Navier–Stokes equations with regularity in two entries of the velocity gradient tensor

Abstract

This paper concerns itself the regularity criteria for the three-dimensional Navier–Stokes equations. In particular, it is proved that if

$${\partial }_1{u}_3\text{,}{\partial }_3{u}_3\in L^{\frac{16}{3}}(0\text{,}T\text{;}{L}^2({\mathbb{R}}^3))\text{,}$$

or

$${\partial }_1{u}_3\text{,}{\partial }_2{u}_3\in L^8(0\text{,}T\text{;}{L}^2({\mathbb{R}}^3))\text{,}$$

then the solution is in fact smooth on $(0\text{,}T)$.

Introduction

In this paper, we consider the Cauchy problem for the three-dimensional (3D) Navier–Stokes equations

$$\left\{\begin{array}{c}{\mathbf{u}}_t+(\mathbf{u}·\nabla )\mathbf{u}-▵\mathbf{u}+\nabla \pi =\mathbf{0}\text{,}\hfill \\ \nabla ·\mathbf{u}=0\text{,}\hfill \\ \mathbf{u}(0)={\mathbf{u}}_0\text{,}\hfill \end{array}\right.$$

where $\mathbf{u}=(u_1\text{,}{u}_2\text{,}{u}_3)$ is the fluid velocity field, $\pi$ is a scalar pressure, and ${\mathbf{u}}_0$ with $\nabla ·{\mathbf{u}}_0=0$ in distributional sense is the prescribed initial velocity field.

For initial data with finite energy, Leray [12] and Hopf [8] established the global existence of weak solution $\mathbf{u}\in L^{\infty }(0\text{,}T\text{;}{L}^2({\mathbb{R}}^3))\cap L^2(0\text{,}T\text{;}{H}^1({\mathbb{R}}^3))$, which nowadays referred to Leray–Hopf weak solutions. However, the issue of uniqueness and regularity of Leray–Hopf solutions is a challenging open problem. The Prodi–Serrin criterions ([19], [20], [5] for $q=3$) state that if the weak solution satisfies

$$\mathbf{u}\in L^p(0\text{,}T\text{;}{L}^q({\mathbb{R}}^3))\text{,}\frac{2}{p}+\frac{3}{q}=1\text{,}3⩽q⩽\infty \text{,}$$

then $\mathbf{u}$ is in fact smooth. Beirão da Veiga [1] then proved the following regularity criterion in terms of $\nabla \mathbf{u}$:

$$\nabla \mathbf{u}\in L^p(0\text{,}T\text{;}{L}^q({\mathbb{R}}^3))\text{,}\frac{2}{p}+\frac{3}{q}=2\text{,}\frac{3}{2}⩽q⩽\infty \text{.}$$

Notice that the limiting case $q=\frac{3}{2}$ follows from the Sobolev imbedding theorem and [5].

Later, regularity conditions via partial components of velocity, velocity gradient, pressure, pressure gradient, vorticity appeared, see [2], [4], [6], [7], [9], [10], [11], [15], [16], [17], [18], [22], [24], [25], [26], [27], [28], [29], [30], [31], [32] and the references cited therein. Here, we list some finest known result. For one component regularity, Zhou and Pokorný [32] show that if

$$u_3\in L^p(0\text{,}T\text{;}{L}^q({\mathbb{R}}^3))\text{,}\frac{2}{p}+\frac{3}{q}=\frac{3}{4}+\frac{1}{2q}\text{,}\frac{10}{3}<q⩽\infty \text{,}$$

then the solution is in fact smooth. For Navier–Stokes equations with regularity in one direction, Kukavica and Ziane [6], [17] and Cao [2] established that the solution is regular provided that

$${\partial }_3\mathbf{u}\in L^p(0\text{,}T\text{;}{L}^q({\mathbb{R}}^3))\text{,}\frac{2}{p}+\frac{3}{q}=2\text{,}\frac{27}{16}⩽q⩽3\text{.}$$

As far as $\nabla u_3$ is concerned, Zhou and Pokorný [31] derive the regularity if

$$\nabla u_3\in L^p(0\text{,}T\text{;}{L}^q({\mathbb{R}}^3))\text{,}\frac{2}{p}+\frac{3}{q}=\frac{23}{12}\text{,}2⩽q⩽3\text{.}$$

Adding conditions only on one entry of the velocity gradient, Cao and Titi [3] obtained the following smoothness conditions

$${\partial }_i{u}_j\in L^p(0\text{,}T\text{;}{L}^q({\mathbb{R}}^3))\text{,}\frac{2}{p}+\frac{3}{q}=\frac{1}{2}+\frac{3}{2q}\text{,}3<q<\infty \text{,}$$

or

$${\partial }_j{u}_j\in L^p(0\text{,}T\text{;}{L}^q({\mathbb{R}}^3))\text{,}\frac{2}{p}+\frac{3}{q}=\frac{3}{4}+\frac{3}{2q}\text{,}2<q<\infty \text{,}$$

with $i\ne j\text{,}1⩽i\text{,}j⩽3$.

In this paper, we would like to continue our study on Navier–Stokes equations with regularity in two components of the velocity gradient ([23], [24]), which is an immediate case between (7), (8) (only one component of $\nabla \mathbf{u}$ is concerned) and (5), (6) (three components of $\nabla \mathbf{u}$ are involved).

Before stating the precise result, let us recall the weak formulation of (1) (see [12] or the books [13], [14], [21]).

Definition 1

Let ${\mathbf{u}}_0\in L^2({\mathbb{R}}^3)$ with $\nabla ·\mathbf{u}=0$, and $T>0$. A measurable ${\mathbb{R}}^3-$valued vector $\mathbf{u}$ is said to be a weak solution of (1) with initial data ${\mathbf{u}}_0$, provided the following three conditions are satisfied:

• $\mathbf{u}\in L^{\infty }(0\text{,}T\text{;}{L}^2({\mathbb{R}}^3))\cap L^2(0\text{,}T\text{;}{H}^1({\mathbb{R}}^3))$;

• $\mathbf{u}$ solves (1) in a distributional sense, i.e.,

  $${\int }_0^T{\int }_{{\mathbb{R}}^3}[{\mathit{\phi }}_t+(\mathbf{u}·\nabla )\mathit{\phi }]·\mathbf{u}\text{d}x\text{d}t+{\int }_{{\mathbb{R}}^3}{\mathbf{u}}_0·\mathit{\phi }\text{d}x={\int }_0^T{\int }_{{\mathbb{R}}^3}\nabla \mathbf{u}:\nabla \mathit{\phi }\text{d}x\text{d}t\text{,}$$

  for all $\mathit{\phi }\in C_c^{\infty }({\mathbb{R}}^3\times [0\text{,}T))$ with $\nabla ·\mathit{\phi }=0$,

  $${\int }_0^T{\int }_{{\mathbb{R}}^3}(\mathbf{u}·\nabla )\psi \text{d}x\text{d}t=0\text{,}$$

  for every $\psi \in C_c^{\infty }({\mathbb{R}}^3\times [0\text{,}T))$;

• the energy inequality

  $${‖\mathbf{u}(t)‖}_{L^2}^2+2{\int }_0^t{‖\nabla \mathbf{u}(s)‖}_{L^2}^2\text{d}s⩽{‖{\mathbf{u}}_0‖}_{L^2}^2\text{,}$$

  for all $0⩽t⩽T$.

Now, our main result reads.

Theorem 2

Let ${\mathbf{u}}_0\in H^1({\mathbb{R}}^3)$ with $\nabla ·\mathbf{u}=0$, and $1⩽p\text{,}q⩽\infty$. Suppose $\mathbf{u}$ is a given weak solution of (1) on $[0\text{,}T)$ with initial data ${\mathbf{u}}_0$. If

$${\partial }_i{u}_i\in L^{\frac{8p}{3}}(0\text{,}T\text{;}{L}^q({\mathbb{R}}^3))\text{,}{\partial }_j{u}_i\in L^{\frac{8p}{3(p-1)}}\left(0\text{,}T\text{;}{L}^{\frac{q}{q-1}}({\mathbb{R}}^3)\right)$$

where $1⩽i\ne j⩽3$; or

$${\partial }_j{u}_i\in L^{4p}(0\text{,}T\text{;}{L}^q({\mathbb{R}}^3))\text{,}{\partial }_k{u}_i\in L^{\frac{4p}{p-1}}\left(0\text{,}T\text{;}{L}^{\frac{q}{q-1}}({\mathbb{R}}^3)\right)\text{,}$$

where $\left\{i\text{,}j\text{,}k\right\}=\left\{1\text{,}2\text{,}3\right\}$, then the solution is in fact smooth on $(0\text{,}T)$.

Taking $p=q=2$ in (9) or (10), we have the following regularity criteria involving only two entries of the velocity gradient, which improves the Corollary in [23].

Corollary 3

Let ${\mathbf{u}}_0\in H^1({\mathbb{R}}^3)$ with $\nabla ·\mathbf{u}=0$, and $T>0$. Suppose $\mathbf{u}$ is a given weak solution of (1) on $[0\text{,}T)$ with initial data ${\mathbf{u}}_0$. If

$${\partial }_i{u}_i\text{,}{\partial }_j{u}_i\in L^{\frac{16}{3}}(0\text{,}T\text{;}{L}^2({\mathbb{R}}^3))\text{,}$$

for some $1⩽i\ne j⩽3$; or

$${\partial }_j{u}_i\text{,}{\partial }_k{u}_i\in L^8(0\text{,}T\text{;}{L}^2({\mathbb{R}}^3))\text{,}$$

with $\left\{i\text{,}j\text{,}k\right\}=\left\{1\text{,}2\text{,}3\right\}$, then the solution is in fact smooth on $(0\text{,}T)$.

The rest of this paper is organized as follows. In Section 2, we fix some notations used throughout the paper, recall an elementary but tricky inequality from [10], and state some nice structures of the nonlinear convective term of (1). Section 3 is devoted to proving Theorem 2 under the condition (9), with the case (10) proved in Section 4.