Extended Regularity Criteria for the Navier–Stokes–Maxwell system

Abstract

We consider the Cauchy problem of the Navier–Stokes–Maxwell system in three dimensions and establish two regularity criteria involving $\varvec{u}$ (or $\nabla \varvec{u}$) in Besov spaces and $\nabla \varvec{B}$ in more flexible Lebesgue spaces. This improves and extends recent results of Ma, Jiang and Zhu.

1 Introduction

In this paper, we study the three-dimensional Navier–Stokes–Maxwell system

$$\begin{aligned} \left\{ \begin{array}{llllllll} \varvec{u}_t+(\varvec{u}\cdot \nabla )\varvec{u}-\Delta \varvec{u}+\nabla P=\varvec{j}\times \varvec{B},\\ \varvec{E}_t-\mathrm {curl\,}\varvec{B}=-\,\varvec{j},\\ \varvec{B}_t+\mathrm {curl\,}\varvec{E}=\varvec{0},\\ \mathrm {div\,}\varvec{u}=\mathrm {div\,}\varvec{B}=0,\\ (\varvec{u},\varvec{E},\varvec{B})|_{t=0}=(\varvec{u}_0,\varvec{E}_0,\varvec{B}_0).\end{array}\right. \end{aligned}$$

(1)

Here $\varvec{u}$, $\varvec{E}$ and $\varvec{B}$ are the fluid velocity field, the electric field and the magnetic field, respectively; $\varvec{j}$ is the electric current density given by the Ohm’s law

$$\begin{aligned} \varvec{j}=\sigma (\varvec{E}+\varvec{u}\times \varvec{B}), \end{aligned}$$

(2)

where $\sigma >0$ represents the electric resistivity; P is the pressure of the fluid, which can be recovered from the velocity field $\varvec{u}$ and the Lorentz force $\varvec{j}\times \varvec{B}$. In fact, taking the divergence of (1)$_1$ gives

$$\begin{aligned} -\,\Delta P=\varvec{F}\equiv \mathrm {div\,}[(\varvec{u}\cdot \nabla )\varvec{u}-\varvec{j}\times \varvec{B}], \end{aligned}$$

and P can be represented by $\varvec{F}$ through the Newton’s potential.

The Navier–Stokes–Maxwell system (1) is a physical model that describes the motion of charged particles in electromagnetic field (see [6, 9]). Mathematically, (1) is a coupled system, constituted by the Navier–Stokes equations from fluid dynamics and the Maxwell’s equations from electromagnetism. Moreover, it can be derived from the Vlasov–Maxwell–Boltzmann system (see [2]). In the 2D case, Masmoudi [8] (see also Kang–Lee [5]) established the global unique strong solution for (1). In the 3D case, Ibrahim–Keraani [4] obtained the global regularity under a smallness assumption, Kang–Lee [5] formulated a blow-up criterion in terms of $\varvec{u}$ and $\varvec{B}$: the maximal existence time of the local strong solution $T^*$ is finite if and only if

$$\begin{aligned} \int _0^{T^*} \left\| \varvec{u}\right\| _{L^\infty }^2 +\left\| \varvec{B}\right\| _{L^\infty }^\frac{8}{3}\mathrm {\,d}t=\infty . \end{aligned}$$

(3)

This was improved by Fan–Zhou [3] to be

$$\begin{aligned} \int _0^{T^*} \left\| \varvec{u}\right\| _{L^\infty }^2 +\left\| \varvec{B}\right\| _{L^\infty }^2\mathrm {\,d}t=\infty . \end{aligned}$$

(4)

Recently, Ma–Jiang–Zhu [7] showed the following three regularity criteria:

$$\begin{aligned}&\varvec{u}\in L^2\left( 0,T;L^\infty (\mathbb {R}^3)\right) ,\quad \text{ and } \quad \nabla \varvec{u}\in L^2(0,T;L^3(\mathbb {R}^3)); \end{aligned}$$

(5)

$$\begin{aligned}&\varvec{u}\in L^p\left( 0,T;L^q(\mathbb {R}^3)\right) \text{ with } \frac{2}{p}+\frac{3}{q}=1,\ 3<q \leqslant \infty \quad \text{ and } \nonumber \\&\qquad \nabla \varvec{B}\in L^2(0,T;L^3(\mathbb {R}^3)); \end{aligned}$$

(6)

$$\begin{aligned}&\nabla \varvec{u}\in L^p\left( 0,T;L^q(\mathbb {R}^3)\right) , \text{ with } \frac{2}{p}+\frac{3}{q}=2,\ \frac{3}{2}<q \leqslant \infty \quad \text{ and } \nonumber \\&\qquad \nabla \varvec{B}\in L^2(0,T;L^3(\mathbb {R}^3)). \end{aligned}$$

(7)

Notice that (5) involves only the velocity field, while all the others involve not only the velocity field, but also the magnetic field.

The purpose of the present paper is to improve and extend (6) and (7) as follows.

Theorem 1

Let $\varvec{u}_0,\varvec{E}_0,\varvec{B}_0\in H^2(\mathbb {R}^3)$ with $\mathrm {div\,}\varvec{u}_0=\mathrm {div\,}\varvec{B}_0=0$. Assume that $\varvec{u},\varvec{E},\varvec{B}$ be the local strong solution of (1) with initial data $\varvec{u}_0,\varvec{E}_0,\varvec{B}_0$. If

$$\begin{aligned} \begin{aligned}&\varvec{u}\in L^\frac{2}{1-r}\left( 0,T;\dot{B}^{-r}_{\infty ,\infty }(\mathbb {R}^3)\right) \text{ with } -\,1<r<1,\\&\quad \text{ and } \nabla \varvec{B}\in L^p(0,T;L^q(\mathbb {R}^3)) \text{ with } \frac{2}{p}+\frac{3}{q}=2 \text{ with } 2\leqslant q\leqslant 3, \end{aligned} \end{aligned}$$

(8)

then

$$\begin{aligned} \varvec{u},\varvec{E},\varvec{B}\in L^\infty \left( 0,T;H^2(\mathbb {R}^3)\right) ,\quad \nabla \varvec{u},\varvec{j}\in L^2(0,T;H^2(\mathbb {R}^3)), \end{aligned}$$

and thus, the solution can be smoothly extended beyond T.

Here, $\dot{B}^s_{\infty ,\infty }(\mathbb {R}^3)$ with $s\in \mathbb {R}$ is the homogeneous Besov spaces. One is referred to Bahouri–Chemin–Danchin [1, Chapter 2] for its definition and relation with the classical Sobolev spaces. In our proof of Theorem 1 in Sect. 2, we shall use the following refined interpolation inequality involving Besov norm [1, Theorem 2.42]:

$$\begin{aligned} \left\| f\right\| _{L^p}\leqslant C \left\| f\right\| _{\dot{B}^{-\alpha }_{\infty ,\infty }}^{1-\theta }\left\| f\right\| _{\dot{B}^\beta _{q,q}}^\theta ,\ \beta =\alpha \left( \frac{p}{q}-1\right) ,\ \theta =\frac{q}{p},\ 1\leqslant q<p<\infty ,\ \alpha {>}0. \end{aligned}$$

(9)

Taking $p=3$, $q=2$ and using the fact that $\dot{B}^s_{2,2}(\mathbb {R}^n)=\dot{H}^s(\mathbb {R}^n)$ yields

$$\begin{aligned} \left\| f\right\| _{L^3}\leqslant C\left\| f\right\| _{\dot{B}^{-\alpha }_{\infty ,\infty }}^\frac{1}{3} \left\| f\right\| _{\dot{H}^\frac{\alpha }{2}}^\frac{2}{3}. \end{aligned}$$

(10)

Remark 2

• (1) Noticing that $L^\frac{3}{r}(\mathbb {R}^3)\subsetneq \dot{B}^{-r}_{\infty ,\infty }(\mathbb {R}^3)$ for $0\leqslant r\leqslant 3$ [1, Propositions 2.39 and 2.20], we see that our Theorem improves (6).

• (2) Noticing that (8)$_1$ is equivalent to

  $$\begin{aligned}&\nabla \varvec{u}\in L^\frac{2}{1-r}\left( 0,T;\dot{B}^{-r-1}_{\infty ,\infty }(\mathbb {R}^3)\right) \ (-1<r<1)\\&\quad \Leftrightarrow \nabla \varvec{u}\in L^\frac{2}{2-\alpha } \left( 0,T;\dot{B}^{-\alpha }_{\infty ,\infty }(\mathbb {R}^3)\right) \ (0<\alpha <2), \end{aligned}$$

  we may argue as item 1 to see that our Theorem improves (7).

• (3) From items 1 and 2, we see that the framework of Besov spaces is more flexible than that of Lebesgue spaces (see [10]).

• (4) The assumptions on $\nabla \varvec{B}$ in our Theorem is more flexible, i.e., the space integrability index q of $\nabla \varvec{B}$ ranges over [2, 3], instead of just 3 in (6) and (7). It seems not so easy to extend the range [2, 3] to be the full range $\displaystyle {\left( \frac{3}{2},\infty \right] }$.

2 Proof of Theorem 1

In this section, we shall prove Theorem 1. Without loss of generality, we may assume $\sigma =1$.

Step 1$L^2$estimate. Taking the inner product of (1)$_{1,2,3}$ with $\varvec{u},\varvec{E},\varvec{B}$ in $L^2(\mathbb {R}^3)$ respectively, and gathering together, we obtain

$$\begin{aligned} \begin{aligned}&\frac{\mathrm {\,d}}{\mathrm {\,d}t}\left\| (\varvec{u},\varvec{E},\varvec{B})\right\| _{L^2}^2 +\left\| \nabla \varvec{u}\right\| _{L^2}^2\\&\quad =\int _{\mathbb {R}^3}(\varvec{j}\times \varvec{B})\cdot \varvec{u}\mathrm {\,d}x +\int _{\mathbb {R}^3}\mathrm {curl\,}\varvec{B}\cdot \varvec{E}\mathrm {\,d}x -\int _{\mathbb {R}^3}\varvec{j}\cdot \varvec{E}\mathrm {\,d}x -\int _{\mathbb {R}^3}\mathrm {curl\,}\varvec{E}\cdot \varvec{B}\mathrm {\,d}x\\&\quad =-\,\int _{\mathbb {R}^3}(\varvec{u}\times \varvec{B})\cdot \varvec{j}\mathrm {\,d}x -\int _{\mathbb {R}^3}\varvec{E}\cdot \varvec{j}\mathrm {\,d}x =-\,\int _{\mathbb {R}^3}|\varvec{j}|^2\mathrm {\,d}x =-\,\left\| \varvec{j}\right\| _{L^2}^2, \end{aligned} \end{aligned}$$

where we have used the Ohm’s law (2). Integrating in time then yields the basic energy estimate

$$\begin{aligned} \left\| (\varvec{u},\varvec{E},\varvec{B})\right\| _{L^2}^2(t) +2\int _0^t \left\| (\nabla \varvec{u},\varvec{j})\right\| _{L^2}^2(s)\mathrm {\,d}s =\left\| (\varvec{u}_0,\varvec{E}_0,\varvec{B}_0)\right\| _{L^2}^2,\quad \forall \ t\geqslant 0. \end{aligned}$$

(11)

Step 2$H^1$estimate. Testing (1)$_{1,2,3}$ by $-\,\Delta \varvec{u}, -\,\Delta \varvec{E}, -\,\Delta \varvec{B}$ in $L^2(\mathbb {R}^3)$ respectively, and putting together, we obtain

$$\begin{aligned} \begin{aligned}&\frac{1}{2}\frac{\mathrm {\,d}}{\mathrm {\,d}t}\left\| (\nabla \varvec{u},\nabla \varvec{E},\nabla \varvec{B})\right\| _{L^2}^2 +\left\| \Delta \varvec{u}\right\| _{L^2}^2\\&\quad =\int _{\mathbb {R}^3}[(\varvec{u}\cdot \nabla )\varvec{u}]\cdot \Delta \varvec{u}\mathrm {\,d}x -\int _{\mathbb {R}^3}(\varvec{j}\times \varvec{B})\cdot \Delta \varvec{u}\mathrm {\,d}x\\&\quad \quad -\int _{\mathbb {R}^3}\mathrm {curl\,}\varvec{B}\cdot \Delta \varvec{E}\mathrm {\,d}x +\int _{\mathbb {R}^3}\varvec{j}\cdot \Delta \varvec{E}\mathrm {\,d}x +\int _{\mathbb {R}^3}\mathrm {curl\,}\varvec{E}\cdot \Delta \varvec{B}\mathrm {\,d}x\\&\quad =-\,\sum _{i=1}^3 \int _{\mathbb {R}^3} [(\partial _i\varvec{u}\cdot \nabla )\varvec{u}]\cdot \partial _i\varvec{u}\mathrm {\,d}x +\sum _{i=1}^3 \int _{\mathbb {R}^3}\partial _i(\varvec{j}\times \varvec{B})\cdot \partial _i\varvec{u}\mathrm {\,d}x\\&\quad \quad -\,\sum _{i=1}^3 \int _{\mathbb {R}^3}\partial _i\varvec{j}\cdot \partial _i\varvec{E}\mathrm {\,d}x. \end{aligned} \end{aligned}$$

Noticing that

$$\begin{aligned} \begin{aligned} -\,\sum _{i=1}^3 \int _{\mathbb {R}^3}\partial _i\varvec{j}\cdot \partial _i\varvec{E}\mathrm {\,d}x&=-\,\sum _{i=1}^3 \int _{\mathbb {R}^3} \partial _i\varvec{j}\cdot \partial _i(\varvec{j}-\varvec{u}\times \varvec{B})\mathrm {\,d}x\\&=-\,\left\| \nabla \varvec{j}\right\| _{L^2}^2 +\sum _{i=1}^3 \int _{\mathbb {R}^3}\partial _i(\varvec{u}\times \varvec{B})\cdot \partial _i\varvec{j}\mathrm {\,d}x, \end{aligned} \end{aligned}$$

we get

$$\begin{aligned}&\frac{1}{2}\frac{\mathrm {\,d}}{\mathrm {\,d}t}\left\| (\nabla \varvec{u},\nabla \varvec{E},\nabla \varvec{B})\right\| _{L^2}^2 +\left\| (\Delta \varvec{u},\nabla \varvec{j})\right\| _{L^2}^2\nonumber \\&\quad =-\,\sum _{i=1}^3 \int _{\mathbb {R}^3} [(\partial _i\varvec{u}\cdot \nabla )\varvec{u}]\cdot \partial _i\varvec{u}\mathrm {\,d}x +\sum _{i=1}^3 \int _{\mathbb {R}^3}\partial _i(\varvec{j}\times \varvec{B})\cdot \partial _i\varvec{u}\mathrm {\,d}x\nonumber \\&\qquad +\,\sum _{i=1}^3 \int _{\mathbb {R}^3}\partial _i(\varvec{u}\times \varvec{B})\cdot \partial _i\varvec{j}\mathrm {\,d}x\nonumber \\&\quad =-\,\sum _{i=1}^3 \int _{\mathbb {R}^3} [(\partial _i\varvec{u}\cdot \nabla )\varvec{u}]\cdot \partial _i\varvec{u}\mathrm {\,d}x +\sum _{i=1}^3 \int _{\mathbb {R}^3} (\partial _i\varvec{j}\times \varvec{B}+\varvec{j}\times \partial _i\varvec{B})\cdot \partial _i\varvec{u}\mathrm {\,d}x\nonumber \\&\qquad +\,\sum _{i=1}^3 \int _{\mathbb {R}^3} (\partial _i\varvec{u}\times \varvec{B}+\varvec{u}\times \partial _i\varvec{B})\cdot \partial _i\varvec{j}\mathrm {\,d}x\\&\quad =-\,\sum _{i=1}^3 \int _{\mathbb {R}^3} [(\partial _i\varvec{u}\cdot \nabla )\varvec{u}]\cdot \partial _i\varvec{u}\mathrm {\,d}x +\sum _{i=1}^3 \int _{\mathbb {R}^3} (\varvec{j}\times \partial _i\varvec{B})\cdot \partial _i\varvec{u}\mathrm {\,d}x\nonumber \\&\qquad +\,\sum _{i=1}^3 \int _{\mathbb {R}^3}(\varvec{u}\times \partial _i\varvec{B})\cdot \partial _i\varvec{j}\mathrm {\,d}x\nonumber \\&\quad \leqslant \left\| \nabla \varvec{u}\right\| _{L^3}^3 +\int _{\mathbb {R}^3}|\varvec{j}|\cdot |\nabla \varvec{B}|\cdot |\nabla \varvec{u}|\mathrm {\,d}x +\int _{\mathbb {R}^3}|\varvec{u}|\cdot |\nabla \varvec{B}|\cdot |\nabla \varvec{j}|\mathrm {\,d}x\nonumber \\&\quad \equiv I_1+I_2+I_3.\nonumber \end{aligned}$$

(12)

The first term $I_1$ can be bounded by (10) as

$$\begin{aligned} \begin{aligned} I_1&\leqslant C\left( \left\| \nabla \varvec{u}\right\| _{\dot{B}^{-(1+r)}_{\infty ,\infty }}^\frac{1}{3} \left\| \nabla \varvec{u}\right\| _{\dot{H}^\frac{1+r}{2}}^\frac{2}{3}\right) ^3\\&\leqslant C\left\| \varvec{u}\right\| _{\dot{B}^{-r}_{\infty ,\infty }} \left\| \nabla \varvec{u}\right\| _{L^2}^{1-r} \left\| \nabla \varvec{u}\right\| _{\dot{H}^1}^{1+r}\quad \left( 0<\frac{1+r}{2}<1\Leftrightarrow -1<r<1\right) \\&\leqslant C\left\| \varvec{u}\right\| _{\dot{B}^{-r}_{\infty ,\infty }} ^\frac{2}{1-r} \left\| \nabla \varvec{u}\right\| _{L^2}^2 +\frac{1}{6}\left\| \Delta \varvec{u}\right\| _{L^2}^2. \end{aligned} \end{aligned}$$

(13)

By the Hölder inequality, the Gagliardo–Nirenberg inequality and the Sobolev inequality, the last two terms $I_2,I_3$ can be dominated as

$$\begin{aligned}&\begin{aligned} I_2&\leqslant \left\| \varvec{j}\right\| _{L^6} \left\| \nabla \varvec{B}\right\| _{L^q} \left\| \nabla \varvec{u}\right\| _{L^\frac{6q}{5q-6}}\\&\leqslant \left\| \nabla \varvec{j}\right\| _{L^2} \left\| \nabla \varvec{B}\right\| _{L^q} \left\| \nabla \varvec{u}\right\| _{L^2}^\frac{2q-3}{q} \left\| \Delta \varvec{u}\right\| _{L^2}^\frac{3-q}{q}\\&\leqslant \left\| \nabla \varvec{B}\right\| _{L^q} \left\| \nabla \varvec{u}\right\| _{L^2}^\frac{2q-3}{q} \left\| (\Delta \varvec{u},\nabla \varvec{j})\right\| _{L^2}^\frac{3}{q}\\&\leqslant C\left\| \nabla \varvec{B}\right\| _{L^q}^\frac{2q}{2q-3} \left\| \nabla \varvec{u}\right\| _{L^2}^2 +\frac{1}{6}\left\| (\Delta \varvec{u},\nabla \varvec{j})\right\| _{L^2}^2, \end{aligned} \end{aligned}$$

(14)

$$\begin{aligned}&\begin{aligned} I_3&\leqslant \left\| \varvec{u}\right\| _{L^\frac{2q}{q-2}} \left\| \nabla \varvec{B}\right\| _{L^q} \left\| \nabla \varvec{j}\right\| _{L^2}\\&\leqslant C\left\| \varvec{u}\right\| _{L^6}^\frac{2q-3}{q} \left\| \nabla \varvec{u}\right\| _{L^6}^\frac{3-q}{q} \left\| \nabla \varvec{B}\right\| _{L^q} \left\| \nabla \varvec{j}\right\| _{L^2}\quad \left( 6\leqslant \frac{2q}{q-2}\leqslant \infty \Leftrightarrow 2\leqslant q\leqslant 3\right) \\&\leqslant C\left\| \nabla \varvec{u}\right\| _{L^2}^\frac{2q-3}{q} \left\| \Delta \varvec{u}\right\| _{L^2}^\frac{3-q}{q} \left\| \nabla \varvec{B}\right\| _{L^q} \left\| \nabla \varvec{j}\right\| _{L^2}\\&\leqslant C\left\| \nabla \varvec{B}\right\| _{L^q}^\frac{2q}{2q-3} \left\| \nabla \varvec{u}\right\| _{L^2}^2 +\frac{1}{6}\left\| (\Delta \varvec{u},\nabla \varvec{j})\right\| _{L^2}^2. \end{aligned} \end{aligned}$$

(15)

Collecting (13)–(15) into (12), and absorbing the diffusion terms, we deduce

$$\begin{aligned} \frac{\mathrm {\,d}}{\mathrm {\,d}t}\left\| (\nabla \varvec{u},\nabla \varvec{E},\nabla \varvec{B})\right\| _{L^2}^2 +\left\| (\Delta \varvec{u},\nabla \varvec{j})\right\| _{L^2}^2 \leqslant C\left( \left\| \varvec{u}\right\| _{\dot{B}^{-r}_{\infty ,\infty }} ^\frac{2}{1-r}+\left\| \nabla \varvec{B}\right\| _{L^q}^\frac{2q}{2q-3}\right) \left\| \nabla \varvec{u}\right\| _{L^2}^2. \end{aligned}$$

Applying the Gronwall inequality, and recalling (8), we have

$$\begin{aligned} \begin{aligned}&\left\| (\nabla \varvec{u},\nabla \varvec{E},\nabla \varvec{B})\right\| _{L^2}^2(t) +\int _0^t \left\| (\Delta \varvec{u},\nabla \varvec{j})\right\| _{L^2}^2(s)\mathrm {\,d}s\\&\quad \leqslant \left\| (\nabla \varvec{u}_0,\nabla \varvec{E}_0,\nabla \varvec{B}_0)\right\| _{L^2}^2 \cdot \exp \left\{ C\int _0^t \left[ \left\| \varvec{u}\right\| _{\dot{B}^{-r}_{\infty ,\infty }} ^\frac{2}{1-r}(s)+\left\| \nabla \varvec{B}\right\| _{L^q}^\frac{2q}{2q-3}(s)\right] \mathrm {\,d}s \right\} {<}\infty , \end{aligned} \end{aligned}$$

(16)

for each $0\leqslant t\leqslant T$.

Step 3$H^2$estimate. Applying the Laplace operator to (1)$_{1,2,3}$, integrating the resulting equations by $\Delta \varvec{u}$, $\Delta \varvec{E}$, $\Delta \varvec{B}$ respectively, and integrating over $\mathbb {R}^3$, we find

$$\begin{aligned} \begin{aligned}&\frac{1}{2}\frac{\mathrm {\,d}}{\mathrm {\,d}t} \left\| (\Delta \varvec{u},\Delta \varvec{E},\Delta \varvec{B})\right\| _{L^2}^2 +\left\| \nabla \Delta \varvec{u}\right\| _{L^2}^2\\&\quad =-\,\int _{\mathbb {R}^3}\Delta [(\varvec{u}\cdot \nabla )\varvec{u}]\cdot \Delta \varvec{u}\mathrm {\,d}x +\int _{\mathbb {R}^3}\Delta (\varvec{j}\times \varvec{B})\cdot \Delta \varvec{u}\mathrm {\,d}x\\&\qquad +\,\int _{\mathbb {R}^3}\mathrm {curl\,}\Delta \varvec{B}\cdot \Delta \varvec{E}\mathrm {\,d}x -\int _{\mathbb {R}^3}\Delta \varvec{j}\cdot \Delta \varvec{E}\mathrm {\,d}x -\int _{\mathbb {R}^3}\mathrm {curl\,}\Delta \varvec{E}\cdot \Delta \varvec{B}\mathrm {\,d}x\\&\quad =-\,\int _{\mathbb {R}^3}\Delta [(\varvec{u}\cdot \nabla )\varvec{u}]\cdot \Delta \varvec{u}\mathrm {\,d}x +\int _{\mathbb {R}^3}\Delta (\varvec{j}\times \varvec{B})\cdot \Delta \varvec{u}\mathrm {\,d}x\\&\qquad -\,\int _{\mathbb {R}^3}\Delta \varvec{j}\cdot \Delta (\varvec{j}-\varvec{u}\times \varvec{B})\mathrm {\,d}x, \end{aligned} \end{aligned}$$

and consequently,

$$\begin{aligned} \begin{aligned}&\frac{1}{2}\frac{\mathrm {\,d}}{\mathrm {\,d}t} \left\| (\Delta \varvec{u},\Delta \varvec{E},\Delta \varvec{B})\right\| _{L^2}^2 +\left\| (\nabla \Delta \varvec{u},\Delta \varvec{j})\right\| _{L^2}^2\\&\quad =-\,\int _{\mathbb {R}^3}\Delta [(\varvec{u}\cdot \nabla )\varvec{u}]\cdot \Delta \varvec{u}\mathrm {\,d}x +\int _{\mathbb {R}^3}\Delta (\varvec{j}\times \varvec{B})\cdot \Delta \varvec{u}+\Delta \varvec{j}\cdot \Delta (\varvec{u}\times \varvec{B})\mathrm {\,d}x\\&\quad \equiv J_1+J_2. \end{aligned} \end{aligned}$$

(17)

Direct computations show

$$\begin{aligned} \begin{aligned} J_1&=-\,\int _{\mathbb {R}^3} (\Delta \varvec{u}\cdot \nabla )\varvec{u}\cdot \Delta \varvec{u}+2\sum _{i=1}^3 [(\partial _i\varvec{u}\cdot \nabla )\partial _i\varvec{u}]\cdot \Delta \varvec{u}\mathrm {\,d}x\\&\leqslant 2\int _{\mathbb {R}^3} |\nabla \varvec{u}|\cdot |\nabla ^2\varvec{u}|^2\mathrm {\,d}x\\&\leqslant C\left\| \nabla \varvec{u}\right\| _{L^3} \left\| \nabla ^2\varvec{u}\right\| _{L^3}^2\\&\leqslant C\left\| \nabla \varvec{u}\right\| _{L^2}^\frac{1}{2} \left\| \Delta \varvec{u}\right\| _{L^2}^\frac{1}{2} \cdot \left\| \Delta \varvec{u}\right\| _{L^2} \left\| \nabla \Delta \varvec{u}\right\| _{L^2}\\&\leqslant C\left\| \nabla \varvec{u}\right\| _{L^2} \left\| \Delta \varvec{u}\right\| _{L^2} \cdot \left\| \Delta \varvec{u}\right\| _{L^2}^2 +\frac{1}{4}\left\| \nabla \Delta \varvec{u}\right\| _{L^2}^2. \end{aligned} \end{aligned}$$

(18)

Meanwhile,

$$\begin{aligned} \begin{aligned} J_2&=\int _{\mathbb {R}^3} \left( \Delta \varvec{j}\times \varvec{B}+2\sum _{i=1}^3 \partial _i\varvec{j}\times \partial _i\varvec{B}+\varvec{j}\times \Delta \varvec{B}\right) \cdot \Delta \varvec{u}\mathrm {\,d}x\\&\quad +\,\int _{\mathbb {R}^3} \left( \Delta \varvec{u}\times \varvec{B}+2\sum _{i=1}^3 \partial _i\varvec{u}\times \partial _i\varvec{B}+\varvec{u}\times \Delta \varvec{B}\right) \cdot \Delta \varvec{j}\mathrm {\,d}x\\&=\int _{\mathbb {R}^3} \left( 2\sum _{i=1}^3 \partial _i\varvec{j}\times \partial _i\varvec{B}+\varvec{j}\times \Delta \varvec{B}\right) \cdot \Delta \varvec{u}\mathrm {\,d}x\\&\quad +\,\int _{\mathbb {R}^3} \left( 2\sum _{i=1}^3 \partial _i\varvec{u}\times \partial _i\varvec{B}+\varvec{u}\times \Delta \varvec{B}\right) \cdot \Delta \varvec{j}\mathrm {\,d}x\\&\leqslant 2\int _{\mathbb {R}^3} |\nabla \varvec{j}|\cdot |\nabla \varvec{B}|\cdot |\Delta \varvec{u}|\mathrm {\,d}x +\int _{\mathbb {R}^3}|\varvec{j}| \cdot |\Delta \varvec{B}|\cdot |\Delta \varvec{u}|\mathrm {\,d}x\\&\quad +\,2\int _{\mathbb {R}^3} |\nabla \varvec{u}|\cdot |\nabla \varvec{B}|\cdot |\Delta \varvec{j}|\mathrm {\,d}x +\int _{\mathbb {R}^3} |\varvec{u}|\cdot |\Delta \varvec{B}|\cdot |\Delta \varvec{j}|\mathrm {\,d}x\\&\equiv J_{21}+J_{22}+J_{23}+J_{24}. \end{aligned} \end{aligned}$$

(19)

These above four terms can be estimated easily as

$$\begin{aligned}&\begin{aligned} J_{21}&\leqslant 2\left\| \nabla \varvec{j}\right\| _{L^6}\left\| \nabla \varvec{B}\right\| _{L^3} \left\| \Delta \varvec{u}\right\| _{L^2}\\&\leqslant C\left\| \nabla \varvec{B}\right\| _{L^2}^\frac{1}{2} \left\| \Delta \varvec{B}\right\| _{L^2}^\frac{1}{2} \left\| \Delta \varvec{u}\right\| _{L^2} \left\| \Delta \varvec{j}\right\| _{L^2}\\&\leqslant C\left( \left\| \nabla \varvec{B}\right\| _{L^2} \left\| \Delta \varvec{B}\right\| _{L^2}\right) \left\| \Delta \varvec{u}\right\| _{L^2}^2 +\frac{1}{16}\left\| \Delta \varvec{j}\right\| _{L^2}^2, \end{aligned} \end{aligned}$$

(20)

$$\begin{aligned}&\begin{aligned} J_{22}&\leqslant \left\| \varvec{j}\right\| _{L^3} \left\| \Delta \varvec{B}\right\| _{L^2} \left\| \Delta \varvec{u}\right\| _{L^6}\\&\leqslant C\left\| \varvec{j}\right\| _{L^2}^\frac{1}{2} \left\| \nabla \varvec{j}\right\| _{L^2}^\frac{1}{2} \left\| \Delta \varvec{B}\right\| _{L^2} \left\| \nabla \Delta \varvec{u}\right\| _{L^2}\\&\leqslant C\left( \left\| \varvec{j}\right\| _{L^2} \left\| \nabla \varvec{j}\right\| _{L^2}\right) \left\| \Delta \varvec{B}\right\| _{L^2}^2 +\frac{1}{16} \left\| \nabla \Delta \varvec{u}\right\| _{L^2}^2, \end{aligned} \end{aligned}$$

(21)

$$\begin{aligned}&\begin{aligned} J_{23}&\leqslant 2\left\| \nabla \varvec{u}\right\| _{L^3} \left\| \nabla \varvec{B}\right\| _{L^6} \left\| \Delta \varvec{j}\right\| _{L^2}\\&\leqslant C\left\| \nabla \varvec{u}\right\| _{L^2}^\frac{1}{2} \left\| \Delta \varvec{u}\right\| _{L^2}^\frac{1}{2} \left\| \Delta \varvec{B}\right\| _{L^2} \left\| \Delta \varvec{j}\right\| _{L^2}\\&\leqslant C\left( \left\| \nabla \varvec{u}\right\| _{L^2} \left\| \Delta \varvec{u}\right\| _{L^2}\right) \left\| \Delta \varvec{B}\right\| _{L^2}^2 +\frac{1}{16}\left\| \Delta \varvec{j}\right\| _{L^2}^2, \end{aligned} \end{aligned}$$

(22)

$$\begin{aligned}&\begin{aligned} J_{24}&\leqslant \left\| \varvec{u}\right\| _{L^\infty } \left\| \Delta \varvec{B}\right\| _{L^2} \left\| \Delta \varvec{j}\right\| _{L^2}\\&\leqslant C\left\| \varvec{u}\right\| _{L^6}^\frac{1}{2} \left\| \nabla \varvec{u}\right\| _{L^6}^\frac{1}{2} \left\| \Delta \varvec{u}\right\| _{L^2} \left\| \Delta \varvec{j}\right\| _{L^2}\\&\leqslant C\left\| \nabla \varvec{u}\right\| _{L^2}^\frac{1}{2} \left\| \Delta \varvec{u}\right\| _{L^2}^\frac{1}{2} \left\| \Delta \varvec{u}\right\| _{L^2} \left\| \Delta \varvec{j}\right\| _{L^2}\\&\leqslant C\left( \left\| \nabla \varvec{u}\right\| _{L^2} \left\| \Delta \varvec{u}\right\| _{L^2}\right) \left\| \Delta \varvec{u}\right\| _{L^2}^2 +\frac{1}{16} \left\| \Delta \varvec{j}\right\| _{L^2}^2. \end{aligned} \end{aligned}$$

(23)

Plugging (20)–(23) into (19), which together with (18) and (17) implies

$$\begin{aligned} \begin{aligned}&\frac{\mathrm {\,d}}{\mathrm {\,d}t} \left\| (\Delta \varvec{u},\Delta \varvec{E},\Delta \varvec{B})\right\| _{L^2}^2 +\left\| (\nabla \Delta \varvec{u},\Delta \varvec{j})\right\| _{L^2}^2\\&\quad \leqslant C\left( \left\| (\nabla \varvec{u},\nabla \varvec{B},\varvec{j})\right\| _{L^2} \left\| (\Delta \varvec{u},\Delta \varvec{B},\nabla \varvec{j})\right\| _{L^2}\right) \left\| (\Delta \varvec{u},\Delta \varvec{E},\Delta \varvec{B})\right\| _{L^2}^2\\&\quad \leqslant C\left( \left\| (\nabla \varvec{u},\nabla \varvec{B},\varvec{j})\right\| _{L^2}^2 + \left\| (\Delta \varvec{u},\Delta \varvec{B},\nabla \varvec{j})\right\| _{L^2}^2\right) \left\| (\Delta \varvec{u},\Delta \varvec{E},\Delta \varvec{B})\right\| _{L^2}^2. \end{aligned} \end{aligned}$$

Applying the Gronwall inequality, and recalling (11) and (16), we obtain

$$\begin{aligned} \begin{aligned}&\left\| (\Delta \varvec{u},\Delta \varvec{E},\Delta \varvec{B})\right\| _{L^2}^2(t) +\int _0^t\left\| (\nabla \Delta \varvec{u},\Delta \varvec{j})\right\| _{L^2}^2(s)\mathrm {\,d}s\\&\quad \leqslant \left\| (\Delta \varvec{u}_0,\Delta \varvec{E}_0,\Delta \varvec{B}_0)\right\| _{L^2}^2 \cdot \exp \\&\qquad \left\{ C\int _0^t \left\| (\nabla \varvec{u},\nabla \varvec{B},\varvec{j})\right\| _{L^2}^2(s) + \left\| (\Delta \varvec{u},\Delta \varvec{B},\nabla \varvec{j})\right\| _{L^2}^2(s) \mathrm {\,d}s \right\} <\infty , \end{aligned} \end{aligned}$$

for any $0\leqslant t\leqslant T$.

The proof of Theorem 1 is completed.