Local existence and blow-up criterion of the ideal density-dependent flows

Abstract

In this paper, we consider two ideal density-dependent flows in a bounded domain, the Euler and magnetohydrodynamics equations. We prove the local existence and a blow-up criterion for each system.

1 Introduction

First, we consider the following 3D density-dependent Euler system:

$$\begin{aligned}& \partial_{t}\rho+\mathbf{u}\cdot\nabla\rho=0, \end{aligned}$$

(1.1)

$$\begin{aligned}& \rho\partial_{t}\mathbf{u}+\rho(\mathbf{u}\cdot\nabla)\mathbf{u}+ \nabla \pi=0, \end{aligned}$$

(1.2)

$$\begin{aligned}& \operatorname {div}\mathbf{u}=0, \end{aligned}$$

(1.3)

$$\begin{aligned}& \mathbf{u}\cdot\mathbf{n}=0 \quad\mbox{on } \partial\Omega\times(0, \infty), \end{aligned}$$

(1.4)

$$\begin{aligned}& (\rho,\mathbf{u}) (\cdot,0)=(\rho_{0},\mathbf{u}_{0}) \quad \mbox{in } \Omega \subset {\mathbb {R}^{3}}. \end{aligned}$$

(1.5)

Here Ω is a bounded domain with smooth boundary $\partial\Omega \in C^{\infty}$, n is the outward unit normal to ∂Ω; the unknowns are the fluid velocity field $\mathbf{u}=\mathbf {u}(x,t)$, the pressure $\pi=\pi(x,t)$, and the density $\rho=\rho(x,t)$.

Beirão da Veiga and Valli [1, 2] and Valli and Zajaczkowski [3] proved the unique solvability, local in time, in some supercritical Sobolev spaces and Hölder spaces in bounded domains. It is worth pointing out that in 1995 Berselli [4] discussed the standard ideal flow.

When $\Omega:={\mathbb {R}^{3}}$, Danchin [5] and Danchin and Fanelli [6] (see also [7, 8]) proved the unique solvability, local in time, in some critical Besov spaces.

The first aim of this paper is to prove the local existence and a blow-up criterion of problem (1.1)-(1.5) in the $L^{p}$ frame work. We will prove the following:

Theorem 1.1

Let $0<\inf\rho_{0}\leq\sup\rho_{0}<\infty$, $\rho_{0},\mathbf{u}_{0}\in W^{s,p}(\Omega)$ with integer $s\geq3$, $s>1+\frac{3}{p}$, and $2< p<\infty$, and $\operatorname {div}\mathbf{u}_{0}=0$ and $\mathbf{u}_{0}\cdot\mathbf{n}=0$ on ∂Ω. Then there exists a positive time $T^{*}>0$ such that problem (1.1)-(1.5) has a unique solution $(\rho,\mathbf {u})$ satisfying

$$ 0< \inf\rho_{0}\leq\rho\leq\sup\rho_{0}< \infty,\quad \rho, \mathbf{u}\in L^{\infty}\bigl(0,T^{*};W^{s,p}\bigr). $$

(1.6)

Furthermore, if u satisfies

$$ \nabla\mathbf{u} \in L^{\infty}\bigl(0,T;L^{\infty}\bigr) $$

(1.7)

with $0< T<\infty$, then the solution $(\rho,\mathbf{u},\pi)$ can be extended beyond $T>0$.

Remark 1.1

When $1< p\leq2$, we can prove a similar result.

We also consider the following ideal density-dependent MHD system:

$$\begin{aligned}& \partial_{t}\rho+\mathbf{u}\cdot\nabla\rho=0, \end{aligned}$$

(1.8)

$$\begin{aligned}& \rho\partial_{t}\mathbf{u}+\rho(\mathbf{u}\cdot\nabla)\mathbf{u}+ \nabla \biggl(\pi+\frac{1}{2} |\mathbf{b}|^{2} \biggr)=(\mathbf{b} \cdot\nabla)\mathbf {b}, \end{aligned}$$

(1.9)

$$\begin{aligned}& \partial_{t}\mathbf{b}+(\mathbf{u}\cdot\nabla)\mathbf{b}=(\mathbf {b} \cdot\nabla)\mathbf{u}, \end{aligned}$$

(1.10)

$$\begin{aligned}& \operatorname {div}\mathbf{u}=\operatorname {div}\mathbf{b}=0, \end{aligned}$$

(1.11)

$$\begin{aligned}& \mathbf{u}\cdot\mathbf{n}=\mathbf{b}\cdot\mathbf{n}=0 \quad\mbox{on } \partial \Omega\times(0,\infty), \end{aligned}$$

(1.12)

$$\begin{aligned}& (\rho,\mathbf{u},\mathbf{b}) (\cdot,0)=(\rho_{0},\mathbf{u}_{0}, \mathbf {b}_{0}) \quad\mbox{in } \Omega\subset {\mathbb {R}^{3}}. \end{aligned}$$

(1.13)

Here Ω is a bounded domain with smooth boundary $\partial\Omega \in C^{\infty}$, n is the outward unit normal to ∂Ω, and the unknowns are the plasma velocity $\mathbf{u}=\mathbf {u}(x,t)$, the magnetic field $\mathbf{b}=\mathbf{b}(x,t)$, the pressure $\pi=\pi(x,t)$, and the density $\rho=\rho(x,t)$. When $\mathbf {b}=0$, system (1.8)-(1.13) reduces to the density-dependent Euler equations (1.1)-(1.5). When $\Omega :={\mathbb {R}^{3}}$, Zhou and Fan [9] proved the local well-posedness of problem (1.8)-(1.13). For other related works, we refer to [10–14] and references therein.

In 1993, Secchi [15] was the first one to consider problem (1.8)-(1.13) and proved the local unique solvability with the main condition that

$$ \|\nabla\rho_{0}\|_{H^{s-1}} \mbox{ is small enough with integer } s\geq 3. $$

(1.14)

The second aim of this paper is to prove the local well-posedness of problem (1.8)-(1.13) without any smallness condition; furthermore, we will also prove a regularity criterion. We will prove the following:

Theorem 1.2

Let $0<\inf\rho_{0}\leq\sup\rho_{0}<\infty$, $\rho_{0},\mathbf{u}_{0},\mathbf {b}_{0}\in H^{s}$ with integer $s\geq3$, $\operatorname {div}\mathbf{u}_{0}=\operatorname {div}\mathbf {b}_{0}=0$ in Ω, and $\mathbf{u}_{0}\cdot\mathbf{n}=\mathbf {b}_{0}\cdot\mathbf{n}=0$ on ∂Ω.

Then there exists a positive time $T^{*}>0$ such that problem (1.8)-(1.13) has a unique solution $(\rho,\mathbf{u},\mathbf {b})$ satisfying

$$ 0< \inf\rho_{0}\leq\rho\leq\sup\rho_{0}< \infty, \quad\rho, \mathbf{u},\mathbf {b}\in L^{\infty}\bigl(0,T^{*};H^{s} \bigr). $$

(1.15)

Furthermore, if u and b satisfy

$$ \nabla\mathbf{u},\nabla\mathbf{b}\in L^{\infty}\bigl(0,T;L^{\infty}\bigr) $$

(1.16)

with $0< T<\infty$, then the solution $(\rho,\mathbf{u},\mathbf{b},\pi)$ can be extended beyond $T>0$.

Remark 1.2

We are unable to prove Theorem 1.1 for the ideal density-dependent MHD system.

We will use the following well-known Osgood lemma in [16].

Lemma 1.3

(Osgood lemma)

Let y be a measurable positive function, f a positive, locally integrable function, and g a continuous increasing function. Assume that, for a positive real number a, the function y satisfies

$$y(t)\leq a+ \int_{t_{0}}^{t} f(s)g\bigl(y(s)\bigr)\,ds. $$

If a is different from zero, then we have

$$-G\bigl(y(t)\bigr)+G(a)\leq \int_{t_{0}}^{t} f(s)\,ds, \quad \textit{where }G(s):= \int_{s}^{1}\frac{dr}{g(r)}. $$

If a is zero and $g(s)$ satisfies $\int_{0}^{1}\frac{dr}{g(r)}=+\infty$, then the function y is identically zero.

We will also use the following bilinear commutator and the product estimate:

• (i) If $f\in W^{s,p}(\Omega)\cap C^{1}(\Omega)$ and $g\in W^{s-1,p}(\Omega)\cap C(\Omega)$, then, for $|\alpha|\leq s$,

  $$\begin{aligned} \bigl\| D^{\alpha}(fg)-fD^{\alpha}g\bigr\| _{L^{p}(\Omega)}\leq C\bigl(\|f \|_{W^{s,p_{1}}(\Omega )}\|g\|_{L^{q_{1}}(\Omega)}+\|\nabla f\|_{L^{p_{2}}(\Omega)}\|g\| _{W^{s-1,q_{2}}(\Omega)}\bigr). \end{aligned}$$

  (1.17)

• (ii) If $f,g\in W^{s,p}(\Omega)\cap C(\Omega)$, then, for $|\alpha|\leq s$,

  $$ \bigl\| D^{\alpha}(fg)\bigr\| _{L^{p}(\Omega)}\leq C\bigl(\|f\|_{W^{s,p_{1}}(\Omega)}\|g\| _{L^{q_{1}}(\Omega)}+\|f\|_{L^{p_{2}}(\Omega)}\|g\|_{W^{s,q_{2}}(\Omega )}\bigr) $$

  (1.18)

  with integer $s>0$, $\frac{1}{p}=\frac{1}{p_{1}}+\frac{1}{q_{1}}=\frac {1}{p_{2}}+\frac{1}{q_{2}}$, and $1< p<\infty$.

The case with $p=2$, $p_{1}=q_{2}=p$, $q_{1}=p_{2}=\infty$ has been proved in [17]. Since the proof of (1.18) is similar to that of (1.17), we will prove (1.17) only in the Appendix.

2 Local existence of the Euler system

This section is devoted to the proof of local existence for the Euler system. We only need to prove a priori estimates (1.6).

First, by the maximum principle, we have the well-known estimates

$$ 0< \inf\rho_{0}\leq\rho\leq\sup\rho_{0}< \infty. $$

(2.1)

Testing (1.2) by u and using (1.1), (1.3), and (1.4), we see that

$$ \int_{\Omega}\rho|\mathbf{u}|^{2}\,dx= \int_{\Omega}\rho_{0}|\mathbf{u}_{0}|^{2}\,dx. $$

(2.2)

Applying $D^{s}$ to (1.1), testing by $|D^{s}\rho|^{p-2}D^{s}\rho$, and using (1.3), (1.4), and (1.17), we derive

$$\begin{aligned} &\frac{1}{p}\frac{d}{dt} \int_{\Omega}\bigl|D^{s}\rho \bigr|^{p}\,dx \\ &\quad=- \int_{\Omega}\bigl(D^{s}(\mathbf{u}\cdot\nabla\rho)- \mathbf{u}\cdot\nabla D^{s}\rho\bigr)\bigl|D^{s} \rho\bigr|^{p-2}D^{s}\rho \,dx \\ &\quad\leq\bigl\| D^{s}(\mathbf{u}\cdot\nabla\rho)-\mathbf{u}\cdot\nabla D^{s}\rho\bigr\| _{L^{p}}\bigl\| D^{s}\rho\bigr\| _{L^{p}}^{p-1} \\ &\quad\leq C\bigl(\|\nabla\mathbf{u}\|_{L^{\infty}}\|\rho\|_{W^{s,p}}+\|\nabla \rho \|_{L^{\infty}}\|\mathbf{u}\|_{W^{s,p}}\bigr)\|\rho\|_{W^{s,p}}^{p-1} \\ &\quad\leq C\|\mathbf{u}\|_{W^{s,p}}\|\rho\|_{W^{s,p}}^{p}, \\ &\quad\leq C\|\mathbf{u}\|_{W^{s,p}}^{p+1}+C\|\rho \|_{W^{s,p}}^{p+1}. \end{aligned}$$

(2.3)

Using (1.1), we rewrite (1.2) as follows:

$$ \partial_{t}\mathbf{u}+\mathbf{u}\cdot\nabla\mathbf{u}+ \frac{1}{\rho}\nabla \pi=0. $$

(2.4)

Applying $D^{s}$ to (2.4), testing by $|D^{s}u|^{p-2}D^{s}u$, and using (1.3), (1.4), (1.17), (1.18), and (2.1), we deduce that

$$\begin{aligned} &\frac{1}{p}\frac{d}{dt} \int_{\Omega}\bigl|D^{s}\mathbf{u}\bigr|^{p} \,dx \\ &\quad\leq- \int_{\Omega}\bigl(D^{s}(\mathbf{u}\cdot\nabla \mathbf{u})-\mathbf{u}\cdot \nabla D^{s}\mathbf{u}\bigr)\bigl|D^{s} \mathbf{u}\bigr|^{p-2}D^{s}\mathbf{u}\,dx- \int_{\Omega}D^{s} \biggl(\frac{1}{\rho}\nabla\pi \biggr)\bigl|D^{s}\mathbf{u}\bigr|^{p-2}D^{s}\mathbf{u}\,dx \\ &\quad\leq\bigl\| D^{s}(\mathbf{u}\cdot\nabla\mathbf{u})-\mathbf{u}\cdot \nabla D^{s}\mathbf{u}\bigr\| _{L^{p}}\bigl\| D^{s}\mathbf{u} \bigr\| _{L^{p}}^{p-1} +\biggl\Vert D^{s} \biggl( \frac{1}{\rho}\nabla\pi \biggr)\biggr\Vert _{L^{p}} \bigl\| D^{s}\mathbf{u}\bigr\| _{L^{p}}^{p-1} \\ &\quad\leq C\|\nabla\mathbf{u}\|_{L^{\infty}}\|\mathbf{u}\|_{W^{s,p}}^{p}+C\bigl( \| \nabla\pi\|_{W^{s,p}} +\|\nabla\pi\|_{L^{\infty}}\|\rho \|_{W^{s,p}}\bigr)\| \mathbf{u}\|_{W^{s,p}}^{p-1}. \end{aligned}$$

(2.5)

Testing (2.4) by ∇π and using (1.3), (1.4), (2.1), and (2.2), we infer that

$$ \|\nabla\pi\|_{L^{2}}\leq C\|\mathbf{u}\cdot\nabla\mathbf{u} \|_{L^{2}}\leq C\|\mathbf{u}\|_{L^{2}}\|\nabla\mathbf{u} \|_{L^{\infty}}\leq C\|\nabla \mathbf{u}\|_{L^{\infty}}. $$

(2.6)

Taking div to (2.4), we observe that

$$ -\Delta\pi=f:=\rho\sum_{i}\nabla \mathbf{u}_{i}\partial_{i}\mathbf {u}-\frac{1}{\rho}\nabla\rho\cdot\nabla\pi. $$

(2.7)

Using (1.1), (1.2), and (1.4), we deduce that

$$ \frac{\partial\pi}{\partial\mathbf{n}}=g:=\rho\mathbf{u}\cdot\nabla \mathbf{n}\cdot\mathbf{u} \quad\mbox{on } \partial\Omega. $$

(2.8)

Using (1.18) and the well-known $W^{s,p}$-estimates of problem (2.7)-(2.8) [18], we have

$$\begin{aligned} &\|\nabla\pi\|_{W^{s,p}(\Omega)} \\ &\quad\leq C\|f\|_{W^{s-1,p}(\Omega)}+C\|g\|_{W^{s-\frac{1}{p},p}(\partial\Omega )} \\ &\quad\leq C\biggl\| \rho\sum_{i}\nabla \mathbf{u}_{i}\partial_{i}\mathbf{u}\biggr\| _{W^{s-1,p}(\Omega)}+C\biggl\Vert \nabla\frac{1}{\rho}\nabla\pi\biggr\Vert _{W^{s-1,p}(\Omega)}+C\|\rho \mathbf{u}\cdot\nabla\mathbf{n}\cdot \mathbf{u}\|_{W^{s-\frac{1}{p},p}(\partial\Omega)} \\ &\quad\leq C\bigl[\|\rho\|_{W^{s-1,p}}\|\nabla\mathbf{u}\|_{L^{\infty}}^{2}+ \|\nabla \mathbf{u}\|_{L^{\infty}}\|\mathbf{u}\|_{W^{s,p}}\bigr] \\ &\qquad{}+C\bigl[\|\nabla\rho\|_{L^{\infty}}\|\nabla\pi\|_{W^{s-1,p}}+\|\nabla \pi\| _{L^{\infty}}\|\rho\|_{W^{s,p}}\bigr]+C\|\rho\mathbf{u}\cdot\nabla\mathbf {n}\cdot\mathbf{u}\|_{W^{s,p}(\Omega)} \\ &\quad\leq C\|\rho\|_{W^{s,p}}\|\mathbf{u}\|_{W^{s,p}}^{2}+C \|\mathbf{u}\| _{W^{s,p}}^{2}+C\|\rho\|_{W^{s,p}}\|\nabla\pi \|_{W^{s-1,p}} \\ &\qquad{}+C\|\rho\mathbf{u}\cdot\mathbf{u}\|_{W^{s,p}}+C\|\rho \mathbf{u}^{2}\| _{L^{\infty}} \\ &\quad\leq C\|\rho\|_{W^{s,p}}\|\mathbf{u}\|_{W^{s,p}}^{2}+C \|\mathbf{u}\| _{W^{s,p}}^{2}+C\|\rho\|_{W^{s,p}}\|\nabla\pi \|_{W^{s-1,p}}, \end{aligned}$$

(2.9)

where we used the estimate [18]

$$ \biggl\Vert \nabla\frac{1}{\rho}\biggr\Vert _{W^{s-1,p}}\leq C\| \rho\| _{W^{s,p}}. $$

By the Gagliardo-Nirenberg inequality

$$ \|\nabla\pi\|_{W^{s-1,p}}\leq C\|\nabla\pi\|_{L^{2}}^{1-\alpha}\| \nabla\pi \|_{W^{s,p}}^{\alpha},\quad 1-\alpha=\frac{1}{s+\frac{3}{2}-\frac{3}{p}}, $$

(2.10)

it follows from (2.6), (2.9), and (2.10) that

$$ \|\nabla\pi\|_{W^{s,p}}\leq C\|\rho\|_{W^{s,p}}\|\mathbf{u}\| _{W^{s,p}}^{2}+C\|\mathbf{u}\|_{W^{s,p}}^{2}+C\| \rho\|_{W^{s,p}}^{s+\frac{3}{2}-\frac{3}{p}}\|\nabla\mathbf{u}\|_{L^{\infty}}. $$

(2.11)

Combining (2.3), (2.5), and (2.11) and using Osgood’s lemma (for some T) and the inequalities

$$\begin{aligned}& \|\nabla\pi\|_{L^{\infty}}\leq C\|\nabla\pi\|_{W^{s,p}},\qquad \|\nabla\mathbf {u}\|_{L^{\infty}}\leq C\|\mathbf{u}\|_{W^{s,p}}, \\& \begin{aligned}[b] \|\rho\|_{W^{s,p}}&\leq C\bigl(\|\rho\|_{L^{p}}+ \bigl\| D^{s}\rho\bigr\| _{L^{p}}\bigr) \\ &\leq C+C\bigl\| D^{s}\rho\bigr\| _{L^{p}}, \end{aligned} \\& \begin{aligned}[b] \|\mathbf{u}\|_{W^{s,p}}&\leq C\bigl(\|\mathbf{u} \|_{L^{p}}+\bigl\| D^{s}\mathbf{u}\bigr\| _{L^{p}}\bigr) \\ &\leq C+C\bigl\| D^{s}\mathbf{u}\bigr\| _{L^{p}}, \end{aligned} \end{aligned}$$

we arrive at

$$ \|\rho\|_{L^{\infty}(0,T;W^{s,p})}+\|\mathbf{u}\|_{L^{\infty}(0,T;W^{s,p})}\leq C. $$

(2.12)

This completes the proof.

3 A blow-up criterion for the Euler system

This section is devoted to the proof of regularity criterion for the Euler system. We only need to establish a priori estimates.

First, we still have (2.1) and (2.2).

Taking ∇ to (1.1), testing by $|\nabla\rho|^{p-2}\nabla\rho$. and using (1.3) and (1.4), we derive

$$\frac{1}{p}\frac{d}{dt} \int_{\Omega}|\nabla\rho|^{p} \,dx\leq\|\nabla\mathbf{u}\| _{L^{\infty}} \int_{\Omega}|\nabla\rho|^{p} \,dx, $$

whence

$$\frac{d}{dt}\|\nabla\rho\|_{L^{p}}\leq\|\nabla\mathbf{u} \|_{L^{\infty}}\| \nabla\rho\|_{L^{p}}. $$

Integrating this inequality and taking the limit as $p\rightarrow+\infty$, we have

$$ \|\nabla\rho\|_{L^{\infty}(0,T;L^{\infty})}\leq C. $$

(3.1)

It follows from (2.6) that

$$ \|\nabla\pi\|_{L^{\infty}(0,T;L^{2})}\leq C. $$

(3.2)

It follows from (2.7), (2.8), (1.7), (3.1), (3.2), and the $W^{2,p}$-estimates of problem (2.7)-(2.8) that

$$\begin{aligned} \|\nabla\pi\|_{W^{1,p}} \leq&C\|f\|_{L^{p}}+C\|g\|_{W^{1-\frac{1}{p},p}(\partial\Omega)} \\ \leq&C\biggl\| \rho\sum_{i}\nabla\mathbf{u}_{i} \partial_{i}\mathbf{u}\biggr\| _{L^{p}}+C\biggl\Vert \nabla \frac{1}{\rho}\nabla\pi\biggr\Vert _{L^{p}}+C\|\rho\mathbf {u} \cdot\nabla n\cdot\mathbf{u}\|_{W^{1-\frac{1}{p},p}(\partial\Omega)} \\ \leq&C+C\|\nabla\pi\|_{L^{p}}+C\|\rho\mathbf{u}\cdot\nabla\mathbf {n} \cdot\mathbf{u}\|_{W^{1,p}} \\ \leq&C+C\|\nabla\pi\|_{L^{2}}^{1-\tilde{\alpha}}\|\nabla\pi\| _{W^{1,p}}^{\tilde{\alpha}} \\ \leq&\frac{1}{2}\|\nabla\pi\|_{W^{1,p}}+C \end{aligned}$$

for any $3< p<\infty$, and thus

$$ \|\nabla\pi\|_{L^{\infty}(0,T;L^{\infty})}\leq C. $$

(3.3)

Similarly to (2.9), we have

$$\begin{aligned} \|\nabla\pi\|_{W^{s,p}} \leq&C\|\rho\|_{W^{s,p}}+C\|\mathbf{u}\| _{W^{s,p}}+C\|\nabla\pi\|_{W^{s-1,p}} \\ \leq&\frac{1}{2}\|\nabla\pi\|_{W^{s,p}}+C+C\|\rho\|_{W^{s,p}}+C \|\mathbf {u}\|_{W^{s,p}}, \end{aligned}$$

and thus

$$ \|\nabla\pi\|_{W^{s,p}}\leq C+C\|\rho\|_{W^{s,p}}+C\|\mathbf{u}\| _{W^{s,p}}. $$

(3.4)

Combining (2.3), (2.5), (3.4), (1.7), (3.3), and (3.1) and using the Gronwall inequality, we arrive at (2.12).

This completes the proof.

4 Local existence for the MHD system

This section is devoted to the proof of local existence for the MHD system. We only need to prove a priori estimates (1.15). Before going to detailed estimates, we write the case with $p=2$, $p_{1}=q_{2}=p$, $q_{1}=p_{2}=\infty$ in (1.17) and (1.18) as follows:

• (i) If $f,g\in H^{s}(\Omega)\cap C(\Omega)$, then

  $$ \|fg\|_{H^{s}(\Omega)}\leq C\bigl(\|f\|_{H^{s}(\Omega)}\|g\|_{L^{\infty}(\Omega )}+\|f \|_{L^{\infty}(\Omega)}\|g\|_{H^{s}(\Omega)}\bigr). $$

  (4.1)

• (ii) If $f\in H^{s}(\Omega)\cap C^{1}(\Omega)$ and $g\in H^{s-1}(\Omega)\cap C(\Omega)$, then, for $|\alpha|\leq s$,

  $$ \bigl\| D^{\alpha}(fg)-fD^{\alpha}g\bigr\| _{L^{2}(\Omega)}\leq C\bigl(\|f \|_{H^{s}(\Omega)}\| g\|_{L^{\infty}(\Omega)}+\|f\|_{W^{1,\infty}(\Omega)}\|g \|_{H^{s-1}(\Omega )}\bigr). $$

  (4.2)

First, by the maximum principle we have the well-known estimates

$$ 0< \inf\rho_{0}\leq\rho\leq\sup\rho_{0}< \infty. $$

(4.3)

Testing (1.2) by u and using (1.8) and (1.11), we see that

$$ \frac{1}{2}\frac{d}{dt} \int_{\Omega}\rho|\mathbf{u}|^{2}\,dx= \int_{\Omega}(\mathbf {b}\cdot\nabla)\mathbf{b}\cdot\mathbf{u}\,dx. $$

(4.4)

Testing (1.10) by b and using (1.4), we find that

$$ \frac{1}{2}\frac{d}{dt} \int_{\Omega}|\mathbf{b}|^{2}\,dx= \int_{\Omega}(\mathbf {b}\cdot\nabla)\mathbf{u}\cdot\mathbf{b}\,dx. $$

(4.5)

Summing up (4.4) and (4.5) and noting the cancellation of the terms on the right-hand sides of (4.4) and (4.5), we get

$$ \int_{\Omega}\bigl(\rho|\mathbf{u}|^{2}+| \mathbf{b}|^{2}\bigr)\,dx= \int_{\Omega}\bigl(\rho |\mathbf{u}_{0}|^{2}+| \mathbf{b}_{0}|^{2}\bigr)\,dx. $$

(4.6)

Applying $D^{s}$ to (1.8), testing by $D^{s}\rho$, and using (1.11) and (4.2), we derive

$$\begin{aligned} \frac{1}{2}\frac{d}{dt} \int_{\Omega}\bigl|D^{s}\rho\bigr|^{2}\,dx &=- \int_{\Omega}\bigl(D^{s}(\mathbf{u}\cdot\nabla\rho)- \mathbf{u}\cdot\nabla D^{s}\rho\bigr)D^{s}\rho \,dx \\ &\leq\bigl\| D^{s}(\mathbf{u}\cdot\nabla\rho)-\mathbf{u}\cdot\nabla D^{s}\rho\bigr\| _{L^{2}}\bigl\| D^{s}\rho\bigr\| _{L^{2}} \\ &\leq C\bigl(\|\nabla\rho\|_{L^{\infty}}\|\mathbf{u}\|_{H^{s}}+\| \mathbf{u}\| _{W^{1,\infty}}\|\nabla\rho\|_{H^{s-1}}\bigr)\bigl\| D^{s}\rho \bigr\| _{L^{2}} \\ &\leq C\|\rho\|_{H^{s}}^{3}+C\|\mathbf{u} \|_{H^{s}}^{3}. \end{aligned}$$

(4.7)

Applying $D^{s}$ to (1.9), testing by $D^{s}\mathbf{u}$, and using (1.11), we get

$$\begin{aligned} &\frac{1}{2}\frac{d}{dt} \int_{\Omega}\rho\bigl|D^{s}\mathbf{u}\bigr|^{2}\,dx \\ &\quad= \int_{\Omega}\bigl(D^{s}(\mathbf{b}\cdot\nabla \mathbf{b})-b\cdot\nabla D^{s}\mathbf{b}\bigr)D^{s}\mathbf{u}\,dx+ \int_{\Omega}\mathbf{b}\cdot\nabla D^{s}\mathbf{b}\cdot D^{s}\mathbf{u}\,dx \\ &\qquad{}- \int_{\Omega}\bigl(D^{s}(\rho\partial_{t} \mathbf{u})-\rho D^{s}\partial_{t}\mathbf {u} \bigr)D^{s} \mathbf{u}\,dx- \int_{\Omega}\bigl(D^{s}(\rho\mathbf{u}\cdot\nabla\mathbf {u})-\rho\mathbf{u}\cdot\nabla D^{s}\mathbf{u}\bigr)D^{s} \mathbf{u}\,dx \\ &\qquad{}- \int_{\Omega}D^{s}\nabla \biggl(\pi+\frac{1}{2} \mathbf{b}^{2} \biggr)\cdot D^{s}\mathbf{u}\,dx=:I_{1}+I_{2}+I_{3}+I_{4}+I_{5}. \end{aligned}$$

(4.8)

Applying $D^{s}$ to (1.10), testing by $D^{s}\mathbf{b}$, and using (1.11), we deduce

$$\begin{aligned} \frac{1}{2}\frac{d}{dt} \int_{\Omega}\bigl|D^{s}\mathbf{b}\bigr|^{2}\,dx ={}& \int_{\Omega}\bigl(D^{s}(\mathbf{b}\cdot\nabla \mathbf{u})-\mathbf{b}\cdot \nabla D^{s}\mathbf{u}\bigr)D^{s} \mathbf{b}\,dx+ \int_{\Omega}\mathbf{b}\cdot\nabla D^{s}\mathbf{u}\cdot D^{s}\mathbf{b}\,dx \\ &{}- \int_{\Omega}\bigl(D^{s}(\mathbf{u}\cdot\nabla \mathbf{b})-\mathbf{u}\cdot \nabla D^{s}\mathbf{b}\bigr)D^{s} \mathbf{b}\,dx=:I_{6}+I_{7}+I_{8}. \end{aligned}$$

(4.9)

Summing up (4.8) and (4.9) and noting that $I_{2}+I_{7}=0$, we find that

$$ \frac{1}{2}\frac{d}{dt} \int_{\Omega}\bigl(\rho\bigl|D^{s}\mathbf{u}\bigr|^{2}+\bigl|D^{s} \mathbf{b}\bigr|^{2}\bigr)\,dx=I_{1}+I_{3}+I_{4}+I_{5}+I_{6}+I_{8}. $$

(4.10)

Using (4.2) and (4.1), we bound $I_{1}$, $I_{3}$, $I_{4}$, $I_{5}$, $I_{6}$, and $I_{8}$ as follows:

$$\begin{aligned} I_{1} \leq&C\|\mathbf{b}\|_{W^{1,\infty}}\|\mathbf{b}\|_{H^{s}} \|\mathbf {u}\|_{H^{s}}\leq C\|\mathbf{b}\|_{H^{s}}^{2}\| \mathbf{u}\|_{H^{s}}, \\ I_{3} \leq&C\bigl(\|\rho\|_{H^{s}}\|\partial_{t}\mathbf{u} \|_{L^{\infty}} +\|\rho\| _{W^{1,\infty}}\|\partial_{t}\mathbf{u} \|_{H^{s-1}}\bigr)\bigl\| D^{s}\mathbf{u}\bigr\| _{L^{2}} \\ \leq&C\|\rho\|_{H^{s}}\|\partial_{t}\mathbf{u} \|_{H^{s-1}}\|\mathbf{u}\| _{H^{s}}, \\ I_{4} \leq&C\bigl(\|\rho\mathbf{u}\|_{H^{s}}\|\nabla\mathbf{u} \|_{L^{\infty}}+\| \rho\mathbf{u}\|_{W^{1,\infty}}\|\nabla\mathbf{u} \|_{H^{s-1}}\bigr)\bigl\| D^{s}\mathbf{u}\bigr\| _{L^{2}} \\ \leq&C\bigl[\bigl(\|\rho\|_{L^{\infty}}\|\mathbf{u}\|_{H^{s}}+\|\mathbf{u} \| _{L^{\infty}}\|\rho\|_{H^{s}}\bigr)\|\nabla\mathbf{u}\|_{L^{\infty}}+\| \rho\| _{W^{1,\infty}}\|\mathbf{u}\|_{W^{1,\infty}}\|\nabla\mathbf{u}\| _{H^{s-1}}\bigr]\bigl\| D^{s}\mathbf{u}\bigr\| _{L^{2}} \\ \leq&C\|\rho\|_{H^{s}}\|\mathbf{u}\|_{H^{s}}^{3}, \\ I_{5} \leq&\biggl\Vert D^{s}\nabla \biggl(\pi+ \frac{1}{2} |\mathbf{b}|^{2} \biggr)\biggr\Vert _{L^{2}} \bigl\| D^{s}\mathbf{u}\bigr\| _{L^{2}}, \\ I_{6} \leq&C\bigl(\|\mathbf{b}\|_{H^{s}}\|\nabla\mathbf{u} \|_{L^{\infty}}+\| \mathbf{b}\|_{W^{1,\infty}}\|\nabla\mathbf{u} \|_{H^{s-1}}\bigr)\bigl\| D^{s}\mathbf {b}\bigr\| _{L^{2}}\leq C\|\mathbf{b} \|_{H^{s}}^{2}\|\mathbf{u}\|_{H^{s}}, \\ I_{8} \leq&C\bigl(\|\mathbf{u}\|_{H^{s}}\|\nabla\mathbf{b} \|_{L^{\infty}}+\| \mathbf{u}\|_{W^{1,\infty}}\|\nabla\mathbf{b} \|_{H^{s-1}}\bigr)\bigl\| D^{s}b\bigr\| _{L^{2}}\leq C\|\mathbf{b} \|_{H^{s}}^{s}\|\mathbf{u}\|_{H^{s}}. \end{aligned}$$

Inserting these estimates into (4.10), we have

$$\begin{aligned} &\frac{1}{2}\frac{d}{dt} \int_{\Omega}\bigl(\rho\bigl|D^{s}\mathbf{u}\bigr|^{2}+\bigl|D^{s} \mathbf {b}\bigr|^{2}\bigr)\,dx \\ &\quad\leq C\|\mathbf{b}\|_{H^{s}}^{2}\|\mathbf{u} \|_{H^{s}}+C\|\rho\|_{H^{s}}\| \partial_{t}\mathbf{u} \|_{H^{s-1}}\|\mathbf{u}\|_{H^{s}} \\ &\qquad{}+C\|\rho\|_{H^{s}}\|\mathbf{u}\|_{H^{s}}^{3}+ \biggl\Vert D^{s}\nabla \biggl(\pi+\frac{1}{2} | \mathbf{b}|^{2} \biggr)\biggr\Vert _{L^{2}}\bigl\| D^{s} \mathbf{u}\bigr\| _{L^{2}}. \end{aligned}$$

(4.11)

Testing (1.9) by $\partial_{t}\mathbf{u}$ and using (1.11), we find that

$$\int_{\Omega}\rho|\partial_{t}\mathbf{u}|^{2}\,dx\leq C\bigl(\bigl\| (\mathbf{b}\cdot \nabla) \mathbf{b}\bigr\| _{L^{2}}+\bigl\| \rho( \mathbf{u}\cdot\nabla) \mathbf{u}\bigr\| _{L^{2}}\bigr)\|\partial_{t} \mathbf{u}\|_{L^{2}}, $$

whence

$$\begin{aligned} \|\partial_{t}\mathbf{u}\|_{L^{2}} \leq&C\bigl(\| \nabla\mathbf{b}\|_{L^{\infty}}+\|\nabla\mathbf{u}\|_{L^{\infty}}\bigr) \\ \leq&C\|\mathbf{b}\|_{H^{s}}+C\|\mathbf{u}\|_{H^{s}}. \end{aligned}$$

(4.12)

Applying $D^{s-1}$ to (1.9), testing by $D^{s-1}\partial_{t}\mathbf {u}$, and using (1.8), we have

$$\begin{aligned} \int_{\Omega}\rho\bigl|D^{s-1}\partial_{t} \mathbf{u}\bigr|^{2}\,dx ={}& \int_{\Omega}D^{s-1}(\mathbf{b}\cdot\nabla \mathbf{b})D^{s-1}\partial _{t} \mathbf{u}\,dx- \int_{\Omega}D^{s-1}(\rho\mathbf{u}\cdot\nabla\mathbf {u}) D^{s-1}\partial_{t} \mathbf{u}\,dx \\ &{}- \int_{\Omega}\bigl(D^{s-1}(\rho\partial_{t} \mathbf{u})-\rho D^{s-1}\partial _{t}\mathbf{u} \bigr)D^{s-1}\partial_{t} \mathbf{u}\,dx \\ &{}- \int_{\Omega}D^{s-1}\nabla \biggl(\pi+\frac{1}{2} \mathbf{b}^{2} \biggr)\cdot D^{s-1}\partial_{t} \mathbf{u}\,dx, \end{aligned}$$

whence

$$\begin{aligned} \bigl\| D^{s-1}\partial_{t}\mathbf{u} \bigr\| _{L^{2}} \leq&C\bigl\| D^{s-1}(\mathbf{b}\cdot \nabla\mathbf{b}) \bigr\| _{L^{2}}+C\bigl\| D^{s-1}(\rho\mathbf{u}\cdot\nabla\mathbf {u}) \bigr\| _{L^{2}} \\ &{}+C\bigl\| D^{s-1}(\rho\partial_{t}\mathbf{u})-\rho D^{s-1}\partial_{t}\mathbf {u}\bigr\| _{L^{2}}+C \biggl\| D^{s-1}\nabla \biggl(\pi+\frac{1}{2} |\mathbf{b}|^{2} \biggr)\biggr\| _{L^{2}} \\ =&:J_{1}+J_{2}+J_{3}+J_{4}. \end{aligned}$$

(4.13)

Using (4.1) and (4.2) again, we bound $J_{1}$, $J_{2}$, and $J_{3}$ as follows:

$$\begin{aligned} J_{1} \leq&C\|\mathbf{b}\|_{L^{\infty}}\|\mathbf{b}\|_{H^{s}} \leq C\|\mathbf {b}\|_{H^{s}}^{2}, \\ J_{2} \leq&C\bigl(\|\rho\mathbf{u}\|_{L^{\infty}}\|\nabla\mathbf{u}\| _{H^{s-1}}+\|\rho\mathbf{u}\|_{H^{s-1}}\|\nabla\mathbf{u} \|_{L^{\infty}}\bigr) \\ \leq&C\|\rho\|_{H^{s-1}}\|\mathbf{u}\|_{H^{s-1}}\|\mathbf{u}\| _{H^{s}}\leq C\|\rho\|_{H^{s}}\|\mathbf{u}\|_{H^{s}}^{2}, \\ J_{3} \leq&C\bigl(\|\rho\|_{H^{s-1}}\|\partial_{t}\mathbf{u} \|_{L^{\infty}}+\|\rho \|_{W^{1,\infty}}\|\partial_{t}\mathbf{u} \|_{H^{s-2}}\bigr) \\ \leq&C\|\rho\|_{H^{s}}\bigl(\|\partial_{t}\mathbf{u} \|_{L^{\infty}}+\|\partial _{t}\mathbf{u}\|_{H^{s-2}}\bigr) \\ \leq&C\|\rho\|_{H^{s}}\bigl(\|\partial_{t}\mathbf{u} \|_{L^{2}}^{\frac{s-5/2}{s-1}}\| \partial_{t}\mathbf{u} \|_{H^{s-1}}^{\frac{3/2}{s-1}}+\|\partial_{t} \mathbf {u} \|_{L^{2}}^{\frac{1}{s-1}}\|\partial_{t}\mathbf{u} \|_{H^{s-1}}^{\frac {s-2}{s-1}}\bigr) \\ \leq&\epsilon\|\partial_{t}\mathbf{u}\|_{H^{s-1}}+C\bigl(\|\rho \| _{H^{s}}^{s-1}+\|\rho\|_{H^{s}}^{\frac{s-1}{s-5/2}}\bigr)\|\partial_{t}\mathbf{u}\|_{L^{2}} \end{aligned}$$

for any $0<\epsilon<1$.

Inserting these estimates into (4.12) and (4.13) and taking ϵ small enough, we have

$$\begin{aligned} \|\partial_{t}\mathbf{u}\|_{H^{s-1}} \leq&C\| \mathbf{b}\|_{H^{s}}^{2} +C\|\rho \|_{H^{s}}\|\mathbf{u} \|_{H^{s}}^{2}+C\|\mathbf{b}\|_{H^{s}}+C\|\mathbf{u}\| _{H^{s}} \\ &{}+C\bigl(\|\rho\|_{H^{s}}^{s-1}+\|\rho\|_{H^{s}}^{\frac{s-1}{s-5/2}}\bigr) \bigl(\|\mathbf {b}\|_{H^{s}}+\|\mathbf{u}\|_{H^{s}}\bigr) \\ &{}+C\biggl\Vert D^{s-1}\nabla \biggl(\pi+\frac{1}{2} \mathbf{b}^{2} \biggr)\biggr\Vert _{L^{2}}. \end{aligned}$$

(4.14)

Using (1.8) and (1.11) and setting $\tilde{\pi}:=\pi+\frac{1}{2} \mathbf{b}^{2}$, we rewrite (1.9) as

$$ \partial_{t}\mathbf{u}+\mathbf{u}\cdot\nabla\mathbf{u}+ \frac{1}{\rho}\nabla \tilde{\pi}=\frac{1}{\rho}\mathbf{b}\cdot\nabla b. $$

(4.15)

Testing (4.15) by ∇π̃ and using (1.11) and (4.3), we infer that

$$ \|\nabla\tilde{\pi}\|_{L^{2}}\leq C\|\mathbf{b}\cdot\nabla\mathbf{b}\| _{L^{2}}+C\|\mathbf{u}\cdot\nabla\mathbf{u}\|_{L^{2}}\leq C\| \mathbf{b}\| _{H^{s}}+C\|\mathbf{u}\|_{H^{s}}. $$

(4.16)

Using (1.8), (1.9), and (1.12), we deduce that

$$ \frac{\partial\tilde{\pi}}{\partial\mathbf{n}}=g:=-\mathbf{b}\cdot\nabla \mathbf{n}\cdot\mathbf{b}+\rho \mathbf{u}\cdot\nabla\mathbf{n}\cdot \mathbf{u} \quad\mbox{on } \partial \Omega. $$

(4.17)

Taking div to (4.15), we observe that

$$ -\Delta\tilde{\pi}=f:=\rho\sum_{i}\nabla \mathbf{u}_{i}\partial _{i}\mathbf{u}-\frac{1}{\rho}( \mathbf{b}\cdot\nabla)\mathbf{b}\cdot\nabla\rho -\sum _{i}\nabla\mathbf{b}_{i}\partial_{i} \mathbf{b}-\frac{1}{\rho}\nabla \rho\cdot\nabla\tilde{\pi}. $$

(4.18)

Using (4.1) and the well-known $H^{s+1}$-estimates of problems (4.18) and (4.17) [18], we have

$$\begin{aligned} \|\nabla\tilde{\pi}\|_{H^{s}} \leq&C\|f\|_{H^{s-1}}+C\|g \|_{H^{s-\frac{1}{2}}(\partial\Omega)} \\ \leq&C\|f\|_{H^{s-1}}+C\|\mathbf{b}\cdot\nabla\mathbf{n}\cdot\mathbf {b} \|_{H^{s}}+C\|\rho\mathbf{u}\cdot\nabla\mathbf{n}\cdot\mathbf{u}\| _{H^{s}} \\ \leq&C\|\rho\|_{H^{s}}\|\mathbf{u}\|_{H^{s}}^{2}+C\| \rho\|_{H^{s}}\|\mathbf {b}\|_{H^{s}}^{2}+C\|\mathbf{b} \|_{H^{s}}+C\|\mathbf{u}\|_{H^{s}} \\ &{}+C\|\mathbf{b}\|_{H^{s}}^{2}+C\|\rho\|_{H^{s}}\| \nabla\tilde{\pi}\|_{\dot{H}^{s-1}}, \end{aligned}$$

(4.19)

whence

$$\begin{aligned} \|\nabla\tilde{\pi}\|_{H^{s}} \leq&C\|\rho\|_{H^{s}}\| \mathbf{u}\|_{H^{s}}^{2}+C\| \rho\|_{H^{s}}\|\mathbf{b} \|_{H^{s}}^{2}+C\|\mathbf{b}\|_{H^{s}} \\ &{}+C\|\mathbf{u}\|_{H^{s}}+C\|\mathbf{b}\|_{H^{s}}^{2}+C \|\rho\|_{H^{s}}^{s}\| \nabla\tilde{\pi}\|_{L^{2}}, \end{aligned}$$

(4.20)

where we used the Gagliardo-Nirenberg inequality

$$ \|\nabla\tilde{\pi}\|_{\dot{H}^{s-1}}\leq C\|\nabla\tilde{\pi}\|_{L^{2}}^{\frac{1}{s}} \|\nabla\tilde{\pi}\|_{H^{s}}^{\frac{s-1}{s}} $$

and the well-known estimate [18]

$$ \biggl\Vert D^{s} \biggl(\frac{1}{\rho}\biggr)\biggr\Vert _{L^{2}}\leq C\|\rho\| _{H^{s}}. $$

Combining (4.7), (4.11), (4.14), and (4.20) and using the Osgood lemma, we arrive at (1.15).

This completes the proof.

5 A blow-up criterion for the MHD system

This section is devoted to the proof of regularity criterion for the MHD system. We only need to establish a priori estimates.

First, we still have (4.3) and (4.6).

Taking ∇ to (1.8), testing by $|\nabla\rho|^{p-2}\nabla \rho$, and using (1.11) and (1.16), we derive

$$\frac{1}{p}\frac{d}{dt} \int_{\Omega}|\nabla\rho|^{p} \,dx\leq\|\nabla\mathbf{u}\| _{L^{\infty}} \int_{\Omega}|\nabla\rho|^{p} \,dx, $$

whence

$$\frac{d}{dt}\|\nabla\rho\|_{L^{p}}\leq\|\nabla\mathbf{u} \|_{L^{\infty}}\| \nabla\rho\|_{L^{p}}. $$

Integrating this inequality and taking the limits as $p\rightarrow +\infty$, we have

$$ \|\nabla\rho\|_{L^{\infty}(0,T;L^{\infty})}\leq C. $$

(5.1)

It follows from (4.6) and (1.16) that

$$ \|\mathbf{u}\|_{L^{\infty}(0,T;W^{1,\infty})}+\|\mathbf{b}\|_{L^{\infty}(0,T;W^{1,\infty})}\leq C. $$

(5.2)

Similarly to (4.16), we find that

$$ \|\nabla\tilde{\pi}\|_{L^{2}}\leq C. $$

(5.3)

It follows from (4.17), (4.18), (5.1), (5.2), (5.3), and the $W^{2,p}$-estimates of problem (4.17)-(4.18) that

$$\begin{aligned} \|\nabla\tilde{\pi}\|_{W^{1,p}(\Omega)} \leq&C\|f\|_{L^{p}(\Omega)}+C\|g\| _{W^{1-\frac{1}{p},p}(\partial\Omega)} \\ \leq&C+C\|\mathbf{b}\cdot\nabla\mathbf{n}\cdot\mathbf{b}\| _{W^{1,p}}+C\| \rho\mathbf{u}\cdot\nabla\mathbf{n}\cdot\mathbf{u}\| _{W^{1,p}} \\ \leq&C \end{aligned}$$

for any $3< p<\infty$, and thus

$$ \|\tilde{\pi}\|_{L^{\infty}(0,T;W^{1,\infty})}\leq C. $$

(5.4)

It follows from (4.15), (4.3), and (5.4) that

$$ \|\partial_{t}\mathbf{u}\|_{L^{\infty}(0,T;L^{\infty})}\leq C.$$

Similarly to (4.19), we have

$$\|\nabla\tilde{\pi}\|_{H^{s}}\leq C\|\mathbf{u}\|_{H^{s}}+C\| \mathbf{b}\| _{H^{s}}+C\|\rho\|_{H^{s}}+C\|\nabla\tilde{\pi}\|_{\dot{H}^{s-1}}, $$

whence

$$ \|\nabla\tilde{\pi}\|_{H^{s}}\leq C\|\mathbf{u}\|_{H^{s}}+C\| \mathbf{b}\| _{H^{s}}+C\|\rho\|_{H^{s}}+C.$$

We still have (4.13), and similarly to (4.14), we have

$$\|\partial_{t}\mathbf{u}\|_{H^{s-1}}\leq C\|\mathbf{b} \|_{H^{s}}+C\|\rho\| _{H^{s}}+C\|\mathbf{u}\|_{H^{s}}+C\| \partial_{t}\mathbf{u}\|_{H^{s-2}}+C\bigl\| D^{s-1}\nabla\tilde{\pi}\bigr\| _{L^{2}}, $$

which gives

$$ \|\partial_{t}\mathbf{u}\|_{H^{s-1}}\leq C\|\rho \|_{H^{s}}+C\|\mathbf{u}\| _{H^{s}}+C\|\mathbf{b} \|_{H^{s}}+C.$$

Similarly to (4.7), we have

$$ \frac{1}{2}\frac{d}{dt} \int_{\Omega}\bigl|D^{s}\rho\bigr|^{2}\,dx\leq C\|\rho \|_{H^{s}}^{2}+C\| \mathbf{u}\|_{H^{s}}^{2}. $$

(5.5)

We still have (4.10). We bound $I_{1}$, $I_{3}$, $I_{4}$, $I_{5}$, $I_{6}$, and $I_{8}$ as follows:

$$\begin{aligned} I_{1} \leq&C\|\mathbf{u}\|_{H^{s}}^{2}+C\|b \|_{H^{s}}^{2}, \\ I_{3} \leq&C\|\mathbf{u}\|_{H^{s}}^{2}+C\|\rho \|_{H^{s}}^{2}+C\|\partial_{t}\mathbf {u} \|_{H^{s-1}}^{2} \\ \leq&C\|\rho\|_{H^{s}}^{2}+C\|\mathbf{u}\|_{H^{s}}^{2}+C \|\mathbf{b}\| _{H^{s}}^{2}+C, \\ I_{4} \leq&C\|\rho\|_{H^{s}}^{2}+C\|\mathbf{u} \|_{H^{s}}^{2}, \\ I_{5} \leq&C\|\rho\|_{H^{s}}^{2}+C\|\mathbf{u} \|_{H^{s}}^{2}+C\|\mathbf{b}\| _{H^{s}}^{2}+C, \\ I_{6} \leq&C\|\mathbf{b}\|_{H^{s}}^{2}+C\|\mathbf{u} \|_{H^{s}}^{2}, \\ I_{8} \leq&C\|\mathbf{b}\|_{H^{s}}^{2}+C\|\mathbf{u} \|_{H^{s}}^{2}. \end{aligned}$$

Inserting these estimates into (4.10) and using (5.5) and the Gronwall inequality, we conclude that

$$\bigl\| (\rho,\mathbf{u},\mathbf{b})\bigr\| _{L^{\infty}(0,T;H^{s})}\leq C. $$

This completes the proof.

Appendix:  Proof of (1.17)

We only prove the case $|\alpha|=s$. We have

$$\begin{aligned} \bigl\| D^{\alpha}(fg)-fD^{\alpha}g\bigr\| _{L^{p}}\leq{}& \sum_{i=1}^{s}C_{i}\bigl\| D^{i}fD^{s-i}g\bigr\| _{L^{p}} \\ \leq{}& C\|\nabla f\|_{L^{p_{2}}}\|g\|_{W^{s-1,q_{2}}}+C\|f \|_{W^{s,p_{1}}}\|g\| _{L^{q_{1}}} \\ &{}+\sum_{i=2}^{s-1}C_{i} \bigl\| D^{i}f\bigr\| _{L^{p_{i}}}\bigl\| D^{s-i}g\bigr\| _{L^{q_{i}}}. \end{aligned}$$

(A.1)

We will use the following two Gagliardo-Nirenberg inequalities:

$$\begin{aligned}& \bigl\| D^{i}f\bigr\| _{L^{p_{i}}}\leq C\|\nabla f\|_{L^{p_{2}}}^{1-\alpha_{i}} \|f\| _{W^{s,p_{1}}}^{\alpha_{i}}, \end{aligned}$$

(A.2)

$$\begin{aligned}& \bigl\| D^{s-i}g\bigr\| _{L^{q_{i}}}\leq C\|g\|_{L^{q_{1}}}^{\alpha_{i}}\|g \| _{W^{s-1,q_{2}}}^{1-\alpha_{i}}, \end{aligned}$$

(A.3)

with $i-\frac{d}{p_{i}}=(1-\alpha_{i}) (1-\frac{d}{p_{2}} )+\alpha _{i} (s-\frac{d}{p_{1}} )$, where d is the dimension number.

Inserting (A.2) and (A.3) into (A.1) and using the Young inequality give (1.17).

This completes the proof.