In this section, we establish the uniform a priori estimates of solutions to the initial boundary value problem (1.1)–(1.4) to extend the local strong solution guaranteed by Proposition 2.1. Then we assume that \((\rho, \mathbf {u}, \theta, \mathbf {B})\) is a smooth solution to (1.1)–(1.4) on \(\Omega \times (0, T)\) for some positive time \(T>0\) with smooth initial data \((\rho _{0}, \mathbf {u}_{0}, \theta _{0}, \mathbf {B}_{0})\) satisfying (1.7) and (1.8). Define
$$\begin{aligned} &A_{1}(T)\triangleq \sup_{0\leqslant t \leqslant T} \bigl\Vert (\nabla \mathbf {u}, \nabla \mathbf {B}) \bigr\Vert _{L^{2}}^{2}+ \int _{0}^{T} \bigl\Vert \bigl(\rho ^{1/2}\mathbf {u}_{t}, \nabla ^{2}\mathbf {B}, \mathbf {B}_{t} \bigr) \bigr\Vert _{L^{2}}^{2} \,dt, \\ &A_{2} (T)\triangleq \sup_{0\leqslant t \leqslant T} \Vert \nabla \theta \Vert _{L^{2}}^{2}+ \int _{0}^{T} \bigl\Vert \rho ^{1/2} \theta _{t} \bigr\Vert _{L^{2}}^{2} \,dt, \\ &A_{3}(T)\triangleq \sup_{0\leqslant t \leqslant T} \bigl\Vert \bigl( \rho ^{1/2} \mathbf {u}_{t}, \mathbf {B}_{t}, \nabla ^{2} \mathbf {B}\bigr) \bigr\Vert _{L^{2}}^{2}+ \int _{0}^{T} \bigl\Vert (\nabla \mathbf {u}_{t}, \nabla \mathbf {B}_{t}) \bigr\Vert _{L^{2}}^{2} \,dt, \\ &A_{4}(T)\triangleq \sup_{0\leqslant t \leqslant T} \bigl\Vert \rho ^{1/2} \theta _{t} \bigr\Vert _{L^{2}}^{2}+ \int _{0}^{T} \Vert \nabla \theta _{t} \Vert _{L^{2}}^{2} \,dt, \\ &A_{5}(T)\triangleq \sup_{0\leqslant t \leqslant T} \Vert \nabla \rho \Vert _{L^{q}}, \\ &A_{6}(T) \triangleq \sup_{0\leqslant t \leqslant T} \bigl( \Vert \nabla \mathbf {u}\Vert _{L^{2}}^{1/2} \Vert \nabla \mathbf {u}\Vert _{H^{1}}^{1/2}+ \Vert \nabla \mathbf {u}\Vert _{L^{2}}^{1/4} \Vert \nabla \mathbf {u}\Vert _{H^{1}}^{3/2} \\ &\phantom{A_{6}(T) \triangleq }{} + \Vert \nabla \mathbf {B}\Vert _{L^{2}}^{1/2} \Vert \nabla \mathbf {B}\Vert _{H^{1}}^{1/2}+ \Vert \nabla \mathbf {B}\Vert _{L^{2}}^{1/4} \Vert \nabla \mathbf {B}\Vert _{H^{1}}^{3/2} \bigr), \\ &A_{7}(T) \triangleq \sup_{0\leqslant t \leqslant T} \bigl\Vert \nabla G_{i j}^{ \alpha \beta}( \rho, \theta ) \bigr\Vert _{L^{q}}^{\frac {q}{q-3}} \Vert \nabla \mathbf {u}\Vert _{L^{2}}^{\frac {1}{2}}, \quad \bigl( \text{see }G_{i j}^{\alpha \beta} \text{in (2.1)} \bigr), \end{aligned}$$
and
$$\begin{aligned} K_{1}\triangleq \bigl\Vert \bigl(\rho ^{1/2}\mathbf {u}_{t}, \mathbf {B}_{t} \bigr) \bigr\Vert _{L^{2}}^{2} |_{t=0}, \qquad K_{2}\triangleq \bigl\Vert \rho ^{1/2} \theta _{t} \bigr\Vert _{L^{2}}^{2} |_{t=0}. \end{aligned}$$
We have the following key a priori estimates on \((\rho, \mathbf {u}, \theta, \mathbf {B})\).
Proposition 3.1
For a constant \(\bar{\rho}>0\) and q satisfying (1.6), assume that \((\rho _{0}, \mathbf{ u}_{0}, \theta _{0}, \mathbf{ B}_{0})\) satisfies (1.7) and (1.8). Let \((\rho, \mathbf{ u}, \theta, \mathbf{ B})\) be a smooth solution of (1.1)–(1.4) on \(\Omega \times (0, T]\) satisfying
$$\begin{aligned} \textstyle\begin{cases} 0\leqslant \rho (x, t)\leqslant 3\bar{\rho} \quad \textit{for all }(x, t) \in \Omega \times [0, T], \\ A_{1}(T)\leqslant 2C_{0}^{1/4}, \qquad A_{2}(T)\leqslant 2C_{0}^{1/2},\qquad A_{3}(T)\leqslant 3K_{1}, \\ A_{4}(T)\leqslant 3K_{2}, \qquad A_{5}(T)\leqslant 4 \Vert \nabla \rho _{0} \Vert _{L^{q}}, \qquad A_{6}(T)+A_{7}(T)\leqslant 2. \end{cases}\displaystyle \end{aligned}$$
(3.1)
Then there exists a constant \(\varepsilon >0\) such that
$$\begin{aligned} \textstyle\begin{cases} 0\leqslant \rho (x, t)\leqslant 2\bar{\rho} \quad\textit{for all }(x, t) \in \Omega \times [0, T], \\ A_{1}(T)\leqslant C_{0}^{1/4}, \qquad A_{2}(T)\leqslant C_{0}^{1/2},\qquad A_{3}(T)\leqslant 2K_{1}, \\ A_{4}(T)\leqslant 2K_{2}, \qquad A_{5}(T)\leqslant 3 \Vert \nabla \rho _{0} \Vert _{L^{q}}, \qquad A_{6}(T)+A_{7}(T)\leqslant 1, \end{cases}\displaystyle \end{aligned}$$
provided that \(C_{0}\leqslant \varepsilon \).
Proof
The proof of Proposition 3.1 will be done by a series of lemmas below. □
Throughout this paper, we denote by C and \(C_{i}\ (i=1, 2, \ldots )\) generic positive constants, which may depend on \(\Omega, \bar{\rho}, \|\rho _{0}\|_{L^{1}}, \kappa, \nu, R, c_{v}, g_{1}\), and \(g_{2}\).
We start with the following uniform estimates for \((\rho, \mathbf {u}, \theta, \mathbf {B})\) under conditions (3.1).
Lemma 3.1
Under condition (3.1), we have
$$\begin{aligned} \sup_{t\in [0, T]} \bigl( \Vert \nabla \theta \Vert _{H^{1}}+ \Vert \theta \Vert _{L^{\infty}}+ \Vert \nabla \theta \Vert _{L^{6}} \bigr)\leqslant C. \end{aligned}$$
(3.2)
Proof
Equation (1.1)3, together with (3.1), (2.2), and (2.4), yields that
$$\begin{aligned} \begin{aligned} \bigl\Vert \nabla ^{2} \theta \bigr\Vert _{L^{2}} \leqslant{}& C \bigl( \Vert \rho \theta _{t} \Vert _{L^{2}}+ \Vert \rho \mathbf {u}\cdot \nabla \theta \Vert _{L^{2}}+ \Vert \rho \theta \operatorname{div} \mathbf {u}\Vert _{L^{2}}+ \Vert \nabla \mathbf {u}\Vert _{L^{4}}^{2}+ \Vert \nabla \mathbf {B}\Vert _{L^{4}}^{2} \bigr) \\ \leqslant{}& C \bigl( \bigl\Vert \rho ^{1/2} \theta _{t} \bigr\Vert _{L^{2}}+ \Vert \mathbf {u}\Vert _{L^{6}} \Vert \nabla \theta \Vert _{L^{3}}+ \Vert \theta \Vert _{L^{6}} \Vert \nabla \mathbf {u}\Vert _{L^{3}}+ \Vert \nabla \mathbf {u}\Vert _{L^{4}}^{2}+ \Vert \nabla \mathbf {B}\Vert _{L^{4}}^{2} \bigr) \\ \leqslant{}& C \bigl( \bigl\Vert \rho ^{1/2} \theta _{t} \bigr\Vert _{L^{2}}+A_{1}^{1/2}(T) \Vert \nabla \theta \Vert _{L^{2}}^{1/2} \Vert \nabla \theta \Vert _{H^{1}}^{1/2}+A_{6}(T) \Vert \nabla \theta \Vert _{L^{2}} \bigr) \\ &{} +CA_{6}(T) \bigl( \Vert \nabla \mathbf {u}\Vert _{L^{2}}^{1/4}+ \Vert \nabla \mathbf {B}\Vert _{L^{2}}^{1/4} \bigr) \\ \leqslant {}&\frac{1}{2} \bigl\Vert \nabla ^{2} \theta \bigr\Vert _{L^{2}}+C \bigl( \bigl\Vert \rho ^{1/2} \theta _{t} \bigr\Vert _{L^{2}}+ \Vert \nabla \theta \Vert _{L^{2}}+ \Vert \nabla \mathbf {u}\Vert _{L^{2}}^{1/4}+ \Vert \nabla \mathbf {B}\Vert _{L^{2}}^{1/4} \bigr), \end{aligned} \end{aligned}$$
and thus
$$\begin{aligned} \Vert \nabla \theta \Vert _{H^{1}}\leqslant C \bigl( \bigl\Vert \rho ^{1/2} \theta _{t} \bigr\Vert _{L^{2}}+ \Vert \nabla \theta \Vert _{L^{2}}+ \Vert \nabla \mathbf {u}\Vert _{L^{2}}^{1/4}+ \Vert \nabla \mathbf {B}\Vert _{L^{2}}^{1/4} \bigr). \end{aligned}$$
(3.3)
On the other hand, from the Sobolev and Poincaré inequalities it follows that
$$\begin{aligned} \Vert \theta \Vert _{L^{\infty}}\leqslant C \Vert \nabla \theta \Vert _{L^{6}} \leqslant C \Vert \nabla \theta \Vert _{H^{1}}, \end{aligned}$$
which, together with (3.3), leads to (3.2). The proof of the lemma is therefore completed. □
Lemma 3.2
Under condition (3.1),
$$\begin{aligned} \sup_{t\in [0, T]} \bigl( \Vert \nabla \rho \Vert _{L^{r}}+ \bigl\Vert (\nabla \mathbf{ u}, \nabla \mathbf{ B}) \bigr\Vert _{H^{1}} \bigr) \leqslant C,\quad r\in [2, q], \end{aligned}$$
(3.4)
where q is defined as in (1.6).
Proof
Thanks to the bounded domain Ω, we obtain
$$\begin{aligned} \Vert \nabla \rho \Vert _{L^{r}}\leqslant C \Vert \nabla \rho \Vert _{L^{q}} \leqslant CA_{5}(T)\leqslant C\quad \text{for }r\in [2, q]. \end{aligned}$$
On the other hand, it follows from Lemma 2.2, (2.2), (2.4), and (3.1) that
$$\begin{aligned} \begin{aligned} \bigl\Vert \nabla ^{2} \mathbf {u}\bigr\Vert _{L^{2}} \leqslant{}& C \bigl( \Vert \rho \mathbf {u}_{t} \Vert _{L^{2}}+ \Vert \rho \mathbf {u}\cdot \nabla \mathbf {u}\Vert _{L^{2}}+ \bigl\Vert \nabla G_{i j}^{\alpha \beta}(\rho, \theta )\nabla \mathbf {u}\bigr\Vert _{L^{2}} \bigr) \\ &{} +C \bigl( \Vert \nabla P \Vert _{L^{2}}+ \bigl\Vert (\operatorname{curl} \mathbf {B})\times \mathbf {B}\bigr\Vert _{L^{2}} \bigr) \\ \leqslant{}& C \bigl( \bigl\Vert \rho ^{1/2} \mathbf {u}_{t} \bigr\Vert _{L^{2}}+ \Vert \mathbf {u}\Vert _{L^{6}} \Vert \nabla \mathbf {u}\Vert _{L^{3}}+ \bigl\Vert \nabla G_{i j}^{\alpha \beta}( \rho, \theta ) \bigr\Vert _{L^{q}} \Vert \nabla \mathbf {u}\Vert _{L^{2q/(q-2)}} \bigr) \\ &{} +C \bigl( \Vert \theta \Vert _{L^{6}} \Vert \nabla \rho \Vert _{L^{3}}+ \Vert \nabla \theta \Vert _{L^{2}}+ \Vert \mathbf {B}\Vert _{L^{6}} \Vert \nabla \mathbf {B}\Vert _{L^{3}} \bigr) \\ \leqslant{}& \frac{1}{2} \bigl\Vert \nabla ^{2} \mathbf {u}\bigr\Vert _{L^{2}}+\frac{1}{4} \bigl\Vert \nabla ^{2} \mathbf {B}\bigr\Vert _{L^{2}}+C \bigl( \bigl\Vert \rho ^{1/2} \mathbf {u}_{t} \bigr\Vert _{L^{2}}+ \Vert \nabla \mathbf {u}\Vert _{L^{2}}^{3}+ \Vert \nabla \mathbf {B}\Vert _{L^{2}}^{3} \bigr) \\ &{} + C \bigl\Vert \nabla G_{i j}^{\alpha \beta}(\rho, \theta ) \bigr\Vert _{L^{q}}^{q/(q-3)} \Vert \nabla \mathbf {u}\Vert _{L^{2}}+C \bigl(1+ \Vert \nabla \rho \Vert _{L^{3}} \bigr) \Vert \nabla \theta \Vert _{L^{2}} \\ \leqslant{}& \frac{1}{2} \bigl\Vert \nabla ^{2} \mathbf {u}\bigr\Vert _{L^{2}}+\frac{1}{4} \bigl\Vert \nabla ^{2} \mathbf {B}\bigr\Vert _{L^{2}}+C \bigl[ \bigl\Vert \rho ^{1/2} \mathbf {u}_{t} \bigr\Vert _{L^{2}}+A_{1}^{5/4}(T) \bigl( \Vert \nabla \mathbf {u}\Vert _{L^{2}}^{1/2}+ \Vert \nabla \mathbf {B}\Vert _{L^{2}}^{1/2} \bigr) \bigr] \\ & {}+ C A_{7}(T) \Vert \nabla \mathbf {u}\Vert _{L^{2}}^{1/2}+C \Vert \nabla \theta \Vert _{L^{2}}+CA_{5}(T) \Vert \nabla \theta \Vert _{L^{2}}, \end{aligned} \end{aligned}$$
and thus
$$\begin{aligned} \begin{aligned} \bigl\Vert \nabla ^{2} \mathbf {u}\bigr\Vert _{L^{2}} &\leqslant \frac{1}{2} \bigl\Vert \nabla ^{2} \mathbf {B}\bigr\Vert _{L^{2}}+C \bigl( \bigl\Vert \rho ^{1/2} \mathbf {u}_{t} \bigr\Vert _{L^{2}}+ \Vert \nabla \mathbf {u}\Vert _{L^{2}}^{1/2}+ \Vert \nabla \mathbf {B}\Vert _{L^{2}}^{1/2}+ \Vert \nabla \theta \Vert _{L^{2}} \bigr) \\ &\leqslant C \bigl(A_{3}^{1/2}(T)+A_{2}^{1/2}(T)+A_{1}^{1/4}(T) \bigr). \end{aligned} \end{aligned}$$
Similarly,
$$\begin{aligned} \begin{aligned} \bigl\Vert \nabla ^{2} \mathbf {B}\bigr\Vert _{L^{2}} &\leqslant C \bigl( \Vert \mathbf {B}_{t} \Vert _{L^{2}}+ \Vert \mathbf {u}\Vert _{L^{6}} \Vert \nabla \mathbf {B}\Vert _{L^{3}}+ \Vert \mathbf {B}\Vert _{L^{\infty}} \Vert \nabla \mathbf {u}\Vert _{L^{2}} \bigr) \\ &\leqslant \frac{1}{2} \bigl\Vert \nabla ^{2} \mathbf {B}\bigr\Vert _{L^{2}}+C \bigl( \Vert \mathbf {B}_{t} \Vert _{L^{2}}+ \Vert \nabla \mathbf {u}\Vert _{L^{2}}^{2} \Vert \nabla \mathbf {B}\Vert _{L^{2}} \bigr) \\ &\leqslant \frac{1}{2} \bigl\Vert \nabla ^{2} \mathbf {B}\bigr\Vert _{L^{2}}+C \Vert \mathbf {B}_{t} \Vert _{L^{2}}+ C \Vert \nabla \mathbf {u}\Vert _{L^{2}}^{1/2}, \end{aligned} \end{aligned}$$
and thus
$$\begin{aligned} \begin{aligned} \bigl\Vert \nabla ^{2} \mathbf {u}\bigr\Vert _{L^{2}}+ \bigl\Vert \nabla ^{2} \mathbf {B}\bigr\Vert _{L^{2}} &\leqslant C \bigl( \bigl\Vert \bigl(\rho ^{1/2} \mathbf {u}_{t}, \mathbf {B}_{t} \bigr) \bigr\Vert _{L^{2}}+ \Vert \nabla \mathbf {u}\Vert _{L^{2}}^{1/2}+ \Vert \nabla \mathbf {B}\Vert _{L^{2}}^{1/2}+ \Vert \nabla \theta \Vert _{L^{2}} \bigr) \\ &\leqslant C \bigl(A_{3}^{1/2}(T)+A_{2}^{1/2}(T)+A_{1}^{1/4}(T) \bigr), \end{aligned} \end{aligned}$$
(3.5)
which gives (3.4). The proof of the lemma is therefore completed. □
Lemma 3.3
There exist a constant \(\varepsilon _{1}>0\) such that
$$\begin{aligned} A_{6}(T)+A_{7}(T)\leqslant 1, \end{aligned}$$
provided that \(C_{0}\leqslant \varepsilon _{1}\).
Proof
From (3.1), (3.2), and (3.4) we obtain
$$\begin{aligned} \begin{aligned} A_{6}(T)+A_{7}(T) \leqslant{}& C \bigl( \Vert \nabla \rho \Vert _{L^{q}} + \Vert \nabla \theta \Vert _{L^{q}} \bigr)^{q/(q-3)} \Vert \nabla \mathbf {u}\Vert _{L^{2}}^{1/2} \\ &{} + \bigl( \Vert \nabla \mathbf {u}\Vert _{L^{2}}^{1/4} \Vert \nabla \mathbf {u}\Vert _{H^{1}}^{1/2}+ \Vert \nabla \mathbf {u}\Vert _{H^{1}}^{3/2} \bigr) \Vert \nabla \mathbf {u}\Vert _{L^{2}}^{1/4} \\ &{} + \bigl( \Vert \nabla \mathbf {B}\Vert _{L^{2}}^{1/4} \Vert \nabla \mathbf {B}\Vert _{H^{1}}^{1/2}+ \Vert \nabla \mathbf {B}\Vert _{H^{1}}^{3/2} \bigr) \Vert \nabla \mathbf {B}\Vert _{L^{2}}^{1/4} \\ \leqslant{}& C_{1}C_{0}^{1/32}\leqslant 1, \end{aligned} \end{aligned}$$
provided that \(C_{0}\leqslant \varepsilon _{1}\triangleq \min \{1, C_{1}^{-32}\}\). The proof of the lemma is therefore completed. □
Lemma 3.4
Let \((\rho, \mathbf{ u}, \theta, \mathbf{ B})\) be a smooth solution of (1.1)–(1.4) satisfying (3.1). Then there exist positive constants C and \(\varepsilon _{2}\), both depending only on \(\kappa, \nu, R, c_{v}, \underline{\mu}\), ρ̄, Ω, \(g_{1}\), and \(g_{2}\) such that
$$\begin{aligned} \sup_{t\in [0, T]}t^{i} \bigl\Vert \bigl( \rho ^{1/2}\mathbf{ u}, \rho ^{1/2} \theta, \mathbf{ B} \bigr) \bigr\Vert _{L^{2}}^{2}+ \int _{0}^{T} t^{i} \bigl\Vert (\nabla \mathbf{ u}, \nabla \theta, \nabla \mathbf{ B}) \bigr\Vert _{L^{2}}^{2} \,dt \leqslant CC_{0} \end{aligned}$$
(3.6)
for \(i=0, 1, \dots, 32\), provided that \(C_{0}\leqslant \varepsilon _{2}\).
Proof
Multiplying (1.1)2 by u in \(L^{2}\), from (2.2) and (3.4) we have that
$$\begin{aligned} \begin{aligned} &\frac{1}{2}\frac{d}{dt} \bigl\Vert \rho ^{1/2}\mathbf {u}\bigr\Vert _{L^{2}}^{2}+ \int \bigl[2\mu (\rho, \theta ) \bigl\vert \mathcal{D}(\mathbf {u}) \bigr\vert ^{2}+\lambda ( \rho, \theta ) (\operatorname{div} \mathbf {u})^{2} \bigr]\,dx \\ &\quad = \int R\rho \theta \operatorname{div} \mathbf {u}\,dx- \int \mathbf {B}\cdot \nabla \mathbf {u}\cdot \mathbf {B}\,dx+\frac{1}{2} \int \vert \mathbf {B}\vert ^{2} \operatorname{div} \mathbf {u}\,dx \\ & \quad\leqslant \frac{1}{2}\underline{\mu} \Vert \nabla \mathbf {u}\Vert _{L^{2}}^{2}+C \Vert \rho \Vert _{L^{3}}^{2} \Vert \theta \Vert _{L^{6}}^{2}+C \Vert \mathbf {B}\Vert _{L^{3}}^{2} \Vert \mathbf {B}\Vert _{L^{6}}^{2} \\ & \quad\leqslant \frac{1}{2}\underline{\mu} \Vert \nabla \mathbf {u}\Vert _{L^{2}}^{2}+C \Vert \nabla \theta \Vert _{L^{2}}^{2}+C \Vert \nabla \mathbf {B}\Vert _{L^{2}}^{2}, \end{aligned} \end{aligned}$$
and thus by Lemma 2.1 we have
$$\begin{aligned} \frac{d}{dt} \bigl\Vert \rho ^{1/2}\mathbf {u}\bigr\Vert _{L^{2}}^{2}+\underline{\mu} \Vert \nabla \mathbf {u}\Vert _{L^{2}}^{2} \leqslant C_{2} \Vert \nabla \theta \Vert _{L^{2}}^{2}+C_{3} \Vert \nabla \mathbf {B}\Vert _{L^{2}}^{2}. \end{aligned}$$
(3.7)
Multiplying (1.1)3 by θ in \(L^{2}\) and using the facts that \(\mu (\rho, \theta ), \lambda (\rho, \theta )\in C^{1}(\mathbb{R}^{2})\), we have from (2.2), (3.1), and (3.4) that
$$\begin{aligned} \begin{aligned} &\frac{1}{2}\frac{d}{dt} \bigl\Vert \rho ^{1/2}\theta \bigr\Vert _{L^{2}}^{2}+ \kappa \Vert \nabla \theta \Vert _{L^{2}}^{2} \\ &\quad=- \int R\rho \theta ^{2} \operatorname{div} \mathbf {u}\,dx+\nu \int \vert \operatorname{curl} \mathbf {B}\vert ^{2} \theta \,dx \\ & \qquad{}+ \int \bigl[2\mu (\rho, \theta ) \bigl\vert \mathcal{D}(\mathbf {u}) \bigr\vert ^{2}+ \lambda (\rho, \theta ) (\operatorname{div} \mathbf {u})^{2} \bigr]\theta \,dx \\ &\quad \leqslant C \Vert \nabla \mathbf {u}\Vert _{L^{2}} \Vert \rho \Vert _{L^{6}} \Vert \theta \Vert _{L^{6}}^{2}+C \Vert \theta \Vert _{L^{6}} \Vert \nabla \mathbf {B}\Vert _{L^{3}} \Vert \nabla \mathbf {B}\Vert _{L^{2}}+C \Vert \theta \Vert _{L^{6}} \Vert \nabla \mathbf {u}\Vert _{L^{3}} \Vert \nabla \mathbf {u}\Vert _{L^{2}} \\ & \quad\leqslant C \Vert \nabla \mathbf {u}\Vert _{L^{2}} \Vert \nabla \theta \Vert _{L^{2}}^{2}+C \Vert \nabla \theta \Vert _{L^{2}} \Vert \nabla \mathbf {B}\Vert _{L^{2}}^{3/2}+C \Vert \nabla \theta \Vert _{L^{2}} \Vert \nabla \mathbf {u}\Vert _{L^{2}}^{3/2} \\ &\quad \leqslant \biggl(C \Vert \nabla \mathbf {u}\Vert _{L^{2}}+ \frac{\kappa}{4} \biggr) \Vert \nabla \theta \Vert _{L^{2}}^{2}+C \Vert \nabla \mathbf {B}\Vert _{L^{2}}^{3}+C \Vert \nabla \mathbf {u}\Vert _{L^{2}}^{3} \\ & \quad\leqslant \biggl(CA_{1}^{1/2}(T)+\frac{\kappa}{4} \biggr) \Vert \nabla \theta \Vert _{L^{2}}^{2}+CA_{1}^{1/2}(T) \Vert \nabla \mathbf {B}\Vert _{L^{2}}^{2}+CA_{1}^{1/2}(T) \Vert \nabla \mathbf {u}\Vert _{L^{2}}^{2} \\ &\quad \leqslant \biggl(C_{4}C_{0}^{1/8}+ \frac{\kappa}{4} \biggr) \Vert \nabla \theta \Vert _{L^{2}}^{2}+CC_{0}^{1/8} \Vert \nabla \mathbf {B}\Vert _{L^{2}}^{2}+CC_{0}^{1/8} \Vert \nabla \mathbf {u}\Vert _{L^{2}}^{2}. \end{aligned} \end{aligned}$$
Choosing \(C_{0}\leqslant \varepsilon _{2, 1} \triangleq \min \{\varepsilon _{1}, (\frac {\kappa}{4C_{4}})^{8}\}\), we obtain
$$\begin{aligned} \frac{d}{dt} \bigl\Vert \rho ^{1/2}\theta \bigr\Vert _{L^{2}}^{2}+\kappa \Vert \nabla \theta \Vert _{L^{2}}^{2} \leqslant CC_{0}^{1/8} \Vert \nabla \mathbf {B}\Vert _{L^{2}}^{2}+CC_{0}^{1/8} \Vert \nabla \mathbf {u}\Vert _{L^{2}}^{2}. \end{aligned}$$
(3.8)
Multiplying (1.1)4 by B in \(L^{2}\), we obtain from (2.2), (3.1), and (3.4) that
$$\begin{aligned} \begin{aligned} \frac{1}{2}\frac{d}{dt} \Vert \mathbf {B}\Vert _{L^{2}}^{2}+\nu \Vert \nabla \mathbf {B}\Vert _{L^{2}}^{2} &=-\frac{1}{2} \int \mathbf {B}^{2} \operatorname{div} \mathbf {u}\,dx+ \int \mathbf {B}\cdot \nabla \mathbf {u}\cdot \mathbf {B}\,dx \\ &\leqslant C \Vert \nabla \mathbf {B}\Vert _{L^{2}} \Vert \mathbf {B}\Vert _{L^{3}} \Vert \nabla \mathbf {u}\Vert _{L^{2}} \\ &\leqslant C \Vert \nabla \mathbf {B}\Vert _{L^{2}}^{2} \Vert \nabla \mathbf {u}\Vert _{L^{2}} \\ &\leqslant CA_{1}^{1/2}(T) \Vert \nabla \mathbf {B}\Vert _{L^{2}}^{2} \\ &\leqslant C_{5}C_{0}^{1/8} \Vert \nabla \mathbf {B}\Vert _{L^{2}}^{2}, \end{aligned} \end{aligned}$$
where we have used the Poincaré inequality and the estimates
$$\begin{aligned} \Vert \mathbf {B}\Vert _{L^{3}}\leqslant C \Vert \mathbf {B}\Vert _{L^{2}}^{1/2} \Vert \nabla \mathbf {B}\Vert _{L^{2}}^{1/2} \leqslant C \Vert \nabla \mathbf {B}\Vert _{L^{2}}. \end{aligned}$$
Thus
$$\begin{aligned} \frac{d}{dt} \Vert \mathbf {B}\Vert _{L^{2}}^{2}+ \nu \Vert \nabla \mathbf {B}\Vert _{L^{2}}^{2} \leqslant 0, \end{aligned}$$
(3.9)
provided that \(C_{0}\leqslant \varepsilon _{2, 2} \triangleq \min \{1, ( \frac {\nu}{2C_{5}})^{8}\}\).
Calculating (3.7)+\(\frac {2C_{2}}{\kappa}\times \)(3.8)+\(\frac {2C_{3}}{\nu}\times \)(3.9) yields
$$\begin{aligned} \begin{aligned} &\frac{d}{dt} \biggl( \bigl\Vert \rho ^{1/2}\mathbf {u}\bigr\Vert _{L^{2}}^{2}+ \frac {2C_{2}}{\kappa} \bigl\Vert \rho ^{1/2}\theta \bigr\Vert _{L^{2}}^{2}+ \frac {2C_{3}}{\nu} \Vert \mathbf {B}\Vert _{L^{2}}^{2} \biggr)+\underline{\mu} \Vert \nabla \mathbf {u}\Vert _{L^{2}}^{2}+C_{2} \Vert \nabla \theta \Vert _{L^{2}}^{2}+C_{3} \Vert \nabla \mathbf {B}\Vert _{L^{2}}^{2} \\ & \quad\leqslant C_{6}C_{0}^{1/8} \Vert \nabla \mathbf {B}\Vert _{L^{2}}^{2}+C_{7}C_{0}^{1/8} \Vert \nabla \mathbf {u}\Vert _{L^{2}}^{2}, \end{aligned} \end{aligned}$$
and thus
$$\begin{aligned} \frac{d}{dt} \bigl\Vert \bigl(\rho ^{1/2}\mathbf {u}, \rho ^{1/2}\theta, \mathbf {B}\bigr) \bigr\Vert _{L^{2}}^{2}+ \bigl\Vert (\nabla \mathbf {u}, \nabla \theta, \nabla \mathbf {B}) \bigr\Vert _{L^{2}}^{2} \leqslant 0, \end{aligned}$$
(3.10)
provided that \(C_{0}\leqslant \varepsilon _{2} \triangleq \min \{ \varepsilon _{2, 1}, \varepsilon _{2, 2}, (\frac {C_{3}}{2C_{6}})^{8}, ( \frac {\underline{\mu}}{2C_{7}})^{8}\}\).
Integrating (3.10) over \([0, T]\) and using the Poincaré inequality, we get
$$\begin{aligned} \begin{aligned} &\sup_{t\in [0, T]} \bigl\Vert \bigl(\rho ^{1/2}\mathbf {u}, \rho ^{1/2} \theta, \mathbf {B}\bigr) \bigr\Vert _{L^{2}}^{2}+ \int _{0}^{T} \bigl\Vert (\nabla \mathbf {u}, \nabla \theta, \nabla \mathbf {B}) \bigr\Vert _{L^{2}}^{2} \,dt \\ &\quad \leqslant C \bigl\Vert (\nabla \mathbf {u}_{0}, \nabla \theta _{0}, \nabla \mathbf {B}_{0}) \bigr\Vert _{L^{2}}^{2} \\ & \quad\leqslant CC_{0}. \end{aligned} \end{aligned}$$
(3.11)
Multiplying (3.10) by t, integrating the result over \([0, T]\), and using the Poincaré inequality again, we have
$$\begin{aligned} \begin{aligned} &\sup_{t\in [0, T]}t \bigl\Vert \bigl(\rho ^{1/2}\mathbf {u}, \rho ^{1/2} \theta, \mathbf {B}\bigr) \bigr\Vert _{L^{2}}^{2}+ \int _{0}^{T}t \bigl\Vert (\nabla \mathbf {u}, \nabla \theta, \nabla \mathbf {B}) \bigr\Vert _{L^{2}}^{2} \,dt \\ &\quad \leqslant C \bigl\Vert (\nabla \mathbf {u}, \nabla \theta, \nabla \mathbf {B}) \bigr\Vert _{L^{2}}^{2} \\ & \quad\leqslant CC_{0}, \end{aligned} \end{aligned}$$
(3.12)
which, together with (3.11), leads to (3.6) for \(i=0, 1\). Similarly to the proof of (3.11) and (3.12), we can obtain (3.6) for \(i=3, 4, \dots, 32\). The proof of the lemma is therefore completed. □
Before stating the following lemma, we define
$$\begin{aligned} \sigma (T)\triangleq \min \{1, T\}. \end{aligned}$$
Then we be establish a uniform upper bound for \(A_{1}(T)\).
Lemma 3.5
Let \((\rho, \mathbf{ u}, \theta, \mathbf{ B})\) be a smooth solution of (1.1)–(1.4) satisfying (3.1). Then there exists a positive constant \(\varepsilon _{3}\), depending only on \(\kappa, \nu, R, c_{v}, \underline{\mu}\), ρ̄, Ω, \(g_{1}\), and \(g_{2}\), such that
$$\begin{aligned} \sup_{t\in [0, T]} \bigl\Vert (\nabla \mathbf{ u}, \nabla \mathbf{ B}) \bigr\Vert _{L^{2}}^{2}+ \int _{0}^{T} \bigl\Vert \bigl(\rho ^{1/2}\mathbf{ u}_{t}, \mathbf{ B}_{t}, \nabla ^{2} \mathbf {B}\bigr) \bigr\Vert _{L^{2}}^{2} \,dt \leqslant C_{0}^{1/4} \end{aligned}$$
(3.13)
and
$$\begin{aligned} \sup_{t\in [0, T]}t^{i} \bigl\Vert (\nabla \mathbf{ u}, \nabla \mathbf{ B}) \bigr\Vert _{L^{2}}^{2}+ \int _{0}^{T} t^{i} \bigl\Vert \bigl( \rho ^{1/2}\mathbf{ u}_{t}, \mathbf{ B}_{t}, \nabla ^{2} \mathbf {B}\bigr) \bigr\Vert _{L^{2}}^{2} \,dt \leqslant CC_{0}^{5/16} \end{aligned}$$
(3.14)
for \(i=1, 2, \dots, 8\), provided that \(C_{0}\leqslant \varepsilon _{3}\).
Proof
We multiply (1.1)2 by \(\mathbf {u}_{t}\) and integrate the result over Ω:
$$\begin{aligned} \begin{aligned} &\frac{d}{dt} \int \biggl[\mu (\rho, \theta ) \bigl\vert \mathcal{D}(\mathbf {u}) \bigr\vert ^{2}+\frac{\lambda (\rho, \theta )}{2}(\operatorname{div} \mathbf {u})^{2} \biggr] \,dx+ \bigl\Vert \rho ^{1/2}\mathbf {u}_{t} \bigr\Vert _{L^{2}}^{2} \\ &\quad = \int \biggl[\mu _{t}(\rho, \theta ) \bigl\vert \mathcal{D}( \mathbf {u}) \bigr\vert ^{2}+ \frac{\lambda _{t}(\rho, \theta )}{2}(\operatorname{div} \mathbf {u})^{2} \biggr] \,dx- \int \rho \mathbf {u}\cdot \nabla \mathbf {u}\cdot \mathbf {u}_{t} \,dx \\ &\qquad{} + \int R\rho \theta \operatorname{div} \mathbf {u}_{t} \,dx- \int \mathbf {B}\cdot \nabla \mathbf {u}_{t} \cdot \mathbf {B}\,dx+ \int \mathbf {B}^{2} \operatorname{div} \mathbf {u}_{t} \,dx \\ &\quad \triangleq \sum_{j=1}^{4} N_{j}. \end{aligned} \end{aligned}$$
(3.15)
The right-hand side terms of (3.15) can be estimated as follows. By (3.4) and the fact that \(\mu (\rho, \theta ), \lambda (\rho, \theta )\in C^{1}(\mathbb{R}^{2})\) we have that
$$\begin{aligned} \begin{aligned} & \biggl\vert \int \biggl[\mu _{\rho}\rho _{t} \bigl\vert \mathcal{D}( \mathbf {u}) \bigr\vert ^{2}+\frac{\lambda _{\rho}\rho _{t} }{2}( \operatorname{div} \mathbf {u})^{2} \biggr] \,dx \biggr\vert \\ & \quad\leqslant C \int \bigl( \vert \nabla \rho \vert \vert \mathbf {u}\vert \vert \nabla \mathbf {u}\vert ^{2}+ \vert \rho \vert \vert \nabla \mathbf {u}\vert ^{3} \bigr)\,dx \\ &\quad \leqslant \Vert \nabla \rho \Vert _{L^{3}} \Vert \mathbf {u}\Vert _{L^{6}} \Vert \nabla \mathbf {u}\Vert _{L^{4}}^{2}+C \Vert \nabla \mathbf {u}\Vert _{L^{3}}^{3} \\ &\quad \leqslant C \Vert \nabla \mathbf {u}\Vert _{L^{2}}^{3/2} \Vert \nabla \mathbf {u}\Vert _{H^{1}}^{3/2} \leqslant C \Vert \nabla \mathbf {u}\Vert _{L^{2}}^{3/2} \end{aligned} \end{aligned}$$
and
$$\begin{aligned} \begin{aligned} \biggl\vert \int \biggl[\mu _{\theta}\theta _{t} \bigl\vert \mathcal{D}( \mathbf {u}) \bigr\vert ^{2}+\frac{\lambda _{\theta}\theta _{t} }{2}( \operatorname{div} \mathbf {u})^{2} \biggr] \,dx \biggr\vert & \leqslant \Vert \theta _{t} \Vert _{L^{2}} \Vert \nabla \mathbf {u}\Vert _{L^{6}} \Vert \nabla \mathbf {u}\Vert _{L^{3}} \\ &\leqslant C \Vert \nabla \theta _{t} \Vert _{L^{2}} \Vert \nabla \mathbf {u}\Vert _{H^{1}}^{1/2} \Vert \nabla \mathbf {u}\Vert _{L^{2}}^{1/2}, \end{aligned} \end{aligned}$$
and thus
$$\begin{aligned} \begin{aligned} N_{1} &\leqslant C \Vert \nabla \mathbf {u}\Vert _{L^{2}}^{3/2}+C \Vert \nabla \theta _{t} \Vert _{L^{2}} \Vert \nabla \mathbf {u}\Vert _{H^{1}}^{1/2} \Vert \nabla \mathbf {u}\Vert _{L^{2}}^{1/2} \\ &\leqslant C A_{1}^{1/4}(T) \Vert \nabla \mathbf {u}\Vert _{L^{2}} +C \Vert \nabla \theta _{t} \Vert _{L^{2}} \Vert \nabla \mathbf {u}\Vert _{H^{1}}^{1/2} \Vert \nabla \mathbf {u}\Vert _{L^{2}}^{1/2} \\ &\leqslant C \Vert \nabla \mathbf {u}\Vert _{L^{2}}+C \Vert \nabla \theta _{t} \Vert _{L^{2}} \Vert \nabla \mathbf {u}\Vert _{L^{2}}^{1/2} \Vert \nabla \mathbf {u}\Vert _{H^{1}}^{1/2}. \end{aligned} \end{aligned}$$
By (3.4) and the Poincaré inequality we have
$$\begin{aligned} N_{2} &=- \int \rho \mathbf {u}\cdot \nabla \mathbf {u}\cdot \mathbf {u}_{t} \,dx \\ &\leqslant \frac{1}{2} \bigl\Vert \rho ^{1/2}\mathbf {u}_{t} \bigr\Vert _{L^{2}}^{2}+C \Vert \mathbf {u}\Vert _{L^{6}}^{2} \Vert \nabla \mathbf {u}\Vert _{L^{3}}^{2} \\ &\leqslant \frac{1}{2} \bigl\Vert \rho ^{1/2}\mathbf {u}_{t} \bigr\Vert _{L^{2}}^{2}+CA_{1}(T) \Vert \nabla \mathbf {u}\Vert _{L^{2}} \\ &\leqslant \frac{1}{2} \bigl\Vert \rho ^{1/2}\mathbf {u}_{t} \bigr\Vert _{L^{2}}^{2}+C \Vert \nabla \mathbf {u}\Vert _{L^{2}}, \\ N_{3}&= \int R\rho \theta \operatorname{div} \mathbf {u}_{t} \,dx\leqslant C \Vert \theta \Vert _{L^{2}} \Vert \nabla \mathbf {u}_{t} \Vert _{L^{2}}\leqslant C \Vert \nabla \theta \Vert _{L^{2}} \Vert \nabla \mathbf {u}_{t} \Vert _{L^{2}}, \end{aligned}$$
and
$$\begin{aligned} \begin{aligned} N_{4} &=- \int \mathbf {B}\cdot \nabla \mathbf {u}_{t} \cdot \mathbf {B}\,dx+ \int \mathbf {B}^{2} \operatorname{div} \mathbf {u}_{t} \,dx \\ &\leqslant C \Vert \mathbf {B}\Vert _{L^{6}} \Vert \mathbf {B}\Vert _{L^{3}} \Vert \nabla \mathbf {u}_{t} \Vert _{L^{2}} \\ &\leqslant C \Vert \nabla \mathbf {B}\Vert _{L^{2}} \Vert \nabla \mathbf {u}_{t} \Vert _{L^{2}}. \end{aligned} \end{aligned}$$
Plugging \(N_{j}\ (j=1, 2, 3, 4)\) into (3.15) yields
$$\begin{aligned} \begin{aligned} &\frac{d}{dt} \int \biggl[\mu (\rho, \theta ) \bigl\vert \mathcal{D}(\mathbf {u}) \bigr\vert ^{2}+\frac{\lambda (\rho, \theta )}{2}(\operatorname{div} \mathbf {u})^{2} \biggr] \,dx+ \bigl\Vert \rho ^{1/2}\mathbf {u}_{t} \bigr\Vert _{L^{2}}^{2} \\ & \quad\leqslant C \Vert \nabla \mathbf {u}\Vert _{L^{2}}+C \Vert \nabla \theta _{t} \Vert _{L^{2}} \Vert \nabla \mathbf {u}\Vert _{L^{2}}^{1/2} \Vert \nabla \mathbf {u}\Vert _{H^{1}}^{1/2} \\ &\qquad{} +C \Vert \nabla \theta \Vert _{L^{2}} \Vert \nabla \mathbf {u}_{t} \Vert _{L^{2}}+C \Vert \nabla \mathbf {B}\Vert _{L^{2}} \Vert \nabla \mathbf {u}_{t} \Vert _{L^{2}}. \end{aligned} \end{aligned}$$
(3.16)
Multiplying (1.1)4 by \(\mathbf {B}_{t}\) in \(L^{2}\), we have from (2.3) and the Poincaré inequality that
$$\begin{aligned} \begin{aligned} &\frac{\nu}{2}\frac{d}{dt} \Vert \nabla \mathbf {B}\Vert _{L^{2}}^{2}+ \Vert \mathbf {B}_{t} \Vert _{L^{2}}^{2} \\ & \quad=- \int \mathbf {u}\cdot \nabla \mathbf {B}\cdot \mathbf {B}_{t} \,dx+ \int \mathbf {B}\cdot \nabla \mathbf {u}\cdot \mathbf {B}_{t} \,dx- \int \mathbf {B}\cdot \mathbf {B}_{t} \operatorname{div} \mathbf {u}\,dx \\ &\quad \leqslant C \Vert \mathbf {u}\Vert _{L^{6}} \Vert \nabla \mathbf {B}\Vert _{L^{3}} \Vert \mathbf {B}_{t} \Vert _{L^{2}}+C \Vert \nabla \mathbf {u}\Vert _{L^{2}} \Vert \mathbf {B}\Vert _{L^{ \infty}} \Vert \mathbf {B}_{t} \Vert _{L^{2}} \\ & \quad\leqslant \frac{1}{2} \Vert \mathbf {B}_{t} \Vert _{L^{2}}^{2}+C \Vert \nabla \mathbf {u}\Vert _{L^{2}}^{2} \Vert \nabla \mathbf {B}\Vert _{H^{1}}^{2}, \end{aligned} \end{aligned}$$
and thus from (3.1) and (3.4) it follows that
$$\begin{aligned} \frac{d}{dt} \Vert \nabla \mathbf {B}\Vert _{L^{2}}^{2}+ \Vert \mathbf {B}_{t} \Vert _{L^{2}}^{2} \leqslant C \Vert \nabla \mathbf {u}\Vert _{L^{2}}^{2} \Vert \nabla \mathbf {B}\Vert _{H^{1}}^{2} \leqslant C \Vert \nabla \mathbf {u}\Vert _{L^{2}}. \end{aligned}$$
(3.17)
On the other hand, from (1.1)4 it follows that
$$\begin{aligned} \begin{aligned} \bigl\Vert \nabla ^{2} \mathbf {B}\bigr\Vert _{L^{2}}^{2} &\leqslant C \bigl( \Vert \mathbf {B}_{t} \Vert _{L^{2}}^{2}+ \Vert \mathbf {u}\Vert _{L^{6}}^{2} \Vert \nabla \mathbf {B}\Vert _{L^{3}}^{2}+ \Vert \mathbf {B}\Vert _{L^{\infty}}^{2} \Vert \nabla \mathbf {u}\Vert _{L^{2}}^{2} \bigr) \\ &\leqslant \frac{1}{2} \bigl\Vert \nabla ^{2} \mathbf {B}\bigr\Vert _{L^{2}}^{2}+C \bigl( \Vert \mathbf {B}_{t} \Vert _{L^{2}}^{2}+ \Vert \nabla \mathbf {u}\Vert _{L^{2}}^{4} \Vert \nabla \mathbf {B}\Vert _{L^{2}}^{2} \bigr) \\ &\leqslant \frac{1}{2} \bigl\Vert \nabla ^{2} \mathbf {B}\bigr\Vert _{L^{2}}^{2}+C \Vert \mathbf {B}_{t} \Vert _{L^{2}}^{2}+ C \Vert \nabla \mathbf {u}\Vert _{L^{2}}. \end{aligned} \end{aligned}$$
(3.18)
Combining (3.16), (3.17), and (3.18), we have
$$\begin{aligned} \begin{aligned} &\frac{d}{dt} \int \biggl[\mu (\rho, \theta ) \bigl\vert \mathcal{D}(\mathbf {u}) \bigr\vert ^{2}+\frac{\lambda (\rho, \theta )}{2}(\operatorname{div} \mathbf {u})^{2}+ \vert \nabla \mathbf {B}\vert ^{2} \biggr] \,dx+ \bigl\Vert \bigl(\rho ^{1/2}\mathbf {u}_{t}, \mathbf {B}_{t}, \nabla ^{2} \mathbf {B}\bigr) \bigr\Vert _{L^{2}}^{2} \\ & \quad\leqslant C \Vert \nabla \mathbf {u}\Vert _{L^{2}}+C \Vert \nabla \theta _{t} \Vert _{L^{2}} \Vert \nabla \mathbf {u}\Vert _{L^{2}}^{1/2} \Vert \nabla \mathbf {u}\Vert _{H^{1}}^{1/2} \\ &\qquad{} +C \Vert \nabla \theta \Vert _{L^{2}} \Vert \nabla \mathbf {u}_{t} \Vert _{L^{2}}+C \Vert \nabla \mathbf {B}\Vert _{L^{2}} \Vert \nabla \mathbf {u}_{t} \Vert _{L^{2}}. \end{aligned} \end{aligned}$$
(3.19)
The Hölder inequality, together with (3.1) and (3.6), yields that for \(i=1, 2, \dots, 8\),
$$\begin{aligned} \begin{aligned} \int _{0}^{T} t^{i} \Vert \nabla \mathbf {u}\Vert _{L^{2}} \,dt \leqslant{}& \biggl( \int _{0}^{\sigma (T)}t^{2i} \Vert \nabla \mathbf {u}\Vert _{L^{2}}^{2} \,dt \biggr)^{1/2} \biggl( \int _{0}^{\sigma (T)} \,dt \biggr)^{1/2} \\ &{} + \biggl( \int _{\sigma (T)}^{T}t^{2i+2} \Vert \nabla \mathbf {u}\Vert _{L^{2}}^{2}\,dt \biggr)^{1/2} \biggl( \int _{\sigma (T)}^{T} t^{-2}\,dt \biggr)^{1/2} \\ \leqslant{}& CC_{0}^{1/2}. \end{aligned} \end{aligned}$$
(3.20)
By (3.1), (3.5), (3.6), and (3.20) we have that
$$\begin{aligned} \begin{aligned} & \int _{0}^{T} t^{i} \Vert \nabla \theta _{t} \Vert _{L^{2}} \Vert \nabla \mathbf {u}\Vert _{L^{2}}^{1/2} \Vert \nabla \mathbf {u}\Vert _{H^{1}}^{1/2} \,dt \\ &\quad \leqslant \biggl( \int _{0}^{T} \Vert \nabla \theta _{t} \Vert _{L^{2}}^{2} \,dt \biggr)^{1/2} \biggl( \int _{0}^{T} \Vert \nabla \mathbf {u}\Vert _{H^{1}}^{2}\,dt \biggr)^{1/4} \biggl( \int _{T}^{T}t^{4i} \Vert \nabla \mathbf {u}\Vert _{L^{2}}^{2}\,dt \biggr)^{1/4} \\ &\quad \leqslant CA_{4}^{1/2}(T)C_{0}^{1/4} \\ &\qquad{}\times \biggl[ \int _{0}^{T} \bigl( \bigl\Vert \nabla ^{2} \mathbf {B}\bigr\Vert _{L^{2}}^{2}+ \bigl\Vert \rho ^{1/2}\mathbf {u}_{t} \bigr\Vert _{L^{2}}^{2}+ \Vert \nabla \mathbf {u}\Vert _{L^{2}}+ \Vert \nabla \mathbf {B}\Vert _{L^{2}}+ \Vert \nabla \theta \Vert _{L^{2}}^{2} \bigr)\,dt \biggr]^{1/4} \\ & \quad\leqslant CC_{0}^{5/16} \end{aligned} \end{aligned}$$
(3.21)
and
$$\begin{aligned} \begin{aligned} \int _{0}^{T} t^{i} \Vert \nabla \theta \Vert _{L^{2}} \Vert \nabla \mathbf {u}_{t} \Vert _{L^{2}} \,dt &\leqslant \biggl( \int _{0}^{T} t^{2i} \Vert \nabla \theta \Vert _{L^{2}}^{2} \,dt \biggr)^{1/2} \biggl( \int _{0}^{T} \Vert \nabla \mathbf {u}_{t} \Vert _{L^{2}}^{2}\,dt \biggr)^{1/2} \\ &\leqslant CA_{3}^{1/2}(T)C_{0}^{1/2} \\ &\leqslant CC_{0}^{1/2}. \end{aligned} \end{aligned}$$
(3.22)
Integrating (3.19) over \([0, T]\) and using Lemma 2.1, (3.1), (3.6), and (3.20)–(3.22) for \(i=0\), we obtain
$$\begin{aligned} \begin{aligned} &\sup_{t\in [0, T]} \bigl\Vert (\nabla \mathbf {u}, \nabla \mathbf {B}) \bigr\Vert _{L^{2}}^{2} \,dx+ \int _{0}^{T} \bigl\Vert \bigl(\rho ^{1/2}\mathbf {u}_{t}, \mathbf {B}_{t}, \nabla ^{2} \mathbf {B}\bigr) \bigr\Vert _{L^{2}}^{2} \,dt \\ &\quad \leqslant C \bigl\Vert (\nabla \mathbf {u}_{0}, \nabla \mathbf {B}_{0}) \bigr\Vert _{L^{2}}^{2}+CC_{0}^{1/2}+CC_{0}^{5/16} \\ & \quad\leqslant C_{8}C_{0}^{5/16}\leqslant C_{0}^{1/4}, \end{aligned} \end{aligned}$$
(3.23)
provided that \(C_{0} \leqslant \varepsilon _{3}\triangleq \min \{\varepsilon _{2}, C_{8}^{-16} \} \). Then we immediately get (3.13).
Next, multiplying (3.19) by t and integrating the result over \([0, T]\), it follows from (3.6) and (3.20)–(3.22) that
$$\begin{aligned} \sup_{t\in [0, T]}t \bigl\Vert (\nabla \mathbf {u}, \nabla \mathbf {B}) \bigr\Vert _{L^{2}}^{2} \,dx+ \int _{0}^{T}t \bigl\Vert \bigl(\rho ^{1/2}\mathbf {u}_{t}, \mathbf {B}_{t}, \nabla ^{2} \mathbf {B}\bigr) \bigr\Vert _{L^{2}}^{2} \,dt \leqslant CC_{0}^{5/16}. \end{aligned}$$
(3.24)
Similarly to the proof of (3.23) and (3.24), we can obtain (3.14) for \(i=2, 3, \dots, 8\). The proof of the lemma is therefore completed. □
Lemma 3.6
Let \((\rho, \mathbf{ u}, \theta, \mathbf{ B})\) be a smooth solution of (1.1)–(1.4) satisfying (3.1). Then there exists a positive constant \(\varepsilon _{4}\), depending only on \(\kappa, \nu, R, c_{v}, \underline{\mu}\), ρ̄, Ω, \(g_{1}\), and \(g_{2}\), such that
$$\begin{aligned} \sup_{t\in [0, T]} \bigl\Vert \bigl(\rho ^{1/2}\mathbf{ u}_{t}, \mathbf{ B}_{t}, \nabla ^{2}\mathbf{ B} \bigr) \bigr\Vert _{L^{2}}^{2}+ \int _{0}^{T} \bigl\Vert (\nabla \mathbf{ u}_{t}, \nabla \mathbf{ B}_{t}) \bigr\Vert _{L^{2}}^{2} \,dt\leqslant 2K_{1}, \end{aligned}$$
(3.25)
provided that \(C_{0}\leqslant \varepsilon _{4}\).
Proof
By (1.1)1 and (1.1)2 we have that
$$\begin{aligned} \begin{aligned} &\rho \mathbf {u}_{tt}+\rho \mathbf {u}\cdot \nabla \mathbf {u}_{t}-\operatorname{div} \bigl(2\mu \mathcal{D}(\mathbf {u}_{t}) \bigr)-\nabla ( \lambda \operatorname{div} \mathbf {u}_{t} ) \\ &\quad =\operatorname{div}(\rho \mathbf {u}) (\mathbf {u}_{t}+\mathbf {u}\cdot \nabla \mathbf {u})-\rho \mathbf {u}_{t} \cdot \nabla \mathbf {u}+ \operatorname{div} \bigl[ (2\mu _{\rho}\rho _{t} +2\mu _{\theta}\theta _{t} ) \mathcal{D}(\mathbf {u}) \bigr] \\ &\qquad{} +\nabla \bigl[ (\lambda _{\rho}\rho _{t}+\lambda _{ \theta}\theta _{t} )\operatorname{div} \mathbf {u}\bigr]-\nabla P_{t}+ \mathbf {B}\cdot \nabla \mathbf {B}_{t}+\mathbf {B}_{t} \cdot \nabla \mathbf {B}- \nabla \mathbf {B}_{t} \cdot \mathbf {B}-\nabla \mathbf {B}\cdot \mathbf {B}_{t}. \end{aligned} \end{aligned}$$
(3.26)
Multiplying (3.26) by \(\mathbf {u}_{t}\) and integrating the resulting equation over \([0, 1]\), we get
$$\begin{aligned} \begin{aligned} &\frac{1}{2}\frac{d}{dt} \bigl\Vert \rho ^{1/2}\mathbf {u}_{t} \bigr\Vert _{L^{2}}^{2}+ \int \bigl[2\mu (\rho, \theta ) \bigl\vert \mathcal{D}(\mathbf {u}_{t}) \bigr\vert ^{2}+ \lambda (\rho, \theta ) ( \operatorname{div} \mathbf {u}_{t})^{2} \bigr]\,dx \\ &\quad = \int \operatorname{div}(\rho \mathbf {u}) (\mathbf {u}_{t}+\mathbf {u}\cdot \nabla \mathbf {u})\cdot \mathbf {u}_{t} \,dx- \int \rho \mathbf {u}_{t} \cdot \nabla \mathbf {u}\cdot \mathbf {u}_{t} \,dx+ \int P_{t} \operatorname{div} \mathbf {u}_{t} \,dx \\ &\qquad{} - \int (\mu _{\rho}\rho _{t} +\mu _{\theta} \theta _{t} ) \bigl[ \bigl\vert \mathcal{D}(\mathbf {u}) \bigr\vert ^{2} \bigr]_{t} \,dx-\frac{1}{2} \int (\lambda _{\rho}\rho _{t}+\lambda _{\theta}\theta _{t} ) \bigl[(\operatorname{div} \mathbf {u})^{2} \bigr]_{t} \,dx \\ &\qquad{} + \int ( \mathbf {B}\cdot \nabla \mathbf {B}_{t}+\mathbf {B}_{t} \cdot \nabla \mathbf {B}-\nabla \mathbf {B}_{t} \cdot \mathbf {B}-\nabla \mathbf {B}\cdot \mathbf {B}_{t} ) \cdot \mathbf {u}_{t} \,dx \\ & \quad\triangleq \sum_{j=1}^{6} M_{j}. \end{aligned} \end{aligned}$$
(3.27)
Now we estimate \(M_{j}\ (j=1, 2, \dots, 6)\). It follows from (2.2)–(2.4) and (3.4) that
$$\begin{aligned} M_{1} \leqslant{}& C \int \vert \rho \operatorname{div} \mathbf {u}+ \mathbf {u}\cdot \nabla \rho \vert \bigl( \vert \mathbf {u}_{t} \vert ^{2}+ \vert \mathbf {u}\vert \vert \nabla \mathbf {u}\vert \vert \mathbf {u}_{t} \vert \bigr)\,dx \\ \leqslant{}& C \bigl( \Vert \rho \Vert _{L^{6}} \Vert \nabla \mathbf {u}\Vert _{L^{2}}+ \Vert \mathbf {u}\Vert _{L^{6}} \Vert \nabla \rho \Vert _{L^{2}} \bigr) \Vert \mathbf {u}_{t} \Vert _{L^{6}}^{2} \\ &{} +C \bigl( \Vert \rho \Vert _{L^{6}} \Vert \nabla \mathbf {u}\Vert _{L^{6}}+ \Vert \mathbf {u}\Vert _{L^{\infty}} \Vert \nabla \rho \Vert _{L^{3}} \bigr) \Vert \mathbf {u}\Vert _{L^{\infty}} \Vert \nabla \mathbf {u}\Vert _{L^{2}} \Vert \mathbf {u}_{t} \Vert _{L^{6}} \\ \leqslant {}&C \Vert \nabla \mathbf {u}\Vert _{L^{2}} \Vert \nabla \mathbf {u}_{t} \Vert _{L^{2}}^{2}+C \Vert \nabla \mathbf {u}\Vert _{H^{1}}^{3/2} \Vert \nabla \mathbf {u}\Vert _{L^{2}}^{3/2} \Vert \nabla \mathbf {u}_{t} \Vert _{L^{2}} \\ \leqslant{}& C A_{1}^{1/2}(T) \Vert \nabla \mathbf {u}_{t} \Vert _{L^{2}}^{2}+C \Vert \nabla \mathbf {u}_{t} \Vert _{L^{2}}^{2} \\ \leqslant{}& C_{9}C_{0}^{1/8} \Vert \nabla \mathbf {u}_{t} \Vert _{L^{2}}^{2}+C \Vert \nabla \mathbf {u}\Vert _{L^{2}}^{2}, \\ M_{2}\leqslant{}& \biggl\vert \int \rho \mathbf {u}_{t} \cdot \nabla \mathbf {u}\cdot \mathbf {u}_{t} \,dx \biggr\vert \leqslant C \Vert \rho \Vert _{L^{6}} \Vert \nabla \mathbf {u}\Vert _{L^{2}} \Vert \mathbf {u}_{t} \Vert _{L^{6}}^{2} \\ \leqslant{}& C_{10} C_{0}^{1/8} \Vert \nabla \mathbf {u}_{t} \Vert _{L^{2}}^{2}, \end{aligned}$$
and
$$\begin{aligned} \begin{aligned} M_{3} \leqslant{}& \biggl\vert \int P_{t} \operatorname{div} \mathbf {u}_{t} \,dx \biggr\vert =R \biggl\vert \int (\rho _{t} \theta +\rho \theta _{t}) \operatorname{div} \mathbf {u}_{t} \,dx \biggr\vert \\ \leqslant{}& C \int \bigl( \vert \rho \operatorname{div} \mathbf {u}+\mathbf {u}\cdot \nabla \rho \vert \vert \theta \vert \vert \nabla \mathbf {u}_{t} \vert +\rho \vert \theta _{t} \vert \vert \nabla \mathbf {u}_{t} \vert \bigr)\,dx \\ \leqslant{}& C \bigl( \Vert \rho \Vert _{L^{6}} \Vert \nabla \mathbf {u}\Vert _{L^{6}}+ \Vert \mathbf {u}\Vert _{L^{\infty}} \Vert \nabla \rho \Vert _{L^{3}} \bigr) \Vert \theta \Vert _{L^{6}} \Vert \nabla \mathbf {u}_{t} \Vert _{L^{2}} \\ &{} +C \bigl\Vert \rho ^{1/2}\theta _{t} \bigr\Vert _{L^{2}} \Vert \nabla \mathbf {u}_{t} \Vert _{L^{2}} \\ \leqslant{}& \frac{1}{8}\underline{\mu} \Vert \nabla \mathbf {u}_{t} \Vert _{L^{2}}^{2}+C \bigl\Vert \rho ^{1/2}\theta _{t} \bigr\Vert _{L^{2}}^{2}+C \Vert \nabla \theta \Vert _{L^{2}}^{2}. \end{aligned} \end{aligned}$$
By (1.1)1 and (3.4) we have
$$\begin{aligned} M_{4} \leqslant{}& \biggl\vert \int (\mu _{\rho}\rho _{t} + \mu _{\theta} \theta _{t} ) \bigl[ \bigl\vert \mathcal{D}(\mathbf {u}) \bigr\vert ^{2} \bigr]_{t} \,dx \biggr\vert \\ \leqslant{}& C \int \bigl[ \vert \mu _{\rho} \vert \vert \rho \operatorname{div} \mathbf {u}+ \mathbf {u}\cdot \nabla \rho \vert \vert \nabla \mathbf {u}\vert \vert \nabla \mathbf {u}_{t} \vert + \vert \mu _{\theta} \vert \vert \theta _{t} \vert \vert \nabla \mathbf {u}\vert \vert \nabla \mathbf {u}_{t} \vert \bigr] \,dx \\ \leqslant {}&C \bigl( \Vert \rho \Vert _{L^{\infty}} \Vert \nabla \mathbf {u}\Vert _{L^{3}}+ \Vert \mathbf {u}\Vert _{L^{\infty}} \Vert \nabla \rho \Vert _{L^{3}} \bigr) \Vert \nabla \mathbf {u}\Vert _{L^{6}} \Vert \nabla \mathbf {u}_{t} \Vert _{L^{2}} \\ &{} +C \Vert \theta _{t} \Vert _{L^{6}} \Vert \nabla \mathbf {u}\Vert _{L^{3}} \Vert \nabla \mathbf {u}_{t} \Vert _{L^{2}} \\ \leqslant{}& C \Vert \nabla \mathbf {u}\Vert _{L^{2}}^{1/2} \Vert \nabla \mathbf {u}_{t} \Vert _{L^{2}}+C \Vert \nabla \mathbf {u}\Vert _{L^{2}}^{1/2} \Vert \nabla \theta _{t} \Vert _{L^{2}} \Vert \nabla \mathbf {u}_{t} \Vert _{L^{2}} \\ \leqslant {}&\frac{1}{8}\underline{\mu} \Vert \nabla \mathbf {u}_{t} \Vert _{L^{2}}^{2}+C \Vert \nabla \mathbf {u}\Vert _{L^{2}}+C \Vert \nabla \mathbf {u}\Vert \Vert \nabla \theta _{t} \Vert _{L^{2}}^{2}, \\ M_{5} \leqslant{}& \biggl\vert \int (\lambda _{\rho}\rho _{t}+ \lambda _{\theta}\theta _{t} ) \bigl[(\operatorname{div} \mathbf {u})^{2} \bigr]_{t} \,dx \biggr\vert \\ \leqslant{}& \frac{1}{8}\underline{\mu} \Vert \nabla \mathbf {u}_{t} \Vert _{L^{2}}^{2}+C \Vert \nabla \mathbf {u}\Vert _{L^{2}}+C \Vert \nabla \mathbf {u}\Vert \Vert \nabla \theta _{t} \Vert _{L^{2}}^{2}, \end{aligned}$$
and
$$\begin{aligned} \begin{aligned} M_{6} &\leqslant \biggl\vert \int ( \mathbf {B}\cdot \nabla \mathbf {B}_{t}+\mathbf {B}_{t} \cdot \nabla \mathbf {B}-\nabla \mathbf {B}_{t} \cdot \mathbf {B}-\nabla \mathbf {B}\cdot \mathbf {B}_{t} ) \cdot \mathbf {u}_{t} \,dx \biggr\vert \\ &\leqslant C \Vert \mathbf {B}\Vert _{L^{3}} \Vert \nabla \mathbf {B}_{t} \Vert _{L^{2}} \Vert \mathbf {u}_{t} \Vert _{L^{6}}+C \Vert \mathbf {B}\Vert _{L^{3}} \Vert \nabla \mathbf {u}_{t} \Vert _{L^{2}} \Vert \mathbf {B}_{t} \Vert _{L^{6}} \\ &\leqslant \frac{1}{8}\underline{\mu} \Vert \nabla \mathbf {u}_{t} \Vert _{L^{2}}^{2}+C \Vert \nabla \mathbf {B}_{t} \Vert _{L^{2}}^{2}. \end{aligned} \end{aligned}$$
Substituting \(M_{1}\)–\(M_{6}\) into (3.27) and using (3.1) and Lemma 2.1, we obtain
$$\begin{aligned} \begin{aligned} &\frac{d}{dt} \bigl\Vert \rho ^{1/2}\mathbf {u}_{t} \bigr\Vert _{L^{2}}^{2}+ \Vert \nabla \mathbf {u}_{t} \Vert _{L^{2}}^{2} \\ & \quad\leqslant C_{11} \Vert \nabla \mathbf {B}_{t} \Vert _{L^{2}}^{2}+C \Vert \nabla \mathbf {u}\Vert _{L^{2}}+C \bigl\Vert \rho ^{1/2}\theta _{t} \bigr\Vert _{L^{2}}^{2}+C \Vert \nabla \theta \Vert _{L^{2}}^{2}+CC_{0}^{1/8} \Vert \nabla \theta _{t} \Vert _{L^{2}}^{2}, \end{aligned} \end{aligned}$$
(3.28)
provided that \(C_{0} \leqslant \varepsilon _{4, 1}\triangleq \min \{1, ( \frac {\underline{\mu}}{8C_{9}})^{8}, ( \frac {\underline{\mu}}{8C_{10}})^{8}\}\).
Differentiating (1.1)4 with respect to t yields
$$\begin{aligned} \mathbf {B}_{t t}-\nu \Delta \mathbf {B}_{t}=- \mathbf {u}_{t} \cdot \nabla \mathbf {B}-\mathbf {u}\cdot \nabla \mathbf {B}_{t}+\mathbf {B}_{t}\cdot \nabla \mathbf {u}+\mathbf {B}\cdot \nabla \mathbf {u}_{t}-\mathbf {B}_{t} \operatorname{div} \mathbf {u}- \mathbf {B}\operatorname{div} \mathbf {u}_{t}. \end{aligned}$$
(3.29)
Multiplying (3.29) by \(\mathbf {B}_{t}\) in \(L^{2}\) and integrating by parts, we obtain
$$\begin{aligned} &\frac{1}{2}\frac{d}{dt} \Vert \mathbf {B}_{t} \Vert _{L^{2}}^{2}+\nu \Vert \nabla \mathbf {B}_{t} \Vert _{L^{2}}^{2}\\ &\quad \leqslant C \bigl( \Vert \nabla \mathbf {u}_{t} \Vert _{L^{2}} \Vert \mathbf {B}\Vert _{L^{3}} \Vert \mathbf {B}_{t} \Vert _{L^{6}}+ \Vert \mathbf {u}_{t} \Vert _{L^{6}} \Vert \mathbf {B}\Vert _{L^{3}} \Vert \nabla \mathbf {B}_{t} \Vert _{L^{2}} \bigr) \\ &\qquad{} +C \Vert \mathbf {u}\Vert _{L^{6}} \Vert \nabla \mathbf {B}_{t} \Vert _{L^{2}} \Vert \mathbf {B}_{t} \Vert _{L^{3}}+C \Vert \mathbf {B}_{t} \Vert _{L^{6}} \Vert \nabla \mathbf {u}\Vert _{L^{2}} \Vert \mathbf {B}_{t} \Vert _{L^{3}} \\ &\quad\leqslant \frac{\nu}{4} \Vert \nabla \mathbf {B}_{t} \Vert _{L^{2}}^{2}+C \bigl( \Vert \mathbf {B}\Vert _{L^{3}}^{2} \Vert \nabla \mathbf {u}_{t} \Vert _{L^{2}}^{2}+ \Vert \mathbf {B}_{t} \Vert _{L^{3}}^{2} \Vert \nabla \mathbf {u}\Vert _{L^{2}}^{2} \bigr) \\ &\quad\leqslant \biggl(\frac{\nu}{4}+C \Vert \nabla \mathbf {u}\Vert _{L^{2}}^{2} \biggr) \Vert \nabla \mathbf {B}_{t} \Vert _{L^{2}}^{2}+C \Vert \nabla \mathbf {B}\Vert _{L^{2}}^{2} \Vert \nabla \mathbf {u}_{t} \Vert _{L^{2}}^{2} \\ &\quad \leqslant \biggl(\frac{\nu}{4}+C_{12}C_{0}^{1/4} \biggr) \Vert \nabla \mathbf {B}_{t} \Vert _{L^{2}}^{2}+C C_{0}^{1/4} \Vert \nabla \mathbf {u}_{t} \Vert _{L^{2}}^{2}, \end{aligned}$$
where we have used (2.2) and the Poincaré inequality. Thus
$$\begin{aligned} \frac{d}{dt} \Vert \mathbf {B}_{t} \Vert _{L^{2}}^{2}+ \Vert \nabla \mathbf {B}_{t} \Vert _{L^{2}}^{2} \leqslant C_{13} C_{0}^{1/4} \Vert \nabla \mathbf {u}_{t} \Vert _{L^{2}}^{2}, \end{aligned}$$
(3.30)
provided that \(C_{0}\leqslant \varepsilon _{4, 2}\triangleq \min \{1, ( \frac {\nu}{4C_{12}})^{4}\}\).
Calculating (3.28)+(3.30)\(\times 2C_{11}\) gives
$$\begin{aligned} \begin{aligned} &\frac{d}{dt} \bigl( \bigl\Vert \rho ^{1/2}\mathbf {u}_{t} \bigr\Vert _{L^{2}}^{2}+ \Vert \mathbf {B}_{t} \Vert _{L^{2}}^{2} \bigr)+ \Vert \nabla \mathbf {u}_{t} \Vert _{L^{2}}^{2}+ \Vert \nabla \mathbf {B}_{t} \Vert _{L^{2}}^{2} \\ & \quad\leqslant C \Vert \nabla \mathbf {u}\Vert _{L^{2}}+C \bigl\Vert \rho ^{1/2}\theta _{t} \bigr\Vert _{L^{2}}^{2}+C \Vert \nabla \theta \Vert _{L^{2}}^{2}+CC_{0}^{1/8} \Vert \nabla \theta _{t} \Vert _{L^{2}}^{2}, \end{aligned} \end{aligned}$$
(3.31)
provided that \(C_{0} \leqslant \varepsilon _{4, 3}\triangleq \min \{1, ( \frac {1}{4C_{11}C_{13}})^{4}\}\). Integrating (3.31) over \([0, T]\) and using (3.18), we have from (3.1), (3.6), and (3.20) that
$$\begin{aligned} \begin{aligned} &\sup_{t\in [0, T]} \bigl( \bigl\Vert \rho ^{1/2}\mathbf {u}_{t} \bigr\Vert _{L^{2}}^{2}+ \Vert \mathbf {B}_{t} \Vert _{L^{2}}^{2}+ \bigl\Vert \nabla ^{2}\mathbf {B}\bigr\Vert _{L^{2}}^{2} \bigr)+ \int _{0}^{T} \bigl( \Vert \nabla \mathbf {u}_{t} \Vert _{L^{2}}^{2}+ \Vert \nabla \mathbf {B}_{t} \Vert _{L^{2}}^{2} \bigr)\,dt \\ &\quad \leqslant K_{1}+C \Vert \nabla \mathbf {u}\Vert _{L^{2}}+CC_{0}^{1/8}A_{4}(T)+CA_{2}(T)+CC_{0}^{1/2} \\ & \quad\leqslant K_{1}+CA_{1}^{1/2}(T)+CC_{0}^{1/8}A_{4}(T)+CA_{2}(T)+CC_{0}^{1/2} \\ & \quad\leqslant K_{1}+C_{14}C_{0}^{1/8} \\ & \quad\leqslant 2K_{1}, \end{aligned} \end{aligned}$$
provided that \(C_{0}\leqslant \varepsilon _{4}\triangleq \min \{\varepsilon _{4, 1}, \varepsilon _{4, 2}, \varepsilon _{4, 3}, (\frac {K_{1}}{C_{14}})^{8} \}\). Thus we immediately obtain (3.25). The proof of the lemma is therefore completed. □
Lemma 3.7
Let \((\rho, \mathbf{ u}, \theta, \mathbf{ B})\) be a smooth solution of (1.1)–(1.4) satisfying (3.1). Then there exists a positive constant \(\varepsilon _{5}\), depending only on \(\kappa, \nu, R, c_{v}, \underline{\mu}\), ρ̄, Ω, \(g_{1}\), and \(g_{2}\), such that
$$\begin{aligned} \sup_{t\in [0, T]} \Vert \nabla \theta \Vert _{L^{2}}^{2}+ \int _{0}^{T} \bigl\Vert \rho ^{1/2} \theta _{t} \bigr\Vert _{L^{2}}^{2} \,dt\leqslant C_{0}^{1/2} \end{aligned}$$
(3.32)
and
$$\begin{aligned} \sup_{t\in [0, T]}t^{2} \Vert \nabla \theta \Vert _{L^{2}}^{2}+ \int _{0}^{T}t^{2} \bigl\Vert \rho ^{1/2}\theta _{t} \bigr\Vert _{L^{2}}^{2} \,dt\leqslant CC_{0}^{9/16}, \end{aligned}$$
(3.33)
provided that \(C_{0}\leqslant \varepsilon _{5}\).
Proof
We multiply (1.1)3 by \(\theta _{t}\) in \(L^{2}\). Then from (2.2)–(2.4), (3.1), (3.2), and (3.4) it follows that
$$\begin{aligned} &\frac{\kappa}{2}\frac{d}{dt} \Vert \nabla \theta \Vert _{L^{2}}^{2}+c_{v} \bigl\Vert \rho ^{1/2} \theta _{t} \bigr\Vert _{L^{2}}^{2} \\ & \quad= \int \bigl[ -c_{v}\rho \mathbf {u}\cdot \nabla \theta -P \operatorname{div} \mathbf {u}+2\mu \bigl\vert \mathcal{D}(\mathbf {u}) \bigr\vert ^{2}+\lambda (\operatorname{div} \mathbf {u})^{2}+ \nu \vert \operatorname{curl} \mathbf {B}\vert ^{2} \bigr]\theta _{t} \,dx \\ &\quad \leqslant \frac{c_{v}}{2} \bigl\Vert \rho ^{1/2}\theta _{t} \bigr\Vert _{L^{2}}^{2}+ C \Vert \nabla \mathbf {u}\Vert _{H^{1}}^{2} \Vert \nabla \theta \Vert _{L^{2}}^{2}+C \Vert \nabla \theta \Vert _{H^{1}}^{2} \Vert \nabla \mathbf {u}\Vert _{L^{2}}^{2} \\ &\qquad{} +C \Vert \nabla \mathbf {u}\Vert _{L^{2}} \Vert \nabla \mathbf {u}\Vert _{L^{3}} \Vert \theta _{t} \Vert _{L^{6}}+C \Vert \nabla \mathbf {B}\Vert _{L^{2}} \Vert \nabla \mathbf {B}\Vert _{L^{3}} \Vert \theta _{t} \Vert _{L^{6}} \\ & \quad\leqslant \frac{c_{v}}{2} \bigl\Vert \rho ^{1/2}\theta _{t} \bigr\Vert _{L^{2}}^{2}+ C \Vert \nabla \theta \Vert _{L^{2}}^{2}+C \Vert \nabla \mathbf {u}\Vert _{L^{2}}^{2}+CA_{1}^{1/4}(T) \bigl( \Vert \nabla \mathbf {u}\Vert _{L^{2}} + \Vert \nabla \mathbf {B}\Vert _{L^{2}} \bigr) \Vert \nabla \theta _{t} \Vert _{L^{2}}, \end{aligned}$$
and thus
$$\begin{aligned} \begin{aligned} &\frac{d}{dt} \Vert \nabla \theta \Vert _{L^{2}}^{2}+ \bigl\Vert \rho ^{1/2} \theta _{t} \bigr\Vert _{L^{2}}^{2} \\ &\quad \leqslant C \Vert \nabla \theta \Vert _{L^{2}}^{2}+C \Vert \nabla \mathbf {u}\Vert _{L^{2}}^{2}+CC_{0}^{1/16} \bigl( \Vert \nabla \mathbf {u}\Vert _{L^{2}} + \Vert \nabla \mathbf {B}\Vert _{L^{2}} \bigr) \Vert \nabla \theta _{t} \Vert _{L^{2}}. \end{aligned} \end{aligned}$$
(3.34)
Integrating (3.34) over \([0, T]\), from (3.1) and (3.6) we have that
$$\begin{aligned} \begin{aligned} &\sup_{t\in [0, T]} \Vert \nabla \theta \Vert _{L^{2}}^{2}+ \int _{0}^{T} \bigl\Vert \rho ^{1/2} \theta _{t} \bigr\Vert _{L^{2}}^{2} \,dt \\ & \quad\leqslant C_{0}+ C C_{0}+CC_{0}^{9/16}A_{4}^{1/2}(T) \\ & \quad\leqslant C_{15} C_{0}+C_{16}C_{0}^{9/16}A_{4}^{1/2}(T) \leqslant C_{0}^{1/2}, \end{aligned} \end{aligned}$$
provided that \(C_{0} \leqslant \varepsilon _{5, 1}\triangleq \min \{ \varepsilon _{4}, \frac {C_{0}^{1/2}}{2C_{15}}, ( \frac {C_{0}^{1/2}}{2C_{16}K_{2}^{1/2}} )^{16/9} \}\), which leads to (3.32).
Next, multiplying (3.34) by \(t^{2}\) and integrating the result over \([0, T]\), from (3.1) and (3.6) we have that
$$\begin{aligned} \begin{aligned} &\sup_{t\in [0, T]}t^{2} \Vert \nabla \theta \Vert _{L^{2}}^{2}+ \int _{0}^{T}t^{2} \bigl\Vert \rho ^{1/2}\theta _{t} \bigr\Vert _{L^{2}}^{2} \,dt \\ & \quad\leqslant C_{17} C_{0}+C C_{0}^{9/16}A_{4}^{1/2}(T) \leqslant CC_{0}^{9/16}, \end{aligned} \end{aligned}$$
provided that \(C_{0} \leqslant \varepsilon _{5, 2}\triangleq \min \{1, \frac {C_{0}^{9/16}}{2C_{17}}\}\). Thus we immediately obtain (3.33). Choosing \(C_{0}\leqslant \varepsilon _{5} \triangleq \min \{\varepsilon _{5, 1}, \varepsilon _{5, 2}\}\), we complete the proof of the lemma. □
Lemma 3.8
Let \((\rho, \mathbf{ u}, \theta, \mathbf{ B})\) be a smooth solution of (1.1)–(1.4) satisfying (3.1). Then there exists a positive constant \(\varepsilon _{6}\), depending only on \(\kappa, \nu, R, c_{v}, \underline{\mu}\), ρ̄, Ω, \(g_{1}\), and \(g_{2}\), such that
$$\begin{aligned} \sup_{t\in [0, T]}t^{2} \bigl\Vert \bigl( \rho ^{1/2}\mathbf{ u}_{t}, \mathbf{ B}_{t}, \nabla ^{2} \mathbf{ B} \bigr) \bigr\Vert _{L^{2}}^{2}+ \int _{0}^{T}t^{2} \bigl\Vert (\nabla \mathbf{ u}_{t}, \nabla \mathbf{ B}_{t}) \bigr\Vert _{L^{2}}^{2} \,dt\leqslant CC_{0}^{5/16}, \end{aligned}$$
(3.35)
provided that \(C_{0}\leqslant \varepsilon _{6}\).
Proof
Multiplying (3.31) by \(t^{2}\) and integrating the resulting equation over \([0, T]\), from (3.6), (3.14), (3.18), (3.20), and (3.33) it follows that
$$\begin{aligned} \begin{aligned} &\sup_{t\in [0, T]}t^{2} \bigl( \bigl\Vert \rho ^{1/2}\mathbf {u}_{t} \bigr\Vert _{L^{2}}^{2}+ \Vert \mathbf {B}_{t} \Vert _{L^{2}}^{2}+ \bigl\Vert \nabla ^{2}\mathbf {B}\bigr\Vert _{L^{2}}^{2} \bigr)+ \int _{0}^{T} t^{2} \bigl( \Vert \nabla \mathbf {u}_{t} \Vert _{L^{2}}^{2}+ \Vert \nabla \mathbf {B}_{t} \Vert _{L^{2}}^{2} \bigr)\,dt \\ &\quad \leqslant C \int _{0}^{T} t \bigl( \bigl\Vert \rho ^{1/2}\mathbf {u}_{t} \bigr\Vert _{L^{2}}^{2}+ \Vert \mathbf {B}_{t} \Vert _{L^{2}}^{2}+ \bigl\Vert \nabla ^{2}\mathbf {B}\bigr\Vert _{L^{2}}^{2} \bigr)\,dt \\ &\qquad{} +C \int _{0}^{T} t^{2} \bigl( \Vert \nabla \mathbf {u}\Vert _{L^{2}}+ \bigl\Vert \rho ^{1/2}\theta _{t} \bigr\Vert _{L^{2}}^{2}+ \Vert \nabla \theta \Vert _{L^{2}}^{2}+C \Vert \nabla \mathbf {u}\Vert _{L^{2}} \Vert \nabla \theta _{t} \Vert _{L^{2}}^{2} \bigr) \,dt \\ &\quad \leqslant CC_{0}^{5/16}+CC_{0}^{1/2}+CC_{0}^{9/16}+CC_{0} \\ & \quad\leqslant CC_{0}^{5/16}, \end{aligned} \end{aligned}$$
provided that \(C_{0}\leqslant \varepsilon _{6}\triangleq \min \{1, \varepsilon _{5} \}\). Thus we immediately obtain (3.35). The proof of the lemma is therefore completed. □
Lemma 3.9
Let \((\rho, \mathbf{ u}, \theta, \mathbf{ B})\) be a smooth solution of (1.1)–(1.4) satisfying (3.1). Then there exists a positive constant \(\varepsilon _{7}\), depending only on \(\kappa, \nu, R, c_{v}, \underline{\mu}\), ρ̄, Ω, \(g_{1}\), and \(g_{2}\), such that
$$\begin{aligned} \sup_{t\in [0, T]} \bigl\Vert \rho ^{1/2} \theta _{t} \bigr\Vert _{L^{2}}^{2}+ \int _{0}^{T} \Vert \nabla \theta _{t} \Vert _{L^{2}}^{2} \,dt\leqslant 2K_{2}, \end{aligned}$$
(3.36)
provided that \(C_{0}\leqslant \varepsilon _{7}\).
Proof
By (1.1)1 and (1.1)3 we have that
$$\begin{aligned} \begin{aligned} &c_{v} [\rho \theta _{tt}+\rho \mathbf {u}\cdot \nabla \theta _{t} ]-\kappa \Delta \theta _{t} \\ &\quad =\operatorname{div}(\rho \mathbf {u}) (c_{v}\theta _{t}+c_{v}\mathbf {u}\cdot \nabla \theta +R\theta \operatorname{div} \mathbf {u})-\rho (c_{v} \mathbf {u}_{t} \cdot \nabla \theta +R\theta _{t} \operatorname{div} \mathbf {u})-R\rho \theta \operatorname{div} \mathbf {u}_{t} \\ & \qquad{}+ (2\mu _{\rho}\rho _{t} +2\mu _{\theta} \theta _{t} ) \bigl\vert \mathcal{D}(\mathbf {u}) \bigr\vert ^{2}+4\mu (\rho, \theta )\mathcal{D}( \mathbf {u}): \mathcal{D}(\mathbf {u}_{t}) \\ &\qquad{} + (\lambda _{\rho}\rho _{t}+\lambda _{\theta}\theta _{t} ) (\operatorname{div} \mathbf {u})^{2}+2\lambda (\rho, \theta )\operatorname{div} \mathbf {u}\operatorname{div} \mathbf {u}_{t}+2\nu \operatorname{curl} \mathbf {B}\cdot \operatorname{curl} \mathbf {B}_{t}. \end{aligned} \end{aligned}$$
(3.37)
Multiplying (3.37) by \(\theta _{t}\) and integrating the resulting equation on Ω, we get
$$\begin{aligned} &\frac{1}{2}\frac{d}{dt} \bigl\Vert \rho ^{1/2}\theta _{t} \bigr\Vert _{L^{2}}^{2}+ \kappa \Vert \nabla \theta _{t} \Vert _{L^{2}}^{2} \\ &\quad= \int \bigl[ \operatorname{div}( \rho \mathbf {u}) (c_{v}\theta _{t}+c_{v}\mathbf {u}\cdot \nabla \theta +R \theta \operatorname{div} \mathbf {u}) \bigr] \theta _{t} \,dx \\ \begin{aligned} & \qquad{}- \int \bigl[ \rho (c_{v} \mathbf {u}_{t} \cdot \nabla \theta +R \theta _{t} \operatorname{div} \mathbf {u}) \bigr]\theta _{t} \,dx- \int R\rho \theta \operatorname{div} \mathbf {u}_{t} \theta _{t} \,dx \\ &\qquad{}+ \int 4\mu (\rho, \theta ) \mathcal{D}(\mathbf {u}): \mathcal{D}(\mathbf {u}_{t})\theta _{t} \,dx \end{aligned} \\ & \qquad{}+ \int (2\mu _{\rho}\rho _{t} +2\mu _{\theta} \theta _{t} ) \bigl\vert \mathcal{D}(\mathbf {u}) \bigr\vert ^{2} \theta _{t} \,dx+ \int ( \lambda _{\rho}\rho _{t}+\lambda _{\theta}\theta _{t} ) (\operatorname{div} \mathbf {u})^{2} \theta _{t} \,dx \\ & \qquad{}+2 \int \lambda (\rho, \theta )\operatorname{div} \mathbf {u}\operatorname{div} \mathbf {u}_{t} \theta _{t} \,dx+2 \int \nu \operatorname{curl} \mathbf {B}\cdot \operatorname{curl} \mathbf {B}_{t} \theta _{t} \,dx \triangleq \sum _{i=1}^{6}I_{i}. \end{aligned}$$
(3.38)
The right-hand side of (3.38) can be estimated as follows. By (2.2)–(2.4), (3.1), (3.2), and (3.4) we have
$$\begin{aligned} I_{1} ={}& \int \bigl[ (\mathbf {u}\cdot \nabla \rho +\rho \operatorname{div} \mathbf {u}) (c_{v}\theta _{t}+c_{v}\mathbf {u}\cdot \nabla \theta +R \theta \operatorname{div} \mathbf {u}) \bigr] \theta _{t} \,dx \\ \leqslant{}& C \bigl( \Vert \mathbf {u}\Vert _{L^{6}} \Vert \nabla \rho \Vert _{L^{2}}+ \Vert \rho \Vert _{L^{6}} \Vert \nabla \mathbf {u}\Vert _{L^{2}} \bigr) \Vert \theta _{t} \Vert _{L^{6}}^{2}+C \Vert \mathbf {u}\Vert _{L^{\infty}}^{2} \Vert \nabla \rho \Vert _{L^{3}} \Vert \nabla \theta \Vert _{L^{2}} \Vert \theta _{t} \Vert _{L^{6}} \\ & {}+C \Vert \nabla \mathbf {u}\Vert _{L^{6}} \Vert \mathbf {u}\Vert _{L^{6}} \Vert \nabla \theta \Vert _{L^{2}} \Vert \theta _{t} \Vert _{L^{6}}+ C \Vert \nabla \mathbf {u}\Vert _{L^{6}} \Vert \theta \Vert _{L^{6}} \Vert \nabla \mathbf {u}\Vert _{L^{2}} \Vert \theta _{t} \Vert _{L^{6}} \\ & {}+C \Vert \mathbf {u}\Vert _{L^{\infty}} \Vert \theta \Vert _{L^{\infty}} \Vert \nabla \rho \Vert _{L^{3}} \Vert \nabla \mathbf {u}\Vert _{L^{2}} \Vert \theta _{t} \Vert _{L^{6}} \\ \leqslant{}& C \Vert \nabla \mathbf {u}\Vert _{L^{2}} \Vert \nabla \rho \Vert _{L^{2}} \Vert \nabla \theta _{t} \Vert _{L^{2}}^{2}+C \Vert \nabla \mathbf {u}\Vert _{H^{1}}^{2} \bigl( \Vert \theta \Vert _{L^{\infty}} + \Vert \nabla \rho \Vert _{L^{3}}+1 \bigr) \Vert \nabla \theta \Vert _{L^{2}} \Vert \nabla \theta _{t} \Vert _{L^{2}} \\ \leqslant{}& \biggl[CC_{0}^{1/8} +\frac{\kappa}{16} \biggr] \Vert \nabla \theta _{t} \Vert _{L^{2}}^{2}+C \Vert \nabla \theta \Vert _{L^{2}}^{2}, \\ I_{2} ={}&{-} \int \bigl[ \rho (c_{v} \mathbf {u}_{t} \cdot \nabla \theta +R\theta _{t} \operatorname{div} \mathbf {u}) \bigr]\theta _{t} \,dx \\ \leqslant{}& C \Vert \rho \Vert _{L^{6}} \Vert \mathbf {u}_{t} \Vert _{L^{6}} \Vert \nabla \theta \Vert _{L^{2}} \Vert \theta _{t} \Vert _{L^{6}}+C \Vert \rho \Vert _{L^{6}} \Vert \nabla \mathbf {u}\Vert _{L^{2}} \Vert \theta _{t} \Vert _{L^{6}}^{2} \\ \leqslant{}& C \Vert \nabla \rho \Vert _{L^{2}} \Vert \nabla \mathbf {u}_{t} \Vert _{L^{2}} \Vert \nabla \theta \Vert _{L^{2}} \Vert \nabla \theta _{t} \Vert _{L^{2}}+C \Vert \nabla \rho \Vert _{L^{2}} \Vert \nabla \mathbf {u}\Vert _{L^{2}} \Vert \nabla \theta _{t} \Vert _{L^{2}}^{2} \\ \leqslant {}& \biggl[CC_{0}^{1/8} +\frac{\kappa}{16} \biggr] \Vert \nabla \theta _{t} \Vert _{L^{2}}^{2}+CC_{0}^{1/8} \Vert \nabla \mathbf {u}_{t} \Vert _{L^{2}}^{2}, \\ I_{3} ={}&{-}R \int \rho \theta \operatorname{div} \mathbf {u}_{t} \theta _{t} \,dx \\ \leqslant{}& \Vert \rho \Vert _{L^{6}} \Vert \theta \Vert _{L^{6}} \Vert \nabla \mathbf {u}_{t} \Vert _{L^{2}} \Vert \theta _{t} \Vert _{L^{6}} \\ \leqslant {}&\Vert \nabla \rho \Vert _{L^{2}} \Vert \nabla \theta \Vert _{L^{2}} \Vert \nabla \mathbf {u}_{t} \Vert _{L^{2}} \Vert \nabla \theta _{t} \Vert _{L^{2}} \\ \leqslant{}& CC_{0}^{1/8} \Vert \nabla \theta _{t} \Vert _{L^{2}}^{2}+CC_{0}^{1/8} \Vert \nabla \mathbf {u}_{t} \Vert _{L^{2}}^{2} \end{aligned}$$
and
$$\begin{aligned} I_{4} ={}& \int 4\mu (\rho, \theta )\mathcal{D}(\mathbf {u}): \mathcal{D}(\mathbf {u}_{t})\theta _{t} \,dx+ \int (2\mu _{\rho}\rho _{t} +2\mu _{\theta} \theta _{t} ) \bigl\vert \mathcal{D}(\mathbf {u}) \bigr\vert ^{2} \theta _{t} \,dx \\ \leqslant{}& C \Vert \nabla \mathbf {u}\Vert _{L^{3}} \Vert \nabla \mathbf {u}_{t} \Vert _{L^{2}} \Vert \theta _{t} \Vert _{L^{6}}+C \Vert \nabla \mathbf {u}\Vert _{L^{3}}^{2} \Vert \theta _{t} \Vert _{L^{6}}^{2} \\ &{} + \bigl( \Vert \mathbf {u}\Vert _{L^{\infty}} \Vert \nabla \rho \Vert _{L^{2}}+ \Vert \nabla \mathbf {u}\Vert _{L^{2}} \bigr) \Vert \nabla \mathbf {u}\Vert _{L^{6}}^{2} \Vert \theta _{t} \Vert _{L^{6}} \\ \leqslant{}& C \Vert \nabla \mathbf {u}\Vert _{L^{2}}^{1/2} \Vert \nabla \mathbf {u}\Vert _{H^{1}}^{1/2} \Vert \nabla \mathbf {u}_{t} \Vert _{L^{2}} \Vert \nabla \theta _{t} \Vert _{L^{2}}+C \Vert \nabla \mathbf {u}\Vert _{L^{2}} \Vert \nabla \mathbf {u}\Vert _{H^{1}} \Vert \nabla \theta _{t} \Vert _{L^{2}}^{2} \\ & {}+ \bigl( \Vert \nabla \mathbf {u}\Vert _{L^{2}}^{1/2} \Vert \nabla \mathbf {u}\Vert _{L^{6}}^{1/2}+ \Vert \nabla \mathbf {u}\Vert _{L^{2}} \bigr) \Vert \nabla \mathbf {u}\Vert _{L^{6}}^{2} \Vert \nabla \theta _{t} \Vert _{L^{2}} \\ \leqslant{}& \frac{\kappa}{8} \Vert \nabla \theta _{t} \Vert _{L^{2}}^{2}+C \Vert \nabla \mathbf {u}\Vert _{L^{2}} \Vert \nabla \theta _{t} \Vert _{L^{2}}^{2}+ \Vert \nabla \mathbf {u}\Vert _{L^{2}} \Vert \nabla \mathbf {u}_{t} \Vert _{L^{2}}^{2}+C \Vert \nabla \mathbf {u}\Vert _{L^{2}}^{2} \\ \leqslant{}& \biggl[CC_{0}^{1/8} +\frac{\kappa}{16} \biggr] \Vert \nabla \theta _{t} \Vert _{L^{2}}^{2}+CC_{0}^{1/8} \Vert \nabla \mathbf {u}_{t} \Vert _{L^{2}}^{2}+C \Vert \nabla \mathbf {u}\Vert _{L^{2}}^{2}. \end{aligned}$$
Similarly,
$$\begin{aligned} \begin{aligned} I_{5} &= \int (\lambda _{\rho}\rho _{t}+\lambda _{ \theta}\theta _{t} ) (\operatorname{div} \mathbf {u})^{2} \theta _{t} \,dx+2 \int \lambda (\rho, \theta )\operatorname{div} \mathbf {u}\operatorname{div} \mathbf {u}_{t} \theta _{t} \,dx \\ &\leqslant \biggl[CC_{0}^{1/8} +\frac{\kappa}{16} \biggr] \Vert \nabla \theta _{t} \Vert _{L^{2}}^{2}+CC_{0}^{1/8} \Vert \nabla \mathbf {u}_{t} \Vert _{L^{2}}^{2}+C \Vert \nabla \mathbf {u}\Vert _{L^{2}}^{2} \end{aligned} \end{aligned}$$
and
$$\begin{aligned} I_{6} &=2 \int \nu \operatorname{curl} \mathbf {B}\cdot \operatorname{curl} \mathbf {B}_{t} \theta _{t} \,dx\leqslant C \Vert \nabla \mathbf {B}\Vert _{L^{3}} \Vert \nabla \mathbf {B}_{t} \Vert _{L^{2}} \Vert \theta _{t} \Vert _{L^{6}} \\ &\leqslant C \Vert \nabla \mathbf {B}\Vert _{L^{2}}^{1/2} \Vert \nabla \mathbf {B}\Vert _{H^{1}}^{1/2} \Vert \nabla \mathbf {B}_{t} \Vert _{L^{2}} \Vert \nabla \theta _{t} \Vert _{L^{2}} \\ &\leqslant C \Vert \nabla \mathbf {B}\Vert _{L^{2}} \Vert \nabla \theta _{t} \Vert _{L^{2}}^{2} +C \Vert \nabla \mathbf {B}\Vert _{L^{2}} \Vert \nabla \mathbf {B}_{t} \Vert _{L^{2}}^{2} \\ &\leqslant CC_{0}^{1/8} \Vert \nabla \theta _{t} \Vert _{L^{2}}^{2} +CC_{0}^{1/8} \Vert \nabla \mathbf {B}_{t} \Vert _{L^{2}}^{2}. \end{aligned}$$
Substituting \(I_{1}\)–\(I_{6}\) into (3.38), we have
$$\begin{aligned} \begin{aligned} &\frac{1}{2}\frac{d}{dt} \bigl\Vert \rho ^{1/2}\theta _{t} \bigr\Vert _{L^{2}}^{2}+ \biggl[\frac{3\kappa}{4}-C_{18}C_{0}^{1/8} \biggr] \Vert \nabla \theta _{t} \Vert _{L^{2}}^{2} \\ &\quad \leqslant C \Vert \nabla \theta \Vert _{L^{2}}^{2}+ CC_{0}^{1/8} \Vert \nabla \mathbf {u}_{t} \Vert _{L^{2}}^{2}+C \Vert \nabla \mathbf {u}\Vert _{L^{2}}^{2}+CC_{0}^{1/8} \Vert \nabla \mathbf {B}_{t} \Vert _{L^{2}}^{2}, \end{aligned} \end{aligned}$$
and thus
$$\begin{aligned} \begin{aligned} \frac{d}{dt} \bigl\Vert \rho ^{1/2}\theta _{t} \bigr\Vert _{L^{2}}^{2}+ \Vert \nabla \theta _{t} \Vert _{L^{2}}^{2} \leqslant C \bigl\Vert (\nabla \theta, \nabla \mathbf {u}) \bigr\Vert _{L^{2}}^{2}+ CC_{0}^{1/8} \bigl\Vert ( \nabla \mathbf {u}_{t}, \nabla \mathbf {B}_{t} ) \bigr\Vert _{L^{2}}^{2}, \end{aligned} \end{aligned}$$
(3.39)
provided that \(C_{0}\leqslant \varepsilon _{7, 1}\triangleq \min \{1, ( \frac {\kappa}{4C_{18}})^{2}\}\).
Integrating (3.39) over \([0, T]\) gives
$$\begin{aligned} \sup_{t\in [0, T]} \bigl\Vert \rho ^{1/2}\theta _{t} \bigr\Vert _{L^{2}}^{2}+ \int _{0}^{T} \Vert \nabla \theta _{t} \Vert _{L^{2}}^{2} \,dt \leqslant K_{2}+C_{19}C_{0}+ C_{20}C_{0}^{1/8}K_{1} \leqslant 2K_{2}, \end{aligned}$$
provided that \(C_{0}\leqslant \varepsilon _{7}\triangleq \min \{\varepsilon _{7, 1}, \frac {K_{2}}{2C_{19}}, (\frac {K_{2}}{2C_{20}K_{1}})^{8}\}\). Thus we immediately obtain (3.36). The proof of the lemma is therefore completed. □
Lemma 3.10
Let \((\rho, \mathbf{ u}, \theta, \mathbf{ B})\) be a smooth solution of (1.1)–(1.4) satisfying (3.1). Then there exists a positive constant \(\varepsilon _{8}\), depending only on \(\kappa, \nu, R, c_{v}, \underline{\mu}\), ρ̄, Ω, \(g_{1}\), and \(g_{2}\), such that
$$\begin{aligned} \sup_{(x, t)\in \Omega \times [0, T]}\rho (x, t)\leqslant 2 \bar{\rho}, \end{aligned}$$
(3.40)
provided that \(C_{0}\leqslant \varepsilon _{8}\).
Proof
Lemma 2.2, together with (1.1)2, (1.6), (3.1), (3.2), (3.4), and the Hölder inequality, gives
$$\begin{aligned} \bigl\Vert \nabla ^{2} \mathbf {u}\bigr\Vert _{L^{3+\beta}} \leqslant{}& C \bigl( \Vert \rho \mathbf {u}_{t} \Vert _{L^{3+\beta}}+ \Vert \rho \mathbf {u}\cdot \nabla \mathbf {u}\Vert _{L^{3+\beta}}+ \bigl\Vert (\nabla \rho +\nabla \theta ) \nabla \mathbf {u}\bigr\Vert _{L^{3+\beta}} \bigr) \\ &{} +C \bigl( \Vert \nabla P \Vert _{L^{3+\beta}}+ \Vert \mathbf {B}\cdot \nabla \mathbf {B}\Vert _{L^{3+\beta}} \bigr) \\ \leqslant {}&C \bigl[ \Vert \rho \mathbf {u}_{t} \Vert _{L^{4}}+ \bigl( \Vert \mathbf {u}\Vert _{L^{q}}+ \Vert \nabla \rho \Vert _{L^{q}}+ \Vert \nabla \theta \Vert _{L^{q}} \bigr) \Vert \nabla \mathbf {u}\Vert _{L^{12}} \bigr] \\ &{} +C \bigl( \Vert \rho \Vert _{L^{12}} \Vert \nabla \theta \Vert _{L^{q}}+ \Vert \nabla \rho \Vert _{L^{q}} \Vert \theta \Vert _{L^{12}}+ \Vert \mathbf {B}\Vert _{L^{q}} \Vert \nabla \mathbf {B}\Vert _{L^{12}} \bigr) \\ \leqslant{}& C \bigl( \Vert \rho \mathbf {u}_{t} \Vert _{L^{2}}^{1/4} \Vert \rho \mathbf {u}_{t} \Vert _{L^{6}}^{3/4}+ \Vert \nabla \mathbf {u}\Vert _{L^{12}}+ \Vert \nabla \theta \Vert _{L^{q}}+ \Vert \theta \Vert _{L^{12}}+ \Vert \nabla \mathbf {B}\Vert _{L^{12}} \bigr) \\ \leqslant{}& C \bigl( \bigl\Vert \rho ^{1/2}\mathbf {u}_{t} \bigr\Vert _{L^{2}}^{1/4} \Vert \nabla \mathbf {u}_{t} \Vert _{L^{2}}^{3/4}+ \bigl\Vert |\nabla \mathbf {u}|^{2} \bigr\Vert _{H^{1}}^{1/2}+ \bigl\Vert |\nabla \mathbf {B}|^{2} \bigr\Vert _{H^{1}}^{1/2} \bigr) \\ & {}+C \bigl( \Vert \nabla \theta \Vert _{H^{1}}+ \bigl\Vert \theta ^{2} \bigr\Vert _{H^{1}}^{1/2} \bigr) \\ \leqslant{}& C \bigl\Vert \rho ^{1/2}\mathbf {u}_{t} \bigr\Vert _{L^{2}}^{1/4} \Vert \nabla \mathbf {u}_{t} \Vert _{L^{2}}^{3/4}+C \Vert \nabla \mathbf {u}\Vert _{L^{4}}+C \Vert \nabla \mathbf {u}\Vert _{L^{2(3+\beta )/(1+\beta )}}^{1/2} \bigl\Vert \nabla ^{2} \mathbf {u}\bigr\Vert _{L^{3+ \beta}}^{1/2} \\ &{} +C \Vert \nabla \mathbf {B}\Vert _{L^{4}}+C \Vert \nabla \mathbf {B}\Vert _{L^{2(3+ \beta )/(1+\beta )}}^{1/2} \bigl\Vert \nabla ^{2} \mathbf {B}\bigr\Vert _{L^{3+\beta}}^{1/2}+C \Vert \nabla \theta \Vert _{H^{1}} \\ \leqslant{}& \frac{1}{2} \bigl\Vert \nabla ^{2} \mathbf {u}\bigr\Vert _{L^{3+\beta}} + \frac{1}{2} \bigl\Vert \nabla ^{2} \mathbf {B}\bigr\Vert _{L^{3+\beta}}+C \bigl\Vert \rho ^{1/2} \mathbf {u}_{t} \bigr\Vert _{L^{2}}^{1/4} \Vert \nabla \mathbf {u}_{t} \Vert _{L^{2}}^{3/4} \\ &{} +C \Vert \nabla \mathbf {u}\Vert _{H^{1}}+C \Vert \nabla \mathbf {B}\Vert _{H^{1}}+C \Vert \nabla \theta \Vert _{H^{1}}, \end{aligned}$$
where we have used the estimates
$$\begin{aligned} \Vert \rho \mathbf {u}_{t} \Vert _{L^{3+\beta}}\leqslant C \biggl( \int \vert \rho \mathbf {u}_{t} \vert ^{(3+\beta )\times \frac {4}{3+\beta}}\,dx \biggr)^{ \frac {1}{3+\beta}\times \frac {3+\beta}{4}} \biggl( \int 1^{ \frac {4}{1-\beta}}\,dx \biggr)^{\frac {1}{3+\beta}\times \frac {1-\beta}{4}}\leqslant C \Vert \rho \mathbf {u}_{t} \Vert _{L^{4}} \end{aligned}$$
and
$$\begin{aligned} \Vert \nabla \mathbf {u}\Vert _{L^{12}}= \bigl\Vert \vert \nabla \mathbf {u}\vert ^{2} \bigr\Vert _{L^{6}}^{1/2} \leqslant C \bigl\Vert \vert \nabla \mathbf {u}\vert ^{2} \bigr\Vert _{H^{1}}^{1/2}. \end{aligned}$$
Thus
$$\begin{aligned} \begin{aligned} \bigl\Vert \nabla ^{2} \mathbf {u}\bigr\Vert _{L^{3+\beta}} &\leqslant \frac{1}{4} \bigl\Vert \nabla ^{2} \mathbf {B}\bigr\Vert _{L^{3+\beta}}+C \bigl\Vert (\nabla \mathbf {u}, \nabla \theta, \nabla \mathbf {B}) \bigr\Vert _{H^{1}}+C \bigl\Vert \rho ^{1/2}\mathbf {u}_{t} \bigr\Vert _{L^{2}}^{1/4} \Vert \nabla \mathbf {u}_{t} \Vert _{L^{2}}^{3/4}. \end{aligned} \end{aligned}$$
(3.41)
On the other hand, from (1.1)4 and (3.4) it follows that
$$\begin{aligned} \begin{aligned} \bigl\Vert \nabla ^{2} \mathbf {B}\bigr\Vert _{L^{3+\beta}} &\leqslant C \bigl( \Vert \mathbf {B}_{t} \Vert _{L^{3+\beta}}+ \Vert \mathbf {u}\cdot \nabla \mathbf {B}\Vert _{L^{3+\beta}}+ \Vert \mathbf {B}\cdot \nabla \mathbf {u}\Vert _{L^{3+\beta}}+ \Vert \mathbf {B}\operatorname{div} \mathbf {u}\Vert _{L^{3+\beta}} \bigr) \\ &\leqslant C \Vert \mathbf {B}_{t} \Vert _{L^{4}}+C \Vert \mathbf {u}\Vert _{L^{q}} \Vert \nabla \mathbf {B}\Vert _{L^{12}}+C \Vert \nabla \mathbf {u}\Vert _{L^{q}} \Vert \mathbf {B}\Vert _{L^{12}} \\ &\leqslant C \Vert \mathbf {B}_{t} \Vert _{L^{2}}^{1/4} \Vert \nabla \mathbf {B}_{t} \Vert _{L^{2}}^{3/4}+C \Vert \nabla \mathbf {B}\Vert _{L^{12}}+C \Vert \mathbf {B}\Vert _{L^{12}} \\ &\leqslant C \Vert \mathbf {B}_{t} \Vert _{L^{2}}^{1/4} \Vert \nabla \mathbf {B}_{t} \Vert _{L^{2}}^{3/4}+C \Vert \nabla \mathbf {B}\Vert _{H^{1}}+C \Vert \nabla \mathbf {B}\Vert _{L^{2(3+\beta )/(1+ \beta )}}^{1/2} \bigl\Vert \nabla ^{2} \mathbf {B}\bigr\Vert _{L^{3+\beta}}^{1/2} \\ &\leqslant \frac{1}{2} \bigl\Vert \nabla ^{2} \mathbf {B}\bigr\Vert _{L^{3+\beta}}+C \Vert \mathbf {B}_{t} \Vert _{L^{2}}^{1/4} \Vert \nabla \mathbf {B}_{t} \Vert _{L^{2}}^{3/4}+C \Vert \nabla \mathbf {B}\Vert _{H^{1}}, \end{aligned} \end{aligned}$$
and thus
$$\begin{aligned} \bigl\Vert \nabla ^{2} \mathbf {B}\bigr\Vert _{L^{3+\beta}}\leqslant C \Vert \mathbf {B}_{t} \Vert _{L^{2}}^{1/4} \Vert \nabla \mathbf {B}_{t} \Vert _{L^{2}}^{3/4}+C \Vert \nabla \mathbf {B}\Vert _{H^{1}}. \end{aligned}$$
(3.42)
Due to (3.1), (3.3), and (3.5), we have
$$\begin{aligned} \bigl\Vert (\nabla \mathbf {u}, \nabla \theta, \nabla \mathbf {B}) \bigr\Vert _{H^{1}} \leqslant C \bigl\Vert \bigl(\rho ^{1/2} \mathbf {u}_{t}, \rho ^{1/2} \theta _{t}, \mathbf {B}_{t} \bigr) \bigr\Vert _{L^{2}}+C \Vert \nabla \theta \Vert _{L^{2}}+C \bigl\Vert (\nabla \mathbf {u}, \nabla \mathbf {B}) \bigr\Vert _{L^{2}}^{1/4}. \end{aligned}$$
(3.43)
Combining (3.41) and (3.42) with (3.43), yields
$$\begin{aligned} \begin{aligned} & \bigl\Vert \nabla ^{2} \mathbf {u}\bigr\Vert _{L^{3+\beta}}+ \bigl\Vert \nabla ^{2} \mathbf {B}\bigr\Vert _{L^{3+\beta}} \\ &\quad \leqslant C \bigl\Vert \rho ^{1/2}\mathbf {u}_{t} \bigr\Vert _{L^{2}}^{1/4} \Vert \nabla \mathbf {u}_{t} \Vert _{L^{2}}^{3/4}+C \Vert \mathbf {B}_{t} \Vert _{L^{2}}^{1/4} \Vert \nabla \mathbf {B}_{t} \Vert _{L^{2}}^{3/4} \\ &\qquad{} +C \bigl\Vert \bigl(\rho ^{1/2}\mathbf {u}_{t}, \rho ^{1/2} \theta _{t}, \mathbf {B}_{t} \bigr) \bigr\Vert _{L^{2}}+C \Vert \nabla \theta \Vert _{L^{2}}+C \bigl\Vert ( \nabla \mathbf {u}, \nabla \mathbf {B}) \bigr\Vert _{L^{2}}^{1/4}. \end{aligned} \end{aligned}$$
(3.44)
Integrating (3.44) over \([0, T]\) gives
$$\begin{aligned} \begin{aligned} & \int _{0}^{T} \bigl( \bigl\Vert \nabla ^{2} \mathbf {u}\bigr\Vert _{L^{3+ \beta}}+ \bigl\Vert \nabla ^{2} \mathbf {B}\bigr\Vert _{L^{3+\beta}} \bigr)\,dt \\ & \quad\leqslant C \int _{0}^{T} \bigl\Vert \rho ^{1/2} \mathbf {u}_{t} \bigr\Vert _{L^{2}}^{1/4} \Vert \nabla \mathbf {u}_{t} \Vert _{L^{2}}^{3/4}\,dt+C \int _{0}^{T} \Vert \mathbf {B}_{t} \Vert _{L^{2}}^{1/4} \Vert \nabla \mathbf {B}_{t} \Vert _{L^{2}}^{3/4}\,dt \\ &\qquad{} +C \int _{0}^{T} \bigl\Vert \bigl(\rho ^{1/2}\mathbf {u}_{t}, \rho ^{1/2} \theta _{t}, \mathbf {B}_{t} \bigr) \bigr\Vert _{L^{2}} \,dt+C \int _{0}^{T} \Vert \nabla \theta \Vert _{L^{2}}\,dt \\ &\qquad{} +C \int _{0}^{T} \bigl\Vert (\nabla \mathbf {u}, \nabla \mathbf {B}) \bigr\Vert _{L^{2}}^{1/4}\,dt \triangleq \sum _{i=1}^{5}J_{5}. \end{aligned} \end{aligned}$$
(3.45)
The right-hand side of (3.45) cab be estimated as follows. By (3.13), (3.14), and (3.25) we have
$$\begin{aligned} \begin{aligned} J_{1} \leqslant {}&C \biggl( \int _{0}^{T} \bigl\Vert \rho ^{1/2} \mathbf {u}_{t} \bigr\Vert _{L^{2}}^{2/5}\,dt \biggr)^{5/8} \biggl( \int _{0}^{T} \Vert \nabla \mathbf {u}_{t} \Vert _{L^{2}}^{2}\,dt \biggr)^{3/8} \\ \leqslant{}& C \biggl( \int _{0}^{\sigma (T)} \bigl\Vert \rho ^{1/2} \mathbf {u}_{t} \bigr\Vert _{L^{2}}^{2/5}\,dt \biggr)^{5/8}+C \biggl( \int _{\sigma (T)}^{T} \bigl\Vert \rho ^{1/2} \mathbf {u}_{t} \bigr\Vert _{L^{2}}^{2/5}\,dt \biggr)^{5/8} \\ \leqslant{}& C \biggl( \int _{0}^{\sigma (T)} \bigl\Vert \rho ^{1/2} \mathbf {u}_{t} \bigr\Vert _{L^{2}}^{2}\,dt \biggr)^{1/8} \biggl( \int _{0}^{\sigma (T)}\,dt \biggr)^{1/2} \\ & {}+ C \biggl( \int _{\sigma (T)}^{T}t^{8} \bigl\Vert \rho ^{1/2} \mathbf {u}_{t} \bigr\Vert _{L^{2}}^{2} \,dt \biggr)^{1/8} \biggl( \int _{\sigma (T)}^{T}t^{-2}\,dt \biggr)^{1/2}\leqslant CC_{0}^{1/32}. \end{aligned} \end{aligned}$$
Similarly,
$$\begin{aligned} \begin{aligned} J_{2} \leqslant{}& C \biggl( \int _{0}^{\sigma (T)} \Vert \mathbf {B}_{t} \Vert _{L^{2}}^{2}\,dt \biggr)^{1/8} \biggl( \int _{0}^{\sigma (T)}\,dt \biggr)^{1/2} \\ & {}+ C \biggl( \int _{\sigma (T)}^{T}t^{8} \Vert \nabla \mathbf {B}_{t} \Vert _{L^{2}}^{2}\,dt \biggr)^{1/8} \biggl( \int _{\sigma (T)}^{T}t^{-2}\,dt \biggr)^{1/2} \leqslant CC_{0}^{1/32}. \end{aligned} \end{aligned}$$
By (3.13), (3.14), (3.32), and (3.33) we have
$$\begin{aligned} J_{3} \leqslant{}& C \biggl( \int _{0}^{\sigma (T)} \bigl\Vert \bigl(\rho ^{1/2} \mathbf {u}_{t}, \rho ^{1/2} \theta _{t}, \mathbf {B}_{t} \bigr) \bigr\Vert _{L^{2}}^{2} \,dt \biggr)^{1/2} \biggl( \int _{0}^{\sigma (T)}\,dt \biggr)^{1/2} \\ &{} + C \biggl( \int _{\sigma (T)}^{T}t^{2} \bigl\Vert \bigl( \rho ^{1/2}\mathbf {u}_{t}, \rho ^{1/2} \theta _{t}, \mathbf {B}_{t} \bigr) \bigr\Vert _{L^{2}}^{2} \,dt \biggr)^{1/2} \biggl( \int _{\sigma (T)}^{T}t^{-2}\,dt \biggr)^{1/2} \\ \leqslant{}& CC_{0}^{1/8}, \\ J_{4} \leqslant {}&C \biggl( \int _{0}^{\sigma (T)} \Vert \nabla \theta \Vert _{L^{2}}^{2}\,dt \biggr)^{1/2} \biggl( \int _{0}^{\sigma (T)}\,dt \biggr)^{1/2} \\ &{} + C \biggl( \int _{\sigma (T)}^{T}t^{2} \Vert \nabla \theta \Vert _{L^{2}}^{2}\,dt \biggr)^{1/2} \biggl( \int _{\sigma (T)}^{T}t^{-2}\,dt \biggr)^{1/2} \\ \leqslant {}&CC_{0}^{1/2}, \end{aligned}$$
and
$$\begin{aligned} J_{5} \leqslant {}&C \biggl( \int _{0}^{\sigma (T)} \bigl\Vert ( \nabla \mathbf {u}, \nabla \mathbf {B}) \bigr\Vert _{L^{2}}^{2}\,dt \biggr)^{1/8} \biggl( \int _{0}^{\sigma (T)}\,dt \biggr)^{7/8} \\ & {}+ C \biggl( \int _{\sigma (T)}^{T}t^{8} \bigl\Vert (\nabla \mathbf {u}, \nabla \mathbf {B}) \bigr\Vert _{L^{2}}^{2}\,dt \biggr)^{1/8} \biggl( \int _{\sigma (T)}^{T}t^{-8/7}\,dt \biggr)^{7/8} \\ \leqslant {}&CC_{0}^{1/8}. \end{aligned}$$
Putting \(J_{1}\)–\(J_{5}\) into (3.45), we obtain
$$\begin{aligned} \int _{0}^{T} \bigl( \bigl\Vert \nabla ^{2} \mathbf {u}\bigr\Vert _{L^{3+\beta}}+ \bigl\Vert \nabla ^{2} \mathbf {B}\bigr\Vert _{L^{3+\beta}} \bigr)\,dt \leqslant CC_{0}^{1/32}. \end{aligned}$$
(3.46)
Thus, combining (2.2)–(2.4) with (3.46), we have
$$\begin{aligned} \begin{aligned} & \int _{0}^{T} \bigl( \Vert \nabla \mathbf {u}\Vert _{L^{\infty}}+ \Vert \nabla \mathbf {B}\Vert _{L^{\infty}} \bigr)\,dt \\ &\quad \leqslant C \int _{0}^{T} \bigl( \Vert \nabla \mathbf {u}\Vert _{L^{4}}+ \Vert \nabla \mathbf {B}\Vert _{L^{4}}+ \bigl\Vert \nabla ^{2} \mathbf {u}\bigr\Vert _{L^{3+\beta}}+ \bigl\Vert \nabla ^{2} \mathbf {B}\bigr\Vert _{L^{3+\beta}} \bigr)\,dt \\ &\quad \leqslant CC_{0}^{1/32}. \end{aligned} \end{aligned}$$
(3.47)
By (1.1)1 we have that
$$\begin{aligned} \frac {d}{dt}\rho \bigl(t, U(t, s, x) \bigr)=-\rho \bigl(t, U(t, s, x) \bigr)\operatorname{div} \mathbf {u}\bigl(t, U(t, s, x) \bigr), \end{aligned}$$
(3.48)
where
$$\begin{aligned} \frac {d}{dt}\rho \bigl(t, U(t, s, x) \bigr)=\rho _{t} \bigl(t, U(t, s, x) \bigr)+\mathbf {u}\bigl(t, U(t, s, x) \bigr)\cdot \nabla \rho \bigl(t, U(t, s, x) \bigr), \end{aligned}$$
and \(U\in C([0, T]\times [0, T])\times \Omega \) is the solution to the initial value problem
$$\begin{aligned} \textstyle\begin{cases} \frac {d}{dt}U(t, s, x)=\mathbf {u}(t, U(t, s, x)), \quad t\in [0, T], \\ U(t, s, x)=x, \quad s\in [0, T], x\in \Omega. \end{cases}\displaystyle \end{aligned}$$
With the help of (3.47) and (3.48), from the Gronwall inequality we get that
$$\begin{aligned} \rho \leqslant \bar{\rho}\exp \biggl\{ \int _{0}^{T} \Vert \nabla \mathbf {u}\Vert _{L^{\infty}}\,dt \biggr\} \leqslant \bar{\rho}\exp \bigl\{ C_{21}C_{0}^{1/32} \bigr\} \leqslant 2\bar{\rho}, \end{aligned}$$
provided that \(C_{0}\leqslant \varepsilon _{8}\triangleq \min \{\varepsilon _{7}, ( \frac {\ln 2}{C_{21}})^{32}\}\). Thus we immediately obtain (3.40). The proof of the lemma is therefore completed. □
Lemma 3.11
Let \((\rho, \mathbf{ u}, \theta, \mathbf{ B})\) be a smooth solution of (1.1)–(1.4) satisfying (3.1). Then there exists a positive constant ε, depending only on \(\kappa, \nu, R, c_{v}, \underline{\mu}\), ρ̄, Ω, \(g_{1}\), and \(g_{2}\), such that
$$\begin{aligned} \sup_{t\in [0, T]} \Vert \nabla \rho \Vert _{L^{q}}\leqslant 3 \Vert \nabla \rho \Vert _{L^{q}}, \end{aligned}$$
(3.49)
and
$$\begin{aligned} \int _{0}^{T} \bigl( \bigl\Vert \nabla ^{2} \mathbf{ u} \bigr\Vert _{L^{q}}+ \bigl\Vert \nabla ^{2} \mathbf{ B} \bigr\Vert _{L^{q}} \bigr)\,dt\leqslant C, \end{aligned}$$
(3.50)
provided that \(C_{0}\leqslant \varepsilon \).
Proof
Differentiating (1.1)1 with respect to \(x_{i}\) and multiplying the results by \(q|\partial _{i}\rho |^{q-2}\partial _{i} \rho \) give
$$\begin{aligned} \begin{aligned} & \bigl( \vert \nabla \rho \vert ^{q} \bigr)_{t}+\operatorname{div} \bigl( \vert \nabla \rho \vert ^{q}\mathbf {u}\bigr)+(q-1) \vert \nabla \rho \vert ^{q}\operatorname{div} \mathbf {u}\\ & \quad{}+q \vert \nabla \rho \vert ^{q-2}(\nabla \rho )^{\mathrm{tr}} \nabla \mathbf {u}( \nabla \rho )+q\rho \vert \nabla \rho \vert ^{q-2} \nabla \rho \cdot \nabla \operatorname{div} \mathbf {u}=0. \end{aligned} \end{aligned}$$
(3.51)
Integrating (3.51) on Ω yields
$$\begin{aligned} \frac{d}{dt} \Vert \nabla \rho \Vert _{L^{q}} \leqslant C \Vert \nabla \mathbf {u}\Vert _{L^{ \infty}} \Vert \nabla \rho \Vert _{L^{q}}+C_{22} \bigl\Vert \nabla ^{2} \mathbf {u}\bigr\Vert _{L^{q}}. \end{aligned}$$
(3.52)
With the help of (1.1)2, Lemma 2.2, (3.2)–(3.4), (3.40), we get
$$\begin{aligned} \begin{aligned} \bigl\Vert \nabla ^{2} \mathbf {u}\bigr\Vert _{L^{q}} \leqslant{}& C \bigl( \Vert \rho \mathbf {u}_{t} \Vert _{L^{q}}+ \Vert \rho \mathbf {u}\cdot \nabla \mathbf {u}\Vert _{L^{q}}+ \bigl\Vert (\nabla \rho +\nabla \theta )\nabla \mathbf {u}\bigr\Vert _{L^{q}} \bigr) \\ &{} +C \bigl( \Vert \nabla P \Vert _{L^{q}}+ \Vert \mathbf {B}\cdot \nabla \mathbf {B}\Vert _{L^{q}} \bigr) \\ \leqslant {}&C \bigl[ \Vert \rho \mathbf {u}_{t} \Vert _{L^{6}}+ \bigl( \Vert \mathbf {u}\Vert _{L^{q}}+ \Vert \nabla \rho \Vert _{L^{q}}+ \Vert \nabla \theta \Vert _{L^{q}} \bigr) \Vert \nabla \mathbf {u}\Vert _{L^{\infty}} \bigr] \\ &{} +C \bigl( \Vert \rho \Vert _{L^{\infty}} \Vert \nabla \theta \Vert _{L^{q}}+ \Vert \nabla \rho \Vert _{L^{q}} \Vert \theta \Vert _{L^{\infty}}+ \Vert \mathbf {B}\Vert _{L^{q}} \Vert \nabla \mathbf {B}\Vert _{L^{\infty}} \bigr) \\ \leqslant{}& C \bigl( \Vert \nabla \mathbf {u}_{t} \Vert _{L^{2}}+ \Vert \nabla \mathbf {u}\Vert _{L^{\infty}}+ \Vert \nabla \theta \Vert _{H^{1}}+ \Vert \nabla \mathbf {B}\Vert _{L^{ \infty}} \bigr) \\ \leqslant{}& C \bigl( \Vert \nabla \mathbf {u}_{t} \Vert _{L^{2}}+ \Vert \nabla \mathbf {u}\Vert _{L^{\infty}}+ \Vert \nabla \mathbf {B}\Vert _{L^{\infty}} \bigr) \\ &{} +C \bigl( \bigl\Vert \rho ^{1/2} \theta _{t} \bigr\Vert _{L^{2}}+ \Vert \nabla \theta \Vert _{L^{2}}+ \Vert \nabla \mathbf {u}\Vert _{L^{2}}^{1/4}+ \Vert \nabla \mathbf {B}\Vert _{L^{2}}^{1/4} \bigr). \end{aligned} \end{aligned}$$
(3.53)
Thus, similarly to (3.45),
$$\begin{aligned} \begin{aligned} \int _{0}^{T} \bigl\Vert \nabla ^{2} \mathbf {u}\bigr\Vert _{L^{q}} \,dt \leqslant {}&C \int _{0}^{T} \Vert \nabla \mathbf {u}_{t} \Vert _{L^{2}} \,dt+ \int _{0}^{T} \bigl( \Vert \nabla \mathbf {u}\Vert _{L^{\infty}}+ \Vert \nabla \mathbf {B}\Vert _{L^{ \infty}} \bigr)\,dt \\ &{} +C \int _{0}^{T} \bigl( \bigl\Vert \rho ^{1/2} \theta _{t} \bigr\Vert _{L^{2}}+ \Vert \nabla \theta \Vert _{L^{2}}+ \Vert \nabla \mathbf {u}\Vert _{L^{2}}^{1/4}+ \Vert \nabla \mathbf {B}\Vert _{L^{2}}^{1/4} \bigr)\,dt \\ \leqslant {}&C \int _{0}^{T} \Vert \nabla \mathbf {u}_{t} \Vert _{L^{2}} \,dt+CC_{0}^{1/32}. \end{aligned} \end{aligned}$$
(3.54)
On the other hand, from (3.25) and (3.35) it follows that
$$\begin{aligned} \begin{aligned} \int _{0}^{T} \Vert \nabla \mathbf {u}_{t} \Vert _{L^{2}} \,dt \leqslant{}& C \biggl( \int _{0}^{s} \Vert \nabla \mathbf {u}_{t} \Vert _{L^{2}}^{2}\,dt \biggr)^{1/2} \biggl( \int _{0}^{s}\,dt \biggr)^{1/2} \\ & {}+ C \biggl( \int _{s}^{T}t^{2} \Vert \nabla \mathbf {u}_{t} \Vert _{L^{2}}^{2}\,dt \biggr)^{1/2} \biggl( \int _{s}^{T}t^{-2}\,dt \biggr)^{1/2} \\ \leqslant{}& Cs^{1/2}+Cs^{-1/2}C_{0}^{5/32} \forall s\in (0, T], \end{aligned} \end{aligned}$$
which, together with (3.52) and (3.54), yields
$$\begin{aligned} C_{22} \int _{0}^{T} \bigl\Vert \nabla ^{2} \mathbf {u}\bigr\Vert _{L^{q}} \,dt \leqslant C_{23}s^{1/2}+C_{24}s^{-1/2}C_{0}^{5/32}+C_{25}C_{0}^{1/32} \quad \forall s\in (0, T]. \end{aligned}$$
Fixing \(s\ll \sigma (T)\) such that \(C_{23}s^{1/2}\leqslant \frac {1}{6}\|\nabla \rho _{0}\|_{L^{q}}\), we have
$$\begin{aligned} C_{22} \int _{0}^{T} \bigl\Vert \nabla ^{2} \mathbf {u}\bigr\Vert _{L^{q}} \,dt \leqslant \frac{1}{2} \Vert \nabla \rho _{0} \Vert _{L^{q}}, \end{aligned}$$
(3.55)
provided that \(C_{0}\leqslant \varepsilon _{9}\triangleq \min \{\varepsilon _{8}, ( \frac {s^{1/2}\|\nabla \rho _{0}\|_{L^{q}}}{6C_{24}})^{32/5}, ( \frac {\|\nabla \rho _{0}\|_{L^{q}}}{6C_{25}})^{32} \}\).
Using (3.47), (3.52), and (3.55), from the Gronwall inequality we obtain that
$$\begin{aligned} \begin{aligned} \sup_{t\in [0, T]} \Vert \nabla \rho \Vert _{L^{q}} &\leqslant \exp \biggl\{ \int _{0}^{T}C \Vert \nabla \mathbf {u}\Vert _{L^{\infty}} \,dt \biggr\} \biggl[ \Vert \nabla \rho _{0} \Vert _{L^{q}}+ \int _{0}^{T}C_{22} \bigl\Vert \nabla ^{2} \mathbf {u}\bigr\Vert _{L^{q}} \,dt \biggr] \\ &\leqslant \frac{3}{2}\exp \bigl\{ C_{26}C_{0}^{1/32} \bigr\} \Vert \nabla \rho _{0} \Vert _{L^{q}} \\ &\leqslant 3 \Vert \nabla \rho _{0} \Vert _{L^{q}}, \end{aligned} \end{aligned}$$
provided that \(C_{0}\leqslant \varepsilon \triangleq \min \{\varepsilon _{9}, ( \frac {\ln 2}{C_{26}})^{32}\}\). Thus we immediately obtain (3.49).
On the other hand, from (3.42) it follows that
$$\begin{aligned} \begin{aligned} \bigl\Vert \nabla ^{2} \mathbf {B}\bigr\Vert _{L^{q}} &\leqslant C \bigl( \Vert \mathbf {B}_{t} \Vert _{L^{q}}+ \Vert \mathbf {u}\cdot \nabla \mathbf {B}\Vert _{L^{q}}+ \Vert \mathbf {B}\cdot \nabla \mathbf {u}\Vert _{L^{q}}+ \Vert \mathbf {B}\operatorname{div} \mathbf {u}\Vert _{L^{q}} \bigr) \\ &\leqslant C \Vert \mathbf {B}_{t} \Vert _{L^{6}}+C \Vert \mathbf {u}\Vert _{L^{q}} \Vert \nabla \mathbf {B}\Vert _{L^{\infty}}+C \Vert \nabla \mathbf {u}\Vert _{L^{\infty}} \Vert \mathbf {B}\Vert _{L^{q}} \\ &\leqslant C \Vert \nabla \mathbf {B}_{t} \Vert _{L^{2}}+C \Vert \nabla \mathbf {B}\Vert _{L^{ \infty}}+C \Vert \nabla \mathbf {u}\Vert _{L^{\infty}}, \end{aligned} \end{aligned}$$
which, together with (3.53), gives
$$\begin{aligned} \begin{aligned} & \bigl\Vert \nabla ^{2} \mathbf {u}\bigr\Vert _{L^{q}}+ \bigl\Vert \nabla ^{2} \mathbf {B}\bigr\Vert _{L^{q}} \\ &\quad \leqslant C \bigl( \Vert \nabla \mathbf {u}_{t} \Vert _{L^{2}}+ \Vert \nabla \mathbf {B}_{t} \Vert _{L^{2}}+ \Vert \nabla \mathbf {u}\Vert _{L^{\infty}}+ \Vert \nabla \mathbf {B}\Vert _{L^{\infty}} \bigr) \\ & \qquad{}+C \bigl( \bigl\Vert \rho ^{1/2} \theta _{t} \bigr\Vert _{L^{2}}+ \Vert \nabla \theta \Vert _{L^{2}}+ \Vert \nabla \mathbf {u}\Vert _{L^{2}}^{1/4}+ \Vert \nabla \mathbf {B}\Vert _{L^{2}}^{1/4} \bigr). \end{aligned} \end{aligned}$$
(3.56)
Similarly to (3.54), from (3.56) we can immediately obtain (3.50). The proof of the lemma is therefore completed. □