligned}$$

By Grönwall’s lemma, we conclude that for every $-4 \le t \le 0$,

$$\begin{aligned} 1 + \psi _1 (t) \int \varphi ^4 \frac{|v| ^2}{2} {\mathrm {d}}x + \int _{-4} ^t \psi _1 (t) \int \varphi ^4 |\nabla v| ^2 {\mathrm {d}}x \le e^{\int _{-4} ^t C {\varPhi } (s) {\mathrm {d}}s} \le e ^{C\eta }. \end{aligned}$$

Therefore by taking the sup over $-1 \le t \le 0$ and $t = 0$ respectively, we conclude that

$$\begin{aligned} \sup _{-1 \le t \le 0} \int |v (t)| ^2 {\mathrm {d}}x \le \eta , \qquad \int _{Q _1} |\nabla v| ^2 {\mathrm {d}}x {\mathrm {d}}t \le \eta . \end{aligned}$$

$\square$

5 Local Study: Part Two, De Giorgi Iteration

In this section, we derive the boundedness of v in $Q _\frac{1}{2}$ which is the following.

Proposition 2

Let v solves (28). If (25) holds for sufficiently small $\eta$, and we have integrability bounds in Lemma 8, then we have

$$\begin{aligned} \Vert v \Vert _{L ^\infty (Q _\frac{1}{2})} = \sup _{t \in (-1, 0)} \Vert v (t) \Vert _{L ^\infty (B _\frac{1}{2})} \le 1. \end{aligned}$$

The proof uses De Giorgi technique and the truncation method. First, we set a dyadically shrinking radius

$$\begin{aligned} r _k ^\flat&= \frac{1}{2}(1 + 8 ^{-k}),&r _k ^\natural&= \frac{1}{2}(1 + 2 \times 8 ^{-k}),&r _k ^\sharp&= \frac{1}{2}(1 + 4 \times 8 ^{-k}). \end{aligned}$$

Then we define dyadically shrinking cylinder $Q _k$’s

$$\begin{aligned} T _k ^\flat&= {r _k ^\flat } ^2,&B _k ^\flat&= B _{r _k ^\flat } (0),&Q _k ^\flat&= (-T _k ^\flat , 0) \times B _k ^\flat , \\ T _k ^\natural&= {r _k ^\natural } ^2,&B _k ^\natural&= B _{r _k ^\natural } (0),&Q _k ^\natural&= (-T _k ^\natural , 0) \times B _k ^\natural , \\ T _k ^\sharp&= {r _k ^\sharp } ^2,&B _k ^\sharp&= B _{r _k ^\sharp } (0),&Q _k ^\sharp&= (-T _k ^\sharp , 0) \times B _k ^\sharp . \end{aligned}$$

We also introduce positive smooth space-time cut-off functions $\rho _k$ and $\rho _k^\sharp$ with

$$\begin{aligned} \mathbf{1} _{Q _k ^\flat } \le \rho _k\le \mathbf{1} _{Q _k ^\natural }, \qquad \mathbf{1} _{Q _k ^\sharp } \le \rho _k^\sharp \le \mathbf{1} _{Q ^\flat _{k - 1}}. \end{aligned}$$

Then, let $c _k$ denote a sequence of rising energy level

$$\begin{aligned} c _k&= 1 - 2 ^{-k},&v _k&= (|v| - c _k) _+,&\beta _k&= \frac{v _k}{|v|}, \\ {\varOmega } _k&= \{ v _k > 0 \},&\mathbf{1} _{k}&= \mathbf{1} _{{\varOmega } _k},&\alpha _k&= 1 - \beta _k. \end{aligned}$$

We define analogous of vector derivative $d _k$ and energy quantity $U_k$ to get

$$\begin{aligned} d _k ^2&= \mathbf{1} _{k} \left( \alpha _k|\nabla |v||^2 + \beta _k|\nabla v| ^2 \right) , \\ U _k&= \Vert v _k \Vert _{L ^\infty (-T _k ^\flat , 0; L ^2 (B _k ^\flat ))} ^2 + \Vert d _k \Vert _{L ^2 (Q _k ^\flat )} ^2. \end{aligned}$$

We have the following truncation estimates:

Lemma 9

$$\begin{aligned} \alpha _kv \le c _k&\le 1, \\ \Vert \beta _kv\Vert _{L ^\infty _t L ^2 _x \cap L ^2 _t \dot{H} ^1 _x(Q _{k-1}^\flat )} ^2&\le 9 U _{k - 1}, \\ \Vert \mathbf{1} _{k} \Vert _{L ^\infty _t L ^2 _x \cap L ^2 _t L ^6 _x (Q _{k-1} ^\flat )} ^2&\le C ^k U _{k - 1}. \end{aligned}$$

Proof

The first estimate follows from the definition. By Lemma 4 in [23], we have $|\nabla v _k| \le d _k$ and $|\nabla (\beta _kv)| \le 3 d _k$. Moreover, since $|\nabla |v|| \le |\nabla v| ^2$, we see $d _k \le d _{k - 1}$, as $v _k$ and $\beta _k$ are monotonously decreasing, so

$$\begin{aligned} \Vert \nabla (\beta _kv) \Vert _{L ^2 (Q _{k - 1} ^\flat )} \le 3\Vert d _k \Vert _{L ^2 (Q _{k - 1} ^\flat )} \le 3\Vert d _{k - 1} \Vert _{L ^2 (Q _{k - 1} ^\flat )}. \end{aligned}$$

Moreover, the truncation gives $|\beta _kv| + 2 ^{-k} \mathbf{1} _{k} = v _k + 2 ^{-k} \mathbf{1} _{k} = \mathbf{1} _{k} v _{k - 1}$, so

$$\begin{aligned} \Vert \beta _kv \Vert _{L ^\infty _t L ^2 _x (Q _{k - 1} ^\flat )}&\le \Vert v _{k - 1}\Vert _{L ^\infty _t L ^2 _x (Q _{k - 1} ^\flat )}, \\ 2 ^{-k} \Vert \mathbf{1} _{k} \Vert _{L ^\infty _t L ^2 _x (Q _{k - 1} ^\flat )}&\le \Vert v _{k - 1}\Vert _{L ^\infty _t L ^2 _x (Q _{k - 1} ^\flat )} , \\ 2 ^{-k} \Vert \mathbf{1} _{k} \Vert _{L ^2 _t L ^6 _x (Q _{k - 1} ^\flat )}&\le \Vert v _{k - 1} \Vert _{L ^2 _t L ^6 _x (Q _{k - 1} ^\flat )} \\&\le \Vert v _{k - 1} \Vert _{L ^\infty _t L ^2 _x (Q _{k - 1} ^\flat )} + \Vert \nabla v _{k - 1} \Vert _{L ^2 (Q ^\flat _{k - 1})} \\&\le \Vert v _{k - 1} \Vert _{L ^\infty _t L ^2 _x (Q _{k - 1} ^\flat )} + \Vert d _{k - 1} \Vert _{L ^2 (Q ^\flat _{k - 1})}. \end{aligned}$$

$\square$

Corollary 4

(Nonlinearization) If $f \in L ^p _t L ^q _x (Q _{k - 1})$, with

$$\begin{aligned} \frac{1}{p} + \gamma \left( \frac{\theta }{2} + \frac{1-\theta }{\infty }\right) = 1, \qquad \frac{1}{q} + \gamma \left( \frac{\theta }{6} + \frac{1-\theta }{2} \right) = 1 \end{aligned}$$

for some $0 \le \theta \le 1$, $0 < \sigma \le \gamma$, then, uniformly in $\sigma$,

$$\begin{aligned} \int _{Q _{k - 1} ^\flat } |\beta _kv| ^\sigma |f| {\mathrm {d}}x {\mathrm {d}}t \le C ^k \Vert f \Vert _{L ^p _t L ^q _x (Q _{k - 1})} U _{k - 1} ^\frac{\gamma }{2}. \end{aligned}$$

Proof

By interpolation,

$$\begin{aligned} \Vert \beta _kv \Vert , \Vert \mathbf{1} _{k} \Vert _{L ^{p _\theta } _tL ^{q _\theta } _x(Q _{k - 1})} \le U _{k - 1} ^\frac{1}{2}, \end{aligned}$$

where

$$\begin{aligned} \frac{1}{p _\theta } = \frac{\theta }{2} + \frac{1-\theta }{\infty }, \qquad \frac{1}{q _\theta } = \frac{\theta }{6} + \frac{1-\theta }{2}. \end{aligned}$$

Therefore, using Hölder’s inequality,

$$\begin{aligned} \int _{Q _{k - 1}} |\beta _kv| ^\sigma |f| {\mathrm {d}}x {\mathrm {d}}t \le \Vert f \Vert _{L ^p _t L ^q _x} \Vert \beta _kv \Vert _{L ^{p _\theta } _tL ^{q _\theta } _x} ^\sigma \Vert \mathbf{1} _{k} \Vert _{L ^{p _\theta } _tL ^{q _\theta } _x} ^{\gamma - \sigma } \le \Vert f \Vert _{L ^p _t L ^q _x} U _{k - 1} ^\frac{\gamma }{2}. \end{aligned}$$

$\square$

First, we recall the following identities from [23]:

$$\begin{aligned} \alpha _kv \cdot \partial _\bullet v&= \partial _{\bullet } \left( \frac{|v| ^2 - v _k ^2}{2}\right) , \end{aligned}$$

(38)

$$\begin{aligned} \alpha _kv \cdot \varDelta v&= \varDelta \left( \frac{|v|^2 - v _k ^2}{2} \right) + d _k ^2 - |\nabla v| ^2. \end{aligned}$$

(39)

Since $\alpha _kv$ is bounded, we can multiply equation (28) by $\alpha _kv$ and obtain

$$\begin{aligned}&\partial _t\left( \frac{|v|^2 - v _k ^2}{2} \right) + \alpha _kv \cdot \nabla {\mathbf {R}}(u \otimes v) \nonumber \\&\quad = \varDelta \left( \frac{|v|^2 - v _k ^2}{2} \right) + d _k ^2 - |\nabla v| ^2 + \alpha _kv \cdot ({\mathbf {B}}+ {\mathbf {L}}+ {\mathbf {W}}), \end{aligned}$$

(40)

using (38) and (39). Denote ${\mathbf {C}}_v= {\mathbf {B}}+ {\mathbf {L}}+ {\mathbf {W}}$. Subtracting (40) from (29), we have

$$\begin{aligned} \partial _t\frac{v _k ^2}{2} + d _k ^2 + {\text {div}}(v {\mathbf {R}}(u \otimes v)) - \alpha _kv \cdot \nabla {\mathbf {R}}(u \otimes v) \le \varDelta \frac{v _k ^2}{2} + \beta _kv \cdot {\mathbf {C}}_v. \end{aligned}$$

Multipling by $\rho _k$, then integrating in space and from $\sigma$ to $\tau$ in time,

$$\begin{aligned}&\left[ \int \rho _k\frac{v _k ^2}{2} {\mathrm {d}}x \right] ^\tau _\sigma + \int _\sigma ^\tau \int \rho _kd _k ^2 {\mathrm {d}}x {\mathrm {d}}t \\&\quad \le \int _\sigma ^\tau \int (\partial _t\rho _k+ \varDelta \rho _k) \frac{v _k ^2}{2} {\mathrm {d}}x {\mathrm {d}}t - \int _\sigma ^\tau \int \rho _k{\text {div}}(v {\mathbf {R}}(u \otimes v)) {\mathrm {d}}x {\mathrm {d}}t \\&\qquad + \int _\sigma ^\tau \int \rho _k\alpha _kv \cdot \nabla {\mathbf {R}}(u \otimes v) {\mathrm {d}}x {\mathrm {d}}t + \int _\sigma ^\tau \int \rho _k\beta _kv \cdot {\mathbf {C}}_v{\mathrm {d}}x {\mathrm {d}}t. \end{aligned}$$

Taking the sup over $\tau > -T _k ^\flat$, and set $\sigma < -T _{k - 1} ^\flat$, we obtain

$$\begin{aligned} U _k&\le \sup _{\tau \in (-T _k ^\flat , 0)} \int \rho _k\frac{v _k ^2}{2} {\mathrm {d}}x + \int _{-T _{k - 1} ^\flat } ^0 \int \rho _kd _k ^2 {\mathrm {d}}x {\mathrm {d}}t \nonumber \\&\le C ^k \int _{Q _k ^\natural } v _k ^2 {\mathrm {d}}x {\mathrm {d}}t + \sup _{\tau \in (-T _k ^\flat , 0)} \bigg \lbrace \int _{-T _k ^\natural } ^\tau \int \rho _k\alpha _kv \cdot \nabla {\mathbf {R}}(u \otimes v) {\mathrm {d}}x {\mathrm {d}}t \nonumber \\&\quad - \int _{-T _k ^\natural } ^\tau \int \rho _k{\text {div}}(v {\mathbf {R}}(u \otimes v)) {\mathrm {d}}x {\mathrm {d}}t \nonumber \\&\quad + \int _{-T _k ^\natural } ^\tau \int \rho _k\beta _kv \cdot {\mathbf {C}}_v{\mathrm {d}}x {\mathrm {d}}t \bigg \rbrace . \end{aligned}$$

(41)

Using Corollary 4, the first one is bounded by

$$\begin{aligned} \int _{Q _k ^\natural } v _k ^2 {\mathrm {d}}x {\mathrm {d}}s \le \int _{Q _{k - 1} ^\flat } |\beta _kv|^2 {\mathrm {d}}x {\mathrm {d}}s \le U _{k - 1} ^\frac{5}{3}. \end{aligned}$$

(42)

Now let us deal with the last few terms. For simplicity, we use $\iint {\mathrm {d}}x {\mathrm {d}}t$ to denote $\int _{-T _k ^\natural } ^\tau \int _{{\mathbb {R}}^3} {\mathrm {d}}x {\mathrm {d}}t$ in the rest of this section.

5.1 Highest Order Nonlinear Term

Define three trilinear forms:

$$\begin{aligned} {\mathbf {T}} _\circ [v _1, v _2, v _3]&= \iint \rho _k{\text {div}}(v _1 {\mathbf {R}}(v _2 \otimes v _3)) {\mathrm {d}}x {\mathrm {d}}t, \\ {\mathbf {T}} _\nabla [v _1, v _2, v _3]&= \iint \rho _kv _1 \cdot \nabla {\mathbf {R}}(v _2 \otimes v _3) {\mathrm {d}}x {\mathrm {d}}t, \\ {\mathbf {T}} _{{\text {div}}}[v _1, v _2, v _3]&= \iint \rho _k{\text {div}}v _1 {\mathbf {R}}(v _2 \otimes v _3) {\mathrm {d}}x {\mathrm {d}}t. \end{aligned}$$

There are symmetric on $v _2$, $v _3$ positions. When we have enough integrability, that is, when

$$\begin{aligned} |\nabla v _1| |v _2| |v _3|, |v _1| |\nabla v _2| |v _3|, |v _1| |v _2| |\nabla v _3| \in L ^1 _{t, x}, \end{aligned}$$

we have Leibniz is rule

$$\begin{aligned} {\mathbf {T}} _\circ = {\mathbf {T}} _\nabla + {\mathbf {T}} _{{\text {div}}}. \end{aligned}$$

The goal is to estimate the first two double integrals in (41),

$$\begin{aligned}&\iint \rho _k\alpha _kv \cdot \nabla {\mathbf {R}}(u \otimes v) {\mathrm {d}}x {\mathrm {d}}t - \iint \rho _k{\text {div}}(v {\mathbf {R}}(u \otimes v)) {\mathrm {d}}x {\mathrm {d}}t \\&\qquad = {\mathbf {T}} _\nabla [\alpha _kv, u, v] - {\mathbf {T}} _\circ [v, u, v]. \end{aligned}$$

We first separate $w \otimes v$ from $u \otimes v$, and we have

$$\begin{aligned} {\mathbf {T}} _\nabla [\alpha _kv, w, v] - {\mathbf {T}} _\circ [v, w, v]&= {\mathbf {T}} _\nabla [\alpha _kv, w, v] - {\mathbf {T}} _\nabla [v, w, v] - {\mathbf {T}} _{{\text {div}}}[v, w, v]\\&= - {\mathbf {T}} _\nabla [\beta _kv, w, v] \\&= -\iint \rho _k\beta _kv \cdot \nabla {\mathbf {R}}(w \otimes v) {\mathrm {d}}x {\mathrm {d}}t. \end{aligned}$$

Denote $-\nabla {\mathbf {R}}(w \otimes v) {=}: {\mathbf {W}}_2$ and we will deal with it later. The remaining $(u - w) \otimes v$ can be separated into interior part and exterior part,

$$\begin{aligned} (u - w) \otimes v = \rho _k^\sharp v \otimes v + (1 - \rho _k^\sharp ) (u - w) \otimes v. \end{aligned}$$

The exterior part is bounded and smooth in space over the support of $\rho _k$:

$$\begin{aligned} \Vert \rho _k{\mathbf {R}}((1 - \rho _k^\sharp ) (u - w) \otimes v) \Vert _{L ^{p _3} _t C ^\infty _x}&\le C \Vert (u - w) \otimes v \Vert _{L ^{p _3} _t L ^2 _x} \\&\le C \Vert u - w \Vert _{L ^{p _1} _t L ^{q _3} _x (Q _2)} \Vert v \Vert _{L ^{p _2} _t L ^{q _4} _x} \le \eta . \end{aligned}$$

Here, we denote

$$\begin{aligned} \frac{1}{p _3} = \frac{1}{p _1} + \frac{1}{p _2} < 1. \end{aligned}$$

Therefore we can use Leibniz is rule similarly as to as w and

$$\begin{aligned}&{\mathbf {T}} _\nabla [\alpha _kv, (1 - \rho _k^\sharp ) (u - w), v] - {\mathbf {T}} _\circ [v, (1 - \rho _k^\sharp ) (u - w), v] \\&\quad = {\mathbf {T}} _\nabla [\alpha _kv, (1 - \rho _k^\sharp ) (u - w), v] - {\mathbf {T}} _\nabla [v, (1 - \rho _k^\sharp ) (u - w), v] \\&\quad = -{\mathbf {T}} _\nabla [\beta _kv, (1 - \rho _k^\sharp ) (u - w), v] \\&\quad = -\iint \rho _k\beta _kv \cdot \nabla {\mathbf {R}}(v \otimes (1 - \rho _k^\sharp ) (u - w)) {\mathrm {d}}x {\mathrm {d}}t \\&\quad \le C ^k U _{k - 1} ^{\frac{5}{3} - \frac{2}{3 p _3}} \end{aligned}$$

by nonlinearization Corollary 4. The interior part is

$$\begin{aligned}&{\mathbf {T}} _\nabla [\alpha _kv, \rho _k^\sharp v, v] - {\mathbf {T}} _\circ [v, \rho _k^\sharp v, v] \\&\quad = {\mathbf {T}} _\nabla [\alpha _kv, \rho _k^\sharp \beta _kv, \beta _kv] + 2{\mathbf {T}} _\nabla [\alpha _kv, \rho _k^\sharp \alpha _kv, \beta _kv] \\&\qquad + {\mathbf {T}} _\nabla [\alpha _kv, \rho _k^\sharp \alpha _kv, \alpha _kv] - {\mathbf {T}} _\circ [v, \rho _k^\sharp v, v] \\&\quad = {\mathbf {T}} _\nabla [\alpha _kv, \rho _k^\sharp \beta _kv, \beta _kv] \\&\qquad + 2{\mathbf {T}} _\circ [\alpha _kv, \rho _k^\sharp \alpha _kv, \beta _kv] - 2{\mathbf {T}} _{{\text {div}}}[\alpha _kv, \rho _k^\sharp \alpha _kv, \beta _kv] \\&\qquad + {\mathbf {T}} _\circ [\alpha _kv, \rho _k^\sharp \alpha _kv, \alpha _kv] - {\mathbf {T}} _{{\text {div}}}[\alpha _kv, \rho _k^\sharp \alpha _kv, \alpha _kv] \\&\qquad - {\mathbf {T}} _\circ [v, \rho _k^\sharp v, v] \\&\quad = {\mathbf {T}} _\nabla [\alpha _kv, \rho _k^\sharp \beta _kv, \beta _kv] \\&\qquad + 2{\mathbf {T}} _{{\text {div}}}[\beta _kv, \rho _k^\sharp \alpha _kv, \beta _kv] + {\mathbf {T}} _{{\text {div}}}[\beta _kv, \rho _k^\sharp \alpha _kv, \alpha _kv] \\&\qquad + 2{\mathbf {T}} _\circ [\alpha _kv, \rho _k^\sharp \alpha _kv, \beta _kv] + {\mathbf {T}} _\circ [\alpha _kv, \rho _k^\sharp \alpha _kv, \alpha _kv] \\&\qquad - {\mathbf {T}} _\circ [v, \rho _k^\sharp v, v] \\&\quad = {\mathbf {T}} _\nabla [\alpha _kv, \rho _k^\sharp \beta _kv, \beta _kv] + {\mathbf {T}} _{{\text {div}}}[\beta _kv, \rho _k^\sharp \alpha _kv, (\beta _k+ 1) v] \\&\qquad - {\mathbf {T}} _\circ [\alpha _kv, \rho _k^\sharp \beta _kv, \beta _kv] - {\mathbf {T}} _\circ [\beta _kv, \rho _k^\sharp v, v]. \end{aligned}$$

Notice that the boundedness of $\alpha _kv$ guarantees enough integrability to switch between trilinear forms. Then

$$\begin{aligned}&|{\mathbf {T}} _\nabla [\alpha _kv, \rho _k^\sharp \beta _kv, \beta _kv]|, |{\mathbf {T}} _{{\text {div}}}[\beta _kv, \rho _k^\sharp \alpha _kv, (\beta _k+ 1) v]| \\&\quad \lesssim \Vert \nabla (\beta _kv) \Vert _{L ^2 (Q _{k - 1})} U _{k - 1} ^\frac{5}{6} \le U _{k - 1} ^\frac{4}{3}, \\&|{\mathbf {T}} _\circ [\alpha _kv, \rho _k^\sharp \beta _kv, \beta _kv]|, |{\mathbf {T}} _\circ [\beta _kv, \rho _k^\sharp v, v]| \lesssim U _{k - 1} ^\frac{5}{3}. \end{aligned}$$

In conclusion,

$$\begin{aligned}&\bigg | \iint \rho _k\alpha _kv \cdot \nabla {\mathbf {R}}(u \otimes v) {\mathrm {d}}x {\mathrm {d}}t - \iint \rho _k{\text {div}}(v {\mathbf {R}}(u \otimes v)) {\mathrm {d}}x {\mathrm {d}}t \nonumber \\&\quad - \iint \rho _k\beta _kv \cdot {\mathbf {W}}_2 {\mathrm {d}}x {\mathrm {d}}t \bigg | \lesssim C ^k U _{k - 1} ^{\min \{\frac{4}{3}, \frac{5}{3}-\frac{2}{3}{p_3}\}}. \end{aligned}$$

(43)

5.2 Lower Order Terms

For the bilinear and linear term, recall that inside $B _1$,

$$\begin{aligned} {\mathbf {B}}&= -{\text {curl}}\varDelta ^{-1}(\nabla \varphi \times (\omega \times u)), \\ {\mathbf {L}}&= {\text {curl}}\varDelta ^{-1}\left( 2 {\text {div}}(\nabla \varphi \otimes \omega ) - (\varDelta \varphi ) \omega \right) . \end{aligned}$$

Therefore,

$$\begin{aligned}&\Vert \rho _k{\mathbf {B}}\Vert _{L ^{p _3} _t L ^\infty _x} \le \Vert \omega \times u \Vert _{L ^{p _3} _t L ^\frac{6}{5} _x (Q _2)} \le \Vert u \Vert _{L ^{p _1} _t L ^{q _3} _x} \Vert \omega \Vert _{L ^{p _2} _t L ^{q _2} _x} \le \eta , \\&\Vert \rho _k{\mathbf {L}}\Vert _{L ^{p _2} _t L ^\infty _x} \le \Vert \omega \Vert _{L ^{p _2} _t L ^{q _2} _x (Q _2)} \le \eta , \end{aligned}$$

Thus

$$\begin{aligned} \iint {\mathbf {B}}\cdot \rho _k\beta _kv {\mathrm {d}}x {\mathrm {d}}t&\le C ^k U _{k - 1} ^{\frac{5}{3} - \frac{2}{3 p _3}}, \end{aligned}$$

(44)

$$\begin{aligned} \iint {\mathbf {L}}\cdot \rho _k\beta _kv {\mathrm {d}}x {\mathrm {d}}t&\le C ^k U _{k - 1} ^{\frac{5}{3} - \frac{2}{3 p _2}}. \end{aligned}$$

(45)

5.3 W Terms

Finally, let us deal with

$$\begin{aligned} {\mathbf {W}}+ {\mathbf {W}}_2&= -\omega \times w + \frac{1}{2} {\mathbb {P}} _{\nabla }\left( \varpi \times u + \omega \times w \right) - \nabla {\mathbf {R}}(w \otimes v). \end{aligned}$$

Here $\nabla {\mathbf {R}}= \frac{1}{2} \nabla {\text {tr}}- {\mathbb {P}} _{\nabla }{\text {div}}$, so

$$\begin{aligned} \nabla {\mathbf {R}}(w \otimes v)&= \frac{1}{2} \nabla (w \cdot v) - {\mathbb {P}} _{\nabla }{\text {div}}(v \otimes w) \\&= \frac{1}{2} \left( w \cdot \nabla v + v \cdot \nabla w + w \times {\text {curl}}v + v \times {\text {curl}}w \right) - {\mathbb {P}} _{\nabla }(v \cdot \nabla w) \\&= \frac{1}{2} \left( w \cdot \nabla v - v \cdot \nabla w \right) + {\mathbb {P}} _{{\text {curl}}}(v \cdot \nabla w) \\&\quad + \frac{1}{2} \left( w \times {\text {curl}}v + v \times {\text {curl}}w \right) , \\ \nabla {\mathbf {R}}(w \otimes v)&= {\mathbb {P}} _{\nabla }(\nabla {\mathbf {R}}(w \otimes v)) \\&= \frac{1}{2} {\mathbb {P}} _{\nabla }\left( w \cdot \nabla v - v \cdot \nabla w \right) + \frac{1}{2} {\mathbb {P}} _{\nabla }\left( w \times {\text {curl}}v + v \times {\text {curl}}w \right) \\&= \frac{1}{2} {\mathbb {P}} _{\nabla }\left( {\text {curl}}(v \times w) - v {\text {div}}w + w {\text {div}}v \right) \\&\quad + \frac{1}{2} {\mathbb {P}} _{\nabla }\left( w \times {\text {curl}}v + v \times {\text {curl}}w \right) \\&= -\frac{1}{2} {\mathbb {P}} _{\nabla }\left( v (u \cdot \nabla \varphi ) \right) + \frac{1}{2} {\mathbb {P}} _{\nabla }\left( w \times {\text {curl}}v + v \times {\text {curl}}w \right) . \end{aligned}$$

Hence

$$\begin{aligned} {\mathbf {W}}+ {\mathbf {W}}_2&= -\omega \times w + \frac{1}{2} {\mathbb {P}} _{\nabla }\left( v (u \cdot \nabla \varphi ) \right) \\&\qquad + \frac{1}{2} {\mathbb {P}} _{\nabla }\left( \varpi \times u + \omega \times w + {\text {curl}}v \times w + {\text {curl}}w \times v \right) . \end{aligned}$$

Again, we separate ${\mathbf {W}}+ {\mathbf {W}}_2$ into exterior and interior part, with

$$\begin{aligned} {\mathbf {W}}+ {\mathbf {W}}_2 = {\mathbf {W}}_{\mathrm {ext}} + {\mathbf {W}}_{\mathrm {int}}, \end{aligned}$$

where

$$\begin{aligned} {\mathbf {W}}_{\mathrm {ext}}&= -(1 - \rho _k^\sharp ) \omega \times w + \frac{1}{2} {\mathbb {P}} _{\nabla }\left( v (u \cdot \nabla \varphi ) \right) \\&\quad + \frac{1}{2} {\mathbb {P}} _{\nabla }\left( (1 - \rho _k^\sharp ) \left( \varpi \times u + \omega \times w + {\text {curl}}v \times w + {\text {curl}}w \times v \right) \right) , \\ {\mathbf {W}}_{\mathrm {int}}&= -\rho _k^\sharp \omega \times w \\&\quad + \frac{1}{2} {\mathbb {P}} _{\nabla }\left( \rho _k^\sharp \left( \varpi \times u + \omega \times w + {\text {curl}}v \times w + {\text {curl}}w \times v \right) \right) \\&= -\rho _k^\sharp {\text {curl}}v \times w - \rho _k^\sharp \varpi \times w \\&\quad + \frac{1}{2} {\mathbb {P}} _{\nabla }\left( \rho _k^\sharp \left( \varpi \times u + {\text {curl}}w \times v + \varpi \times w \right) \right) \\&\quad + \frac{1}{2} {\mathbb {P}} _{\nabla }\left( \rho _k^\sharp \left( \omega \times w + {\text {curl}}v \times w - \varpi \times w \right) \right) \\&= -\rho _k^\sharp {\text {curl}}v \times w - \rho _k^\sharp \varpi \times w \\&\quad + {\mathbb {P}} _{\nabla }\left( \rho _k^\sharp \varpi \times u \right) + {\mathbb {P}} _{\nabla }\left( \rho _k^\sharp {\text {curl}}v \times w \right) \\&= -{\mathbb {P}} _{{\text {curl}}}(\rho _k^\sharp {\text {curl}}v \times w) - {\mathbb {P}} _{{\text {curl}}}(\rho _k^\sharp \varpi \times w) + {\mathbb {P}} _{\nabla }\left( \rho _k^\sharp \varpi \times v \right) . \end{aligned}$$

Similarly as to the bilinear terms, $\rho _k{\mathbf {W}}_{\mathrm {ext}}$ is small in $L ^{p _3} _t L ^\infty _x$. Among the three terms in ${\mathbf {W}}_{\mathrm {int}}$, $\rho _k^\sharp \varpi \times w$ is bounded in $L ^{p _3} _t L ^\infty _x$, and $\rho _k^\sharp \varpi$ is in $L ^{p _2} _t L ^\infty _x$. Finally, for the first term,

$$\begin{aligned} {\mathbb {P}} _{{\text {curl}}}({\text {curl}}v \times \rho _k^\sharp w)&= -{\mathbb {P}} _{{\text {curl}}}({\text {curl}}\rho _k^\sharp w \times v) + {\mathbb {P}} _{{\text {curl}}}(v \cdot \nabla \rho _k^\sharp w + \rho _k^\sharp w \cdot \nabla v), \\ {\mathbb {P}} _{{\text {curl}}}(\rho _k^\sharp w \cdot \nabla v)&= {\mathbb {P}} _{{\text {curl}}}( {\text {curl}}(v \times \rho _k^\sharp w) + v \cdot \nabla \rho _k^\sharp w - v {\text {div}}\rho _k^\sharp w ) \\&= {\text {curl}}(v \times \rho _k^\sharp w) + {\mathbb {P}} _{{\text {curl}}}( v \cdot \nabla \rho _k^\sharp w - v {\text {div}}\rho _k^\sharp w ), \\ {\text {curl}}(v \times \rho _k^\sharp w)&= v {\text {div}}\rho _k^\sharp w + \rho _k^\sharp w \cdot \nabla v - v \cdot \nabla \rho _k^\sharp w. \end{aligned}$$

Every term is a product of v and $\nabla \rho _k^\sharp w$ (possibly with a Riesz transform) except $\rho _k^\sharp w \cdot \nabla v$. Because in ${\varOmega } _k$, $\nabla |v| = \nabla v _k$ are the same, we have

$$\begin{aligned} \int \rho _k\beta _kv \cdot (\rho _k^\sharp w \cdot \nabla ) v {\mathrm {d}}x&= \int \rho _k\beta _k(w \cdot \nabla ) \frac{|v| ^2}{2} {\mathrm {d}}x \\&= \int \rho _k\beta _k|v| (w \cdot \nabla ) |v| {\mathrm {d}}x \\&= \int \rho _kv _k (w \cdot \nabla ) v _k {\mathrm {d}}x \\&= \int \rho _k(w \cdot \nabla ) \frac{v _k ^2}{2} {\mathrm {d}}x \\&= -\int \frac{v _k ^2}{2} {\text {div}}(\rho _kw) {\mathrm {d}}x. \end{aligned}$$

Therefore, every term of ${\mathbb {P}} _{{\text {curl}}}({\text {curl}}v \times \rho _k^\sharp w)$ is a product of v and $\nabla \rho _kw$ or $\nabla \rho _k^\sharp w$. Inside $B _1$, $w \in L ^{p _1} _t C ^\infty _x$. In conclusion,

$$\begin{aligned} \iint \rho _k\beta _kv \cdot {\mathbf {W}}_{\text {ext}} {\mathrm {d}}x {\mathrm {d}}t&\le C ^k U _{k - 1} ^{\frac{5}{3} - \frac{2}{3 p _3}}, \\ \iint \rho _k\beta _kv \cdot {\mathbb {P}} _{{\text {curl}}}(\rho _k^\sharp {\text {curl}}v \times w) {\mathrm {d}}x {\mathrm {d}}t&\le C ^k U _{k - 1} ^{\frac{5}{3} - \frac{2}{3 p _1}}, \\ \iint \rho _k\beta _kv \cdot {\mathbb {P}} _{{\text {curl}}}(\rho _k^\sharp \varpi \times w) {\mathrm {d}}x {\mathrm {d}}t&\le C ^k U _{k - 1} ^{\frac{5}{3} - \frac{2}{3 p _3}}, \\ \iint \rho _k\beta _kv \cdot {\mathbb {P}} _{\nabla }(\rho _k^\sharp \varpi \times v) {\mathrm {d}}x {\mathrm {d}}t&\le C ^k U _{k - 1} ^{\frac{5}{3} - \frac{2}{3 p _2}}, \end{aligned}$$

so the sum is bounded in

$$\begin{aligned} \iint \rho _k\beta _kv \cdot ({\mathbf {W}}+ {\mathbf {W}}_2) {\mathrm {d}}x {\mathrm {d}}t&= \iint \rho _k\beta _kv \cdot ({\mathbf {W}}_{\text {int}} + {\mathbf {W}}_{\text {ext}}) {\mathrm {d}}x {\mathrm {d}}t \le C ^k U _{k - 1} ^{\frac{5}{3} - \frac{2}{3 p _3}}, \end{aligned}$$

(46)

provided that $U _{k - 1} < 1$.

5.4 Proof of Proposition 2

Proof of Proposition 2

Coming back to (41), by estimates (42) on the first term, (43) on the trilinear terms, (44), (45) on the ${\mathbf {B}}, {\mathbf {L}}$ terms and (46) on the ${\mathbf {W}}$ terms, we conclude that

$$\begin{aligned} U _{k} \le C ^k U _{k - 1} ^{\min \{\frac{5}{3} - \frac{2}{3 p _3}, \frac{4}{3}\}}, \end{aligned}$$

provided that $U _{k - 1} < 1$. Here $p _3 > 1$ ensures the index is strictly greater than 1. Since

$$\begin{aligned} U _0&= \sup _{t \in (-1, 0)} \int |v _0| ^2 {\mathrm {d}}x + \int _{-1} ^0 \int _{B _1} d _0 ^2 {\mathrm {d}}x {\mathrm {d}}t \\&= \sup _{t \in (-1, 0)} \int |v| ^2 {\mathrm {d}}x + \int _{-1} ^0 \int _{B _1} |\nabla v| ^2 {\mathrm {d}}x {\mathrm {d}}t \le \eta \end{aligned}$$

by Proposition 1, we know that if $\eta$ is small enough, $U _k \rightarrow 0$ as $k \rightarrow \infty$. Thus in $Q _\frac{1}{2}$, $|v| \le 1$ a.e.. This finishes the proof of Proposition 2. $\square$

6 Local Study: Part Three, More Regularity

In this section, we will show that the vorticity $\omega$ is smooth in space. We will only work with the vorticity equation from now on. After the previous two steps, in $B _\frac{1}{2}$ we should always decompose $u = v + w$, because v is bounded and w is harmonic.

For convenience, given a vector $\omega$, we denote

$$\begin{aligned} \omega ^0 :{=} \frac{\omega }{|\omega |}, \qquad \omega ^\alpha :{=} |\omega | ^\alpha \omega ^0, \alpha \in {\mathbb {R}}. \end{aligned}$$

Let $\partial _\bullet$ be the partial derivative in any space direction or time. Then we have

$$\begin{aligned} \partial _\bullet (|\omega | ^\alpha )&= \alpha \omega ^{\alpha - 1} \cdot \partial _\bullet \omega , \\ \partial _\bullet (\omega ^\alpha )&= |\omega | ^{\alpha - 1} \partial _\bullet \omega + (\alpha - 1) (\omega ^{\alpha - 2} \cdot \partial _\bullet \omega ) \omega , \\ \frac{1}{\alpha }\partial _\bullet \partial _\bullet (|\omega | ^\alpha )&= |\omega | ^{\alpha - 2} |\partial _\bullet \omega | ^2 + (\alpha - 2) (\omega ^{\frac{\alpha }{2} - 1} \cdot \partial _\bullet \omega ) ^2 + \omega ^{\alpha - 1} \cdot \partial _\bullet \partial _\bullet \omega \\&\ge (\alpha - 1) (\omega ^{\frac{\alpha }{2} - 1} \cdot \partial _\bullet \omega ) ^2 + \omega ^{\alpha - 1} \cdot \partial _\bullet \partial _\bullet \omega \\&= \frac{4(\alpha - 1)}{\alpha ^2} \left| \partial _\bullet \omega ^{\frac{\alpha }{2}} \right| ^2 + \omega ^{\alpha - 1} \cdot \partial _\bullet \partial _\bullet \omega . \end{aligned}$$

6.1 Bound Vorticity in the Energy Space

We will first show $\omega$ is bounded in the energy space.

Proposition 3

If $u = v + w$ in $Q _\frac{1}{2}$, where v, w are bounded in

$$\begin{aligned} \Vert v \Vert _{L ^\infty (Q _\frac{1}{2})} + \Vert \nabla v \Vert _{L ^2 (Q _\frac{1}{2})} \le 2, \end{aligned}$$

(47)

$$\begin{aligned} \Vert {\text {curl}}w \Vert _{L ^2 _t L ^\frac{3}{2} _x (Q _\frac{1}{2})} + \Vert w \Vert _{L ^\frac{4}{3} _t \mathrm {Lip}_x (Q _\frac{1}{2})} \le 2, \end{aligned}$$

(48)

$\omega = {\text {curl}}u$ solves the vorticity equation (6), then

• (a) $\Vert \omega ^\frac{3}{4} \Vert _{{\mathcal {E}} (Q _\frac{1}{4})} \le C$,

• (b) $\Vert \omega \Vert _{{\mathcal {E}} (Q _\frac{1}{8})} \le C$,

Proof of Proposition 3 (a)

We fix a pair of smooth space-time cut-off functions $\varrho$ and $\varsigma$ which satisfy

$$\begin{aligned} \mathbf{1} _{Q _\frac{1}{8}} \le \varsigma \le \mathbf{1} _{Q _\frac{1}{4}} \le \varrho \le \mathbf{1} _{Q _\frac{1}{2}}. \end{aligned}$$

Take the dot product of the vorticity equation (6) with $\frac{3}{2} \omega ^\frac{1}{2}$:

$$\begin{aligned} \frac{3}{2} \omega ^\frac{1}{2} \cdot \partial _t\omega&= \partial _t(|\omega | ^\frac{3}{2}), \\ \frac{3}{2} \omega ^\frac{1}{2} \cdot (u \cdot \nabla ) \omega&= (u \cdot \nabla ) (|\omega | ^\frac{3}{2}), \\ \frac{3}{2} \omega ^\frac{1}{2} \cdot \varDelta \omega&\le \varDelta (|\omega | ^\frac{3}{2}) - \frac{4}{3} |\nabla \omega ^\frac{3}{4}| ^2. \end{aligned}$$

Therefore,

$$\begin{aligned} (\partial _t+ u \cdot \nabla - \varDelta ) (|\omega |^\frac{3}{2}) + \frac{3}{2} \omega \cdot \nabla u \cdot \omega ^\frac{1}{2} + \frac{4}{3} |\nabla \omega ^\frac{3}{4}| ^2 \le 0. \end{aligned}$$

Multiply by $\varrho ^6$ then integrate over space to get

$$\begin{aligned} \int \varrho ^6 (\partial _t+ u \cdot \nabla - \varDelta ) (|\omega |^\frac{3}{2}) {\mathrm {d}}x + \frac{4}{3} \int \varrho ^6 |\nabla \omega ^\frac{3}{4}| ^2 {\mathrm {d}}x&\le -\frac{3}{2} \int \varrho ^6 \omega \cdot \nabla u \cdot \omega ^\frac{1}{2} {\mathrm {d}}x. \end{aligned}$$

(49)

For the left hand side, we can integrate by parts to get

$$\begin{aligned}&\int \varrho ^6 (\partial _t+ u \cdot \nabla - \varDelta ) (|\omega |^\frac{3}{2}) {\mathrm {d}}x \nonumber \\&\quad = \frac{{\mathrm {d}}}{{\mathrm {d}}t} \int \varrho ^6 |\omega | ^\frac{3}{2} {\mathrm {d}}x - \int \left( (\partial _t+ u \cdot \nabla + \varDelta ) \varrho ^6 \right) |\omega |^\frac{3}{2} {\mathrm {d}}x, \end{aligned}$$

(50)

where the latter can be controlled by

$$\begin{aligned} \int \left( (\partial _t+ u \cdot \nabla + \varDelta ) \varrho ^6 \right) |\omega |^\frac{3}{2} {\mathrm {d}}x \le C \left( 1 + \Vert u \Vert _{L ^\infty (B _\frac{1}{2})} \right) \int \varrho ^4 |\omega | ^\frac{3}{2} {\mathrm {d}}x. \end{aligned}$$

(51)

For the right hand side, using $u = v + w$ over the support of $\varrho$, we can separate

$$\begin{aligned} \int \varrho ^6 \omega \cdot \nabla u \cdot \omega ^\frac{1}{2} {\mathrm {d}}x = \int \varrho ^6 \omega \cdot \nabla v \cdot \omega ^\frac{1}{2} {\mathrm {d}}x + \int \varrho ^6 \omega \cdot \nabla w \cdot \omega ^\frac{1}{2} {\mathrm {d}}x, \end{aligned}$$

(52)

The $\nabla v$ term can be controlled by

$$\begin{aligned} \int \varrho ^6 \omega \cdot \nabla v \cdot \omega ^\frac{1}{2} {\mathrm {d}}x&= -\int \omega \cdot \nabla (\varrho ^6 \omega ^\frac{1}{2}) \cdot v {\mathrm {d}}x \nonumber \\&= -\int \varrho ^6 \omega \cdot \nabla (\omega ^\frac{1}{2}) \cdot v {\mathrm {d}}x - \int \omega \cdot (\omega ^\frac{1}{2} \otimes \nabla \varrho ^6) \cdot v {\mathrm {d}}x, \end{aligned}$$

(53)

where

$$\begin{aligned} \omega \cdot \nabla (\omega ^\frac{1}{2})&= |\omega | ^{-\frac{1}{2}} \omega \cdot \nabla \omega - \frac{1}{2} (\omega \cdot \nabla \omega \cdot \omega ^{-\frac{3}{2}}) \omega = \omega ^\frac{1}{2} \cdot \nabla \omega - \frac{1}{2} (\omega ^\frac{1}{2} \cdot \nabla \omega \cdot \omega ^0) \omega ^0 \\ \Rightarrow |\omega \cdot \nabla (\omega ^\frac{1}{2})|&\le \left| \frac{3}{2} \omega ^\frac{1}{2} \cdot \nabla \omega \right| = 2 |\omega | ^\frac{3}{4} \left| \frac{3}{4} \omega ^{-\frac{1}{4}} \cdot \nabla \omega \right| = 2 |\omega | ^\frac{3}{4} \left| \nabla |\omega | ^\frac{3}{4} \right| \\&\le 2 |\omega | ^\frac{3}{4} |\nabla \omega ^\frac{3}{4}| \le |\omega | ^\frac{3}{2} + |\nabla \omega ^\frac{4}{3}| ^2. \end{aligned}$$

Here the second to the last inequality is due to $\partial _i |\omega | ^\frac{3}{4} = \partial _i \omega ^\frac{3}{4} \cdot \omega ^0$. Since $|v| \le 1$ over the support of $\varrho$,

$$\begin{aligned} \int \varrho ^6 \omega \cdot \nabla (\omega ^\frac{1}{2}) \cdot v {\mathrm {d}}x&\le \int \varrho ^6 |\omega | ^\frac{3}{2} {\mathrm {d}}x + \int \varrho ^6 | \nabla \omega ^\frac{3}{4} | ^2 {\mathrm {d}}x. \end{aligned}$$

(54)

By using (50)–(54) in (49), we conclude that

$$\begin{aligned}&\frac{{\mathrm {d}}}{{\mathrm {d}}t} \int \varrho ^6 |\omega | ^\frac{3}{2} {\mathrm {d}}x + \frac{4}{3} \int \varrho ^6 |\nabla \omega ^\frac{3}{4}| ^2 {\mathrm {d}}x \\&\quad \le \int \left[ (\partial _t+ u \cdot \nabla + \varDelta ) \varrho ^6 \right] |\omega |^\frac{3}{2} {\mathrm {d}}x \\&\qquad + \int \varrho ^6 \omega \cdot \nabla w \cdot \omega ^\frac{1}{2} {\mathrm {d}}x \\&\qquad + \int \omega \cdot (\omega ^\frac{1}{2} \otimes \nabla \varrho ^6) \cdot v {\mathrm {d}}x \\&\qquad + \int \varrho ^6 |\omega | ^\frac{3}{2} {\mathrm {d}}x + \int \varrho ^6 | \nabla \omega ^\frac{3}{4} | ^2 {\mathrm {d}}x\\&\frac{{\mathrm {d}}}{{\mathrm {d}}t} \int \varrho ^6 |\omega | ^\frac{3}{2} {\mathrm {d}}x + \frac{1}{3} \int \varrho ^6 |\nabla \omega ^\frac{3}{4}| ^2 {\mathrm {d}}x\\&\quad \le C \left( 1 + \Vert u (t) \Vert _{L ^{\infty } (B _\frac{1}{2})} + \Vert \nabla w (t) \Vert _{L ^{\infty } (B _\frac{1}{2})} \right) \int \varrho ^4 |\omega | ^\frac{3}{2} {\mathrm {d}}x. \end{aligned}$$

By Hölder’s inequality,

$$\begin{aligned} \int \varrho ^4 |\omega | ^\frac{3}{2} {\mathrm {d}}x \le \Vert \omega (t) \Vert _{L ^\frac{3}{2} (B _\frac{1}{2})} ^\frac{1}{2} \left( \int \varrho ^6 |\omega | ^\frac{3}{2} {\mathrm {d}}x \right) ^\frac{2}{3}. \end{aligned}$$

Therefore we can write

$$\begin{aligned}&\frac{{\mathrm {d}}}{{\mathrm {d}}t} \int \varrho ^6 |\omega | ^\frac{3}{2} {\mathrm {d}}x + \frac{1}{3} \int \varrho ^6 |\nabla \omega ^\frac{3}{4}| ^2 {\mathrm {d}}x \le C {\varPhi } (t) \left( 1 + \int \varrho ^6 |\omega | ^\frac{3}{2} {\mathrm {d}}x \right) , \end{aligned}$$

where

$$\begin{aligned} {\varPhi } (t)&= \left( 1 + \Vert u (t) \Vert _{L ^{\infty } (B _\frac{1}{2})} + \Vert \nabla w (t) \Vert _{L ^{\infty } (B _\frac{1}{2})} \right) \Vert \omega (t) \Vert _{L ^\frac{3}{2} (B _\frac{1}{2})} ^\frac{1}{2} \\&\le \left( 2 + \Vert w (t) \Vert _{L ^{\infty } (B _\frac{1}{2})} + \Vert \nabla w (t) \Vert _{L ^{\infty } (B _\frac{1}{2})} \right) \\&\quad \times \left( \Vert {\text {curl}}v (t) \Vert _{L ^\frac{3}{2} (B _\frac{1}{2})} ^\frac{1}{2} + \Vert {\text {curl}}w (t) \Vert _{L ^\frac{3}{2} (B _\frac{1}{2})} ^\frac{1}{2} \right) . \end{aligned}$$

since $u = w + v$, and $|v| \le 1$ inside $B _\frac{1}{2}$. By (47),

$$\begin{aligned} \int _{-\frac{1}{4}} ^0 {\varPhi } (t) {\mathrm {d}}t \lesssim \left( 1 + \Vert w \Vert _{L ^\frac{4}{3} _t \mathrm {Lip}_x (Q _\frac{1}{2})} \right) \left( \Vert \nabla v \Vert _{L ^2 (Q _\frac{1}{2})} ^\frac{1}{2} + \Vert {\text {curl}}w (t) \Vert _{L ^2 _t L ^\frac{3}{2} _x (Q _\frac{1}{2})} ^\frac{1}{2} \right) \le C. \end{aligned}$$

Thus, by Grönwall’s inequality,

$$\begin{aligned} \Vert \omega ^\frac{3}{4} \Vert _{L ^\infty _t L ^2 _x \cap L ^2 _t \dot{H} ^1 _x(Q _\frac{1}{4})} ^2 \le e ^C - 1. \end{aligned}$$

$\square$

Proof of Proposition 3(b)

From Proposition 3(a) and Sobolev embedding,

$$\begin{aligned} \Vert \omega \Vert _{L ^\infty _t L ^\frac{3}{2} _x \cap L ^\frac{3}{2} _t L ^\frac{9}{2} _x (Q _\frac{1}{4})} \le C, \end{aligned}$$

this interpolates the space

$$\begin{aligned} \Vert \omega \Vert _{L ^4 _t L ^2 _x (Q _\frac{1}{4})} \le C. \end{aligned}$$

Multiply the vorticity equation (6) by $\varsigma ^2 \omega$ then integrate over ${\mathbb {R}}^3$ to get

$$\begin{aligned} \frac{{\mathrm {d}}}{{\mathrm {d}}t} \int \varsigma ^2 \frac{|\omega | ^2}{2} {\mathrm {d}}x + \int \varsigma ^2 |\nabla \omega | ^2 {\mathrm {d}}x&= \int (\partial _t\varsigma ^2 + \varDelta \varsigma ^2) \frac{|\omega | ^2}{2} {\mathrm {d}}x \\&\quad - \int (u \cdot \nabla \omega ) \cdot \varsigma ^2 \omega {\mathrm {d}}x \\&\quad + \int (\omega \cdot \nabla u) \cdot \varsigma ^2 \omega {\mathrm {d}}x. \end{aligned}$$

The first integral is $L ^1$ in time because $\omega \in L ^4 _t L ^2 _x$. For the second,

$$\begin{aligned} \int (u \cdot \nabla \omega ) \cdot \varsigma ^2 \omega {\mathrm {d}}x&= \int \varsigma ^2 u \cdot \nabla \frac{|\omega | ^2}{2} {\mathrm {d}}x \\&= -\int \frac{|\omega | ^2}{2} u \cdot \nabla \varsigma ^2 {\mathrm {d}}x \\&= -\int \varsigma |\omega | ^2 u \cdot \nabla \varsigma \\&\le \Vert \varsigma \omega \Vert _{L ^2} \Vert u \cdot \nabla \varsigma |\omega | \Vert _{L ^2}; \end{aligned}$$

the latter is bounded $L ^1$ in time, by $u \in L ^\frac{4}{3} _t L ^\infty _x$ and $\omega \in L ^4 _t L ^2 _x$. For the third integral,

$$\begin{aligned} \int (\omega \cdot \nabla u) \cdot \varsigma ^2 \omega {\mathrm {d}}x = \int (\omega \cdot \nabla v) \cdot \varsigma ^2 \omega {\mathrm {d}}x + \int (\omega \cdot \nabla w) \cdot \varsigma ^2 \omega {\mathrm {d}}x. \end{aligned}$$

w is bounded in $L ^\frac{4}{3} _t \mathrm {Lip}_x$, and for v,

$$\begin{aligned} \int (\omega \cdot \nabla v) \cdot \varsigma ^2 \omega {\mathrm {d}}x&= \int v \cdot (\omega \cdot \nabla ) (\varsigma ^2 \omega ) {\mathrm {d}}x \\&= \int v \cdot \omega (\omega \cdot \nabla \varsigma ^2) {\mathrm {d}}x + \int v \cdot (\varsigma ^2 \omega \cdot \nabla \omega ) {\mathrm {d}}x. \end{aligned}$$

The former is $L ^1$ in time, while the latter can be bounded by Cauchy-Schwartz,

$$\begin{aligned} \int v \cdot (\varsigma ^2 \omega \cdot \nabla \omega ) {\mathrm {d}}x \le \frac{1}{2} \int |v \otimes \varsigma \omega | ^2 {\mathrm {d}}x + \frac{1}{2} \int \varsigma ^2 |\nabla \omega | ^2 {\mathrm {d}}x. \end{aligned}$$

In conclusion,

$$\begin{aligned}&\frac{{\mathrm {d}}}{{\mathrm {d}}t} \int \varsigma ^2 \frac{|\omega | ^2}{2} {\mathrm {d}}x + \frac{1}{2} \int \varsigma ^2 |\nabla \omega | ^2 {\mathrm {d}}x \\&\quad \le C \Vert \omega (t) \Vert _{L ^2 (B _\frac{1}{4})} ^2 + C \Vert u (t) \Vert _{L ^\infty (B _\frac{1}{4})} \Vert \omega (t) \Vert _{L ^2 (B _\frac{1}{4})} \Vert \varsigma \omega (t) \Vert _{L ^2} \\&\qquad + C \Vert \nabla w \Vert _{L ^\infty (B _\frac{1}{4})} \Vert \varsigma \omega (t) \Vert _{L ^2} ^2 \\&\quad \le C{\varPhi } (t) \left( 1 + \int \varsigma ^2 \frac{|\omega | ^2}{2} {\mathrm {d}}x \right) , \end{aligned}$$

where

$$\begin{aligned} {\varPhi } (t) = \Vert \omega (t) \Vert _{L ^2 (B _\frac{1}{4})} ^2 + \Vert u (t) \Vert _{L ^\infty (B _\frac{1}{4})} \Vert \omega (t) \Vert _{L ^2 (B _\frac{1}{4})} + \Vert \nabla w (t) \Vert _{L ^\infty (B _\frac{1}{4})}, \end{aligned}$$

whose integral is bounded using (47),

$$\begin{aligned} \int _{-\frac{1}{16}} ^0 {\varPhi } (t) {\mathrm {d}}t \le \Vert \omega \Vert _{L ^2 (Q _\frac{1}{4})} ^2 + \Vert u \Vert _{L ^{\frac{4}{3}} _t L ^\infty _x (Q _\frac{1}{4})} \Vert \omega \Vert _{L ^4 _t L ^2 _x (Q _\frac{1}{4})} + \Vert \nabla w \Vert _{L ^{\frac{4}{3}} _t L ^\infty _x (Q _\frac{1}{4})} \le C. \end{aligned}$$

By a Grönwall is argument, we have

$$\begin{aligned} \Vert \omega \Vert _{L ^\infty _t L ^2 _x \cap L ^2 _t \dot{H} ^1 _x(Q _\frac{1}{8})} ^2 \le e ^C - 1. \end{aligned}$$

$\square$

6.2 Bound Higher Derivatives in the Energy Space

Now we bootstrap to higher regularity of $\omega$ using similar ideas as in the proof of Proposition 3.

Proposition 4

For any $n \ge 1$, if $u = v + w$ in $Q _{8 ^{-n}}$, where v, w are bounded in

$$\begin{aligned}&\Vert v \Vert _{L ^\infty (Q _{8 ^{-n} / 2})} + \Vert v \Vert _{L ^2 _t H ^{n + 1} _x (Q _{8 ^{-n} / 2})} \le c _n, \end{aligned}$$

(55)

$$\begin{aligned}&\Vert w \Vert _{L ^\frac{4}{3} _t C ^{n + 1} _x (Q _{8 ^{-n} / 2})} \le c _n, \end{aligned}$$

(56)

for some constant $c _n$, $\omega = {\text {curl}}u$ solves the vorticity equation (6), and is bounded in

$$\begin{aligned} \Vert \omega \Vert _{L ^\infty _t H ^{n - 1} _x \cap L ^2 _t H ^n _x (Q _{8 ^{-n} / 2})} \le c _n, \end{aligned}$$

(57)

then for any multiindex $\alpha$ with $|\alpha | = n$,

• (a) $\Vert \nabla ^\alpha \omega ^\frac{3}{4} \Vert _{{\mathcal {E}} (Q _{8 ^{-n} / 4})} \le C _n$

• (b) $\Vert \nabla ^\alpha \omega \Vert _{{\mathcal {E}} (Q _{8 ^{-n-1}})} \le C _n$

for some $C _n$ depending on $c _n$ and n.

Proof of Proposition 4(a)

Similarly, we fix smooth cut-off functions $\varrho _n$ and $\varsigma _n$ which satisfy

$$\begin{aligned} \mathbf{1} _{Q _{8 ^{-n-1}}} \le \varsigma _n\le \mathbf{1} _{Q _{8 ^{-n} / 4}} \le \varrho _n\le \mathbf{1} _{Q _{8 ^{-n} / 2}}. \end{aligned}$$

Differentiate (6) by $\nabla ^\alpha$ to get

$$\begin{aligned} \partial _t\nabla ^\alpha \omega + u \cdot \nabla \nabla ^\alpha \omega - \nabla ^\alpha \omega \cdot \nabla u + {\mathbf {P}}_{\alpha } = \varDelta \nabla ^\alpha \omega , \end{aligned}$$

(58)

where

$$\begin{aligned} {\mathbf {P}}_\alpha = \sum _{\beta < \alpha } \begin{pmatrix} \alpha \\ \beta \end{pmatrix} {\text {curl}}\left( \nabla ^\beta \omega \times \nabla ^{\alpha - \beta } u \right) . \end{aligned}$$

Multiply (58) by $\frac{3}{2}\varrho _n^6 (\nabla ^\alpha \omega ) ^\frac{1}{2}$ then integrate in space to get

$$\begin{aligned}&\frac{{\mathrm {d}}}{{\mathrm {d}}t} \int \varrho _n^6 |\nabla ^\alpha \omega | ^\frac{3}{2} {\mathrm {d}}x + \frac{4}{3} \int \varrho _n^6 |\nabla \nabla ^\alpha \omega ^\frac{3}{4}| ^2 {\mathrm {d}}x \\&\quad \le \int \left[ (\partial _t+ u \cdot \nabla + \varDelta ) \varrho _n^6 \right] |\nabla ^\alpha \omega |^\frac{3}{2} {\mathrm {d}}x \\&\qquad + \int \varrho _n^6 \nabla ^\alpha \omega \cdot \nabla w \cdot (\nabla ^\alpha \omega ) ^\frac{1}{2} {\mathrm {d}}x \\&\qquad + \int \nabla ^\alpha \omega \cdot ((\nabla ^\alpha \omega ) ^\frac{1}{2} \otimes \nabla \varrho _n^6) \cdot v {\mathrm {d}}x \\&\qquad + \Vert v\Vert _{L ^\infty (Q _{8 ^{-n}})} ^2 \int \varrho _n^6 |\nabla ^\alpha \omega | ^\frac{3}{2} {\mathrm {d}}x + \int \varrho _n^6 | \nabla \nabla ^\alpha \omega ^\frac{3}{4} | ^2 {\mathrm {d}}x \\&\qquad + \frac{3}{2} \int \varrho _n^6 (\nabla ^\alpha \omega ) ^\frac{1}{2} \cdot {\mathbf {P}}_\alpha {\mathrm {d}}x, \end{aligned}$$

the same as in the proof of Proposition 3(a). Therefore,

$$\begin{aligned}&\frac{{\mathrm {d}}}{{\mathrm {d}}t} \int \varrho _n^6 |\nabla ^\alpha \omega | ^\frac{3}{2} {\mathrm {d}}x + \frac{1}{3} \int \varrho _n^6 |\nabla \nabla ^\alpha \omega ^\frac{3}{4}| ^2 {\mathrm {d}}x \\&\quad \le C \left( 1 + \Vert u (t) \Vert _{L ^{\infty } (B _{8 ^{-n}})} + \Vert \nabla w (t) \Vert _{L ^{\infty } (B _{8 ^{-n}})} \right) \int \varrho _n^4 |\nabla ^\alpha \omega | ^\frac{3}{2} {\mathrm {d}}x \\&\qquad + \frac{3}{2} \int \varrho _n^6 (\nabla ^\alpha \omega ) ^\frac{1}{2} \cdot {\mathbf {P}}_\alpha {\mathrm {d}}x. \end{aligned}$$

Terms other than ${\mathbf {P}}_\alpha$ are dealt with in the same way as in Proposition 3:

$$\begin{aligned} \int \varrho _n^4 |\nabla ^\alpha \omega | ^\frac{3}{2} {\mathrm {d}}x \le \Vert \nabla ^\alpha \omega (t) \Vert _{L ^\frac{3}{2} (B _{8^{-n}})} ^\frac{1}{2} \left( \int \varrho _n^6 |\nabla ^\alpha \omega | ^\frac{3}{2} {\mathrm {d}}x \right) ^\frac{2}{3}. \end{aligned}$$

The induction condition (57) ensures that $\Vert \nabla ^\alpha \omega \Vert _{L ^2 (Q _{8 ^{-n}})} \le c _n$. Therefore

$$\begin{aligned}&\int _{-8 ^{-2n}} ^0 \left( 1 + \Vert u (t) \Vert _{L ^{\infty } (B _{8 ^{-n}})} + \Vert \nabla w (t) \Vert _{L ^{\infty } (B _{8 ^{-n}})} \right) \Vert \nabla ^\alpha \omega (t) \Vert _{L ^\frac{3}{2} (B _{8^{-n}})} ^\frac{1}{2} {\mathrm {d}}t \\&\qquad \lesssim \left( 1 + \Vert v \Vert _{L ^\infty (B _{8 ^{-n}})} + \Vert w \Vert _{L ^\frac{4}{3} _t C ^1 _x (B _{8 ^{-n}})} \right) \Vert \nabla ^\alpha \omega \Vert _{L ^2 (Q _{8 ^{-n}})} ^{\frac{1}{2}} \le C _n. \end{aligned}$$

Now let’s focus on ${\mathbf {P}}_\alpha$:

$$\begin{aligned} | {\mathbf {P}}_\alpha |&\lesssim \sum _{k = 0} ^n |\nabla ^k \omega | |\nabla ^{n - k + 1} u| \le \sum _{k = 0} ^n |\nabla ^k \omega | |\nabla ^{n - k + 1} v| + \sum _{k = 0} ^n |\nabla ^k \omega | |\nabla ^{n - k + 1} w|. \end{aligned}$$

We denote

$$\begin{aligned} {\mathbf {P}}_{v, k}= |\nabla ^k \omega | |\nabla ^{n - k + 1} v|, \qquad {\mathbf {P}}_{w, k}= |\nabla ^k \omega | |\nabla ^{n - k + 1} w|. \end{aligned}$$

First we estimate ${\mathbf {P}}_{v, k}$. By (55) and (57), when $k = 0$,

$$\begin{aligned} \Vert {\mathbf {P}}_{v, 0}\Vert _{L ^1 _t L ^\frac{3}{2} _x (Q _{8 ^{-n}})} \le \Vert \omega \Vert _{L ^2 _t L ^6 _x (Q _{8 ^{-n}})} \Vert \nabla ^{n + 1} v \Vert _{L ^2 _t L ^2 _x (Q _{8 ^{-n}})} \le C _n, \end{aligned}$$

and when $0 < k \le n$,

$$\begin{aligned} \Vert {\mathbf {P}}_{v, k}\Vert _{L ^1 _t L ^\frac{3}{2} _x (Q _{8 ^{-n}})} \le \Vert \nabla ^{k} \omega \Vert _{L ^2 _t L ^2 _x (Q _{8 ^{-n}})} \Vert \nabla ^{n + 1 - k} v \Vert _{L ^2 _t L ^6 _x (Q _{8 ^{-n}})} \le C _n. \end{aligned}$$

Next we estimate ${\mathbf {P}}_{w, k}$. When $0 \le k < n$,

$$\begin{aligned} \Vert {\mathbf {P}}_{w, k}\Vert _{L ^1 _t L ^\frac{3}{2} _x (Q _{8 ^{-n}})} \le \Vert \nabla ^{k} \omega \Vert _{L ^\infty _t L ^2 _x (Q _{8 ^{-n}})} \Vert \nabla ^{n + 1 - k} w \Vert _{L ^\frac{4}{3} _t L ^\infty _x (Q _{8 ^{-n}})} \le C _n. \end{aligned}$$

Finally, when $k = n$,

$$\begin{aligned} | {\mathbf {P}}_{w, n} | _{L ^\frac{3}{2} _x (B _{8 ^{-n}})} \le |\nabla ^n \omega | | \nabla w |. \end{aligned}$$

Therefore,

$$\begin{aligned}&\int \varrho _n^6 (\nabla ^\alpha \omega ) ^\frac{1}{2} \cdot {\mathbf {P}}_\alpha {\mathrm {d}}x \le \left( 1 + \int \varrho _n^6 |\nabla ^\alpha \omega | ^\frac{3}{2} {\mathrm {d}}x \right) \\&\quad \times \left( \sum _{k = 0} ^n \Vert {\mathbf {P}}_{v, k}\Vert _{L ^\frac{3}{2} _x (B _{8 ^{-n}})} + \sum _{k = 0} ^{n - 1} \Vert {\mathbf {P}}_{w, k}\Vert _{L ^\frac{3}{2} _x (B _{8 ^{-n}})} + \Vert \nabla w \Vert _{L ^\infty _x (B _{8 ^{-n}})} \right) . \end{aligned}$$

In conclusion, we have shown that

$$\begin{aligned}&\frac{{\mathrm {d}}}{{\mathrm {d}}t} \int \varrho _n^6 |\nabla ^\alpha \omega | ^\frac{3}{2} {\mathrm {d}}x + \frac{1}{3} \int \varrho _n^6 |\nabla \nabla ^\alpha \omega ^\frac{3}{4}| ^2 {\mathrm {d}}x \le C {\varPhi } (t) \left( 1 + \int \varrho _n^6 |\nabla ^n \omega | ^\frac{3}{2} {\mathrm {d}}x \right) , \end{aligned}$$

where

$$\begin{aligned} {\varPhi } (t)&= \left( 1 + \Vert u (t) \Vert _{L ^{\infty } (B _{8 ^{-n}})} + \Vert \nabla w (t) \Vert _{L ^{\infty } (B _{8 ^{-n}})} \right) \Vert \nabla ^\alpha \omega (t) \Vert _{L ^\frac{3}{2} (B _{8^{-n}})} ^\frac{1}{2} \\&\quad + \sum _{k = 0} ^n \Vert {\mathbf {P}}_{v, k}\Vert _{L ^\frac{3}{2} _x (B _{8 ^{-n}})} + \sum _{k = 0} ^{n - 1} \Vert {\mathbf {P}}_{w, k}\Vert _{L ^\frac{3}{2} _x (B _{8 ^{-n}})} + \Vert \nabla w \Vert _{L ^\infty _x (B _{8 ^{-n}})} \end{aligned}$$

with integral

$$\begin{aligned} \int _{-8 ^{-2n} / 4} ^0 {\varPhi } (t) {\mathrm {d}}t \le C _n. \end{aligned}$$

Taking the sum over all multi-index $\alpha$ with size $|\alpha | = n$, we have

$$\begin{aligned}&\frac{{\mathrm {d}}}{{\mathrm {d}}t} \int \varrho _n^6 |\nabla ^{n} \omega | ^\frac{3}{2} {\mathrm {d}}x + \frac{1}{3} \int \varrho _n^6 |\nabla ^{n + 1} \omega ^\frac{3}{4}| ^2 {\mathrm {d}}x \le C {\varPhi } (t) \left( 1 + \int \varrho _n^6 |\nabla ^{n + 1} \omega | ^\frac{3}{2} {\mathrm {d}}x \right) . \end{aligned}$$

Finally, Grönwall inequality gives

$$\begin{aligned} \Vert |\nabla ^{n + 1} \omega | ^\frac{3}{4}\Vert _{L ^\infty _t L ^2 _x \cap L ^2 _t \dot{H} ^1 _x(Q _{8 ^{-n} / 4})} \le C _n. \end{aligned}$$

$\square$

Proof of Proposition 4(b)

Now we multiply (58) by $\varsigma _n^2 \nabla ^\alpha \omega$ then integrate over ${\mathbb {R}}^3$ to get that

$$\begin{aligned} \frac{{\mathrm {d}}}{{\mathrm {d}}t} \int \varsigma _n^2 \frac{|\nabla ^\alpha \omega | ^2}{2} {\mathrm {d}}x + \int \varsigma _n^2 |\nabla \nabla ^\alpha \omega | ^2 {\mathrm {d}}x&= \int (\partial _t\varsigma _n^2 + \varDelta \varsigma _n^2) \frac{|\nabla ^\alpha \omega | ^2}{2} {\mathrm {d}}x \\&\quad - \int (u \cdot \nabla \nabla ^\alpha \omega ) \cdot \varsigma _n^2 \nabla ^\alpha \omega {\mathrm {d}}x \\&\quad + \int (\nabla ^\alpha \omega \cdot \nabla u) \cdot \varsigma _n^2 \nabla ^\alpha \omega {\mathrm {d}}x \\&\quad + \int \varsigma _n^2 \nabla ^\alpha \omega \cdot {\mathbf {P}}_\alpha {\mathrm {d}}x. \end{aligned}$$

For the same reason, the only term that we need to take care of is ${\mathbf {P}}_\alpha$ term, and the others are dealt the same as with Proposition 3(b):

$$\begin{aligned}&\int (\partial _t\varsigma _n^2 + \varDelta \varsigma _n^2) \frac{|\nabla ^\alpha \omega | ^2}{2} {\mathrm {d}}x - \int (u \cdot \nabla \nabla ^\alpha \omega ) \cdot \varsigma _n^2 \nabla ^\alpha \omega {\mathrm {d}}x + \int (\nabla ^\alpha \omega \cdot \nabla u) \cdot \varsigma _n^2 \nabla ^\alpha \omega {\mathrm {d}}x \\&\quad \lesssim _n \Vert \nabla ^\alpha \omega \Vert _{L ^2 (Q _{8 ^{-n} / 4})} ^2 + \Vert u \Vert _{L ^\infty (Q _{8 ^{-n} / 4})} \Vert \nabla ^\alpha \omega \Vert _{L ^2 (Q _{8 ^{-n} / 4})} \left( \int \varsigma _n^2 \frac{| \nabla ^\alpha \omega | ^2}{2} {\mathrm {d}}x \right) ^\frac{1}{2} \\&\qquad + \Vert \nabla w \Vert _{L ^\infty (Q _{8 ^{-n} / 4})} \int \varsigma _n^2 \frac{| \nabla ^\alpha \omega | ^2}{2} {\mathrm {d}}x + \Vert v \Vert _{L ^\infty (Q _{8 ^{-n} / 4})} \Vert \nabla ^\alpha \omega \Vert _{L ^2 (Q _{8 ^{-n} / 4})} ^2 \\&\qquad + \frac{1}{\varepsilon }\Vert v \Vert _{L ^\infty (Q _{8 ^{-n} / 4})} ^2 \int \varsigma _n^2 \frac{| \nabla ^\alpha \omega | ^2}{2} {\mathrm {d}}x + \varepsilon \int \varsigma _n^2 |\nabla \nabla ^\alpha \omega | ^2 {\mathrm {d}}x. \end{aligned}$$

The last term can be absorbed into the left, and we will use Grönwall on the remaining terms.

Now we shall focus on the ${\mathbf {P}}_\alpha$ term. From Proposition 4(a), we have

$$\begin{aligned} \Vert \nabla ^n \omega \Vert _{L ^\infty _t L ^\frac{3}{2} _x \cap L ^\frac{3}{2} _x L ^\frac{9}{2} _t (Q _{8 ^{-n} / 4})} \le C _n. \end{aligned}$$

(59)

Again, by interpolation,

$$\begin{aligned} \Vert \nabla ^n \omega \Vert _{L ^4 _t L ^2 _x (Q _{8 ^{-n} / 4})} \le C _n, \qquad \Vert \nabla ^n \omega \Vert _{L ^2 _t L ^3 _x (Q _{8 ^{-n} / 4})} \le C _n. \end{aligned}$$

First we estimate ${\mathbf {P}}_{w, k}$. In this case, for any $0 \le k \le n$,

$$\begin{aligned} \Vert {\mathbf {P}}_{w, k}\Vert _{L ^1 _t L ^2 _x (Q _{8 ^{-n} / 4})} \le \Vert \nabla ^{k} \omega \Vert _{L ^4 _t L ^2 _x (Q _{8 ^{-n} / 4})} \Vert \nabla ^{n + 1 - k} w \Vert _{L ^\frac{4}{3} _t L ^\infty _x (Q _{8 ^{-n}})} \le C _n. \end{aligned}$$

Then we estimate ${\mathbf {P}}_{v, k}$. When $0 < k \le n$,

$$\begin{aligned} \Vert {\mathbf {P}}_{v, k}\Vert _{L ^1 _t L ^2 _x (Q _{8 ^{-n}})} \le \Vert \nabla ^{k} \omega \Vert _{L ^2 _t L ^3 _x (Q _{8 ^{-n}})} \Vert \nabla ^{n + 1 - k} v \Vert _{L ^2 _t L ^6 _x (Q _{8 ^{-n}})} \le C _n. \end{aligned}$$

For the case $k = 0$ of the $v$ term, we put the curl on $\nabla ^\alpha \omega$ to obtain

$$\begin{aligned}&\int \varsigma _n^2 \nabla ^\alpha \omega \cdot {\text {curl}}\left( \omega \times \nabla ^\alpha v\right) {\mathrm {d}}x \\&\quad = \int \left( \omega \times \nabla ^\alpha v\right) \cdot {\text {curl}}(\varsigma _n^2 \nabla ^\alpha \omega ) {\mathrm {d}}x \\&\quad \le \int \varsigma _n^2 |\omega | |\nabla ^\alpha v| |\nabla \nabla ^\alpha \omega | + \varsigma _n|\nabla \varsigma _n| |\omega | |\nabla ^\alpha v| |\nabla ^\alpha \omega | {\mathrm {d}}x \\&\quad \le \int \varsigma _n^2 |\omega | ^2 |\nabla ^\alpha v| ^2 {\mathrm {d}}x + \varepsilon \int \varsigma _n^2 |\nabla \nabla ^\alpha \omega | ^2 {\mathrm {d}}x + \frac{1}{\varepsilon }\int |\nabla \varsigma _n| ^2 |\nabla ^\alpha \omega | ^2 {\mathrm {d}}x, \end{aligned}$$

where $|\nabla \nabla ^\alpha \omega |$ term can be absorbed to the left. By (59) and Sobolev embedding,

$$\begin{aligned} \Vert \omega \Vert _{L ^\infty _t L ^3 _x (Q _{8 ^{-n} / 4})} \le C _n. \end{aligned}$$

Therefore

$$\begin{aligned} \iint \varsigma _n^2 |\omega | ^2 |\nabla ^\alpha v| ^2 {\mathrm {d}}x {\mathrm {d}}t \le \Vert \omega \Vert _{L ^\infty _t L ^3 _x (Q _{8 ^{-n} / 4})} ^2 \Vert \nabla ^\alpha v\Vert _{L ^2 _t L ^6 _x (Q _{8 ^{-n} / 4})} ^2 \le C _n. \end{aligned}$$

In conclusion,

$$\begin{aligned} \frac{{\mathrm {d}}}{{\mathrm {d}}t} \int \varsigma _n^2 \frac{|\nabla ^\alpha \omega | ^2}{2} {\mathrm {d}}x + \int \varsigma _n^2 |\nabla \nabla ^\alpha \omega | ^2 {\mathrm {d}}x \le C {\varPhi } (t) \left( 1 + \int \varsigma _n^2 \frac{|\nabla ^\alpha \omega | ^2}{2} {\mathrm {d}}x \right) , \end{aligned}$$

where

$$\begin{aligned} {\varPhi } (t)&= \Vert \nabla ^\alpha \omega (t) \Vert _{L ^2 (B _{8 ^{-n} / 4})} ^2 + \Vert u \Vert _{L ^\infty (B _{8 ^{-n} / 4})} \Vert \nabla ^\alpha \omega \Vert _{L ^2 (B _{8 ^{-n} / 4})} \\&\quad + \Vert \nabla w \Vert _{L ^\infty (B _{8 ^{-n} / 4})} + \Vert v \Vert _{L ^\infty (B _{8 ^{-n} / 4})} \Vert \nabla ^\alpha \omega \Vert _{L ^2 (B _{8 ^{-n} / 4})} ^2 \\&\quad + \frac{1}{\varepsilon }\Vert v \Vert _{L ^\infty (B _{8 ^{-n} / 4})} ^2 + \sum _{k = 0} ^n \Vert {\mathbf {P}}_{w, k}\Vert _{L ^2 (B _{8 ^{-n} / 4})} + \sum _{k = 0} ^{n - 1} \Vert {\mathbf {P}}_{v, k}\Vert _{L ^2 (B _{8 ^{-n} / 4})} \\&\quad + \Vert \omega \Vert _{L ^3 (B _{8 ^{-n} / 4})} ^2 \Vert \nabla ^\alpha v\Vert _{L ^6 (B _{8 ^{-n} / 4})} ^2 + \frac{1}{\varepsilon }\Vert \nabla ^\alpha \omega (t) \Vert _{L ^2 (B _{8 ^{-n} / 4})} ^2 \end{aligned}$$

has the integral $\int _{-8 ^{-2n} / 16} ^0 {\varPhi } (t) {\mathrm {d}}t \le C _n$. Finally, Grönwall is inequality gives

$$\begin{aligned} \Vert \nabla ^\alpha \omega \Vert _{L ^\infty _t L ^2 _x \cap L ^2 _t \dot{H} ^1 _x(Q _{8 ^{-n-1}})} \le C _{n + 1}. \end{aligned}$$

$\square$

6.3 Proof of the Local Theorem

Proof of the Local Theorem 2

First, Proposition 1 gives

$$\begin{aligned} \Vert v \Vert _{{\mathcal {E}} (Q _1)} \le \eta , \end{aligned}$$

where $\eta$ can be chosen arbitrarily small if we pick $\eta _1$ small. Next, by Proposition 2, we know that

$$\begin{aligned} \Vert v \Vert _{L ^\infty (Q _\frac{1}{2})} \le 1. \end{aligned}$$

These two steps implies (47). As for (48), ${\text {curl}}w = \varpi$ in $B _1$, so we use interpolation in (37) to get that

$$\begin{aligned} \Vert {\text {curl}}w \Vert _{L ^2 _t L ^\frac{3}{2} _x (Q _\frac{1}{2})} \le \Vert \varpi \Vert _{L ^2 _t L ^{\frac{12}{7}} _x} \le \Vert \varpi \Vert _{L ^{p _1} _t L ^{q _3} _x} ^\frac{1}{2} \Vert \varpi \Vert _{L ^{p _2} _t L ^{q _4} _x} ^\frac{1}{2} \le \eta , \end{aligned}$$

w is harmonic inside $B _1$, therefore

$$\begin{aligned} \Vert w \Vert _{L ^\frac{4}{3} _t C ^n _x (Q _\frac{1}{2})} \lesssim _n \Vert w \Vert _{L ^\frac{4}{3} _t L ^1 _x (Q _1)} \le \eta , \end{aligned}$$

due to (36) and $p _1 \ge \frac{4}{3}$. Therefore, we can use Proposition 3 to obtain

$$\begin{aligned} \Vert \omega \Vert _{{\mathcal {E}} (Q _\frac{1}{8})} \le C. \end{aligned}$$

The next step is to use Proposition 4 iteratively. Suppose that for $n \ge 1$ we know that

$$\begin{aligned} \Vert \nabla ^{n - 1} \omega \Vert _{{\mathcal {E}} (Q _{8 ^{-n}})} \le c _{n}, \end{aligned}$$

which is equivalent to (57). Let $\varphi _n$ and $\varphi ^\sharp _n$ be a pair of smooth spatial cut-off functions, with

$$\begin{aligned} \mathbf{1} _{B _\frac{1}{8 ^n + 4}} \le \varphi _n\le \mathbf{1} _{B _\frac{1}{8 ^n + 3}}, \qquad \mathbf{1} _{B _\frac{1}{8 ^n + 2}} \le \varphi ^\sharp _n\le \mathbf{1} _{B _\frac{1}{8 ^n + 1}}, \end{aligned}$$

and set

$$\begin{aligned} v _n :{=} -{\text {curl}}\varphi ^\sharp _n\varDelta ^{-1}\varphi _n\omega , \qquad w _n = \varphi _nu - v _n. \end{aligned}$$

On the one hand, $\nabla v _n$ is a Riesz transform of $\varphi _n \omega$ up to lower order terms, so by the boundedness of Riesz transform we know that

$$\begin{aligned} \Vert \nabla ^{n + 1} v _n \Vert _{L ^2 (Q _{8 ^{-n} / 2})} \le \Vert \nabla ^{n} \omega \Vert _{L ^2 (Q _{8 ^{-n}})} \le c _{n - 1}. \end{aligned}$$

On the other hand, we have similar boundedness estimates to those of Proposition 2 as before, so

$$\begin{aligned} \Vert v _n \Vert _{L ^\infty (Q _{8 ^{-n} / 2})} \le 1. \end{aligned}$$

$w _n$ is harmonic in $B _{\frac{1}{8^n + 4}}$, so we also have

$$\begin{aligned} \Vert w _n \Vert _{L ^\frac{4}{3} _t C ^{n + 1} _x (Q _{8 ^{-n} / 2})} \lesssim _n \Vert w _n \Vert _{L ^\frac{4}{3} _t L ^1 _x (Q _{\frac{1}{8^n + 4}})} \le \eta . \end{aligned}$$

Therefore, by Proposition 4,

$$\begin{aligned} \Vert \nabla ^n \omega \Vert _{{\mathcal {E}} (Q _{8 ^{-n - 1}})} \le C _n. \end{aligned}$$

By induction, we have

$$\begin{aligned} \Vert \nabla ^n \omega \Vert _{L ^\infty _t L ^2 _x \cap L ^2 _t \dot{H} ^1 _x(Q _{8 ^{-n-1}})} \le C _n \end{aligned}$$

for any n. By Sobolev embedding, this implies, for any n, that

$$\begin{aligned} \Vert \nabla ^n \omega \Vert _{L ^\infty (Q _{8 ^{-n-3}})} \lesssim \Vert \nabla ^n \omega \Vert _{L ^\infty _t L ^2 _x (Q _{8 ^{-n-3}})} + \Vert \nabla ^{n + 2} \omega \Vert _{L ^\infty _t L ^2 _x (Q _{8 ^{-n-3}})} \le C _n . \end{aligned}$$

$\square$

Suitability of Solutions

Theorem 5

Let u be a suitable weak solution to the Navier–Stokes equation in ${\mathbb {R}}^3$. That is, $u \in L ^\infty _t L ^2 _x \cap L ^2 _t \dot{H} ^1 _x$ solves the following equation in the sense of distribution:

$$\begin{aligned} \partial _tu + u \cdot \nabla u + \nabla P&= \varDelta u, \qquad {\text {div}}u = 0, \end{aligned}$$

(60)

where P is the pressure, and u satisfies the following local energy inequality in the sense of distribution:

$$\begin{aligned} \partial _t\frac{|u|^2}{2}+ {\text {div}}\left( u \left( \frac{|u|^2}{2}+ P \right) \right) + |\nabla u| ^2 \le \varDelta \frac{|u|^2}{2}. \end{aligned}$$

(61)

Suppose $v \in L ^\infty _t L ^2 _x \cap L ^2 _t \dot{H} ^1 _x$ is compactly supported in space and solves the following equation,

$$\begin{aligned} \partial _tv + \omega \times v + \nabla {\mathbf {R}}(u \otimes v)&= \varDelta v + {\mathbf {C}}_v, \qquad {\text {div}}v = 0 \end{aligned}$$

(62)

where $\omega = {\text {curl}}u$ is the vorticity, ${\mathbf {C}}_v\in L ^1 _t L ^2 _{\mathrm {loc},x} + L ^2 _t L ^\frac{6}{5} _{\mathrm {loc},x}$ is a force term, and

$$\begin{aligned} {\mathbf {R}}= \frac{1}{2} {\text {tr}}-\varDelta ^{-1}{\text {div}}{\text {div}}\end{aligned}$$

is a symmetric Riesz operator. Moreover, suppose v differs from $\varphi u$ by

$$\begin{aligned} \varphi u - v = w \in L ^\infty _t H ^1 _x \cap L ^2 _t H ^2 _x \end{aligned}$$

for some fixed $\varphi \in C _c ^\infty ({\mathbb {R}}^3)$. Then v satisfies the following local energy inequality:

$$\begin{aligned} \partial _t\frac{|v|^2}{2}+ {\text {div}}\left( v {\mathbf {R}}(u \otimes v) \right) + |\nabla v| ^2&\le \varDelta \frac{|v|^2}{2}+ v \cdot {\mathbf {C}}_v. \end{aligned}$$

(63)

Proof

It is well-known that the pressure P can be recovered from u by

$$\begin{aligned} P = -\varDelta ^{-1}{\text {div}}{\text {div}}(u \otimes u). \end{aligned}$$

Since

$$\begin{aligned} u \cdot \nabla u + \nabla P&= \nabla \frac{|u|^2}{2}+ \omega \times u - \nabla \varDelta ^{-1}{\text {div}}{\text {div}}(u \otimes u) \\&= \omega \times u + \nabla {\mathbf {R}}(u \otimes u), \end{aligned}$$

the Navier–Stokes equation (60) can be rewritten as

$$\begin{aligned} \partial _tu + \omega \times u + \nabla {\mathbf {R}}(u \otimes u)&= \varDelta u, \end{aligned}$$

(64)

and local energy inequality (61) can be rewritten as

$$\begin{aligned} \partial _t\frac{|u|^2}{2}+ {\text {div}}\left( u {\mathbf {R}}(u \otimes u) \right) + |\nabla u| ^2 \le \varDelta \frac{|u|^2}{2}. \end{aligned}$$

(65)

First, multiply (64) by $\varphi$, to get

$$\begin{aligned} \partial _t\varphi u + \omega \times \varphi u + \nabla {\mathbf {R}}(u \otimes \varphi u) = \varDelta (\varphi u) + [\nabla {\mathbf {R}}, \varphi ] (u \otimes u) + [\varphi , \varDelta ] u. \end{aligned}$$

Denote

$$\begin{aligned} {\mathbf {C}}_u= [\nabla {\mathbf {R}}, \varphi ] (u \otimes u) + [\varphi , \varDelta ] u \end{aligned}$$

for these commutator terms. Subtracting the equation of v from this equation of $\varphi u$, we will have the equation for w. In summary,

$$\begin{aligned} \partial _t\varphi u + \omega \times \varphi u + \nabla {\mathbf {R}}(u \otimes \varphi u)&= \varDelta (\varphi u) + {\mathbf {C}}_u, \end{aligned}$$

(66)

$$\begin{aligned} \partial _tv + \omega \times v + \nabla {\mathbf {R}}(u \otimes v)&= \varDelta v + {\mathbf {C}}_v, \end{aligned}$$

(67)

$$\begin{aligned} \partial _tw + \omega \times w + \nabla {\mathbf {R}}(u \otimes w)&= \varDelta w + {\mathbf {C}}_u- {\mathbf {C}}_v. \end{aligned}$$

(68)

Recall from [22] that $\varDelta u \in L ^{\frac{4}{3}-\varepsilon } _{\mathrm {loc}(t, x)}$. Since $\varDelta w \in L ^2 _{t, x}$, we have $\varDelta v \in L ^{\frac{4}{3}-\varepsilon } _{\mathrm {loc}(t, x)}$. Moreover, ${\mathbf {C}}_u, {\mathbf {C}}_v\in L ^1 _t L ^2 _{\mathrm {loc},x} + L ^2 _t L ^\frac{6}{5} _{\mathrm {loc},x}$, and $\varphi u, v \in {\mathcal {E}}$ are compactly supported. Therefore, we can multiply (66) and (67) by w, and (68) by $\varphi u$ and v, to get

$$\begin{aligned} w \cdot \partial _t(\varphi u) + w \cdot \omega \times \varphi u + w \cdot \nabla {\mathbf {R}}(u \otimes \varphi u)&= w \cdot \varDelta (\varphi u) + w \cdot {\mathbf {C}}_u, \end{aligned}$$

(69)

$$\begin{aligned} w \cdot \partial _tv + w \cdot \omega \times v + w \cdot \nabla {\mathbf {R}}(u \otimes v)&= w \cdot \varDelta v + w \cdot {\mathbf {C}}_v\end{aligned}$$

(70)

$$\begin{aligned} \varphi u \cdot \partial _tw + \varphi u \cdot \omega \times w + \varphi u \cdot \nabla {\mathbf {R}}(u \otimes w)&= \varphi u \cdot \varDelta w + \varphi u \cdot ({\mathbf {C}}_u- {\mathbf {C}}_v). \end{aligned}$$

(71)

$$\begin{aligned} v \cdot \partial _tw + v \cdot \omega \times w + v \cdot \nabla {\mathbf {R}}(u \otimes w)&= v \cdot \varDelta w + v \cdot ({\mathbf {C}}_u- {\mathbf {C}}_v). \end{aligned}$$

(72)

Now take the sum of (69)–(72). The $\partial _t$ terms are

$$\begin{aligned}&\varphi u \cdot \partial _tw + w \cdot \partial _t(\varphi u) + v \cdot \partial _tw + w \cdot \partial _tv \\&\quad = \partial _t(\varphi u \cdot w) + \partial _t(w \cdot v) \\&\quad = \partial _t(|\varphi u| ^2 - |v| ^2). \end{aligned}$$

The $\omega \times$ terms are

$$\begin{aligned} w \cdot \omega \times \varphi u + \varphi u \cdot \omega \times w + w \cdot \omega \times v + v \cdot \omega \times w = 0. \end{aligned}$$

The $\nabla {\mathbf {R}}$ terms are

$$\begin{aligned}&w \cdot \nabla {\mathbf {R}}(u \otimes \varphi u) + v \cdot \nabla {\mathbf {R}}(u \otimes w) + \varphi u \cdot \nabla {\mathbf {R}}(u \otimes w) + w \cdot \nabla {\mathbf {R}}(u \otimes v) \\&\quad = {\text {div}}(w {\mathbf {R}}(u \otimes \varphi u)) + {\text {div}}(v {\mathbf {R}}(u \otimes w)) \\&\qquad + {\text {div}}(\varphi u {\mathbf {R}}(u \otimes w)) + {\text {div}}(w {\mathbf {R}}(u \otimes v)) \\&\qquad - {\text {div}}(w) \nabla {\mathbf {R}}(u \otimes \varphi u) - {\text {div}}(v) \nabla {\mathbf {R}}(u \otimes w) \\&\qquad - {\text {div}}(\varphi u) \nabla {\mathbf {R}}(u \otimes w) - {\text {div}}(\varphi ) \nabla {\mathbf {R}}(u \otimes v) \\&\quad = 2 {\text {div}}(\varphi u {\mathbf {R}}(u \otimes \varphi u) - v {\mathbf {R}}(u \otimes v)) \\&\qquad - (u \cdot \nabla \varphi ) \left( \nabla {\mathbf {R}}(u \otimes \varphi u) + \nabla {\mathbf {R}}(u \otimes w) + \nabla {\mathbf {R}}(u \otimes v) \right) \\&\quad = 2 {\text {div}}(\varphi u {\mathbf {R}}(u \otimes \varphi u) - v {\mathbf {R}}(u \otimes v)) - 2 (u \cdot \nabla \varphi ) {\mathbf {R}}(u \otimes \varphi u). \end{aligned}$$

Here we use ${\text {div}}v = 0, {\text {div}}(\varphi u) = {\text {div}}w = u \cdot \nabla \varphi$. The $\varDelta$ terms are

$$\begin{aligned}&\varphi u \cdot \varDelta w + w \cdot \varDelta (\varphi u) + v \cdot \varDelta w + w \cdot \varDelta v \\&\quad = \varDelta (u \cdot w) - 2 \nabla (\varphi u) : \nabla w + \varDelta (v \cdot w) - 2 \nabla v : \nabla w \\&\quad = \varDelta (|\varphi u| ^2 - |v| ^2) - 2 (|\nabla (\varphi u)| ^2 - |\nabla v| ^2). \end{aligned}$$

The commutator terms are

$$\begin{aligned} w \cdot {\mathbf {C}}_u+ \varphi u \cdot ({\mathbf {C}}_u- {\mathbf {C}}_v) + w \cdot {\mathbf {C}}_v+ v \cdot ({\mathbf {C}}_u- {\mathbf {C}}_v) = 2 \varphi u \cdot {\mathbf {C}}_u- 2 v \cdot {\mathbf {C}}_v. \end{aligned}$$

In summary, half the sum of these four identities (69)–(72) gives

$$\begin{aligned}&\partial _t\frac{|\varphi u| ^2 - |v| ^2}{2} + {\text {div}}(\varphi u {\mathbf {R}}(u \otimes \varphi u) - v {\mathbf {R}}(u \otimes v)) + |\nabla (\varphi u)| ^2 - |\nabla v| ^2 \nonumber \\&\quad = \varDelta \frac{|\varphi u| ^2 - |v| ^2}{2} + \varphi u \cdot {\mathbf {C}}_u- v \cdot {\mathbf {C}}_v+ (u \cdot \nabla \varphi ) {\mathbf {R}}(\varphi u \otimes u). \end{aligned}$$

(73)

Next, multiply local energy inequality of u (65) by $\varphi ^2$. Then

$$\begin{aligned}&\partial _t\frac{|\varphi u| ^2}{2} + |\varphi \nabla u| ^2 + {\text {div}}\left( \varphi ^2 u {\mathbf {R}}(u \otimes u) \right) \nonumber \\&\quad \le \varDelta \frac{|\varphi u| ^2}{2} + [\varphi ^2, \varDelta ] \frac{|u|^2}{2}+ [{\text {div}}, \varphi ^2] \left( u {\mathbf {R}}(u \otimes u) \right) , \nonumber \\&\partial _t\frac{|\varphi u| ^2}{2} + |\nabla (\varphi u)| ^2 + {\text {div}}\left( \varphi u {\mathbf {R}}(u \otimes \varphi u) \right) \nonumber \\&\quad \le \varDelta \frac{|\varphi u| ^2}{2} + [\varphi ^2, \varDelta ] \frac{|u|^2}{2}+ |u \otimes \nabla \varphi | ^2 + 2 (u \otimes \nabla \varphi ) : (\varphi \nabla u) \nonumber \\&\qquad + [{\text {div}}, \varphi ^2] \left( u {\mathbf {R}}(u \otimes u) \right) + {\text {div}}(\varphi u [{\mathbf {R}}, \varphi ] (u \otimes u)). \end{aligned}$$

(74)

The quadratic commutator terms in (74) are

$$\begin{aligned}&[\varphi ^2, \varDelta ] \frac{|u|^2}{2}+ |u \otimes \nabla \varphi | ^2 + 2 (u \otimes \nabla \varphi ) : (\varphi \nabla u) \\&\quad = [\varphi ^2, \varDelta ] \frac{|u|^2}{2}+ |u| ^2 |\nabla \varphi | ^2 + 2 \nabla \varphi \cdot \varphi \nabla u \cdot u \\&\quad = -2 \nabla (\varphi ^2) \cdot \nabla \frac{|u|^2}{2}- \varDelta (\varphi ^2) \frac{|u|^2}{2}+ |u| ^2 |\nabla \varphi | ^2 + 2 \nabla \varphi \cdot \nabla u \cdot \varphi u \\&\quad = -4 \varphi \nabla \varphi \cdot \nabla \frac{|u|^2}{2}- \frac{1}{2} \varDelta (\varphi ^2) |u| ^2 + |u| ^2 |\nabla \varphi | ^2 + 2 \nabla \varphi \cdot \nabla u \cdot \varphi u \\&\quad = -2 \varphi \nabla \varphi \cdot \nabla u \cdot u- \varphi \varDelta \varphi |u| ^2 \\&\quad = \varphi u \cdot (-2 \nabla \varphi \cdot \nabla u - (\varDelta \varphi ) u) \\&\quad = \varphi u \cdot [\varphi , \varDelta ] u. \end{aligned}$$

The cubic commutator terms in (74) are

$$\begin{aligned}&[{\text {div}}, \varphi ^2] \left( u {\mathbf {R}}(u \otimes u) \right) + {\text {div}}(\varphi u [{\mathbf {R}}, \varphi ] (u \otimes u)) \\&\quad = 2 \varphi \nabla \varphi \cdot u {\mathbf {R}}(u \otimes u) + \varphi u \cdot \nabla [{\mathbf {R}}, \varphi ] (u \otimes u) + {\text {div}}(\varphi u) [{\mathbf {R}}, \varphi ] (u \otimes u) \\&\quad = 2 \varphi (u \cdot \nabla \varphi ) {\mathbf {R}}(u \otimes u) + \varphi u \cdot \nabla [{\mathbf {R}}, \varphi ] (u \otimes u) + (u \cdot \nabla \varphi ) [{\mathbf {R}}, \varphi ] (u \otimes u) \\&\quad = 2 \varphi (u \cdot \nabla \varphi ) {\mathbf {R}}(u \otimes u) + \varphi u \cdot \nabla [{\mathbf {R}}, \varphi ] (u \otimes u) \\&\qquad + (u \cdot \nabla \varphi ) {\mathbf {R}}(\varphi u \otimes u) - (u \cdot \nabla \varphi ) \varphi {\mathbf {R}}( u \otimes u) \\&\quad = \varphi u \cdot \nabla \varphi {\mathbf {R}}(u \otimes u) + \varphi u \cdot \nabla [{\mathbf {R}}, \varphi ] (u \otimes u) + (u \cdot \nabla \varphi ) {\mathbf {R}}(\varphi u \otimes u) \\&\quad = \varphi u \cdot [\nabla , \varphi ] {\mathbf {R}}(u \otimes u) + \varphi u \cdot \nabla [{\mathbf {R}}, \varphi ] (u \otimes u) + (u \cdot \nabla \varphi ) {\mathbf {R}}(\varphi u \otimes u) \\&\quad = \varphi u \cdot \left( [\nabla , \varphi ] {\mathbf {R}}- \nabla [\varphi , {\mathbf {R}}] \right) (u \otimes u) + (u \cdot \nabla \varphi ) {\mathbf {R}}(\varphi u \otimes u) \\&\quad = \varphi u \cdot [\nabla {\mathbf {R}}, \varphi ] (u \otimes u) + (u \cdot \nabla \varphi ) {\mathbf {R}}(\varphi u \otimes u). \end{aligned}$$

Therefore, local energy inequality for $\varphi u$ can be simplified as

$$\begin{aligned}&\partial _t\frac{|\varphi u| ^2}{2} + |\nabla (\varphi u)| ^2 + {\text {div}}\left( \varphi u {\mathbf {R}}(u \otimes \varphi u) \right) \\&\quad \le \varDelta \frac{|\varphi u| ^2}{2} + \varphi u \cdot {\mathbf {C}}_u+ (u \cdot \nabla \varphi ) {\mathbf {R}}(\varphi u \otimes u). \end{aligned}$$

Subtracting (73) from this, we obtain (63). $\square$