The local existence has been established by Chen et al. [9]. Hence, it only remains to establish some necessary a priori bounds for the strong solutions \((\rho, \mathrm {u},\mathrm {H})\) to the initial-boundary value problem (1.1)-(1.4).
Let \(T>0\) be a fixed time and \((\rho, \mathrm {u}, \mathrm {H})\) be the strong solution to (1.1)-(1.4) defined on \(\Omega\times(0,T]\). Throughout this paper, we will denote by C the various generic positive constants, which may depend on the initial data and ρ̄, \(\underline{\rho}\), f, and Ω but are independent of t. A special dependence will be pointed out explicitly in this paper if necessary.
3.1 Uniform estimate
First, by the method of characteristics and standard energy we have the following uniform estimates:
$$ \underline{\rho}\leq\rho(x,t)\leq\bar{\rho} $$
(3.1)
and
$$\begin{aligned} \sup_{0\leq t\leq \infty} \int \biggl(| \mathrm {u}|^{2}+\frac{1}{2}| \mathrm {H}|^{2} \biggr) \,\mathrm{d} x+ \int_{0}^{\infty}\bigl(\| \nabla \mathrm {u}\|_{L^{2}}^{2}+\|\nabla \mathrm {H}\| _{L^{2}}^{2} \bigr) \,\mathrm{d} t\leq C. \end{aligned}$$
(3.2)
The next lemma is the crucial estimate in this paper. Higher-order estimates of the density, velocity and magnetic field can be obtained in a standard way provided that \(\|\mathrm {u}\|_{H^{1}}\) and \(\|\mathrm {H}\|_{H^{1}}\) are uniformly bounded with respect to time.
Lemma 3.1
Let
\((\rho, \mathrm {u}, \mathrm {H})\)
be a smooth solution of (1.1)-(1.6) on
\(\Omega\times(0,\infty)\). Then there exists a constant
C
such that
$$\begin{aligned} \sup_{0\leq t\leq \infty} \bigl(\|\mathrm {u}\|_{H^{1}}^{2}+\|\mathrm {H}\|_{H^{1}}^{2} \bigr)+ \int_{0}^{\infty}\bigl(\bigl\| \rho^{1/2} \mathrm {u}_{t}\bigr\| _{L^{2}}^{2}+\|\mathrm {H}_{t} \|_{L^{2}}^{2}+\|\mathrm {H}\| _{H^{2}}^{2}+\|\mathrm {u}\|_{H^{2}}^{2} \bigr)\,\mathrm{d} t\leq C. \end{aligned}$$
(3.3)
Proof
Multiplying (1.2) by \(\mathrm {u}_{t}\) and integrating by parts over Ω, one obtains
$$\begin{aligned} \frac{1}{2}\frac{\mathrm {d}}{\mathrm {d}t}\|\nabla \mathrm {u}\|_{L^{2}}^{2}+\bigl\| \rho^{1/2}\mathrm {u}_{t}\bigr\| _{L^{2}}^{2} = \int (\mathrm {H}\cdot\nabla \mathrm {H}\cdot \mathrm {u}_{t}-\rho \mathrm {u}\cdot\nabla \mathrm {u}\cdot \mathrm {u}_{t}+\rho \mathrm {f}\cdot u_{t} ) \,\mathrm {d}x. \end{aligned}$$
(3.4)
Similarly, it follows from (1.3) that
$$\begin{aligned} \frac{\mathrm {d}}{\mathrm {d}t}\|\nabla \mathrm {H}\|_{L^{2}}^{2}+ \bigl(\| \mathrm {H}_{t}\|_{L^{2}}^{2}+\|\triangle \mathrm {H}\|_{L^{2}} \bigr)= \int|\mathrm {H}_{t}-\Delta \mathrm {H}|^{2}\,\mathrm {d}x= \int|\mathrm {H}\cdot\nabla \mathrm {u}-\mathrm {u}\cdot\nabla \mathrm {H}|^{2}\,\mathrm {d}x. \end{aligned}$$
(3.5)
Putting (3.4) and (3.5) together leads to
$$\begin{aligned} &\frac{\mathrm {d}}{\mathrm {d}t} \biggl(\frac{1}{2}\|\nabla \mathrm {u}\|_{L^{2}}^{2}+\|\nabla \mathrm {H}\|_{L^{2}}^{2} \biggr)+ \bigl(\bigl\| \rho^{1/2} \mathrm {u}_{t}\bigr\| _{L^{2}}^{2}+ \|\mathrm {H}_{t}\|_{L^{2}}^{2}+\|\triangle \mathrm {H}\|_{L^{2}}^{2} \bigr) \\ & \quad\leq \int|\mathrm {H}||\nabla \mathrm {H}||\mathrm {u}_{t}|\,\mathrm {d}x+ \int\rho|\mathrm {u}||\nabla \mathrm {u}||\mathrm {u}_{t}| \,\mathrm {d}x+2 \int|\mathrm {H}|^{2}|\nabla \mathrm {u}|^{2}\,\mathrm {d}x \\ &\qquad{} +2 \int|\mathrm {u}|^{2}|\nabla \mathrm {H}|^{2}\,\mathrm {d}x+ \int\rho| \mathrm {f}||\mathrm {u}_{t}| \,\mathrm {d}x=\sum _{i=1}^{5}I_{i}. \end{aligned}$$
(3.6)
Next, we will estimate all the terms on the right-hand side of (3.6) term by term. Using the Young inequality and (3.1), we have
$$\begin{aligned} I_{1} \leq&\varepsilon\bigl\| \rho^{1/2}\mathrm {u}_{t} \bigr\| ^{2}_{L^{2}}+C_{1}\|\mathrm {H}\| ^{2}_{L^{4}} \|\nabla \mathrm {H}\|^{2}_{L^{4}} \\ \leq&\varepsilon\bigl\| \rho^{1/2}\mathrm {u}_{t}\bigr\| ^{2}_{L^{2}}+ \varepsilon\|\Delta \mathrm {H}\|_{L^{2}}+C(\varepsilon)\|\nabla \mathrm {H}\|^{4}_{L^{2}}, \end{aligned}$$
where we use the fact
$$\begin{aligned} \|\mathrm {H}\|^{2}_{L^{4}}\|\nabla \mathrm {H}\|^{2}_{L^{4}} \leq& C\bigl(\|\mathrm {H}\|^{2}_{L^{2}}+ \|\mathrm {H}\|_{L^{2}}\|\nabla \mathrm {H}\|_{L^{2}}\bigr) \bigl(\|\nabla \mathrm {H}\|_{L^{2}}\|\Delta \mathrm {H}\|_{L^{2}}+\|\nabla \mathrm {H}\|^{2}_{L^{2}} \bigr) \\ \leq&\|\mathrm {H}\| ^{2}_{L^{2}}\|\nabla \mathrm {H}\|_{L^{2}}\|\Delta \mathrm {H}\|_{L^{2}}+\|\mathrm {H}\|^{2}_{L^{2}}\| \nabla \mathrm {H}\|^{2}_{L^{2}} \\ &{}+\|\mathrm {H}\|_{L^{2}}\|\nabla \mathrm {H}\| ^{2}_{L^{2}}\|\Delta \mathrm {H}\|_{L^{2}}+\|\mathrm {H}\|^{2}_{L^{2}}\|\nabla \mathrm {H}\|^{3}_{L^{2}} \\ \leq&\frac{\varepsilon}{C_{1}}\|\Delta \mathrm {H}\|_{L^{2}}+C\| \nabla \mathrm {H}\|^{4}_{L^{2}}, \end{aligned}$$
(3.7)
due to (3.2), (2.5), and (2.6). Similarly, we have
$$\begin{aligned} I_{2} \leq&\varepsilon\bigl\| \rho^{1/2}\mathrm {u}_{t} \bigr\| ^{2}_{L^{2}}+C(\varepsilon)\| \mathrm {u}\|^{2}_{L^{4}} \|\nabla \mathrm {u}\|^{2}_{L^{4}} \\ \leq&\varepsilon\bigl\| \rho^{1/2}\mathrm {u}_{t}\bigr\| ^{2}_{L^{2}}+ \varepsilon\bigl\| \nabla ^{2} u\bigr\| _{L^{2}}+C\|\nabla \mathrm {u}\|^{4}_{L^{2}}. \end{aligned}$$
Next, we turn to estimating \(\|\nabla^{2} u\|_{L^{2}}\). From (1.2), we know that u satisfies the following Stokes equations:
$$ \left \{ \textstyle\begin{array}{@{}l} -\triangle \mathrm {u}+\nabla P=-\rho \mathrm {u}_{t}-\rho \mathrm {u}\cdot\nabla \mathrm {u}+\frac{1}{2}\nabla|\mathrm {H}|^{2}-\mathrm {H}\cdot\nabla \mathrm {H}+\rho \mathrm {f},\\ \operatorname {div}\mathrm {u}=0, \quad\mbox{in } \Omega,\\ \mathrm {u}=0, \quad\mbox{on } \partial\Omega. \end{array}\displaystyle \right . $$
By the well-known regularity theory on the Stokes equations (see [23]) and using (2.5), (3.2), and (3.7), we have
$$\begin{aligned} \|\mathrm {u}\|_{H^{2}} \leq& C\bigl(\|\rho \mathrm {u}_{t}\|_{L^{2}}+\| \rho \mathrm {u}\cdot\nabla \mathrm {u}\|_{L^{2}}+\|\nabla|\mathrm {H}|^{2} \|_{L^{2}}+\|\mathrm {H}\cdot\nabla \mathrm {H}\| _{L^{2}}+\|\mathrm {f}\|_{L^{2}} \bigr) \\ \leq& C\bigl(\bigl\| \rho^{1/2} \mathrm {u}_{t}\bigr\| _{L^{2}}+\|\mathrm {u}\|_{L^{4}}\|\nabla \mathrm {u}\| _{L^{4}}+\|\mathrm {H}\|_{L^{4}}\|\nabla \mathrm {H}\|_{L^{4}}+\|\mathrm {f}\|_{L^{2}}\bigr) \\ \leq& C\bigl(\bigl\| \rho^{1/2} \mathrm {u}_{t}\bigr\| _{L^{2}}+\|\nabla \mathrm {u}\|_{L^{2}}\bigl\| \nabla^{2} \mathrm {u}\bigr\| ^{1/2}_{L^{2}} \\ &{}+\| \nabla \mathrm {u}\|^{2}_{L^{2}}+\|\nabla \mathrm {H}\|_{L^{2}}\| \Delta \mathrm {H}\|^{1/2}_{L^{2}}+\|\nabla \mathrm {H}\|^{2}_{L^{2}}+\|\mathrm {f}\| _{L^{2}}\bigr) \\ \leq& C\bigl(\bigl\| \rho^{1/2} \mathrm {u}_{t}\bigr\| _{L^{2}}+\|\nabla \mathrm {u}\|^{2}_{L^{2}}+\|\nabla \mathrm {H}\|^{2}_{L^{2}}+\| \mathrm {f}\|_{L^{2}}\bigr)+\frac{1}{2}\bigl\| \nabla^{2} \mathrm {u}\bigr\| ^{1/2}_{L^{2}}+\|\Delta \mathrm {H}\|^{1/2}_{L^{2}}, \end{aligned}$$
which immediately leads to
$$\begin{aligned} \|\mathrm {u}\|_{H^{2}}\leq C\bigl(\bigl\| \rho^{1/2} \mathrm {u}_{t} \bigr\| _{L^{2}}+\|\nabla \mathrm {u}\| ^{2}_{L^{2}}+\|\nabla \mathrm {H}\|^{2}_{L^{2}}+\|\mathrm {f}\|_{L^{2}}\bigr)+2\|\Delta \mathrm {H}\|_{L^{2}}. \end{aligned}$$
Substituting the above inequality into \(I_{2}\), we have
$$\begin{aligned} I_{2}\leq C\varepsilon\bigl\| \rho^{1/2}\mathrm {u}_{t} \bigr\| ^{2}_{L^{2}}+\varepsilon\|\Delta \mathrm {H}\|^{2}_{L^{2}}+C \|\nabla \mathrm {u}\|^{4}_{L^{2}}+C\|\nabla \mathrm {u}\|^{2}_{L^{2}} \| \nabla \mathrm {H}\|^{2}_{L^{2}}+C\|\mathrm {f}\|^{2}_{L^{2}}. \end{aligned}$$
Similarly, for \(I_{3}\) and \(I_{4}\), we have
$$\begin{aligned} I_{3}\leq\varepsilon\bigl\| \rho^{1/2}\mathrm {u}_{t} \bigr\| ^{2}_{L^{2}}+\varepsilon\|\Delta \mathrm {H}\|^{2}_{L^{2}}+C \|\nabla \mathrm {H}\|^{2}_{L^{2}}\|\nabla \mathrm {u}\|^{2}_{L^{2}}+C \| \nabla \mathrm {u}\|^{4}_{L^{2}}+C\|\nabla \mathrm {H}\|^{4}_{L^{2}} \end{aligned}$$
and
$$\begin{aligned} I_{4} \leq& C\|\mathrm {u}\|^{2}_{L^{4}}\|\nabla \mathrm {H}\|^{2}_{L^{4}}\leq C\|\mathrm {u}\|_{L^{2}}\| \nabla \mathrm {u}\|_{L^{2}}\bigl(\|\nabla \mathrm {H}\|^{2}_{L^{2}}+\|\nabla \mathrm {H}\|_{L^{2}}\| \Delta \mathrm {H}\|_{L^{2}}\bigr) \\ \leq& C\|\nabla \mathrm {H}\|^{2}_{L^{2}}\|\nabla \mathrm {u}\|^{2}_{L^{2}}+\varepsilon\| \Delta \mathrm {H}\|^{2}_{L^{2}}, \end{aligned}$$
due to (3.2).
Using the Young inequality immediately leads to
$$\begin{aligned} I_{5}\leq\varepsilon\bigl\| \rho^{1/2}\mathrm {u}_{t} \bigr\| ^{2}_{L^{2}}+C(\varepsilon)\| \mathrm {f}\|^{2}_{L^{2}}. \end{aligned}$$
Taking ε small enough and substituting \(I_{1}\)-\(I_{5}\) into (3.6), one obtains
$$\begin{aligned} &\frac{\mathrm {d}}{\mathrm {d}t} \bigl(\|\nabla \mathrm {u}\|_{L^{2}}^{2}+\|\nabla \mathrm {H}\|_{L^{2}}^{2} \bigr)+ \bigl(\bigl\| \rho^{1/2} \mathrm {u}_{t}\bigr\| _{L^{2}}^{2}+\|\mathrm {H}_{t} \|_{L^{2}}^{2}+\|\triangle \mathrm {H}\|_{L^{2}}^{2} \bigr) \\ &\quad\leq C\bigl(\|\nabla \mathrm {H}\|^{2}_{L^{2}}+\|\nabla \mathrm {u}\|^{2}_{L^{2}}\bigr)^{2}+\|\mathrm {f}\|^{2}_{L^{2}}, \end{aligned}$$
which, together with Gronwall’s inequality, immediately leads to
$$\begin{aligned} \sup_{0\leq t\leq \infty} \bigl(\|\nabla \mathrm {u}\|_{L^{2}}^{2}+ \|\nabla \mathrm {H}\|_{L^{2}}^{2} \bigr)+ \int_{0}^{\infty}\bigl(\bigl\| \rho^{1/2} \mathrm {u}_{t}\bigr\| _{L^{2}}^{2}+\|\mathrm {H}_{t} \|_{L^{2}}^{2}+\bigl\| \nabla^{2} \mathrm {H}\bigr\| _{L^{2}}^{2} \bigr)\,\mathrm{d} t\leq C. \end{aligned}$$
(3.8)
We get the desired estimate (3.3) by (3.2) and (3.8). □
The following lemma is concerned with the \(L^{2}\)-estimate of \(\rho^{1/2} \mathrm {u}_{t}\) and \(\mathrm {H}_{t}\).
Lemma 3.2
Let
\((\rho, \mathrm {u}, \mathrm {H})\)
be a strong solution of (1.1)-(1.6) on
\(\Omega\times(0,\infty)\). Then there exists a constant
C
such that
$$\begin{aligned} \sup_{0\leq t\leq \infty} \bigl(\bigl\| \rho^{1/2}\mathrm {u}_{t} \bigr\| _{L^{2}}^{2}+\|\mathrm {H}_{t}\|_{L^{2}}^{2} \bigr)+ \int_{0}^{\infty}\|\nabla \mathrm {u}_{t} \|_{L^{2}}^{2}+\|\nabla \mathrm {H}_{t}\| _{L^{2}}^{2} \,\mathrm{d} t \leq C. \end{aligned}$$
(3.9)
Proof
Differentiating the momentum equations (1.2) with respect to t yields
$$\begin{aligned} &\rho \mathrm {u}_{tt}+\rho \mathrm {u}\cdot\nabla \mathrm {u}_{t}+\rho \mathrm {u}_{t}\cdot\nabla \mathrm {u}+\rho_{t} (\mathrm {u}_{t}+\mathrm {u}\cdot \nabla \mathrm {u})+\nabla P_{t}\\ &\quad=\triangle \mathrm {u}_{t}+\biggl(\mathrm {H}\cdot \nabla \mathrm {H}-\frac{1}{2}\nabla|\mathrm {H}|^{2}\biggr)_{t}+(\rho \mathrm {f})_{t}. \end{aligned}$$
Multiplying the equation above with \(\mathrm {u}_{t}\) and integrating by parts over Ω, one gets
$$\begin{aligned} &\frac{1}{2}\frac{\mathrm {d}}{\mathrm {d}t}\bigl\| \rho^{1/2} \mathrm {u}_{t}\bigr\| _{L^{2}}^{2}+\|\nabla \mathrm {u}_{t} \|_{L^{2}}^{2} \\ &\quad=- \int\rho_{t}|\mathrm {u}_{t}|^{2}\,\mathrm {d}x- \int\rho(\mathrm {u}_{t}\cdot\nabla \mathrm {u})\cdot \mathrm {u}_{t}\,\mathrm{d}x- \int\rho_{t}(\mathrm {u}\cdot\nabla \mathrm {u})\cdot \mathrm {u}_{t}\,\mathrm {d}x \\ &\qquad{} + \int(\mathrm {H}_{t}\cdot\nabla \mathrm {H}+\mathrm {H}\cdot\nabla \mathrm {H}_{t}+ \rho _{t}\mathrm {f}+\rho \mathrm {f}_{t})\cdot \mathrm {u}_{t}\,\mathrm {d}x. \end{aligned}$$
(3.10)
Differentiating (1.3) with respect to t and multiplying the resulting equation by \(\mathrm {H}_{t}\), we obtain after integrating by parts
$$\begin{aligned} &\frac{1}{2}\frac{\mathrm {d}}{\mathrm {d}t}\|\mathrm {H}_{t}\|_{L^{2}}^{2}+ \|\nabla \mathrm {H}_{t}\| _{L^{2}}^{2} \\ &\quad= - \int \mathrm {u}_{t}\cdot\nabla \mathrm {H}\cdot \mathrm {H}_{t} \,\mathrm {d}x + \int \mathrm {H}_{t}\cdot\nabla \mathrm {u}\cdot \mathrm {H}_{t}+\mathrm {H}\cdot \nabla \mathrm {u}_{t}\cdot \mathrm {H}_{t}\,\mathrm {d}x. \end{aligned}$$
(3.11)
Putting (3.10) and (3.11) together leads to
$$\begin{aligned} &\frac{1}{2}\frac{\mathrm {d}}{\mathrm {d}t} \bigl(\|\mathrm {H}_{t} \|_{L^{2}}^{2}+\bigl\| \rho ^{1/2}\mathrm {u}_{t} \bigr\| _{L^{2}}^{2} \bigr)+\|\nabla \mathrm {u}_{t} \|_{L^{2}}^{2}+\|\nabla \mathrm {H}_{t}\|_{L^{2}}^{2} \\ &\quad = - \int\rho_{t}|\mathrm {u}_{t}|^{2}\,\mathrm {d}x- \int\rho(\mathrm {u}_{t}\cdot\nabla \mathrm {u})\cdot \mathrm {u}_{t}\,\mathrm {d}x- \int\rho_{t}(\mathrm {u}\cdot\nabla \mathrm {u})\cdot \mathrm {u}_{t}\,\mathrm {d}x \\ &\qquad{}+ \int\rho_{t}\mathrm {f}\cdot \mathrm {u}_{t}\,\mathrm {d}x+ \int\rho \mathrm {f}_{t}\cdot \mathrm {u}_{t}\,\mathrm {d}x \\ &\qquad{} + \int \mathrm {H}_{t}\cdot\nabla \mathrm {H}\cdot \mathrm {u}_{t}- \mathrm {u}_{t}\cdot\nabla \mathrm {H}\cdot \mathrm {H}_{t} \,\mathrm {d}x + \int \mathrm {H}_{t}\cdot\nabla \mathrm {u}\cdot \mathrm {H}_{t}\,\mathrm {d}x \\ &\qquad{}+ \int \mathrm {H}\cdot\nabla \mathrm {u}_{t}\cdot \mathrm {H}_{t}+\mathrm {H}\cdot \nabla \mathrm {H}_{t}\cdot \mathrm {u}_{t}\,\mathrm {d}x \\ &\quad\triangleq\sum_{i=1}^{8}R_{i}. \end{aligned}$$
(3.12)
We now estimate each term on the right-hand side of (3.12). First, using the Hölder and Young inequalities, we obtain
$$\begin{aligned} |R_{1}| =&\biggl\vert \int\rho \mathrm {u}\cdot\nabla|\mathrm {u}_{t}|^{2}\,\mathrm {d}x\biggr\vert \leq C\|\nabla \mathrm {u}_{t}\|_{L^{2}}\|\mathrm {u}\|_{L^{4}}\| \mathrm {u}_{t}\|_{L^{4}} \\ \leq& C\|\nabla \mathrm {u}_{t}\|_{L^{2}}\| \mathrm {u}\|^{1/2}_{L^{2}} \|\nabla \mathrm {u}\| ^{1/2}_{L^{2}}\| \mathrm {u}_{t} \|^{1/2}_{L^{2}}\|\nabla \mathrm {u}_{t}\| ^{1/2}_{L^{2}} \\ \leq& C\| \mathrm {u}_{t}\|^{1/2}_{L^{2}}\|\nabla \mathrm {u}_{t}\|^{3/2}_{L^{2}}\leq \varepsilon\|\nabla \mathrm {u}_{t}\|^{2}_{L^{2}}+C(\varepsilon)\bigl\| \rho^{1/2} \mathrm {u}_{t}\bigr\| ^{2}_{L^{2}}, \\ |R_{2}| \leq& C \int|\mathrm {u}_{t}|^{2}|\nabla \mathrm {u}|\,\mathrm {d}x\leq C\|\nabla \mathrm {u}\| _{L^{2}}\|\mathrm {u}_{t}\|^{2}_{L^{4}} \\ \leq& C\|\mathrm {u}_{t}\|_{L^{2}}\|\nabla \mathrm {u}_{t} \|_{L^{2}}\leq\varepsilon\| \nabla \mathrm {u}_{t}\|^{2}_{L^{2}}+C( \varepsilon)\bigl\| \rho^{1/2}\mathrm {u}_{t}\bigr\| ^{2}_{L^{2}}, \end{aligned}$$
where (3.1), (3.2), and (3.3) are all used.
The estimate of \(R_{3}\) is given as follows:
$$\begin{aligned} |R_{3}| =& \biggl\vert \int\rho \mathrm {u}\cdot\nabla(\mathrm {u}\cdot\nabla \mathrm {u}\cdot \mathrm {u}_{t}) \,\mathrm {d}x \biggr\vert \\ \leq& C \int|\mathrm {u}||\nabla \mathrm {u}|^{2}|\mathrm {u}_{t}|+| \mathrm {u}|^{2}|\nabla^{2} \mathrm {u}||\mathrm {u}_{t}|+| \mathrm {u}|^{2}|\nabla \mathrm {u}||\nabla \mathrm {u}_{t}|\,\mathrm {d}x\triangleq\sum _{i=1}^{3}J_{i}. \end{aligned}$$
Using (2.4) and (3.3), we can make the reduction
$$\begin{aligned} |J_{1}| \leq& C \int|\mathrm {u}||\nabla \mathrm {u}|^{2}|\mathrm {u}_{t}| \,\mathrm {d}x\leq C\| \nabla \mathrm {u}\|^{2}_{L^{4}}\|\mathrm {u}\|_{L^{4}}\| \mathrm {u}_{t}\|_{L^{4}} \\ \leq& C\bigl(\|\nabla \mathrm {u}\|^{2}_{L^{2}}+\|\nabla \mathrm {u}\|_{L^{2}}\bigl\| \nabla^{2} \mathrm {u}\bigr\| _{L^{2}}\bigr)\|\nabla \mathrm {u}\|_{L^{2}}\|\nabla \mathrm {u}_{t}\|_{L^{2}} \\ \leq& C\bigl(1+\bigl\| \nabla^{2} \mathrm {u}\bigr\| _{L^{2}}\bigr)\|\nabla \mathrm {u}\|_{L^{2}}\|\nabla \mathrm {u}_{t}\|_{L^{2}} \\ \leq&\varepsilon\|\nabla \mathrm {u}_{t}\|^{2}_{L^{2}}+C( \varepsilon)\|\nabla \mathrm {u}\|^{2}_{L^{2}}+ C(\varepsilon)\bigl\| \nabla^{2} \mathrm {u}\bigr\| ^{2}_{L^{2}}, \\ |J_{2}| \leq& C\bigl\| \nabla^{2} \mathrm {u}\bigr\| _{L^{2}}\|\mathrm {u}\|^{2}_{L^{8}}\|\mathrm {u}_{t}\| _{L^{4}} \\ \leq& C\bigl\| \nabla^{2} \mathrm {u}\bigr\| _{L^{2}}\|\nabla \mathrm {u}\|^{2}_{L^{2}}\|\nabla \mathrm {u}_{t}\| _{L^{2}} \\ \leq&\varepsilon\|\nabla \mathrm {u}_{t}\|^{2}_{L^{2}}+C( \varepsilon)\bigl\| \nabla ^{2} \mathrm {u}\bigr\| ^{2}_{L^{2}}, \\ |J_{3}| \leq& C\|\nabla \mathrm {u}\|_{L^{4}}\|\mathrm {u}\|^{2}_{L^{8}} \|\nabla \mathrm {u}_{t}\| _{L^{2}}\leq\varepsilon\|\nabla \mathrm {u}_{t}\|^{2}_{L^{2}}+C(\varepsilon)\bigl\| \nabla^{2} \mathrm {u}\bigr\| ^{2}_{L^{2}}+C(\varepsilon)\|\nabla \mathrm {u}\|^{2}_{L^{2}}. \end{aligned}$$
From the estimate of \(J_{1}\)-\(J_{3}\), we obtain
$$\begin{aligned} |R_{3}|\leq3\varepsilon\|\nabla \mathrm {u}_{t}\|^{2}_{L^{2}}+C \bigl\| \nabla^{2} \mathrm {u}\bigr\| ^{2}_{L^{2}}+C\|\nabla \mathrm {u}\|^{2}_{L^{2}}. \end{aligned}$$
For \(R_{4}\)-\(R_{7}\), we have
$$\begin{aligned} |R_{4}| =&\biggl\vert \int\rho \mathrm {u}\cdot\nabla(\mathrm {f}\cdot \mathrm {u}_{t})\,\mathrm {d}x\biggr\vert \\ \leq& C\|\mathrm {u}\|_{L^{4}}\|\mathrm {u}_{t}\|_{L^{4}}\|\nabla \mathrm {f}\|_{L^{2}}+C\|\mathrm {u}\| _{L^{4}}\|\nabla \mathrm {u}_{t} \|_{L^{2}}\|\mathrm {f}\|_{L^{4}} \\ \leq& C\|\nabla \mathrm {u}_{t}\|_{L^{2}}\|\nabla \mathrm {f}\|_{L^{2}}+C\|\mathrm {f}\|_{H^{1}}\| \nabla \mathrm {u}_{t} \|_{L^{2}} \\ \leq&\varepsilon\|\nabla \mathrm {u}_{t}\|^{2}_{L^{2}}+C( \varepsilon)\|\mathrm {f}\| ^{2}_{H^{1}}, \\ |R_{5}| \leq&\bigl\| \rho^{1/2}\mathrm {u}_{t} \bigr\| ^{2}_{L^{2}}+\|\mathrm {f}_{t}\|^{2}_{L^{2}}, \\ |R_{6}| \leq& C \int|\mathrm {H}_{t}||\nabla \mathrm {u}||\mathrm {u}_{t}|\,\mathrm {d}x \\ \leq& C\|\nabla \mathrm {u}\|_{L^{4}}\|\mathrm {H}_{t}\|_{L^{2}}\| \mathrm {u}_{t}\|_{L^{4}} \\ \leq& C\|\nabla \mathrm {u}\|^{1/2}_{H^{1}}\|\mathrm {H}_{t} \|_{L^{2}}\|\nabla \mathrm {u}_{t}\| _{L^{2}} \\ \leq&\varepsilon\|\nabla \mathrm {u}_{t}\|^{2}_{L^{2}}+C( \varepsilon)\|\nabla \mathrm {u}\|_{H^{1}}\|\mathrm {H}_{t}\|^{2}_{L^{2}} \\ \leq&\varepsilon\|\nabla \mathrm {u}_{t}\|^{2}_{L^{2}}+C( \varepsilon)\|\mathrm {H}_{t}\| ^{4}_{L^{2}}+C(\varepsilon)\| \mathrm {u}\|^{2}_{H^{2}}, \\ |R_{7}| \leq& C \int|\mathrm {H}_{t}|^{2}|\nabla \mathrm {u}|\,\mathrm {d}x\leq C\| \mathrm {H}_{t}\|^{2}_{L^{4}}\| \nabla \mathrm {u}\|_{L^{2}} \\ \leq& C\bigl(\|\mathrm {H}_{t}\|^{2}_{L^{2}}+\| \mathrm {H}_{t}\|_{L^{2}}\|\nabla \mathrm {H}_{t}\| _{L^{2}} \bigr) \\ \leq&\varepsilon\|\nabla \mathrm {H}_{t}\|^{2}_{L^{2}}+C( \varepsilon)\| \mathrm {H}_{t}\|^{2}_{L^{2}}. \end{aligned}$$
It is easy to prove that \(R_{8}=0\). Taking ε small enough and substituting \(R_{1}\)-\(R_{8}\) into (3.12), one obtains
$$\begin{aligned} &\frac{\mathrm {d}}{\mathrm {d}t} \bigl(\|\mathrm {H}_{t}\|_{L^{2}}^{2}+\bigl\| \rho^{1/2}\mathrm {u}_{t}\bigr\| _{L^{2}}^{2} \bigr)+ \bigl(\|\nabla \mathrm {u}_{t}\|_{L^{2}}^{2}+ \|\nabla \mathrm {H}_{t}\|_{L^{2}}^{2} \bigr) \\ &\quad \leq C\|\nabla \mathrm {u}\|^{2}_{L^{2}}+ C\|\mathrm {u}\|^{2}_{H^{2}}+C\| \mathrm {H}_{t}\| ^{2}_{L^{2}}+C \| \mathrm {H}_{t}\|^{4}_{L^{2}}+C\bigl\| \rho^{1/2} \mathrm {u}_{t}\bigr\| ^{2}_{L^{2}}+C\| \nabla \mathrm {f}\|^{2}_{L^{2}}+C\|\mathrm {f}_{t}\|^{2}_{L^{2}}, \end{aligned}$$
which, together with Gronwall’s inequality and (3.3), immediately leads to the desired estimate (3.9). □
3.2 Time-dependent estimate for higher derivative
It suffices to prove the large-time behavior with the help of the uniform estimates in Lemmas 3.1 and 3.2. Lemma 3.3 below deals with the higher-order estimates of the solutions which are needed to guarantee the extension of a local classical solution to a global one.
Lemma 3.3
Let
\((\rho, \mathrm {u}, \mathrm {H})\)
be a strong solution of (1.1)-(1.6) on
\(\Omega\times(0,T)\). Then there exists a constant
\(C(T)\)
such that
$$\begin{aligned} \sup_{0\leq t\leq T} \bigl(\|\rho\|_{H^{2}}+\|\mathrm {u}\|_{H^{2}}+\| \mathrm {H}\|_{H^{2}} \bigr)+ \int _{0}^{T}\|\mathrm {u}\|_{H^{3}}^{2}+\| \mathrm {H}\|_{H^{3}}^{2}\,\mathrm{d} t \leq C(T). \end{aligned}$$
(3.13)
Proof
The proof of this lemma is standard; details are omitted. □