Global strong solutions of 3D compressible isothermal magnetohydrodynamics

Abstract

This paper establishes the global existence of strong solutions for the three-dimensional compressible isothermal magnetohydrodynamic (MHD) flows. It is essentially shown that for the regular data with small energy but possible large oscillations, the global well-posedness of strong solutions is established.

1 Introduction

In this paper, we are concerned with the 3D compressible isothermal magnetohydrodynamic (MHD) equations, which are a combination of the compressible Navier–Stokes equations of fluid dynamics and Maxwells equations of electromagnetism (see [1, 13]):

$$ \textstyle\begin{cases} \rho _{t} +\operatorname{div}(\rho \mathbf{u})=0, \\ (\rho \mathbf{u})_{t}+\operatorname{div}(\rho \mathbf{u}\otimes \mathbf{u})- \mu \Delta \mathbf{ u}-(\mu +\lambda )\nabla \operatorname{div}\mathbf{ u}+ \nabla P(\rho )=(\operatorname{curl}\mathbf{B})\times \mathbf{B}, \\ \mathbf{B}_{t}-\operatorname{curl} ~(\mathbf{u}\times \mathbf{B})=\nu \Delta \mathbf{B},\qquad \operatorname{div}\mathbf{B}=0, \end{cases} $$

(1.1)

where $t\geqslant 0$ is the time and $x\in {\mathbb{R}}^3$ is the spatial coordinate. The unknown functions $\rho , \mathbf{u}=(u^{1}, u^{2}, u^{3})^{\mathrm{tr}}$, $\mathbf{B}=(b^{1}, b^{2}, b^{3})^{\mathrm{tr}}$, and P denote the fluid density, velocity, magnetic field, and pressure, respectively. The viscosity coefficients μ and λ satisfy the physical restrictions

$$ \mu >0, \qquad 2\mu +3\lambda \geqslant 0. $$

(1.2)

Positive constant ν is the magnetic diffusion coefficient. The pressure $P(\rho )$ is determined through the equations of states. Here, we consider the isothermal gas dynamics

$$ P(\rho )\triangleq a\rho \quad \text{with } a>0. $$

(1.3)

The main aim of this paper is to study the Cauchy problem of (1.1)–(1.3) with the initial data

$$(\rho ,\mathbf{u},\mathbf{B}){|}_{t=0}=({\rho }_0,{\mathbf{u}}_0,{\mathbf{B}}_0)(x),x\in {\mathbb{R}}^3,$$

(1.4)

and the far field behavior

$$ (\rho , \mathbf{u}, \mathbf{B}) (x, t) \to (\tilde{\rho}, 0, 0) \quad \text{as } \vert x \vert \to \infty , t>0, $$

(1.5)

where ρ̃ is a given nonnegative constant. Without loss of generality, we assume that $\tilde{\rho}=1$.

The compressible MHD system (1.1) describes the relationship between the Navier–Stokes equations of fluid dynamics and Maxwell’s equations of electromagnetism, and it has been studied by many articles [2, 3, 7, 9] and the references cited therein. For the one-dimensional isentropic compressible Navier–Stokes equations, Hoff [4], Kazhikhov and Shelukhin [12], and Serre [21, 22] have studied the isothermal problems respectively. For the multi-dimensional case, Itaya [11], Nash [19], Serrin [23], and Tani [24] investigated the local existence and uniqueness of classical solutions with vacuum, respectively. Matsumura and Nishida in [16–18] proved the global smooth solutions when the initial data are close to a non-vacuum equilibrium. Later, Huang, Li, and Xin [10] investigated the vacuum and non-vacuum state for the three-dimensional case, and they obtained the global existence and uniqueness of classical solutions. Yu in [25] studied the 3D compressible isothermal Navier–Stokes equations with a vacuum at infinity and proved the global existence of strong solutions. For the isentropic MHD system, Hu and Wang [8, 9] obtained the global existence of renormalized solutions with general large initial data. Li, Xu, and Zhang [15] considered the Cauchy problem of the 3D case and obtained the global well-posedness of classical solution with small energy.

For the isothermal Navier–Stokes system away from vacuum, Nishida in [20] proved the global existence of BV solutions for one-dimensional MHD equations. Hoff in [5, 6] obtained the global weak solutions for three-dimensional case. Matsumura and Nishida in [17] obtained the global smooth solutions. A natural question to ask is whether or not smooth solutions exist globally in three-dimensional MHD equations. Therefore, the main purpose of this paper is to investigate the global existence of strong solutions for the 3D compressible isothermal MHD system.

Before stating the main results, we explain the notation and conventions used throughout this paper. We denote

$$\int f(x)dx\triangleq {\int }_{{\mathbb{R}}^3}f(x)dx,$$

and the standard homogeneous and inhomogeneous Sobolev spaces as follows:

$$\{\begin{array}{c}{L}^r=L^r({\mathbb{R}}^3),D^{k,r}=\{\mathbf{u}\in L_{\mathrm{loc}}^1|{\parallel {\mathrm{\nabla }}^k\mathbf{u}\parallel }_{L^r}<\mathrm{\infty }\},{\parallel \mathbf{u}\parallel }_{D^{k,r}}={\parallel {\mathrm{\nabla }}^k\mathbf{u}\parallel }_{L^r},\hfill \\ W^{k,r}=L^r\cap D^{k,r},H^k=W^{k,2},D^k=D^{k,2},D^1=\{\mathbf{u}\in L^6|{\parallel \mathrm{\nabla }\mathbf{u}\parallel }_{L^2}<\mathrm{\infty }\},\hfill \end{array}$$

for $1< r<\infty$ and $k\in \mathbb{Z}$. The total energy is defined as

$$ m_{0}\triangleq \int \biggl(\frac{1}{2}\rho _{0} \vert \mathbf{u}_{0} \vert ^{2}+G( \rho _{0})+ \frac{1}{2} \vert \mathbf{B}_{0} \vert ^{2} \biggr) (x)\,dx, $$

where $G(\rho )$ denotes the potential energy density

$$ G(\rho )\triangleq \rho \int _{1}^{\rho}\frac{P(s)-P(1)}{s^{2}}\,ds, $$

and it is clear that $G(\rho )\sim (\rho -1)^{2}$.

We now state the definition of strong solution of (1.1)–(1.5) as follows.

Definition 1.1

A triple of functions $(\rho , \mathbf{ u}, \mathbf{ B})$ is said to be a strong solution of (1.1)–(1.5) provided that $(\rho , \mathbf{ u}, \mathbf{ B})$ satisfies equations (1.1)–(1.5) almost everywhere and belongs to the class of functions (1.8) in which the uniqueness can be shown to hold.

The main result of this paper is formulated as follows.

Theorem 1.1

For any given numbers $M>0$ (not necessary small), $\bar{\rho}\geqslant 2$ and $p\in [2, 6]$, assume that

$$ \textstyle\begin{cases} \rho _{0} \vert \mathbf{ u}_{0} \vert ^{2}+(\rho _{0}-1)^{2}+ \vert \mathbf{ B}_{0} \vert ^{2} \in L^{1}, \quad (\mathbf{ u}_{0}, \mathbf{ B}_{0})\in D^{1}\mathop{ \cap} D^{2}, \\ 0\leqslant \inf \rho _{0}\leqslant \rho _{0}(x)\leqslant \sup \rho _{0} \leqslant \bar{\rho},\quad (\rho _{0}-1)\in H^{1} \mathop{\cap} W^{1, p}, \\ \Vert \nabla \mathbf{ u}_{0} \Vert _{L^{2}}^{2}+ \Vert \nabla \mathbf{ B}_{0} \Vert _{L^{2}}^{2} \leqslant M, \end{cases} $$

(1.6)

and the compatibility condition holds

$$ -\mu \Delta \mathbf{ u}_{0}-(\lambda +\mu )\nabla \operatorname{div} \mathbf{ u}_{0}+\nabla P(\rho _{0})-( \nabla \times \mathbf{ B}_{0}) \times \mathbf{ B}_{0}= \rho _{0}^{1/2} g $$

(1.7)

for some $g\in L^{2}$. Then there exists a positive constant ε depending only on μ, ν, λ, a, M, and ρ̄ such that if

$$ m_{0}\leqslant \varepsilon , $$

then for any $0< T<\infty$, there exists a unique global strong solution $(\rho , \mathbf{ u}, \mathbf{ B})$ of problem (1.1)–(1.5) on ${\mathbb{R}}^3\times [0,T]$ satisfying

$$0⩽\rho ⩽2\overline{\rho }\mathit{\text{for}}\mathit{\text{all}}x\in {\mathbb{R}}^3,t⩾0,$$

and

$$ \textstyle\begin{cases} (\rho -1)\in C([0,T]; H^{1} \mathop{\cap} W^{1, p}),\quad \rho _{t} \in C([0, T]; L^{p}), \\ \mathbf{ u}\in C([0,T]; D^{1} \mathop{\cap} D^{2}) \mathop{\cap} L^{2}(0,T; D^{2, p}), \\ \sqrt{\rho}\mathbf{ u}_{t}\in L^{\infty}(0, T; L^{2}),\quad \mathbf{ u}_{t}\in L^{2}(0,T; D^{1}), \\ \mathbf{ B}\in C([0,T]; H^{2}) \mathop{\cap} L^{2}(0,T; H^{2}),\quad \mathbf{ B}_{t}\in L^{2}(0,T; H^{1}). \end{cases} $$

(1.8)

To obtain the strong solutions globally in time, we need global a priori estimates on smooth solutions for $(\rho , \mathbf{u}, \mathbf{B})$. The main difficulty is due to the appearance of the strong coupling between the velocity field and the magnetic field. Another difficulty is the weaker compatibility condition (1.7). Therefore, to overcome the two difficulties, we first give some known inequalities and facts in Sect. 2 and then establish the estimates of the global strong solutions that are independent of time t to problem (1.1)–(1.5) in Sect. 3. Finally, with the help of global (uniform) estimates at hand, in Sect. 4 we prove that Theorem 1.1 holds.

2 Preliminaries

In this section, we recall some known facts and elementary inequalities which will be used later. Firstly, we give the following local existence due to [17].

Proposition 2.1

Assume that the initial data $(\rho _{0}, \mathbf{ u}_{0}, \mathbf{ B}_{0})$ satisfies (1.6) and (1.7). Then there exist a small time $T_{*}>0$ and a strong solution $(\rho , \mathbf{ u}, \mathbf{ B})$ to problem (1.1)–(1.5) on ${\mathbb{R}}^3\times (0,T_{\ast }]$.

Lemma 2.1

([26])

Let $y\in W^{1,1}(0,T)$ satisfy the ODE system

$$ y'=g(y)+b'(t)\quad \textit{on } [0,T], y(0)=y_{0}, $$

where $b\in W^{1,1}(0,T)$, $g\in C(\mathbb{R})$, and $g(+\infty )=-\infty$. Assume that there are two constants $N_{0}\geqslant 0$ and $N_{1}\geqslant 0$ such that for all $0\leqslant t_{1}< t_{2}\leqslant T$,

$$ b(t_{2})-b(t_{1})\leqslant N_{0}+N_{1}(t_{2}-t_{1}). $$

(2.1)

Then

$$ y(t)\leqslant \max \bigl\{ y_{0}, \xi ^{*}\bigr\} +N_{0}< +\infty \quad \textit{on } [0,T], $$

where ${\xi }^{\ast }\in \mathbb{R}$ is a constant such that

$$ g(\xi )\leqslant -N_{1}\quad \textit{for}\ \xi \geqslant \xi ^{*}. $$

(2.2)

The following well-known Gagliardo–Nirenberg inequality can be found in [14].

Lemma 2.2

For $p\in [2, 6]$, $q\in (1, \infty )$, and $r\in (3, \infty )$, assume that $f\in H^1({\mathbb{R}}^3)$ and $g\in L^q({\mathbb{R}}^3)\cap D^{1,r}({\mathbb{R}}^3)$. Then there exists a generic constant $C>0$, depending only on q and r, such that

$$\begin{aligned}& \Vert f \Vert _{L^{p}}\leqslant C \Vert f \Vert _{L^{2}}^{\frac {6-p}{2p}} \Vert \nabla f \Vert _{L^{2}}^{ \frac {3p-6}{2p}}, \end{aligned}$$

(2.3)

$$\begin{aligned}& \Vert g \Vert _{L^{\infty}}\leqslant C \Vert g \Vert _{L^{q}}^{ \frac {q(r-3)}{3r+q(r-3)}} \Vert \nabla g \Vert _{L^{r}}^{\frac {3r}{3r+q(r-3)}}. \end{aligned}$$

(2.4)

Finally, we introduce the effective viscous flux F, the vorticity ω, and the material derivative “⋅”, which are defined as follows:

$$ F \triangleq (2\mu +\lambda )\operatorname{div} \mathbf{u}-\bigl(P(\rho )-P(1) \bigr)- \frac{1}{2} \vert \mathbf{B} \vert ^{2}, \quad \omega \triangleq \nabla \times \mathbf{u}, \dot{\mathbf{u}} \triangleq \mathbf{u}_{t}+ \mathbf{u}\cdot \nabla \dot{\mathbf{u}}, $$

then

$$ \Delta F=\operatorname{div}(\rho \dot{\mathbf{u}})-\operatorname{div} \operatorname{div}( \mathbf{B}\otimes \mathbf{B}) \quad \text{and} \quad \mu \Delta \omega =\nabla \times \bigl(\rho \dot{\mathbf{u}}-\operatorname{div}( \mathbf{B} \otimes \mathbf{B})\bigr). $$

Thus, it follows from Lemma 2.2 and the standard $L^{p}$-estimates of elliptic equations that we have the following lemma.

Lemma 2.3

([15])

Let $(\rho , \mathbf{ u}, \mathbf{ B})$ be a smooth solution of (1.1)–(1.5). Then there exists a generic constant $C>0$ such that for any $p\in [2, 6]$,

$$\begin{aligned}& \Vert \nabla F \Vert _{L^{p}}+ \Vert \nabla \omega \Vert _{L^{p}}\leqslant C \bigl( \Vert \rho \dot{\mathbf{ u}} \Vert _{L^{p}}+ \Vert \nabla \mathbf{ B}\cdot \mathbf{ B} \Vert _{L^{p}}\bigr), \end{aligned}$$

(2.5)

$$\begin{aligned}& \begin{aligned}[b] \Vert F \Vert _{L^{p}}+ \Vert \omega \Vert _{L^{p}}\leqslant{}& C\bigl( \Vert \nabla \mathbf{ u} \Vert _{L^{2}}+ \bigl\Vert P(\rho )-P(1) \bigr\Vert _{L^{2}}+ \bigl\Vert \vert \mathbf{ B} \vert ^{2} \bigr\Vert _{L^{2}}\bigr)^{(6-p)/2p} \\ &{} \times \bigl( \Vert \rho \dot{\mathbf{ u}} \Vert _{L^{2}}+ \Vert \nabla \mathbf{ B}\cdot \mathbf{ B} \Vert _{L^{2}} \bigr)^{(3p-6)/2p}, \end{aligned} \end{aligned}$$

(2.6)

$$\begin{aligned}& \Vert \nabla \mathbf{ u} \Vert _{L^{p}}\leqslant C \bigl( \Vert F \Vert _{L^{p}}+ \bigl\Vert P(\rho )-P(1) \bigr\Vert _{L^{p}}+ \bigl\Vert \vert \mathbf{ B} \vert ^{2} \bigr\Vert _{L^{p}}+ \Vert \omega \Vert _{L^{p}}\bigr). \end{aligned}$$

(2.7)

3 A priori estimates

In this section, we establish the uniform a priori estimates of solutions to problem (1.1)–(1.5) to extend the local strong solution guaranteed by Proposition 2.1. Assume that $(\rho , \mathbf{u}, \mathbf{B})$ is a smooth solution to (1.1)–(1.5) on ${\mathbb{R}}^3\times (0,T)$ for some positive time $T>0$ with smooth initial data $(\rho _{0}, \mathbf{u}_{0}, \mathbf{B}_{0})$ satisfying (1.6) and (1.7). Set $\sigma (t)\triangleq \min \{1, t\}$ and define

$$\begin{aligned}& A_{1}(T)\triangleq \sup_{0\leqslant t \leqslant T}\sigma \bigl( \Vert \nabla \mathbf{u} \Vert _{L^{2}}^{2}+ \Vert \nabla \mathbf{B} \Vert _{L^{2}}^{2} \bigr),\\& A_{2} (T)\triangleq \sup_{0\leqslant t \leqslant T} \bigl( \Vert \nabla \mathbf{u} \Vert _{L^{2}}^{2}+ \Vert \nabla \mathbf{B} \Vert _{L^{2}}^{2} \bigr). \end{aligned}$$

We have the following key a priori estimates on $(\rho , \mathbf{u}, \mathbf{B})$.

Proposition 3.1

For given constant $\bar{\rho}>0$ and M (not necessarily small), assume that $(\rho _{0}, \mathbf{ u}_{0}, \mathbf{ B}_{0})$ satisfies (1.6) and (1.7). Then there exist positive constants K and $\varepsilon _{0}$, depending only on μ, ν, λ, a, ρ̄, and M, such that if $(\rho , \mathbf{ u}, \mathbf{ B})$ is a smooth solution of (1.1)–(1.5) on ${\mathbb{R}}^3\times (0,T]$ satisfying

$$\{\begin{array}{c}0⩽\rho (x,t)⩽2\overline{\rho }\mathit{\text{for all }}(x,t)\in {\mathbb{R}}^3\times [0,T],\hfill \\ A_1(T)⩽2m_0^{1/2},A_2(T)⩽3K,\hfill \end{array}$$

(3.1)

then

$$\{\begin{array}{c}0⩽\rho (x,t)⩽\frac{7}{4}\overline{\rho }\mathit{\text{for all }}(x,t)\in {\mathbb{R}}^3\times [0,T],\hfill \\ A_1(T)⩽m_0^{1/2},A_2(T)⩽2K,\hfill \end{array}$$

(3.2)

provided $m_{0}\leqslant \varepsilon _{0}$.

Proof

The proof of Proposition 3.1 will be done by a series of lemmas below. □

Throughout this paper, we denote by C, $C_{i}$ ($i=1, 2, \ldots$) the generic positive constants that may depend on μ, ν, λ, a, ρ̄, and M, but are independent of time $T>0$. We also use $C(\alpha )$ to emphasize the dependence on α.

We first begin with the following standard energy estimates, which can be easily deduced from (1.1)–(1.5).

Lemma 3.1

Let $(\rho , \mathbf{ u}, \mathbf{ B})$ be a smooth solution of (1.1)–(1.5) on ${\mathbb{R}}^3\times [0,T]$. Then

$$ \sup_{t\in [0, T]} \int \bigl((\rho -1)^{2}+\rho \vert \mathbf{ u} \vert ^{2}+ \vert \mathbf{ B} \vert ^{2} \bigr)\,dx+ \int _{0}^{T} \bigl( \Vert \nabla \mathbf{ u} \Vert _{L^{2}}^{2}+ \Vert \nabla \mathbf{ B} \Vert _{L^{2}}^{2} \bigr)\,dt\leqslant C m_{0}. $$

(3.3)

Proof

Multiplying (1.1)₁, (1.1)₂, and (1.1)₃ by $G'(\rho )$, u, and B, respectively, integrating the resulting equations by parts over ${\mathbb{R}}^3$, and adding them together, one has

$$ \frac{d}{dt} \int \biggl(G(\rho )+\frac{1}{2}\rho \vert \mathbf{ u} \vert ^{2}+ \frac{1}{2} \vert \mathbf{ B} \vert ^{2} \biggr)\,dx+\mu \Vert \nabla \mathbf{ u} \Vert _{L^{2}}^{2}+( \mu +\lambda )\|\operatorname{div} \mathbf{ u} \|_{L^{2}}^{2}+\nu \|\nabla \mathbf{ B}\|_{L^{2}}^{2}=0, $$

which, integrated over $(0, T)$, immediately leads to (3.3). □

Lemma 3.2

Under the conditions of Proposition 3.1, one has

$$ \sup_{t\in [0, T]} \Vert \mathbf{ B} \Vert _{L^{3}}^{3}+ \int _{0}^{T} \Vert \mathbf{ B} \Vert _{L^{9}}^{3}\,dt\leqslant C \Vert \mathbf{ B}_{0} \Vert _{L^{3}}^{3}. $$

(3.4)

Proof

By virtue of (3.1) and (3.3), we infer from Lemma 2.2 ($p=6$ in (2.3))that

$$ \begin{aligned}[b] & \int _{0}^{T} \bigl( \Vert \mathbf{ u} \Vert _{L^{6}}^{4}+ \Vert \nabla \mathbf{ u} \Vert _{L^{2}}^{4}+ \Vert \nabla \mathbf{ B} \Vert _{L^{2}}^{4} \bigr)\,dt \\ &\quad \leqslant C \int _{0}^{\sigma (T)} \bigl( \Vert \nabla \mathbf{ u} \Vert _{L^{2}}^{4}+ \Vert \nabla \mathbf{ B} \Vert _{L^{2}}^{4} \bigr)\,dt+C \int _{\sigma (T)}^{T} \sigma \bigl( \Vert \nabla \mathbf{ u} \Vert _{L^{2}}^{4}+ \Vert \nabla \mathbf{ B} \Vert _{L^{2}}^{4} \bigr)\,dt \\ &\quad \leqslant C\sup_{t\in [0, \sigma (T)]} \bigl( \Vert \nabla \mathbf{ u} \Vert _{L^{2}}^{2}+ \Vert \nabla \mathbf{ B} \Vert _{L^{2}}^{2}\bigr) \int _{0}^{\sigma (T)} \bigl( \Vert \nabla \mathbf{ u} \Vert _{L^{2}}^{2}+ \Vert \nabla \mathbf{ B} \Vert _{L^{2}}^{2} \bigr)\,dt \\ &\qquad{} +C\sup_{t\in [\sigma (T), T]} \sigma \bigl( \Vert \nabla \mathbf{ u} \Vert _{L^{2}}^{2}+ \Vert \nabla \mathbf{ B} \Vert _{L^{2}}^{2}\bigr) \int _{\sigma (T)}^{T} \bigl( \Vert \nabla \mathbf{ u} \Vert _{L^{2}}^{2}+ \Vert \nabla \mathbf{ B} \Vert _{L^{2}}^{2} \bigr)\,dt \\ &\quad \leqslant Cm_{0}. \end{aligned} $$

(3.5)

Multiplying (1.1)₃ by $3|\mathbf{B}|\mathbf{B}$ and integrating by parts over ${\mathbb{R}}^3$, we obtain

$$ \begin{aligned}[b] &\frac{d}{dt} \Vert \mathbf{B} \Vert _{L^{3}}^{3}+3\nu \int \bigl( \vert \mathbf{B} \vert \vert \nabla \mathbf{B} \vert ^{2}+ \vert \mathbf{B} \vert \bigl\vert \nabla \bigl( \vert \mathbf{B} \vert \bigr) \bigr\vert ^{2}\bigr)\,dx \\ &\quad \leqslant \nu \int \vert \mathbf{B} \vert \vert \nabla \mathbf{B} \vert ^{2}\,dx+C \Vert \nabla \mathbf{u} \Vert _{L^{2}}^{2} \Vert \mathbf{B} \Vert _{L^{9/2}}^{3}, \end{aligned} $$

(3.6)

where the last term on the right-hand side in (3.6) comes from the following inequality:

$$ \begin{aligned} \int \vert \nabla \mathbf{ u} \vert \vert \mathbf{ B} \vert ^{3}\,dx & \leqslant C \Vert \nabla \mathbf{ u} \Vert _{L^{2}} \Vert \mathbf{ B} \Vert _{L^{9/2}}^{3/2} \Vert \mathbf{ B} \Vert _{L^{9}}^{3/2} \\ &\leqslant C \Vert \nabla \mathbf{ u} \Vert _{L^{2}} \Vert \mathbf{ B} \Vert _{L^{9/2}}^{3/2} \bigl\Vert \vert \mathbf{ B} \vert ^{3/2} \bigr\Vert _{L^{6}} \\ &\leqslant C \bigl( \Vert \nabla \mathbf{ u} \Vert _{L^{2}}^{2} \Vert \mathbf{ B} \Vert _{L^{9/2}}^{3} \bigr)^{1/2} \bigl\Vert \vert \nabla \mathbf{ B} \vert \vert \mathbf{ B} \vert ^{1/2} \bigr\Vert _{L^{2}}. \end{aligned} $$

To deal with the right-hand side of (3.6), we notice that

$$ \Vert \mathbf{B} \Vert _{L^{9}}^{3}\leqslant C \bigl\Vert \vert \mathbf{B} \vert ^{3/2} \bigr\Vert _{L^{6}}^{2} \leqslant C \bigl\Vert \vert \nabla \mathbf{B} \vert \vert \mathbf{B} \vert ^{1/2} \bigr\Vert _{L^{2}}^{2}, $$

then

$$ \Vert \mathbf{B} \Vert _{L^{9/2}}\leqslant C \Vert \mathbf{B} \Vert _{L^{3}}^{1/2} \Vert \mathbf{B} \Vert _{L^{9}}^{1/2}\leqslant C \Vert \mathbf{B} \Vert _{L^{3}}^{1/2} \bigl\Vert \nabla \vert \mathbf{B} \vert ^{3/2} \bigr\Vert _{L^{2}}^{1/3}, $$

which together with (3.6) yields

$$ \frac{d}{dt} \Vert \mathbf{B} \Vert _{L^{3}}^{3}+ \Vert \mathbf{B} \Vert _{L^{9}}^{3} \leqslant C \Vert \nabla \mathbf{u} \Vert _{L^{2}}^{4} \Vert \mathbf{B} \Vert _{L^{3}}^{3}, $$

which together with (3.5) and Gronwall’s inequality yields the desired estimate (3.4). □

Lemma 3.3

Under the conditions of Proposition 3.1,

$$\begin{aligned} & \bigl(\mu \beta \Vert \nabla \mathbf{ u} \Vert _{L^{2}}^{2}+(\mu + \lambda )\beta \Vert \operatorname{div}\mathbf{ u} \Vert _{L^{2}}^{2}+ \nu \beta \Vert \nabla \mathbf{ B} \Vert _{L^{2}}^{2} \bigr)_{t}+\beta \bigl\Vert \rho ^{1/2} \dot{\mathbf{ u}} \bigr\Vert _{L^{2}}^{2}+\beta \Vert \mathbf{ B}_{t} \Vert _{L^{2}}^{2}+ \beta \bigl\Vert \nabla ^{2} \mathbf{ B} \bigr\Vert _{L^{2}}^{2} \\ &\quad \leqslant \biggl(\beta \int a(\rho -1)\operatorname{div}\mathbf{ u}\,dx+ \beta \int \biggl(\mathbf{ B}\cdot \nabla \mathbf{ B}-\frac{1}{2} \nabla \vert \mathbf{ B} \vert ^{2}\biggr)\cdot \mathbf{ u}\,dx \biggr)_{t} \\ &\qquad{} +C\bigl(\beta + \bigl\vert \beta ' \bigr\vert \bigr) \bigl( \Vert \nabla \mathbf{ u} \Vert _{L^{2}}^{2}+ \Vert \nabla \mathbf{ B} \Vert _{L^{2}}^{2} \bigr) \\ &\qquad{}+C \bigl(\beta + \bigl\vert \beta ' \bigr\vert \bigr) \bigl( \Vert \nabla \mathbf{ u} \Vert _{L^{2}}^{6}+ \Vert \nabla \mathbf{ B} \Vert _{L^{2}}^{6} \bigr)+Cm_{0}, \end{aligned}$$

(3.7)

where $\beta =\beta (t)\geqslant 0$ is a piecewise smooth function.

Proof

Multiplying (1.1)₂ by $\beta (t) \dot{\mathbf{u}}$ and integrating by parts over ${\mathbb{R}}^3$, one has

$$\begin{aligned} \int \beta \rho \vert \dot{\mathbf{u}} \vert ^{2}\,dx ={}&{-} \int \beta \nabla \bigl(P(\rho )-P(1) \bigr) \cdot \dot{\mathbf{u}}\,dx+ \mu \beta \int \Delta \mathbf{u}\cdot \dot{\mathbf{u}}\,dx+\beta (\mu +\lambda ) \int \nabla \operatorname{div}\mathbf{u}\cdot \dot{\mathbf{u}} \\ &{} + \int \beta \biggl(\mathbf{B}\cdot \nabla \mathbf{B}-\frac{1}{2} \nabla \vert \mathbf{B} \vert ^{2}\biggr)\cdot \dot{\mathbf{u}}\,dx \triangleq \sum_{i=1}^{4} I_{i}. \end{aligned}$$

(3.8)

The right-hand side terms of (3.8) can be estimated as follows. First, noting that

$$ \bigl(P(\rho )-P(1) \bigr)_{t}+\mathbf{u}\cdot \nabla \bigl(P(\rho )-P(1) \bigr)+P(\rho )\operatorname{div}\mathbf{u}=0, $$

(3.9)

we obtain from (3.3), (3.9) and Cauchy–Schwarz’s inequality that

$$ \begin{aligned} I_{1} ={}& \biggl( \int \beta \bigl(P(\rho )-P(1) \bigr) \operatorname{div} \mathbf{u}\,dx \biggr)_{t}-\beta ' \int \bigl(P(\rho )-P(1) \bigr) \operatorname{div} \mathbf{u}\,dx \\ &{} - \int \beta \bigl(P(\rho )-P(1) \bigr)_{t} \operatorname{div} \mathbf{u}\,dx+ \int \beta \bigl(P(\rho )-P(1) \bigr) \partial _{i} \bigl(u^{j} \partial _{j} u^{i}\bigr)\,dx \\ \leqslant{}& \biggl( \int \beta \bigl(P(\rho )-P(1) \bigr) \operatorname{div} \mathbf{u}\,dx \biggr)_{t}+C\beta ' \Vert \nabla \mathbf{u} \Vert _{L^{2}} \bigl\Vert P(\rho )-P(1) \bigr\Vert _{L^{2}} \\ &{} + \int \beta P(1) (\operatorname{div}\mathbf{u})^{2}\,dx+\beta \int \bigl(P( \rho )-P(1) \bigr) \partial _{i} u^{i} \partial _{j} u^{j}\,dx \\ \leqslant{}& a \biggl( \int \beta (\rho -1) \operatorname{div}\mathbf{u}\,dx \biggr)_{t}+C \bigl( \beta + \bigl\vert \beta ' \bigr\vert \bigr) \Vert \nabla \mathbf{u} \Vert _{L^{2}}^{2}+C \vert \beta \vert 'm_{0}. \end{aligned} $$

By virtue of Cauchy-Schwarz’s inequality, one has

$$ \begin{aligned} I_{2} &=-\mu \int \beta \partial _{i} u^{j} \partial _{i} \bigl(u_{t}^{j}+u^{k} \partial _{k} u^{j} \bigr)\,dx \\ &=-\frac{\mu}{2} \biggl( \int \beta \vert \nabla \mathbf{u} \vert ^{2}\,dx \biggr)_{t}+ \frac{\mu}{2}\beta ' \Vert \nabla \mathbf{u} \Vert _{L^{2}}^{2}-\mu \beta \int \biggl(\partial _{i} u^{j} \partial _{i} u^{k} \partial _{k} u^{j}- \frac{1}{2} \vert \nabla \mathbf{u} \vert ^{2} \operatorname{div}\mathbf{u} \biggr)\,dx \\ &\leqslant -\frac{\mu}{2} \biggl( \int \beta \vert \nabla \mathbf{u} \vert ^{2}\,dx \biggr)_{t}+C \bigl\vert \beta ' \bigr\vert \Vert \nabla \mathbf{u} \Vert _{L^{2}}^{2}+C_{1} \beta \Vert \nabla \mathbf{u} \Vert _{L^{3}}^{3}. \end{aligned} $$

Similarly,

$$ I_{3}\leqslant -\frac{\mu +\lambda}{2} \biggl( \int \beta \vert \operatorname{div} \mathbf{u} \vert ^{2}\,dx \biggr)_{t}+C \bigl\vert \beta ' \bigr\vert \Vert \nabla \mathbf{u} \Vert _{L^{2}}^{2}+C_{1} \beta \Vert \nabla \mathbf{u} \Vert _{L^{3}}^{3}. $$

For $I_{4}$, it follows from (2.3) and (2.4) that

$$ \begin{aligned} I_{4} &= \biggl( \int \beta \biggl(\mathbf{B}\cdot \nabla \mathbf{B}-\frac{1}{2} \nabla \vert \mathbf{B} \vert ^{2}\biggr) \cdot \mathbf{u}\,dx \biggr)_{t}-\beta ' \int \biggl(\mathbf{B}\cdot \nabla \mathbf{B}-\frac{1}{2} \nabla \vert \mathbf{B} \vert ^{2}\biggr) \cdot \mathbf{u}\,dx \\ &\quad -\beta \int \biggl(\mathbf{B}\cdot \nabla \mathbf{B}-\frac{1}{2} \nabla \vert \mathbf{B} \vert ^{2}\biggr)_{t} \cdot \mathbf{u}\,dx+\beta \int \biggl( \mathbf{B}\cdot \nabla \mathbf{B}-\frac{1}{2} \nabla \vert \mathbf{B} \vert ^{2}\biggr) \cdot \mathbf{u}\cdot \nabla \mathbf{u}\,dx \\ &\leqslant \biggl( \int \beta \biggl(\mathbf{B}\cdot \nabla \mathbf{B}- \frac{1}{2}\nabla \vert \mathbf{B} \vert ^{2}\biggr) \cdot \mathbf{u}\,dx \biggr)_{t}+C \beta ' \Vert \mathbf{B} \Vert _{L^{2}}^{1/2} \Vert \nabla \mathbf{B} \Vert _{L^{2}}^{3/2} \Vert \nabla \mathbf{u} \Vert _{L^{2}} \\ &\quad +C\beta \Vert \nabla \mathbf{B} \Vert _{L^{2}}^{1/2} \bigl\Vert \nabla ^{2} \mathbf{B} \bigr\Vert _{L^{2}}^{1/2} \Vert \mathbf{B}_{t} \Vert _{L^{2}} \Vert \nabla \mathbf{u} \Vert _{L^{2}}+C\beta \Vert \nabla \mathbf{B} \Vert _{L^{2}} \bigl\Vert \nabla ^{2} \mathbf{B} \bigr\Vert _{L^{2}} \Vert \nabla \mathbf{u} \Vert _{L^{2}}^{2} \\ &\leqslant \biggl( \int \beta \biggl(\mathbf{B}\cdot \nabla \mathbf{B}- \frac{1}{2}\nabla \vert \mathbf{B} \vert ^{2}\biggr) \cdot \mathbf{u}\,dx \biggr)_{t}+ \frac{\beta}{4} \Vert \mathbf{B}_{t} \Vert _{L^{2}}^{2}+ \frac{\nu \beta}{8} \bigl\Vert \nabla ^{2} \mathbf{B} \bigr\Vert _{L^{2}}^{2} \\ &\quad +C\bigl(\beta + \bigl\vert \beta ' \bigr\vert \bigr) \Vert \nabla \mathbf{u} \Vert _{L^{2}}^{6}+C\bigl( \beta + \bigl\vert \beta ' \bigr\vert \bigr) \Vert \nabla \mathbf{B} \Vert _{L^{2}}^{6}+C \bigl\vert \beta ' \bigr\vert \Vert \nabla \mathbf{B} \Vert _{L^{2}}^{2}+C \bigl\vert \beta ' \bigr\vert m_{0}^{6}. \end{aligned} $$

On the other hand, it follows from (2.7) and Cauchy-Schwarz’s inequality that

$$\begin{aligned} \Vert \nabla \mathbf{u} \Vert _{L^{3}}^{3} \leqslant{}& \bigl( \bigl\Vert \rho ^{1/2} \dot{\mathbf{u}} \bigr\Vert _{L^{2}}+ \Vert \mathbf{B}\cdot \nabla \mathbf{B} \Vert _{L^{2}} \bigr)^{3/2} \bigl( \Vert \nabla \mathbf{u} \Vert _{L^{2}}+ \bigl\Vert P( \rho )-P(1) \bigr\Vert _{L^{2}}+ \bigl\Vert \vert \mathbf{B} \vert ^{2} \bigr\Vert _{L^{2}} \bigr)^{3/2} \\ &{} + \bigl\Vert P(\rho )-P(1) \bigr\Vert _{L^{3}}^{3}+ \bigl\Vert \vert \mathbf{B} \vert ^{2} \bigr\Vert _{L^{3}}^{3} \\ \leqslant{}& \frac{1}{2C_{1}} \bigl\Vert \rho ^{1/2} \dot{\mathbf{u}} \bigr\Vert _{L^{2}}^{2}+ \frac{\nu}{8C_{1}} \bigl\Vert \nabla ^{2} \mathbf{B} \bigr\Vert _{L^{2}}^{2}+C \bigl( \Vert \nabla \mathbf{u} \Vert _{L^{2}}^{6}+ \Vert \nabla \mathbf{B} \Vert _{L^{2}}^{6} \bigr)+Cm_{0}. \end{aligned}$$

(3.10)

Substituting $I_{1}$, $I_{1}$, $I_{3}$, $I_{4}$ into (3.8) and using (3.10), we immediately obtain

$$ \begin{aligned}[b] & \bigl(\mu \beta \Vert \nabla \mathbf{ u} \Vert _{L^{2}}^{2}+(\mu + \lambda )\beta \Vert \operatorname{div}\mathbf{ u} \Vert _{L^{2}}^{2} \bigr)_{t}+\beta \bigl\Vert \rho ^{1/2}\dot{\mathbf{ u}} \bigr\Vert _{L^{2}}^{2} \\ &\quad \leqslant \biggl(\beta \int a(\rho -1)\operatorname{div}\mathbf{ u}\,dx+ \beta \int \biggl(\mathbf{ B}\cdot \nabla \mathbf{ B}-\frac{1}{2} \nabla \vert \mathbf{ B} \vert ^{2}\biggr)\cdot \mathbf{ u}\,dx \biggr)_{t}+Cm_{0} \\ &\qquad{} +C\bigl(\beta + \bigl\vert \beta ' \bigr\vert \bigr) \bigl( \Vert \nabla \mathbf{ u} \Vert _{L^{2}}^{2}+ \Vert \nabla \mathbf{ B} \Vert _{L^{2}}^{2} \bigr)+C \bigl(\beta + \bigl\vert \beta ' \bigr\vert \bigr) \bigl( \Vert \nabla \mathbf{ u} \Vert _{L^{2}}^{6}+ \Vert \nabla \mathbf{ B} \Vert _{L^{2}}^{6} \bigr). \end{aligned} $$

(3.11)

We infer from (1.1)₃ and (2.3) that

$$ \begin{aligned} & \bigl(\nu \beta \Vert \nabla \mathbf{B} \Vert _{L^{2}}^{2} \bigr)_{t}+ \nu ^{2} \beta \bigl\Vert \nabla ^{2} \mathbf{B} \bigr\Vert _{L^{2}}^{2}+\beta \Vert \mathbf{B}_{t} \Vert _{L^{2}}^{2} = \beta \int (\mathbf{B}_{t}-\nu \Delta \mathbf{B})^{2}\,dx-\nu \beta ' \Vert \nabla \mathbf{B} \Vert _{L^{2}}^{2} \\ &\quad =\beta \int \vert \mathbf{B}\cdot \nabla \mathbf{u}-\mathbf{u} \cdot \nabla \mathbf{B}-\mathbf{B}\operatorname{div}\mathbf{u} \vert ^{2}\,dx -\nu \beta ' \Vert \nabla \mathbf{B} \Vert _{L^{2}}^{2} \\ &\quad \leqslant C\beta \Vert \nabla \mathbf{B} \Vert _{L^{2}}^{2}+C \beta \Vert \mathbf{u} \Vert _{L^{2}}^{2} \Vert \nabla \mathbf{B} \Vert _{L^{3}}^{2} \\ &\quad \leqslant \frac{\beta \nu }{2} \bigl\Vert \nabla ^{2} \mathbf{B} \bigr\Vert _{L^{2}}^{2}+C \bigl\vert \beta ' \bigr\vert \Vert \nabla \mathbf{B} \Vert _{L^{2}}^{2}+C\beta \bigl( \Vert \nabla \mathbf{ u} \Vert _{L^{2}}^{6}+ \Vert \nabla \mathbf{ B} \Vert _{L^{2}}^{6} \bigr), \end{aligned} $$

thus

$$ \bigl(\nu \beta \Vert \nabla \mathbf{B} \Vert _{L^{2}}^{2} \bigr)_{t}+\nu ^{2} \beta \bigl\Vert \nabla ^{2} \mathbf{B} \bigr\Vert _{L^{2}}^{2}+\beta \Vert \mathbf{B}_{t} \Vert _{L^{2}}^{2} \leqslant C \bigl\vert \beta ' \bigr\vert \Vert \nabla \mathbf{B} \Vert _{L^{2}}^{2}+C\beta \bigl( \Vert \nabla \mathbf{ u} \Vert _{L^{2}}^{6}+ \Vert \nabla \mathbf{ B} \Vert _{L^{2}}^{6} \bigr), $$

which, together with (3.11), yields (3.7). □

Lemma 3.4

Under the conditions of Proposition 3.1, there exist positive constants $K\geqslant M+1$ and $\varepsilon _{1}<1$, depending only on μ, ν, λ, a, ρ̄, and M, such that

$$ A_{2}\bigl(\sigma (T)\bigr)+ \int _{0}^{\sigma (T)} \bigl( \bigl\Vert \rho ^{1/2} \dot{\mathbf{ u}} \bigr\Vert _{L^{2}}^{2}+ \Vert \mathbf{ B}_{t} \Vert _{L^{2}}^{2}+ \bigl\Vert \nabla ^{2} \mathbf{ B} \bigr\Vert _{L^{2}}^{2} \bigr)\,dt\leqslant 2K, $$

(3.12)

provided $m_{0}\leqslant \varepsilon _{1}$.

Proof

Taking $\beta =1$ in (3.7) and integrating it over $(0, \sigma (T))$, we deduce from (1.6), (1.7), (3.1), (3.3), (2.7), (3.3), and (3.4) that

$$ \begin{aligned} &A_{2}\bigl(\sigma (T)\bigr) + \int _{0}^{\sigma (T)} \bigl( \bigl\Vert \rho ^{1/2} \dot{\mathbf{u}} \bigr\Vert _{L^{2}}^{2}+ \Vert \mathbf{B}_{t} \Vert _{L^{2}}^{2}+ \bigl\Vert \nabla ^{2} \mathbf{B} \bigr\Vert _{L^{2}}^{2} \bigr)\,dt \\ &\quad \leqslant C(M+1)+a \int (\rho -1)\operatorname{div}\mathbf{u}\,dx|_{0}^{ \sigma (T)}+ \int \biggl(\mathbf{B}\cdot \nabla \mathbf{B}-\frac{1}{2} \nabla \vert \mathbf{B} \vert ^{2}\biggr)\,dx|_{0}^{\sigma (T)} \\ &\qquad{} +C \int _{0}^{\sigma (T)} \bigl( \Vert \nabla \mathbf{u} \Vert _{L^{2}}^{6}+ \Vert \nabla \mathbf{B} \Vert _{L^{2}}^{6} \bigr)\,dt \\ &\quad \leqslant \frac{1}{2}A_{2}\bigl(\sigma (T) \bigr)+Cm_{0}^{1/2}A_{2}^{3/2} \bigl( \sigma (T)\bigr)+C(M+1)^{2}, \end{aligned} $$

thus

$$ \begin{aligned} &A_{2}\bigl(\sigma (T)\bigr) + \int _{0}^{\sigma (T)} \bigl( \bigl\Vert \rho ^{1/2}\dot{\mathbf{u}} \bigr\Vert _{L^{2}}^{2}+ \Vert \mathbf{B}_{t} \Vert _{L^{2}}^{2}+ \bigl\Vert \nabla ^{2} \mathbf{B} \bigr\Vert _{L^{2}}^{2} \bigr)\,dt \\ &\quad \leqslant C_{2}m_{0}^{1/2}K^{3/2}+C_{3}(M+1)^{2} \leqslant 2K, \end{aligned} $$

provided

$$ m_{0}\leqslant \varepsilon _{1}\triangleq \min \biggl\{ 1, \frac {K}{C_{2}^{2}} \biggr\} \quad \text{with } K \triangleq C_{3}(M+1)^{2}. $$

Thus, we immediately obtain the desired estimate (3.12). The proof of Lemma 3.4 is therefore completed. □

Lemma 3.5

Under the conditions of Proposition 3.1, there exists a positive constant $\varepsilon _{2}$, depending only on μ, ν, λ, a, ρ̄, and M, such that

$$ A_{1}(T)+ \int _{i-1}^{i+1}\sigma _{i} \bigl( \bigl\Vert \rho ^{1/2} \dot{\mathbf{ u}} \bigr\Vert _{L^{2}}^{2}+ \Vert \mathbf{ B}_{t} \Vert _{L^{2}}^{2}+ \bigl\Vert \nabla ^{2} \mathbf{ B} \bigr\Vert _{L^{2}}^{2} \bigr)\,dt\leqslant m_{0}^{1/2}, $$

(3.13)

provided $m_{0}\leqslant \varepsilon _{2}$, where $\sigma _{i} \triangleq \sigma (t+1-i)$ and i is an integer satisfying $1\leqslant i \leqslant [T]-1$ with $[T]$ denoting the largest integer less than or equal to T.

Proof

Without loss of generality, assume that $T\geqslant 2$. Otherwise, things can be done by choosing a suitable small step size. For integer $i(1\leqslant i \leqslant [T]-1)$, taking $\beta =\sigma _{i}(t)$ in (3.7) and integrating the results over $(i-1, i+1]$, one deduces from (3.1), (3.3), and (3.5) that

$$ \begin{aligned} &\sup_{t\in [i-1, i+1]} \bigl(\sigma _{i} \Vert \nabla \mathbf{u} \Vert _{L^{2}}^{2}+ \sigma _{i} \Vert \nabla \mathbf{B} \Vert _{L^{2}}^{2} \bigr)+ \int _{i-1}^{i+1}\sigma _{i} \bigl( \bigl\Vert \rho ^{1/2}\dot{\mathbf{u}} \bigr\Vert _{L^{2}}^{2}+ \Vert \mathbf{B}_{t} \Vert _{L^{2}}^{2}+ \bigl\Vert \nabla ^{2} \mathbf{B} \bigr\Vert _{L^{2}}^{2} \bigr)\,dt \\ &\quad \leqslant Cm_{0}+\sigma _{i} \int \bigl(P(\rho )-P(1)\bigr)\operatorname{div} \mathbf{u}\,dx +\sigma _{i} \int \biggl(\mathbf{B}\cdot \nabla \mathbf{B}- \frac{1}{2} \nabla \vert \mathbf{B} \vert ^{2}\biggr)\cdot \mathbf{u}\,dx \\ &\qquad{} +C \int _{i-1}^{i+1}\sigma _{i} \bigl( \Vert \nabla \mathbf{u} \Vert _{L^{2}}^{6}+ \Vert \nabla \mathbf{B} \Vert _{L^{2}}^{6} \bigr)\,dt+C \int _{i-1}^{i+1}\sigma _{i} \bigl( \Vert \nabla \mathbf{u} \Vert _{L^{2}}^{2}+ \Vert \nabla \mathbf{B} \Vert _{L^{2}}^{2} \bigr)\,dt \\ &\quad \leqslant \frac{\sigma _{i}}{2} \Vert \nabla \mathbf{u} \Vert _{L^{2}}^{2}+C_{4} m_{0}^{1/2}M^{1/2} \bigl(\sigma _{i} \Vert \nabla \mathbf{B} \Vert _{L^{2}}^{2} \bigr)+Cm_{0}, \end{aligned} $$

thus

$$ \begin{aligned} &\sup_{t\in [i-1, i+1]} \bigl(\sigma _{i} \Vert \nabla \mathbf{u} \Vert _{L^{2}}^{2}+ \sigma _{i} \Vert \nabla \mathbf{B} \Vert _{L^{2}}^{2} \bigr)+ \int _{i-1}^{i+1}\sigma _{i} \bigl( \bigl\Vert \rho ^{1/2}\dot{\mathbf{u}} \bigr\Vert _{L^{2}}^{2}+ \Vert \mathbf{B}_{t} \Vert _{L^{2}}^{2}+ \bigl\Vert \nabla ^{2} \mathbf{B} \bigr\Vert _{L^{2}}^{2} \bigr)\,dt \\ &\quad \leqslant C_{5}m_{0}\leqslant Cm_{0}^{1/2}, \end{aligned} $$

provided

$$ m_{0}\leqslant \varepsilon _{2} \triangleq \min \biggl\{ 1, \frac{1}{2C_{4}^{2}M}, \frac{1}{C_{5}^{2}} \biggr\} . $$

 □

Lemma 3.6

Under the conditions of Proposition 3.1, it holds that

$$ \begin{aligned}[b] & \bigl( \xi \bigl\Vert \rho ^{1/2} \dot{\mathbf{ u}} \bigr\Vert _{L^{2}}^{2}+ \xi \Vert \mathbf{ B}_{t} \Vert _{L^{2}}^{2} \bigr)_{t}+\xi \bigl( \Vert \nabla \dot{\mathbf{ u}} \Vert _{L^{2}}^{2}+ \Vert \nabla \mathbf{ B}_{t} \Vert _{L^{2}}^{2} \bigr) \\ &\quad \leqslant \bigl\vert \xi ' \bigr\vert \bigl( \bigl\Vert \rho ^{1/2}\dot{\mathbf{ u}} \bigr\Vert _{L^{2}}^{2}+ \Vert \mathbf{ B}_{t} \Vert _{L^{2}}^{2} \bigr)+C\xi \bigl( \Vert \nabla \mathbf{ u} \Vert _{L^{2}}^{4}+ \Vert \nabla \mathbf{ B} \Vert _{L^{2}}^{4} \bigr) \bigl( \Vert \mathbf{ B}_{t} \Vert _{L^{2}}^{2}+ \bigl\Vert \nabla ^{2} \mathbf{ B} \bigr\Vert _{L^{2}}^{2} \bigr) \\ &\qquad{} +C\xi \Vert \nabla \mathbf{ u} \Vert _{L^{2}}^{2}+C \xi \Vert \nabla \mathbf{ u} \Vert _{L^{2}}^{4} \Vert \nabla \mathbf{ B} \Vert _{L^{2}}^{3/2} \bigl\Vert \nabla ^{2} \mathbf{B} \bigr\Vert _{L^{2}}^{3/2}+C\xi m_{0} \Vert \nabla \mathbf{ B} \Vert _{L^{2}}^{3} \bigl\Vert \nabla ^{2} \mathbf{B} \bigr\Vert _{L^{2}}^{3} \\ &\qquad{} +C\xi \bigl( \Vert \nabla \mathbf{ u} \Vert _{L^{2}}+m_{0}^{1/2}+m_{0}^{1/4} \Vert \nabla \mathbf{ B} \Vert _{L^{2}}^{3/2} \bigr) \bigl( \bigl\Vert \rho ^{1/2} \dot{\mathbf{ u}} \bigr\Vert _{L^{2}}^{3}+ \bigl\Vert \nabla ^{2} \mathbf{B} \bigr\Vert _{L^{2}}^{3} \bigr)+C\xi m_{0}, \end{aligned} $$

(3.14)

where $\xi =\xi (t)\geqslant 0$ is a piecewise smooth function.

Proof

Operating $\xi \dot{u}^{j}[\partial _{t}+\operatorname{div}(\mathbf{u}\cdot )]$ to $\text{(1.1)}_{2}^{j}$, summing with respect to j, and integrating the resulting equation over ${\mathbb{R}}^3$, one has after integration by parts

$$ \begin{aligned}[b] & \biggl(\frac{\xi}{2} \int \rho \vert \dot{\mathbf{u}} \vert ^{2}\,dx \biggr)_{t}-\frac{\xi '}{2} \int \rho \vert \dot{\mathbf{u}} \vert ^{2}\,dx \\ &\quad =-\xi \int \dot{u}^{j}\bigl[\partial _{j}P_{t}+ \operatorname{div}(\mathbf{u} \partial _{j}P)\bigr]\,dx+\xi \mu \int \dot{u}^{j}\bigl[\Delta {u}^{j}_{t}+ \operatorname{div}\bigl( \mathbf{u}\Delta u^{j}\bigr)\bigr]\,dx \\ &\qquad{} +\xi (\lambda +\mu ) \int \dot{u}^{j}\bigl[\partial _{j}\partial _{t}(\operatorname{div}\mathbf{u})+\operatorname{div}\bigl(\mathbf{u} \partial _{j}(\operatorname{div} \mathbf{u})\bigr)\bigr]\,dx \\ &\qquad{} +\xi \int \dot{u}^{j}\bigl[\partial _{t}\bigl( \mathbf{B}\cdot \nabla B^{j}\bigr)+\operatorname{div}\bigl(\mathbf{u} \mathbf{B}\cdot \nabla B^{j}\bigr)\bigr]\,dx \\ &\qquad {}-\frac{\xi}{2} \int \dot{u}^{j}\bigl[\partial _{t}\partial _{j}\bigl( \vert \mathbf{B} \vert ^{2}\bigr)+ \operatorname{div}\bigl(\mathbf{u}\partial _{j}\bigl( \vert \mathbf{B} \vert ^{2}\bigr)\bigr)\bigr]\,dx \triangleq \sum _{i=1}^{5}J_{i}, \end{aligned} $$

(3.15)

where the first term on the right-hand side of (3.15) can be estimated as follows. Based on integrating by parts and (3.1), we obtain

$$ \begin{aligned} J_{1} &=\xi \int \bigl(\partial _{j} \dot{u}^{j} P(\rho )_{t}+ \partial _{k} \dot{u}^{j} u^{k}\partial _{j}P(\rho )\bigr)\,dx \\ &=\xi \int \bigl(-a \operatorname{div}\mathbf{u}\partial _{j} \dot{u}^{j}-a\operatorname{div}\bigl((\rho -1)\mathbf{u}\bigr)\partial _{j}\dot{u}^{j} -\partial _{j}\bigl( \partial _{k}\dot{u}^{j} u^{k}\bigr) \bigl(P( \rho )-P(1)\bigr) \bigr)\,dx \\ &=\xi \int \bigl(-a \operatorname{div}\mathbf{u}\partial _{j} \dot{u}^{j}- \partial _{k}\dot{u}^{j} \partial _{j} u^{k}\bigl(P(\rho )-P(1)\bigr) \bigr)\,dx \\ &\leqslant \frac{\mu}{8}\xi \Vert \nabla \dot{\mathbf{u}} \Vert _{L^{2}}^{2}+C( \bar{\rho})\xi ( \Vert \nabla \mathbf{u} \Vert _{L^{2}}^{2}. \end{aligned} $$

Similarly,

$$ J_{2}=\mu \xi \int \dot{u}^{j}\bigl[\Delta {u}^{j}_{t}+ \operatorname{div}\bigl( \mathbf{u}\Delta u^{j}\bigr)\bigr]\,dx \leqslant -\frac{3\mu}{4}\xi \Vert \nabla \dot{\mathbf{u}} \Vert _{L^{2}}^{2}+C\xi \Vert \nabla \mathbf{u} \Vert _{L^{4}}^{4} $$

and

$$ J_{3}\leqslant -\frac{\lambda +\mu}{2}\xi \Vert \operatorname{div} \dot{\mathbf{u}} \Vert _{L^{2}}^{2}+\frac{\mu}{4} \xi \Vert \nabla \dot{\mathbf{u}} \Vert _{L^{2}}^{2}+C \xi \Vert \nabla \mathbf{u} \Vert _{L^{4}}^{4}. $$

Keeping in mind that $\operatorname{div} \mathbf{B}=0$ and integrating by parts over ${\mathbb{R}}^3$, one has

$$ \begin{aligned} J_{4} &= \xi \int \bigl(\dot{u}^{j}\bigl(B_{t}^{i} \partial _{i}B^{j}+B^{i} \partial _{i}B_{t}^{j}\bigr)-\partial _{k}\dot{u}^{j}u^{k}B^{i} \partial _{i}B^{j} \bigr)\,dx \\ &\leqslant -\xi \int \bigl(B^{j}\partial _{i} \dot{u}^{j}B_{t}^{i}+B_{t}^{j} \partial _{i}\dot{u}^{j}B^{i}\,dx+ \partial _{k}\dot{u}^{j}u^{k}B^{i} \partial _{i}B^{j} \bigr)\,dx \\ &\leqslant \frac{\mu}{8}\xi \Vert \nabla \dot{\mathbf{u}} \Vert _{L^{2}}^{2}+C \xi \Vert \mathbf{B} \Vert _{L^{3}}^{2} \Vert \nabla \mathbf{B}_{t} \Vert _{L^{2}}^{2}+ C \xi \Vert \nabla \mathbf{B} \Vert _{L^{2}}^{2} \bigl\Vert \nabla ^{2}\mathbf{B} \bigr\Vert _{L^{2}}^{2} \Vert \nabla \mathbf{u} \Vert _{L^{2}}^{2} \\ &\leqslant \frac{\mu}{8}\xi \Vert \nabla \dot{\mathbf{u}} \Vert _{L^{2}}^{2}+C \xi m_{0}^{1/2} \Vert \nabla \mathbf{B}_{t} \Vert _{L^{2}}^{2}+ C\xi \bigl( \Vert \nabla \mathbf{B} \Vert _{L^{2}}^{4}+ \Vert \nabla \mathbf{u} \Vert _{L^{2}}^{4} \bigr) \bigl\Vert \nabla ^{2} \mathbf{B} \bigr\Vert _{L^{2}}^{2}. \end{aligned} $$

Similarly,

$$ J_{5}\leqslant \frac{\mu}{8}\xi \Vert \nabla \dot{ \mathbf{u}} \Vert _{L^{2}}^{2}+C \xi m_{0}^{1/2} \Vert \nabla \mathbf{B}_{t} \Vert _{L^{2}}^{2}+ C\xi \bigl( \Vert \nabla \mathbf{B} \Vert _{L^{2}}^{4}+ \Vert \nabla \mathbf{u} \Vert _{L^{2}}^{4} \bigr) \bigl\Vert \nabla ^{2} \mathbf{B} \bigr\Vert _{L^{2}}^{2}. $$

It follows from (2.6), (2.7), (3.4) and Cauchy–Schwarz’s inequality that

$$ \begin{aligned}[b] \Vert \nabla \mathbf{u} \Vert _{L^{4}}^{4} \leqslant{}& \bigl( \bigl\Vert \rho ^{1/2} \dot{\mathbf{u}} \bigr\Vert _{L^{2}}^{3}+ \Vert \mathbf{B}\cdot \nabla \mathbf{B} \Vert _{L^{2}}^{3} \bigr) \bigl( \Vert \nabla \mathbf{u} \Vert _{L^{2}}+ \bigl\Vert P( \rho )-P(1) \bigr\Vert _{L^{2}}+ \bigl\Vert \vert \mathbf{B} \vert ^{2} \bigr\Vert _{L^{2}} \bigr) \\ &{} + \bigl\Vert P(\rho )-P(1) \bigr\Vert _{L^{4}}^{4}+ \bigl\Vert \vert \mathbf{B} \vert ^{2} \bigr\Vert _{L^{4}}^{4} \\ \leqslant{}& \bigl( \Vert \nabla \mathbf{u} \Vert _{L^{2}}+m_{0}^{1/2}+ \Vert \mathbf{B} \Vert _{L^{2}}^{1/2} \Vert \nabla \mathbf{B} \Vert _{L^{2}}^{3/2} \bigr) \bigl( \bigl\Vert \rho ^{1/2} \dot{\mathbf{u}} \bigr\Vert _{L^{2}}^{3}+ \Vert \mathbf{B} \Vert _{L^{3}}^{3} \bigl\Vert \nabla ^{2} \mathbf{B} \bigr\Vert _{L^{2}}^{3} \bigr) \\ &\quad +C(\bar{\rho})m_{0}+ \Vert \mathbf{B} \Vert _{L^{2}}^{2} \Vert \mathbf{B} \Vert _{L^{ \infty}}^{6} \\ \leqslant{}& \bigl( \Vert \nabla \mathbf{u} \Vert _{L^{2}}+m_{0}^{1/2}+m_{0}^{1/4} \Vert \nabla \mathbf{B} \Vert _{L^{2}}^{3/2} \bigr) \bigl( \bigl\Vert \rho ^{1/2} \dot{\mathbf{u}} \bigr\Vert _{L^{2}}^{3}+ \bigl\Vert \nabla ^{2} \mathbf{B} \bigr\Vert _{L^{2}}^{3} \bigr) \\ &{} +C(\bar{\rho})m_{0}+ \Vert \nabla \mathbf{B} \Vert _{L^{2}}^{3} \bigl\Vert \nabla ^{2}\mathbf{B} \bigr\Vert _{L^{2}}^{3}. \end{aligned} $$

(3.16)

Substituting $J_{1}, J_{2}, \ldots , J_{5}$ into (3.15) and using (3.16), we obtain

$$ \begin{aligned}[b] & \bigl( \xi \bigl\Vert \rho ^{1/2} \dot{\mathbf{ u}} \bigr\Vert _{L^{2}}^{2} \bigr)_{t}+\xi \Vert \nabla \dot{\mathbf{ u}} \Vert _{L^{2}}^{2} \\ &\quad \leqslant \frac{1}{2}\xi \Vert \nabla \mathbf{B}_{t} \Vert _{L^{2}}^{2}+ \bigl\vert \xi ' \bigr\vert \bigl\Vert \rho ^{1/2}\dot{\mathbf{ u}} \bigr\Vert _{L^{2}}^{2}+C\xi \bigl( \Vert \nabla \mathbf{ u} \Vert _{L^{2}}^{4}+ \Vert \nabla \mathbf{ B} \Vert _{L^{2}}^{4} \bigr) \bigl\Vert \nabla ^{2} \mathbf{ B} \bigr\Vert _{L^{2}}^{2} \\ &\qquad{} +C\xi \Vert \nabla \mathbf{ u} \Vert _{L^{2}}^{2}+C \xi m_{0} \Vert \nabla \mathbf{ B} \Vert _{L^{2}}^{3} \bigl\Vert \nabla ^{2} \mathbf{B} \bigr\Vert _{L^{2}}^{3}+C \xi m_{0} \\ &\qquad{} +C\xi \bigl( \Vert \nabla \mathbf{ u} \Vert _{L^{2}}+m_{0}^{1/2}+m_{0}^{1/4} \Vert \nabla \mathbf{ B} \Vert _{L^{2}}^{3/2} \bigr) \bigl( \bigl\Vert \rho ^{1/2} \dot{\mathbf{ u}} \bigr\Vert _{L^{2}}^{3}+ \bigl\Vert \nabla ^{2} \mathbf{B} \bigr\Vert _{L^{2}}^{3} \bigr). \end{aligned} $$

(3.17)

On the other hand, it follows from (1.1)₃ that

$$ \mathbf{B}_{tt}-\nu \Delta \mathbf{B}_{t}= (\mathbf{B}\cdot \nabla \mathbf{u}-\mathbf{u}\cdot \nabla \mathbf{B}-\mathbf{B} \operatorname{div} \mathbf{u})_{t}. $$

(3.18)

Multiplying (3.18) by $\xi \mathbf{B}_{t}$ and integrating over ${\mathbb{R}}^3$ yield

$$ \begin{aligned}[b] & \biggl(\frac{\xi}{2} \Vert \mathbf{B}_{t} \Vert _{L^{2}}^{2} \biggr)_{t}-\frac{\xi '}{2} \Vert \mathbf{B}_{t} \Vert _{L^{2}}^{2} +\nu \xi \Vert \nabla \mathbf{B}_{t} \Vert _{L^{2}}^{2} \\ &\quad = \xi \int ( \mathbf{B}_{t} \cdot \nabla \mathbf{u}-\mathbf{u} \cdot \nabla \mathbf{B}_{t}-\mathbf{B}_{t} \cdot \operatorname{div}\mathbf{u}) \cdot \mathbf{B}_{t}\,dx \\ &\qquad{} +\xi \int \bigl(-\mathbf{B}\cdot \nabla (\mathbf{u}\cdot \nabla \mathbf{u})+( \mathbf{u}\cdot \nabla \mathbf{u})\cdot \nabla \mathbf{B}+\mathbf{B} \operatorname{div}(\mathbf{u}\cdot \nabla \mathbf{u}) \bigr) \cdot \mathbf{B}_{t}\,dx \\ &\qquad{} +\xi \int ( \mathbf{B}\cdot \nabla \dot{\mathbf{u}}- \dot{\mathbf{u}}\cdot \nabla \mathbf{B}-\mathbf{B}\cdot \operatorname{div} \dot{\mathbf{u}} )\cdot \mathbf{B}_{t}\,dx\triangleq \sum_{i=1}^{3}N_{i}. \end{aligned} $$

(3.19)

Now, we estimate $N_{i}$ ($i=1, 2, 3$) as follows. By using (2.1), (2.2) and integrating by parts, we have

$$ N_{1}\leqslant C\xi \Vert \mathbf{B}_{t} \Vert _{L^{3}} \Vert \nabla \mathbf{u} \Vert _{L^{2}} \Vert \nabla \mathbf{B}_{t} \Vert _{L^{2}}\leqslant \frac{\nu}{6}\xi \Vert \nabla \mathbf{B}_{t} \Vert _{L^{2}}^{2}+C \Vert \mathbf{B}_{t} \Vert _{L^{2}}^{2} \Vert \nabla \mathbf{u} \Vert _{L^{2}}^{4}. $$

Due to (2.3), (2.4) and Cauchy–Schwarz’s inequality, we have

$$ \begin{aligned} N_{2} &=\xi \int \bigl( u^{k} \partial _{k} u^{j} B^{i} \partial _{i}B_{t}^{j} + u^{k} \partial _{k} u^{i} \partial _{i}B^{j} B_{t}^{j}-u^{k} \partial _{k} u^{i} \partial _{i}B^{j} B_{t}^{j} -u^{k} \partial _{k} u^{i} B^{j}\partial _{i}B_{t}^{j} \bigr)\,dx \\ &\leqslant C\xi \Vert \mathbf{B} \Vert _{L^{\infty}} \Vert \nabla \mathbf{u} \Vert _{L^{2}} \Vert \nabla \mathbf{u} \Vert _{L^{3}} \Vert \nabla \mathbf{B}_{t} \Vert _{L^{2}}+C\xi \Vert \nabla \mathbf{B} \Vert _{L^{3}} \Vert \nabla \mathbf{u} \Vert _{L^{2}} \Vert \nabla \mathbf{u} \Vert _{L^{3}} \Vert \mathbf{B}_{t} \Vert _{L^{6}} \\ &\leqslant \frac{\nu}{6}\xi \Vert \nabla \mathbf{B}_{t} \Vert _{L^{2}}^{2}+C \xi \Vert \nabla \mathbf{B} \Vert _{L^{2}}^{3/2} \bigl\Vert \nabla ^{2} \mathbf{B} \bigr\Vert _{L^{2}}^{3/2} \Vert \nabla \mathbf{u} \Vert _{L^{2}}^{4}+C\xi \Vert \nabla \mathbf{u} \Vert _{L^{4}}^{4}, \end{aligned} $$

and inequality (3.4) gives

$$ N_{3}\leqslant C\xi \Vert \mathbf{B} \Vert _{L^{3}} \Vert \nabla \mathbf{B}_{t} \Vert _{L^{2}} \Vert \nabla \dot{\mathbf{u}} \Vert _{L^{2}}\leqslant C\xi m_{0}^{1/2} \Vert \nabla \dot{\mathbf{u}} \Vert _{L^{2}}^{2}+\frac{\nu}{6}\xi \Vert \nabla \mathbf{B}_{t} \Vert _{L^{2}}^{2}. $$

Thus, substituting $N_{1}$, $N_{2}$, $N_{3}$ into (3.19), we infer from (3.16) that

$$ \begin{aligned} & \bigl( \xi \Vert \mathbf{ B}_{t} \Vert _{L^{2}}^{2} \bigr)_{t}+\xi \Vert \nabla \mathbf{ B}_{t} \Vert _{L^{2}}^{2} \\ &\quad \leqslant \frac{1}{2}\xi \Vert \nabla \dot{\mathbf{u}} \Vert _{L^{2}}^{2}+ \bigl\vert \xi ' \bigr\vert \Vert \mathbf{ B}_{t} \Vert _{L^{2}}^{2}+C \xi \Vert \nabla \mathbf{ u} \Vert _{L^{2}}^{4} \Vert \mathbf{ B}_{t} \Vert _{L^{2}}^{2} \\ &\qquad{} +C\xi \Vert \nabla \mathbf{ u} \Vert _{L^{2}}^{4} \Vert \nabla \mathbf{ B} \Vert _{L^{2}}^{3/2} \bigl\Vert \nabla ^{2} \mathbf{B} \bigr\Vert _{L^{2}}^{3/2}+C \xi m_{0} \Vert \nabla \mathbf{ B} \Vert _{L^{2}}^{3} \bigl\Vert \nabla ^{2} \mathbf{B} \bigr\Vert _{L^{2}}^{3} \\ &\qquad{} +C\xi \bigl( \Vert \nabla \mathbf{ u} \Vert _{L^{2}}+m_{0}^{1/2}+m_{0}^{1/4} \Vert \nabla \mathbf{ B} \Vert _{L^{2}}^{3/2} \bigr) \bigl( \bigl\Vert \rho ^{1/2} \dot{\mathbf{ u}} \bigr\Vert _{L^{2}}^{3}+ \bigl\Vert \nabla ^{2} \mathbf{B} \bigr\Vert _{L^{2}}^{3} \bigr)+C\xi m_{0}, \end{aligned} $$

which together with (3.19) gives (3.14). □

Lemma 3.7

Under the conditions of Proposition 3.1, it holds that

$$ \sup_{t\in [0, T]}\sigma ^{2} \bigl( \bigl\Vert \rho ^{1/2} \dot{\mathbf{u}} \bigr\Vert _{L^{2}}^{2}+ \Vert \mathbf{ B}_{t} \Vert _{L^{2}}^{2}+ \bigl\Vert \nabla ^{2} \mathbf{ B} \bigr\Vert _{L^{2}}^{2} \bigr)\leqslant Cm_{0}^{1/2}. $$

(3.20)

Moreover, for any $0 \leqslant t_{1}< t_{2}\leqslant T$, it holds that

$$ \int _{t_{1}}^{t_{2}}\sigma ^{2} \bigl( \Vert \nabla \dot{\mathbf{u}} \Vert _{L^{2}}^{2}+ \Vert \nabla \mathbf{ B}_{t} \Vert _{L^{2}}^{2} \bigr)\,dt\leqslant Cm_{0}^{1/2}+Cm_{0}(t_{2}-t_{1}). $$

(3.21)

Proof

For any integer $1\leqslant i \leqslant [T]-1$, taking $\xi =\sigma _{i}^{2}$ with $\sigma _{i}(t)\triangleq \sigma (t+1-i)$ in (3.14) and integrating it over $(i-1, i+1]$, we obtain from (3.1), (3.3), and (3.13) that

$$\begin{aligned} &\sup_{t\in [i-1, i+1]}\sigma _{i}^{2} \bigl( \bigl\Vert \rho ^{1/2} \dot{\mathbf{u}} \bigr\Vert _{L^{2}}^{2}+ \Vert \mathbf{B}_{t} \Vert _{L^{2}}^{2} \bigr)+ \int _{i-1}^{i+1}\sigma _{i}^{2} \bigl( \Vert \nabla \dot{\mathbf{u}} \Vert _{L^{2}}^{2}+ \Vert \nabla \mathbf{ B}_{t} \Vert _{L^{2}}^{2} \bigr)\,dt \\ &\quad \leqslant Cm_{0}+C \int _{i-1}^{i+1}\sigma _{i} \sigma _{i}' \bigl( \bigl\Vert \rho ^{1/2} \dot{\mathbf{u}} \bigr\Vert _{L^{2}}^{2}+ \Vert \mathbf{ B}_{t} \Vert _{L^{2}}^{2} \bigr)\,dt+C \int _{i-1}^{i+1}\sigma _{i}^{2} \Vert \nabla \mathbf{u} \Vert _{L^{2}}^{2}\,dt \\ &\qquad{} +C \int _{i-1}^{i+1} \sigma _{i}^{2} \bigl( \Vert \nabla \mathbf{ u} \Vert _{L^{2}}^{4}+ \Vert \nabla \mathbf{ B} \Vert _{L^{2}}^{4} \bigr) \bigl( \Vert \mathbf{ B}_{t} \Vert _{L^{2}}^{2}+ \bigl\Vert \nabla ^{2} \mathbf{ B} \bigr\Vert _{L^{2}}^{2} \bigr)\,dt \\ &\qquad{} +C \int _{i-1}^{i+1}\sigma _{i}^{2} \Vert \nabla \mathbf{ u} \Vert _{L^{2}}^{4} \Vert \nabla \mathbf{ B} \Vert _{L^{2}}^{3/2} \bigl\Vert \nabla ^{2} \mathbf{B} \bigr\Vert _{L^{2}}^{3/2}\,dt+Cm_{0} \int _{i-1}^{i+1}\sigma _{i}^{2} \Vert \nabla \mathbf{ B} \Vert _{L^{2}}^{3} \bigl\Vert \nabla ^{2} \mathbf{B} \bigr\Vert _{L^{2}}^{3}\,dt \\ &\qquad{} +C \int _{i-1}^{i+1}\sigma _{i}^{2} \bigl( \Vert \nabla \mathbf{ u} \Vert _{L^{2}}+m_{0}^{1/2}+m_{0}^{1/4} \Vert \nabla \mathbf{ B} \Vert _{L^{2}}^{3/2} \bigr) \bigl( \bigl\Vert \rho ^{1/2}\dot{\mathbf{ u}} \bigr\Vert _{L^{2}}^{3}+ \bigl\Vert \nabla ^{2} \mathbf{B} \bigr\Vert _{L^{2}}^{3} \bigr)\,dt \\ &\quad \leqslant \sup_{t\in [i-1, i+1]} \bigl( \Vert \nabla \mathbf{ u} \Vert _{L^{2}}+m_{0}^{1/2}+m_{0}^{1/4} \Vert \nabla \mathbf{ B} \Vert _{L^{2}}^{3/2} \bigr) \int _{i-1}^{i+1}\sigma _{i}^{2} \bigl( \bigl\Vert \rho ^{1/2}\dot{\mathbf{ u}} \bigr\Vert _{L^{2}}^{3}+ \bigl\Vert \nabla ^{2} \mathbf{B} \bigr\Vert _{L^{2}}^{3} \bigr)\,dt \\ &\qquad{} +Cm_{0} \Bigl(\sup_{t\in [i-1, i+1]} \Vert \nabla \mathbf{ B} \Vert _{L^{2}}^{3} \Bigr) \int _{i-1}^{i+1}\sigma _{i}^{2} \bigl\Vert \nabla ^{2} \mathbf{B} \bigr\Vert _{L^{2}}^{3}\,dt+Cm_{0}^{1/2} \\ &\quad \leqslant Cm_{0}^{1/2}\sup_{t\in [i-1, i+1]} \sigma _{i}^{2} \bigl( \bigl\Vert \rho ^{1/2}\dot{\mathbf{ u}} \bigr\Vert _{L^{2}}+ \bigl\Vert \nabla ^{2} \mathbf{B} \bigr\Vert _{L^{2}} \bigr)+Cm_{0}^{1/2}. \end{aligned}$$

(3.22)

On the other hand, we obtain from (1.1)₃ that

$$ \bigl\Vert \nabla ^{2} \mathbf{B} \bigr\Vert _{L^{2}}\leqslant C \Vert \mathbf{B}_{t} \Vert _{L^{2}}+C \Vert \nabla \mathbf{u} \Vert _{L^{2}}^{2} \Vert \nabla \mathbf{B} \Vert _{L^{2}}, $$

(3.23)

which together with (3.22) yields (3.20).

Next, to prove (3.21), taking $\beta =\sigma$ in (3.7) and integrating the results over $[t_{1}, t_{2}]\subseteq [0, T]$, we deduce from (3.1) and (3.3) that

$$ \begin{aligned}[b] & \int _{t_{1}}^{t_{2}}\sigma \bigl( \bigl\Vert \rho ^{1/2} \dot{\mathbf{u}} \bigr\Vert _{L^{2}}^{2}+ \Vert \mathbf{ B}_{t} \Vert _{L^{2}}^{2}+ \bigl\Vert \nabla ^{2} \mathbf{ B} \bigr\Vert _{L^{2}}^{2} \bigr)\,dt \\ &\quad \leqslant C\sup_{t\in [t_{1}, t_{2}]}\sigma \bigl( \Vert \nabla \mathbf{u} \Vert _{L^{2}}^{2}+ \Vert \nabla \mathbf{B} \Vert _{L^{2}}^{2} \bigr)^{2} \int _{t_{1}}^{t_{2}}\sigma \bigl( \Vert \nabla \mathbf{u} \Vert _{L^{2}}^{2}+ \Vert \nabla \mathbf{B} \Vert _{L^{2}}^{2} \bigr)\,dt \\ &\qquad{} +C \bigl(m_{0}+A_{1}(T) \bigr)+ \Vert \mathbf{B} \Vert _{L^{3}}A_{1}(T)+Cm_{0}+Cm_{0}(t_{2}-t_{1}) \\ &\quad \leqslant Cm_{0}^{1/2}+Cm_{0}(t_{2}-t_{1}). \end{aligned} $$

(3.24)

Taking $\xi =\sigma ^{2}$ in (3.14) and integrating it over $[t_{1}, t_{2}]\subseteq [0, T]$, we deduce from (3.1), (3.3), (3.20), and (3.24) that

$$ \begin{aligned} & \int _{t_{1}}^{t_{2}}\sigma ^{2} \bigl( \Vert \nabla \dot{\mathbf{u}} \Vert _{L^{2}}^{2}+ \Vert \nabla \mathbf{ B}_{t} \Vert _{L^{2}}^{2} \bigr)\,dt \\ &\quad \leqslant C\sup_{t\in [t_{1}, t_{2}]}\sigma \bigl( \bigl\Vert \rho ^{1/2} \dot{\mathbf{u}} \bigr\Vert _{L^{2}}^{2}+ \bigl\Vert \nabla ^{2} \mathbf{B} \bigr\Vert _{L^{2}}^{2} \bigr)^{2} \int _{t_{1}}^{t_{2}}\sigma \bigl( \bigl\Vert \rho ^{1/2} \dot{\mathbf{u}} \bigr\Vert _{L^{2}}^{2}+ \bigl\Vert \nabla ^{2} \mathbf{B} \bigr\Vert _{L^{2}}^{2} \bigr)\,dt \\ &\qquad{} +Cm_{0}^{1/2}+Cm_{0}(t_{2}-t_{1}) \leqslant Cm_{0}^{1/2}+Cm_{0}(t_{2}-t_{1}). \end{aligned} $$

The proof of Lemma 3.7 is therefore completed. □

Lemma 3.8

Under the conditions of Proposition 3.1, it holds that

$$ \begin{gathered}[b] \sup_{t\in [0, \sigma (T)]}\sigma \bigl( \bigl\Vert \rho ^{1/2} \dot{\mathbf{u}} \bigr\Vert _{L^{2}}^{2}+ \Vert \mathbf{ B}_{t} \Vert _{L^{2}}^{2}+ \bigl\Vert \nabla ^{2} \mathbf{ B} \bigr\Vert _{L^{2}}^{2} \bigr)\\ \quad {}+ \int _{0}^{\sigma (T)} \sigma \bigl( \Vert \nabla \dot{\mathbf{u}} \Vert _{L^{2}}^{2}+ \Vert \nabla \mathbf{ B}_{t} \Vert _{L^{2}}^{2} \bigr)\,dt \leqslant C. \end{gathered} $$

(3.25)

Proof

Taking $\xi =\sigma$ in (3.14) and integrating it over $[0, \sigma (T)]$, we deduce from (3.1), (3.3), and (3.20) that

$$\begin{aligned} &\sup_{t\in [0, \sigma (T)]}\sigma \bigl( \bigl\Vert \rho ^{1/2} \dot{\mathbf{u}} \bigr\Vert _{L^{2}}^{2}+ \Vert \mathbf{B}_{t} \Vert _{L^{2}}^{2} \bigr)+ \int _{0}^{\sigma (T)}\sigma \bigl( \Vert \nabla \dot{\mathbf{u}} \Vert _{L^{2}}^{2}+ \Vert \nabla \mathbf{ B}_{t} \Vert _{L^{2}}^{2} \bigr)\,dt \\ &\quad \leqslant Cm_{0}+C \int _{0}^{\sigma (T)}\sigma ' \bigl( \bigl\Vert \rho ^{1/2} \dot{\mathbf{u}} \bigr\Vert _{L^{2}}^{2}+ \Vert \mathbf{ B}_{t} \Vert _{L^{2}}^{2} \bigr)\,dt+C \int _{0}^{\sigma (T)}\sigma \Vert \nabla \mathbf{u} \Vert _{L^{2}}^{2}\,dt \\ &\qquad{} +C \int _{0}^{\sigma (T)} \sigma \bigl( \Vert \nabla \mathbf{ u} \Vert _{L^{2}}^{4}+ \Vert \nabla \mathbf{ B} \Vert _{L^{2}}^{4} \bigr) \bigl( \Vert \mathbf{ B}_{t} \Vert _{L^{2}}^{2}+ \bigl\Vert \nabla ^{2} \mathbf{ B} \bigr\Vert _{L^{2}}^{2} \bigr)\,dt \\ &\qquad{} +C \int _{0}^{\sigma (T)}\sigma \Vert \nabla \mathbf{ u} \Vert _{L^{2}}^{4} \Vert \nabla \mathbf{ B} \Vert _{L^{2}}^{3/2} \bigl\Vert \nabla ^{2} \mathbf{B} \bigr\Vert _{L^{2}}^{3/2}\,dt+ \int _{0}^{\sigma (T)}\sigma \Vert \nabla \mathbf{ B} \Vert _{L^{2}}^{3} \bigl\Vert \nabla ^{2} \mathbf{B} \bigr\Vert _{L^{2}}^{3}\,dt \\ &\qquad{} +C \int _{0}^{\sigma (T)}\sigma \bigl( \Vert \nabla \mathbf{ u} \Vert _{L^{2}}+m_{0}^{1/2}+m_{0}^{1/4} \Vert \nabla \mathbf{ B} \Vert _{L^{2}}^{3/2} \bigr) \bigl( \bigl\Vert \rho ^{1/2} \dot{\mathbf{ u}} \bigr\Vert _{L^{2}}^{3}+ \bigl\Vert \nabla ^{2} \mathbf{B} \bigr\Vert _{L^{2}}^{3} \bigr)\,dt \\ &\quad \leqslant \sup_{t\in [0, \sigma (T)]} ( \Vert \nabla \mathbf{ u} \Vert _{L^{2}}+m_{0}^{1/2}+m_{0}^{1/4} \Vert \nabla \mathbf{ B} \Vert _{L^{2}}^{3/2} \int _{0}^{\sigma (T)}\sigma \bigl( \bigl\Vert \rho ^{1/2}\dot{\mathbf{ u}} \bigr\Vert _{L^{2}}^{3}+ \bigl\Vert \nabla ^{2} \mathbf{B} \bigr\Vert _{L^{2}}^{3} \bigr)\,dt \\ &\qquad{} +Cm_{0} \Bigl(\sup_{t\in [0, \sigma (T)]} \Vert \nabla \mathbf{ B} \Vert _{L^{2}}^{3} \Bigr) \int _{0}^{\sigma (T)}\sigma \bigl\Vert \nabla ^{2} \mathbf{B} \bigr\Vert _{L^{2}}^{3}\,dt+C \\ &\quad \leqslant \frac{1}{2}\sup_{t\in [0, \sigma (T)]}\sigma \bigl( \bigl\Vert \rho ^{1/2}\dot{\mathbf{ u}} \bigr\Vert _{L^{2}}+ \bigl\Vert \nabla ^{2} \mathbf{B} \bigr\Vert _{L^{2}} \bigr)+C, \end{aligned}$$

which together with (3.23) yields (3.24). □

Lemma 3.9

Let $(\rho , \mathbf{ u}, \mathbf{ B})$ be a smooth solution of (1.1)–(1.5) on ${\mathbb{R}}^3\times (0,T]$ satisfying (3.1). Then there exists a positive constant $\varepsilon >0$, depending only on μ, λ, ν, a, ρ̄, and M, such that

$$0⩽\rho (x,t)⩽\frac{7}{4}\overline{\rho },\mathrm{\forall }(x,t)\in {\mathbb{R}}^3\times [0,T),$$

(3.26)

provided $m_{0}\leqslant \varepsilon$.

Let $D_{t}\triangleq \partial _{t}+\mathbf{u}\cdot \nabla$ denote the material derivation operator. Then we can rewrite (3.1)₁ as follows:

$$ D_{t} \rho =g(\rho )+b'(t), $$

where

$$ g(\rho )\triangleq -\frac{a \rho}{2\mu +\lambda}(\rho -1), \qquad b(t)\triangleq - \frac{1}{2\mu +\lambda} \int _{0}^{t} \biggl( \rho F+ \frac{1}{2}\rho \vert \mathbf{B} \vert ^{2} \biggr)\,ds. $$

Obviously, it holds that $g(\infty )=-\infty$. So, to apply Lemma 2.1, we still need to deal with $b(t)$. To do this, we first use (2.4)–(2.6), (3.3), (3.4), (3.12), (3.13), and (3.25) to deduce that for any $0\leqslant t_{1} \leqslant t_{2} \leqslant \sigma (T)\leqslant 1$,

$$ \begin{aligned} & \bigl\vert b(t_{2})-b(t_{1}) \bigr\vert \\ &\quad\leqslant C(\bar{\rho}) \int _{0}^{ \sigma (T)} \bigl( \Vert F \Vert _{L^{\infty}}+ \Vert \mathbf{B} \Vert _{L^{\infty}}^{2} \bigr)\,dt \\ & \quad\leqslant C \int _{0}^{\sigma (T)} \bigl( \Vert F \Vert _{L^{2}}^{1/4} \Vert \nabla F \Vert _{L^{6}}^{3/4}+ \Vert \mathbf{B} \Vert _{L^{6}} \Vert \nabla \mathbf{B} \Vert _{L^{6}} \bigr)\,dt \\ & \quad\leqslant C \int _{0}^{\sigma (T)} \bigl( \Vert \nabla \mathbf{u} \Vert _{L^{2}}+m_{0}^{1/2}+ \Vert \mathbf{B} \Vert _{L^{2}}^{2/3} \Vert \mathbf{B} \Vert _{L^{3}}^{1/2} \bigr)^{1/4} \bigl( \Vert \nabla \dot{\mathbf{u}} \Vert _{L^{2}}+ \Vert \mathbf{B}\cdot \nabla \mathbf{B} \Vert _{L^{2}} \bigr)^{3/4}\,dt \\ &\qquad{}+C \int _{0}^{\sigma (T)} \Vert \nabla \mathbf{B} \Vert _{L^{2}} \bigl\Vert \nabla ^{2} \mathbf{B} \bigr\Vert _{L^{2}}\,dt \\ & \quad\leqslant Cm_{0}^{1/16} \biggl( \int _{0}^{\sigma (T)} \sigma ^{-4/5}\,dt \biggr)^{5/8} \biggl( \int _{0}^{\sigma (T)} \sigma \Vert \nabla \dot{\mathbf{u}} \Vert _{L^{2}}^{2}\,dt \biggr)^{3/8} \\ &\qquad{}+Cm_{0}^{1/12} \biggl( \int _{0}^{\sigma (T)} \sigma ^{-3/5}\,dt \biggr)^{5/8} \biggl( \int _{0}^{\sigma (T)} \sigma \Vert \nabla \dot{\mathbf{u}} \Vert _{L^{2}}^{2}\,dt \biggr)^{3/8} \\ &\qquad{}+Cm_{0}^{7/64}+Cm_{0}^{1/2} \biggl( \int _{0}^{\sigma (T)} \bigl\Vert \nabla ^{2} \mathbf{B} \bigr\Vert _{L^{2}}^{2}\,dt \biggr)^{1/2}\leqslant C_{6}m_{0}^{1/16}. \end{aligned} $$

Therefore, for $t\in [0, \sigma (T)]$, we can choose $N_{0}$ and $N_{1}$ in (2.1) as

$$ N_{1}=0, \qquad N_{0}=C_{6}m_{0}^{1/16}, $$

and $\xi ^{*}=1$ in (2.2), then we obtain from Lemma 2.1 that

$$ \sup_{t\in [0, \sigma (T)]} \Vert \rho \Vert _{L^{\infty}}\leqslant \max \bigl\{ \bar{\rho}, \xi ^{*} \bigr\} +N_{0}\leqslant \bar{\rho}+C_{6}m_{0}^{1/16} \leqslant \frac{3\bar{\rho}}{2}, $$

(3.27)

provided that

$$ m_{0}\leqslant \varepsilon _{3}\triangleq \biggl\{ \varepsilon _{1}, \varepsilon _{2}, \biggl( \frac{\bar{\rho}}{2C_{6}} \biggr)^{16} \biggr\} . $$

For $t\in [\sigma (T), T]$, we obtain from (2.4)–(2.6), (3.3), (3.4), (3.12), (3.13), and (3.25) that, for all $\sigma (T)\leqslant t_{1}< t_{2} \leqslant T$,

$$ \begin{aligned} & \bigl\vert b(t_{2})-b(t_{1}) \bigr\vert \\ &\quad \leqslant C(\bar{\rho}) \int _{t_{1}}^{t_{2}} \bigl( \Vert F \Vert _{L^{\infty}}+ \Vert \mathbf{B} \Vert _{L^{\infty}}^{2} \bigr)\,dt \\ &\quad \leqslant \frac{a}{4\mu +2\lambda}(t_{2}-t_{1})+ C \int _{t_{1}}^{t_{2}} \bigl( \Vert F \Vert _{L^{2}}^{2/3} \Vert \nabla F \Vert _{L^{6}}^{2}+ \Vert \mathbf{B} \Vert _{L^{6}} \Vert \nabla \mathbf{B} \Vert _{L^{6}} \bigr)\,dt \\ &\quad \leqslant \frac{a}{4\mu +2\lambda}(t_{2}-t_{1})+ C \int _{t_{1}}^{t_{2}} \bigl( \Vert \nabla \mathbf{u} \Vert _{L^{2}}+m_{0}^{1/3} \bigr)^{2/3} \bigl( \Vert \nabla \dot{\mathbf{u}} \Vert _{L^{2}}^{2}+ \Vert \mathbf{B}\cdot \nabla \mathbf{B} \Vert _{L^{2}}^{2} \bigr)\,dt \\ &\quad \leqslant \biggl(\frac{a}{4\mu +2\lambda}+C_{7} m_{0}^{7/6} \biggr) (t_{2}-t_{1})+C_{8} m_{0}^{2/3} \\ &\quad \leqslant \frac{a}{2\mu +\lambda}(t_{2}-t_{1})+C_{8} m_{0}^{2/3}, \end{aligned} $$

provided that

$$ m_{0}\leqslant \varepsilon _{4}\triangleq \biggl\{ \varepsilon _{3}, \biggl(\frac{a}{C_{7}(4\mu +2\lambda )} \biggr)^{6/7} \biggr\} . $$

Therefore, for $t\in [\sigma (T), T]$, we can choose $N_{0}$ and $N_{1}$ in (2.1) as

$$ N_{1}=\frac{a}{2\mu +\lambda}(t_{2}-t_{1}), \qquad N_{0}=C_{8}m_{0}^{2/3}, $$

and $\xi ^{*}=2$ in (2.2), then we obtain from Lemma 2.1 that

$$ \sup_{t\in [\sigma (T), T]} \Vert \rho \Vert _{L^{\infty}}\leqslant \max \biggl\{ \frac{3}{2}\bar{\rho}, \xi ^{*} \biggr\} +N_{0}\leqslant \frac{3}{2}\bar{ \rho}+C_{8}m_{0}^{2/3}\leqslant \frac{7\bar{\rho}}{4}, $$

(3.28)

provided that

$$ m_{0}\leqslant \varepsilon \triangleq \biggl\{ \varepsilon _{4}, \biggl( \frac{\bar{\rho}}{4C_{8}} \biggr)^{3/2} \biggr\} . $$

(3.29)

The combination of (3.27) with (3.28) yields (3.26).

4 Proof of Theorem 1.1

In this section, we always assume that the initial energy $m_{0}$ satisfies (3.29) and the positive constant C may depend on T and g besides μ, λ, ν, a, ρ̄, and M.

Lemma 4.1

Under the conditions of Theorem 1.1, the following estimates hold:

$$ \sup_{t\in [0, \sigma (T)]} \bigl( \bigl\Vert \rho ^{1/2} \dot{\mathbf{u}} \bigr\Vert _{L^{2}}^{2}+ \Vert \mathbf{ B}_{t} \Vert _{L^{2}}^{2}+ \bigl\Vert \nabla ^{2} \mathbf{ B} \bigr\Vert _{L^{2}}^{2} \bigr)+ \int _{0}^{\sigma (T)} \bigl( \Vert \nabla \dot{\mathbf{u}} \Vert _{L^{2}}^{2}+ \Vert \nabla \mathbf{ B}_{t} \Vert _{L^{2}}^{2} \bigr)\leqslant C. $$

(4.1)

Proof

Taking $\xi =1$ in (3.14) and integrating the resulting equation over $(0, \sigma (T)]$, we deduce from (3.23) that

$$ \begin{aligned} &\sup_{t\in [0, \sigma (T)]} \bigl( \bigl\Vert \rho ^{1/2} \dot{\mathbf{u}} \bigr\Vert _{L^{2}}^{2}+ \Vert \mathbf{ B}_{t} \Vert _{L^{2}}^{2}+ \bigl\Vert \nabla ^{2} \mathbf{ B} \bigr\Vert _{L^{2}}^{2} \bigr)+ \int _{0}^{\sigma (T)} \bigl( \Vert \nabla \dot{\mathbf{u}} \Vert _{L^{2}}^{2}+ \Vert \nabla \mathbf{ B}_{t} \Vert _{L^{2}}^{2} \bigr) \\ &\quad \leqslant \bigl\Vert \rho _{0}^{1/2}g \bigr\Vert _{L^{2}}^{2}+Cm_{0}+C \int _{0}^{ \sigma (T)} \bigl( \Vert \nabla \mathbf{ u} \Vert _{L^{2}}^{4}+ \Vert \nabla \mathbf{ B} \Vert _{L^{2}}^{4} \bigr) \bigl( \bigl\Vert \rho ^{1/2}\dot{\mathbf{ u}} \bigr\Vert _{L^{2}}^{2}+ \Vert \mathbf{ B}_{t} \Vert _{L^{2}}^{2} \bigr)\,dt+C \\ &\quad \leqslant C, \end{aligned} $$

which leads to (4.1). □

Lemma 4.2

Under the conditions of Theorem 1.1, the following estimates hold:

$$ \sup_{t\in [0, T]} \bigl( \bigl\Vert \rho ^{1/2} \dot{\mathbf{u}} \bigr\Vert _{L^{2}}^{2}+ \Vert \mathbf{ B}_{t} \Vert _{L^{2}}^{2}+ \bigl\Vert \nabla ^{2} \mathbf{ B} \bigr\Vert _{L^{2}}^{2} \bigr)+ \int _{0}^{T} \bigl( \Vert \nabla \dot{\mathbf{u}} \Vert _{L^{2}}^{2}+ \Vert \nabla \mathbf{ B}_{t} \Vert _{L^{2}}^{2} \bigr)\leqslant C, $$

(4.2)

and for any $p \in [2, 6]$,

$$ \sup_{0 \leqslant t \leqslant T} \bigl( \Vert \nabla \rho \Vert _{{L^{2}}{\cap} {L^{p}}}+ \Vert \nabla \mathbf{ u} \Vert _{H^{1}} \bigr) + \int _{0}^{T} \Vert \nabla \mathbf{ u} \Vert _{L^{\infty}}\,dt\leqslant C(T). $$

(4.3)

Proof

Inequality (4.2) can be obtained directly from Lemma 3.7 and Lemma 4.1. Similar to the proof of [10], we can obtain (4.3). □

Proof of Theorem 1.1

To prove Theorem 1.1, by virtue of Proposition 2.1, we know that there exists a positive time $T_{*}>0$ such that problem (1.1)–(1.5) possesses a strong solution $(\rho , \mathbf{u}, \mathbf{B})$ in ${\mathbb{R}}^3\times (0,T_{\ast }]$. Next, with all the a priori estimates established in Sect. 3, we will extend the local strong solution to all time.

First, in view of the definitions of $A_{1}(T)$ and $A_{2}(T)$, it is easily deduced from (3.1) that

$$ A_{1}(0)\leqslant m_{0}^{1/2}, \qquad A_{2}(0)\leqslant 2K, \quad 0 \leqslant \rho \leqslant \frac{7}{4}\bar{\rho}, $$

due to $m_{0}\leqslant \varepsilon$. Thus, there exists a time $T_{1}\in (0, T_{*}]$ such that (3.1) holds for $T=T_{1}$.

Set

$$ T^{*}\triangleq \sup \bigl\{ T|(\rho , \mathbf{u}, \mathbf{B}) \text{ is a strong solution on } [0, T] \bigr\} $$

and

$$ T^{*}_{1}\triangleq \sup \bigl\{ T|(\rho , \mathbf{u}, \mathbf{B}) \text{ is a strong solution on } [0, T] \text{ satisfying (3.1)} \bigr\} . $$

Thus, $T_{1}^{*}\geqslant T_{1}>0$. By virtue of Proposition 3.1, we know

$$ T^{*}=T_{1}^{*}, $$

provided $m_{0}\leqslant \varepsilon$.

Next, similar to the proof of [10, Sect. 4], we can claim that $T^{*}=\infty$. Thus, the proof of Theorem 1.1 is complete. □