In the section, we will establish the uniform a priori bounds of local solutions \((\rho , \mathbf {u}, \mathbf {B})\) to the Cauchy problems (1.1)–(1.5) whose existence is guaranteed by Lemma 2.4. Thus, let \(T> 0\) be a fixed time and \((\rho , \mathbf {u}, \mathbf {B})\) be the smooth solution of (1.1)–(1.5) on \(\mathbb{R}^{3}\times [0, T]\) with smooth initial data \((\rho _{0}, \mathbf {u}_{0}, \mathbf {B}_{0})\) satisfying (1.8). To estimate this solution, we define
$$\begin{aligned}& A_{1}(T) \triangleq \sup_{t\in [0,T]} \bigl\Vert (\nabla \mathbf {u}, \nabla \mathbf {B}) \bigr\Vert _{L^{2}}^{2}+ \int _{0}^{T} \bigl\Vert \bigl(\sqrt{\rho }\dot{ \mathbf {u}}, \nabla ^{2} \mathbf {B}, \mathbf {B}_{t} \bigr) \bigr\Vert _{L^{2}}^{2}\,dt, \\& A_{2}(T) \triangleq \sup_{t\in [0,T]} \bigl\Vert \bigl( \sqrt{\rho }\dot{\mathbf {u}}, \nabla ^{2} \mathbf {B}, \mathbf {B}_{t} \bigr) \bigr\Vert _{L^{2}}^{2}+ \int _{0}^{T} \bigl\Vert ( \nabla \dot{\mathbf {u}}, \nabla \mathbf {B}_{t}) \bigr\Vert _{L^{2}}^{2} \,dt. \end{aligned}$$
Here, \(\|(f, g)\|_{L^{p}}\triangleq \|f\|_{L^{p}}+\|g\|_{L^{p}}\).
The proof of Theorem 1.1 is based on the following key a priori estimates of \((\rho , \mathbf {u}, \mathbf {B})\).
Proposition 3.1
Let the conditions (1.8) and (1.9) be in force. Assume that \((\rho , \mathbf{ u}, \mathbf{ B})\) is a smooth solution of (1.1)–(1.5) on \(\mathbb{R}^{3} \times [0, T]\) with \(T>0\). Then there exist positive constants K and ε, depending only on μ, λ, ν, γ, A, \(E_{0}\), \(M_{1}\), and \(M_{2}\), such that if
$$ \textstyle\begin{cases} 0\leqslant \rho (x, t)\leqslant 2M_{0}, \quad \forall (x, t)\in \mathbb{R}^{3}\times [0, T], \\ A_{1}(t)+A_{2}(t)\leqslant 2K, \end{cases} $$
(3.1)
then one has
$$ \textstyle\begin{cases} 0\leqslant \rho (x, t)\leqslant \frac {7}{4}M_{0}, \quad \forall (x, t) \in \mathbb{R}^{3}\times [0, T], \\ A_{1}(t)+A_{2}(t)\leqslant K, \end{cases} $$
(3.2)
provided
$$ \Vert \mathbf{ B}_{0} \Vert _{L^{3}}^{\frac {1}{2\gamma }} \leqslant M_{0} \leqslant \varepsilon \quad \textit{and}\quad \varepsilon \in (1, 6). $$
(3.3)
The proof of Proposition 3.1 will be presented by a series of lemmas below. For simplicity, we will use the conventions that C and \(C_{i}\) (\(i = 1, 2, \dots \)) denote various positive constants, which may depend on μ, λ, ν, γ, A, \(E_{0}\), \(M_{1}\), and \(M_{2}\), but are independent of T and \(M_{0}\). Sometimes we also write \(C(\alpha )\) to emphasize the dependence on α.
We first begin with the following standard energy estimates, which can be easily deduced from (1.1)–(1.5).
Lemma 3.1
Let \((\rho , \mathbf{ u}, \mathbf{ B})\) be a smooth solution of (1.1)–(1.5) on \(\mathbb{R}^{3}\times [0,T]\). Then
$$ E(t)+ \int _{0}^{T} \bigl(\mu \Vert \nabla \mathbf{ u} \Vert _{L^{2}}^{2}+(\mu + \lambda ) \Vert \operatorname{div} \mathbf{ u} \Vert _{L^{2}}^{2}+\nu \Vert \nabla \mathbf{ B} \Vert _{L^{2}}^{2} \bigr)\,dt\leqslant E_{0}, $$
(3.4)
where \(E(t)\geqslant 0\) and \(E_{0}\) are as given in (1.6) and (1.7).
Proof
Multiplying (1.1)1, (1.1)2, and (1.1)3 by \(\frac {\gamma }{\gamma -1}A\rho ^{\gamma -1}\), \(\mathbf{ u,}\) and B, respectively, and integrating the resulting equations by parts over \(\mathbb{R}^{3}\), we obtain after adding them together that
$$ \frac {d}{dt}E(t)+\mu \Vert \nabla \mathbf{ u} \Vert _{L^{2}}^{2}+(\mu + \lambda )\|\operatorname{div} \mathbf{ u}\|_{L^{2}}^{2}+\nu \|\nabla \mathbf{ B} \|_{L^{2}}^{2}=0, $$
which, integrated over \((0, t)\), \(\forall t\in [0, T]\), immediately leads to (3.4). □
By virtue of (3.1) and (3.4), we infer from Lemma 2.1 (\(p=6\) in (2.1)) that
$$ \begin{aligned} & \int _{0}^{T} \bigl( \Vert \mathbf{ u} \Vert _{L^{6}}^{4}+ \Vert \nabla \mathbf{ u} \Vert _{L^{2}}^{4}+ \Vert \nabla \mathbf{ B} \Vert _{L^{2}}^{4} \bigr)\,dt \\ &\quad \leqslant C \int _{0}^{T} \bigl( \Vert \nabla \mathbf{ u} \Vert _{L^{2}}^{4}+ \Vert \nabla \mathbf{ B} \Vert _{L^{2}}^{4} \bigr)\,dt \\ &\quad \leqslant C\sup_{t\in [0, T]} \bigl( \Vert \nabla \mathbf{ u} \Vert _{L^{2}}^{2}+ \Vert \nabla \mathbf{ B} \Vert _{L^{2}}^{2} \bigr) \int _{0}^{T} \bigl( \Vert \nabla \mathbf{ u} \Vert _{L^{2}}^{2}+ \Vert \nabla \mathbf{ B} \Vert _{L^{2}}^{2} \bigr)\,dt \leqslant C(E_{0})K. \end{aligned} $$
(3.5)
Here the constant \(C>0\) comes from the Gagliardo–Nirenberg–Sobolev inequality in (2.1), and we use the constant \(C(E_{0})>0\) to emphasize the dependence on \(E_{0}\). Thus, combining (3.4) and (3.5) yields the following lemma:
Lemma 3.2
Let \((\rho , \mathbf{ u}, \mathbf{ B})\) be a smooth solution of (1.1)–(1.5) on \(\mathbb{R}^{3}\times [0,T]\) satisfying (3.1). Then
$$ \sup_{t\in [0, T]} \Vert \mathbf{ B} \Vert _{L^{3}}^{3}+ \int _{0}^{T} \Vert \mathbf{ B} \Vert _{L^{9}}^{3}\,dt\leqslant e^{CK} \Vert \mathbf{ B}_{0} \Vert _{L^{3}}^{3}, $$
(3.6)
where the constant \(C>0\) depends on ν, \(E_{0}\), and the coefficients of the Gagliardo–Nirenberg–Sobolev inequality in Lemma 2.1, but is independent of \(M_{0}\).
Proof
Multiplying the third equation of (1.1) by \(3|\mathbf {B}|\mathbf {B}\) and integrating by parts over \(\mathbb{R}^{3}\), we have
$$\begin{aligned}& \frac {d}{dt} \Vert \mathbf {B}\Vert _{L^{3}}^{3}+3 \nu \int \bigl( \vert \mathbf {B}\vert \vert \nabla \mathbf {B}\vert ^{2}+ \vert \mathbf {B}\vert \bigl\vert \nabla \bigl( \vert \mathbf {B}\vert \bigr) \bigr\vert ^{2} \bigr)\,dx \\& \quad \leqslant \nu \int \vert \mathbf {B}\vert \vert \nabla \mathbf {B}\vert ^{2}\,dx+C \Vert \nabla \mathbf {u}\Vert _{L^{2}}^{2} \Vert \mathbf {B}\Vert _{L^{9/2}}^{3}, \end{aligned}$$
(3.7)
where the last term on the right-hand in (3.7) comes from the following inequality:
$$ \begin{aligned} \int \vert \nabla \mathbf{ u} \vert \vert \mathbf{ B} \vert ^{3}\,dx & \leqslant C \Vert \nabla \mathbf{ u} \Vert _{L^{2}} \Vert \mathbf{ B} \Vert _{L^{9/2}}^{3/2} \Vert \mathbf{ B} \Vert _{L^{9}}^{3/2} \\ &\leqslant C \Vert \nabla \mathbf{ u} \Vert _{L^{2}} \Vert \mathbf{ B} \Vert _{L^{9/2}}^{3/2} \bigl\Vert \vert \mathbf{ B} \vert ^{3/2} \bigr\Vert _{L^{6}} \\ &\leqslant C \bigl( \Vert \nabla \mathbf{ u} \Vert _{L^{2}}^{2} \Vert \mathbf{ B} \Vert _{L^{9/2}}^{3} \bigr)^{1/2} \bigl\Vert \vert \nabla \mathbf{ B} \vert \vert \mathbf{ B} \vert ^{1/2} \bigr\Vert _{L^{2}}. \end{aligned} $$
To deal with the right-hand side of (3.7), we notice that
$$ \Vert \mathbf {B}\Vert _{L^{9}}^{3}\leqslant C \bigl\Vert \vert \mathbf {B}\vert ^{3/2} \bigr\Vert _{L^{6}}^{2} \leqslant C \bigl\Vert \vert \nabla \mathbf {B}\vert \vert \mathbf {B}\vert ^{1/2} \bigr\Vert _{L^{2}}^{2}, $$
then
$$ \Vert \mathbf {B}\Vert _{L^{9/2}}\leqslant C \Vert \mathbf {B}\Vert _{L^{3}}^{1/2} \Vert \mathbf {B}\Vert _{L^{9}}^{1/2} \leqslant C \Vert \mathbf {B}\Vert _{L^{3}}^{1/2} \bigl\Vert \nabla \vert \mathbf {B}\vert ^{3/2} \bigr\Vert _{L^{2}}^{1/3}, $$
which, together with (3.7), yields
$$ \frac {d}{dt} \Vert \mathbf {B}\Vert _{L^{3}}^{3}+ \Vert \mathbf {B}\Vert _{L^{9}}^{3} \leqslant C \Vert \nabla \mathbf {u}\Vert _{L^{2}}^{4} \Vert \mathbf {B}\Vert _{L^{3}}^{3}. $$
As a result, we deduce from (3.5) and the Gronwall’s inequality that (3.6) holds. □
Now, to estimate \(A_{1}(T)\) and \(A_{2}(T)\), we first prove the following lemma.
Lemma 3.3
Let \((\rho , \mathbf{ u}, \mathbf{ B})\) be a smooth solution of (1.1)–(1.5) on \(\mathbb{R}^{3}\times [0,T]\) satisfying (3.1). Then
$$\begin{aligned}& A_{1}(T) \leqslant C \Bigl(\sup_{t\in [0, T]} \Vert \mathbf{ B} \Vert _{L^{3}}^{2} \Bigr)A_{1}(T)+ C \bigl(1+M_{0}^{\gamma } \bigr) +C \int _{0}^{T} \Vert \nabla \mathbf{ u} \Vert _{L^{4}}^{4}\,dt, \end{aligned}$$
(3.8)
$$\begin{aligned}& \begin{aligned} &A_{2}(T) \leqslant C+ \frac {1}{4}A_{1}(T)+C \Bigl[(M_{0}^{\frac {1}{2}}K+M_{0}^{ \frac {5\gamma }{6}}K^{1/2}+ \Bigl(\sup_{t\in [0, T]} \Vert \mathbf{ B} \Vert _{L^{3}} \Bigr)K \Bigr]A_{1}(T) \\ &\hphantom{A_{2}(T) \leqslant}{} +C \Bigl(\sup_{t\in [0, T]} \Vert \mathbf{ B} \Vert _{L^{3}}^{2}+K\sup_{t \in [0, T]} \Vert \mathbf{ B} \Vert _{L^{3}}^{4} \Bigr)A_{2}(T) \\ &\hphantom{A_{2}(T) \leqslant}{}+C \int _{0}^{T} \bigl( \Vert \nabla \mathbf{ u} \Vert _{L^{4}}^{4}+ \bigl\Vert P(\rho ) \bigr\Vert _{L^{4}}^{4} \bigr)\,dt, \end{aligned} \end{aligned}$$
(3.9)
where the constant \(C>0\) depends on μ, λ, ν, γ, A, \(E_{0}\), \(M_{1}\), \(M_{2}\), and the coefficients of the Gagliardo–Nirenberg–Sobolev inequality in Lemma 2.1, but is independent of \(M_{0}\).
Proof
In order to prove (3.8), using Lemma 2.1 and equation (1.1)3, we have
$$ \begin{aligned} &\nu \bigl( \Vert \nabla \mathbf {B}\Vert _{L^{2}}^{2} \bigr)_{t}+\nu ^{2} \bigl\Vert \nabla ^{2} \mathbf {B}\bigr\Vert _{L^{2}}^{2}+ \Vert \mathbf {B}_{t} \Vert _{L^{2}}^{2} \\ &\quad = \int (\mathbf {B}_{t}-\nu \Delta \mathbf {B})^{2}\,dx \\ &\quad = \int \vert \mathbf {B}\cdot \nabla \mathbf {u}-\mathbf {u}\cdot \nabla \mathbf {B}- \mathbf {B}\operatorname{div} \mathbf {u}\vert ^{2}\,dx \\ &\quad \leqslant C \bigl( \Vert \mathbf {B}\Vert _{L^{6}}^{2} \Vert \nabla \mathbf {u}\Vert _{L^{3}}^{2}+ \Vert \mathbf {u}\Vert _{L^{\infty }}^{2} \Vert \nabla \mathbf {B}\Vert _{L^{2}}^{2} \bigr) \\ &\quad \leqslant C \Vert \mathbf {B}\Vert _{L^{2}} \bigl\Vert \nabla ^{2} \mathbf {B}\bigr\Vert _{L^{2}} \bigl( \Vert \nabla \mathbf {u}\Vert _{L^{3}}^{2}+ \Vert \mathbf {u}\Vert _{L^{\infty }}^{2} \bigr) \\ &\quad \leqslant \frac {\nu ^{2}}{2} \bigl\Vert \nabla ^{2} \mathbf {B}\bigr\Vert _{L^{2}}^{2}+C(E_{0}) \bigl( \Vert \nabla \mathbf {u}\Vert _{L^{3}}^{4}+ \Vert \mathbf {u}\Vert _{L^{\infty }}^{4} \bigr), \end{aligned} $$
which, together with the Cauchy–Schwarz inequality, one obtains
$$ \bigl(\nu \Vert \nabla \mathbf {B}\Vert _{L^{2}}^{2} \bigr)_{t}+\nu ^{2} \bigl\Vert \nabla ^{2} \mathbf {B}\bigr\Vert _{L^{2}}^{2}+ \Vert \mathbf {B}_{t} \Vert _{L^{2}}^{2} \leqslant C \bigl( \Vert \nabla \mathbf {u}\Vert _{L^{3}}^{4}+ \Vert \mathbf {u}\Vert _{L^{\infty }}^{4} \bigr). $$
(3.10)
Next, multiplying (1.1)2 by \(\dot{\mathbf {u}}\) and integrating by parts, we deduce
$$ \begin{aligned} \int \rho \vert \dot{\mathbf {u}} \vert ^{2}\,dx ={}& \int \biggl(-\nabla P \cdot \dot{\mathbf {u}}+\mu \Delta \mathbf {u}\cdot \dot{ \mathbf {u}}+(\lambda + \mu )\nabla \operatorname{div} \mathbf {u}\cdot \dot{\mathbf {u}} \\ &{} +\mathbf {B}\cdot \nabla \mathbf {B}\cdot \dot{\mathbf {u}}-\frac {1}{2} \dot{ \mathbf {u}}\cdot \nabla \bigl( \vert \mathbf {B}\vert ^{2} \bigr) \biggr) \,dx\triangleq \sum_{i=1}^{5}I_{i}. \end{aligned} $$
(3.11)
The right-hand side of (3.11) can be estimated as follows. It holds by (1.1)1 that
$$ P(\rho )_{t}+\mathbf {u}\cdot \nabla P(\rho )+\gamma P( \rho )\operatorname{div} \mathbf {u}=0, $$
(3.12)
which, together with (3.1), yields
$$ \begin{aligned} I_{1}&=- \int \nabla P\cdot (\mathbf {u}_{t}+\mathbf {u}\cdot \nabla \mathbf {u})\,dx \\ &= \int \bigl(P(\rho )\operatorname{div} \mathbf {u}_{t}-(\mathbf {u}\cdot \nabla \mathbf {u}) \cdot \nabla P(\rho ) \bigr)\,dx \\ &= \biggl( \int P(\rho )\operatorname{div} \mathbf {u}\,dx \biggr)_{t}- \int \bigl(P(\rho )_{t}\operatorname{div} \mathbf {u}+\mathbf {u}\cdot \nabla \mathbf {u}\cdot \nabla P(\rho ) \bigr)\,dx \\ &= \biggl( \int P(\rho )\operatorname{div} \mathbf {u}\,dx \biggr)_{t}+ \int \bigl((\gamma -1)P( \rho ) (\operatorname{div} \mathbf {u})^{2}+P( \rho ) \vert \nabla \mathbf {u}\vert ^{2} \bigr)\,dx \\ &\leqslant C \biggl( \int P(\rho )\operatorname{div} \mathbf {u}\,dx \biggr)_{t}+C(A, \gamma )M_{0}^{\gamma } \Vert \nabla \mathbf {u}\Vert _{L^{2}}^{2}. \end{aligned} $$
(3.13)
Thanks to (3.4), we find (keep in mind that \(0\leqslant \rho \leqslant 2M_{0}\))
$$ \begin{aligned} \int P(\rho )\operatorname{div} \mathbf {u}\,dx &\leqslant \frac {\mu }{8} \Vert \nabla \mathbf {u}\Vert _{L^{2}}^{2}+C \bigl\Vert P( \rho ) \bigr\Vert _{L^{\infty }} \bigl\Vert P(\rho ) \bigr\Vert _{L^{1}} \\ &\leqslant \frac {\mu }{8} \Vert \nabla \mathbf {u}\Vert _{L^{2}}^{2}+C(E_{0})M_{0}^{\gamma }. \end{aligned} $$
(3.14)
Integrating by parts, one has
$$ \begin{aligned} I_{2}&=\mu \int \Delta \mathbf {u}\cdot (\mathbf {u}_{t}+ \mathbf {u}\cdot \nabla \mathbf {u})\,dx=-\frac {\mu }{2} \bigl( \Vert \nabla \mathbf {u}\Vert _{L^{2}}^{2} \bigr)_{t}+ \mu \int \partial _{i} u^{j}\partial _{i} \bigl(u^{k}\partial _{k}u^{j} \bigr)\,dx \\ &\leqslant -\frac {\mu }{2} \bigl( \Vert \nabla \mathbf {u}\Vert _{L^{2}}^{2} \bigr)_{t}+\mu \int \partial _{i} u^{j}\partial _{i} u^{k}\partial _{k}u^{j}\,dx- \frac {1}{2} \int \vert \nabla \mathbf {u}\vert ^{2} \operatorname{div} \mathbf {u}\,dx \\ &\leqslant -\frac {\mu }{2} \bigl( \Vert \nabla \mathbf {u}\Vert _{L^{2}}^{2} \bigr)_{t}+C \Vert \nabla \mathbf {u}\Vert ^{3}_{L^{3}}. \end{aligned} $$
(3.15)
Similarly,
$$ I_{3}=(\lambda +\mu ) \int \dot{\mathbf {u}}\cdot \nabla \operatorname{div} \mathbf {u}\,dx \leqslant - \frac {\lambda +\mu }{2} \bigl( \Vert \operatorname{div} \mathbf {u}\Vert _{L^{2}}^{2} \bigr)_{t}+C \Vert \nabla \mathbf {u}\Vert _{L^{3}}^{3}. $$
(3.16)
Integration by parts also gives
$$ \begin{aligned} I_{4}={}& \int \mathbf {B}\cdot \nabla \mathbf {B}\cdot (\mathbf {u}_{t}+ \mathbf {u}\cdot \nabla \mathbf {u})\,dx \\ ={}&- \int \mathbf {B}\cdot \nabla \mathbf {u}_{t} \cdot \mathbf {B}\,dx+ \int B^{i}\partial _{i}B^{j} u^{k} \partial _{k} u^{j}\,dx \\ ={}& - \biggl( \int \mathbf {B}\cdot \nabla \mathbf {u}\cdot \mathbf {B}\,dx \biggr)_{t}+ \int \bigl(\mathbf {B}_{t} \cdot \nabla \mathbf {u}\cdot \mathbf {B}+ \mathbf {B}\cdot \nabla \mathbf {u}\cdot \mathbf {B}_{t}+B^{i} \partial _{i}B^{j} u^{k} \partial _{k} u^{j} \bigr)\,dx \\ \leqslant{}& - \biggl( \int \mathbf {B}\cdot \nabla \mathbf {u}\cdot \mathbf {B}\,dx \biggr)_{t} \\ &{}+C \Vert \mathbf {B}_{t} \Vert _{L^{2}} \Vert \nabla \mathbf {u}\Vert _{L^{3}} \Vert \mathbf {B}\Vert _{L^{6}}+C \Vert \mathbf {B}\Vert _{L^{2}} \Vert \nabla \mathbf {B}\Vert _{L^{6}} \Vert \mathbf {u}\Vert _{L^{\infty }} \Vert \nabla \mathbf {u}\Vert _{L^{3}} \\ \leqslant{}& - \biggl( \int \mathbf {B}\cdot \nabla \mathbf {u}\cdot \mathbf {B}\,dx \biggr)_{t} \\ &{}+ \frac {1}{4} \bigl( \Vert \mathbf {B}_{t} \Vert _{L^{2}}^{2}+\nu ^{2} \bigl\Vert \nabla ^{2}\mathbf {B}\bigr\Vert _{L^{2}}^{2} \bigr)+C (E_{0}) \bigl( \Vert \nabla \mathbf {u}\Vert _{L^{3}}^{4}+ \Vert \mathbf {u}\Vert _{L^{\infty }}^{4} \bigr), \end{aligned} $$
(3.17)
where we also used (2.1), (3.4), and the simple fact
$$ \Vert \mathbf {B}\Vert _{L^{6}}\leqslant C \Vert \mathbf {B}\Vert _{L^{2}}^{1/2} \Vert \nabla \mathbf {B}\Vert _{L^{6}}^{1/2}\leqslant C \Vert \mathbf {B}\Vert _{L^{2}}^{1/2} \bigl\Vert \nabla ^{2} \mathbf {B}\bigr\Vert _{L^{2}}^{1/2}, $$
which, together with Cauchy–Schwarz inequality, gives
$$ \int \mathbf {B}\cdot \nabla \mathbf {u}\cdot \mathbf {B}\,dx \leqslant \frac { \mu }{8} \Vert \nabla \mathbf {u}\Vert _{L^{2}}^{2}+C \Vert \mathbf {B}\Vert _{L^{3}}^{2} \Vert \nabla \mathbf {B}\Vert _{L^{2}}^{2}. $$
(3.18)
Similarly,
$$ \begin{aligned} I_{5}={}&-\frac {1}{2} \int \nabla \bigl( \vert \mathbf {B}\vert ^{2} \bigr)\cdot ( \mathbf {u}_{t}+\mathbf {u}\cdot \nabla \mathbf {u})\,dx \\ \leqslant{}& \frac {1}{2} \biggl( \int \vert \mathbf {B}\vert ^{2} \operatorname{div} \mathbf {u}\,dx \biggr)_{t} \\ &{}+\frac {1}{4} \bigl( \Vert \mathbf {B}_{t} \Vert _{L^{2}}^{2}+\nu ^{2} \bigl\Vert \nabla ^{2}\mathbf {B}\bigr\Vert _{L^{2}}^{2} \bigr)+C (E_{0}) \bigl( \Vert \nabla \mathbf {u}\Vert _{L^{3}}^{4}+ \Vert \mathbf {u}\Vert _{L^{\infty }}^{4} \bigr) \end{aligned} $$
(3.19)
and
$$ \frac {1}{2} \int \vert \mathbf {B}\vert ^{2} \operatorname{div} \mathbf {u}\,dx \leqslant \frac { \mu }{8} \Vert \nabla \mathbf {u}\Vert _{L^{2}}^{2}+C \Vert \mathbf {B}\Vert _{L^{3}}^{2} \Vert \nabla \mathbf {B}\Vert _{L^{2}}^{2}. $$
(3.20)
Thus, substituting (3.13), (3.15)–(3.17), and (3.19) into (3.11), using (3.10), we obtain
$$ \begin{aligned} & \biggl(\frac {\mu }{2} \Vert \nabla \mathbf {u}\Vert _{L^{2}}^{2}+ \frac {\mu +\lambda }{2} \Vert \operatorname{div} \mathbf {u}\Vert _{L^{2}}^{2}+\nu \Vert \nabla \mathbf {B}\Vert _{L^{2}}^{2} \biggr)_{t} \\ &\qquad {}+ \Vert \sqrt{\rho }\dot{\mathbf {u}} \Vert _{L^{2}}^{2}+ \Vert \mathbf {B}_{t} \Vert _{L^{2}}^{2}+ \nu ^{2} \bigl\Vert \nabla ^{2} \mathbf {B}\bigr\Vert _{L^{2}}^{2} \\ &\quad \leqslant \bigg( \int \biggl[ \biggl(P(\rho )\operatorname{div} \mathbf {u}-\mathbf {B}\cdot \nabla \mathbf {u}\cdot \mathbf {B}+\frac {1}{2} \vert \mathbf {B}\vert ^{2} \operatorname{div} \mathbf {u}\biggr]\,dx \biggr)_{t}+C M_{0}^{\gamma } \Vert \nabla \mathbf {u}\Vert _{L^{2}}^{2} \\ & \qquad {}+C \bigl( \Vert \nabla \mathbf {u}\Vert _{L^{3}}^{3}+ \Vert \nabla \mathbf {u}\Vert _{L^{3}}^{4}+ \Vert \mathbf {u}\Vert _{L^{\infty }}^{4} \bigr). \end{aligned} $$
(3.21)
Thanks to (2.1) and (2.2), we find from (3.21) that
$$ \begin{aligned} & \int _{0}^{T} \bigl( \Vert \nabla \mathbf {u}\Vert _{L^{3}}^{3}+ \Vert \nabla \mathbf {u}\Vert _{L^{3}}^{4}+ \Vert \mathbf {u}\Vert _{L^{\infty }}^{4} \bigr)\,dt \\ &\quad \leqslant C \int _{0}^{T} \bigl( \Vert \nabla \mathbf {u}\Vert _{L^{2}}^{2}+ \Vert \nabla \mathbf {u}\Vert _{L^{4}}^{4}+ \Vert \nabla \mathbf {u}\Vert _{L^{2}}^{4/3} \Vert \nabla \mathbf {u}\Vert _{L^{4}}^{8/3} \bigr)\,dt \\ &\quad \leqslant C+\delta \int _{0}^{T} \Vert \nabla \mathbf {u}\Vert _{L^{2}}^{4}\,dt+C( \delta ) \int _{0}^{T} \Vert \nabla \mathbf {u}\Vert _{L^{4}}^{4}\,dt \\ &\quad \leqslant C+\delta E_{0}/\mu \sup_{t\in [0, T]} \Vert \nabla \mathbf {u}\Vert _{L^{2}}^{2}+C(\delta ) \int _{0}^{T} \Vert \nabla \mathbf {u}\Vert _{L^{4}}^{4}\,dt, \end{aligned} $$
(3.22)
where \(\delta >0\) is an undetermined small constant and the constant \(C(\delta )>0\) depends on δ and the coefficients in (2.1). Moreover, the derivation of inequality (3.22) also uses the following inequalities:
$$ \Vert \nabla \mathbf {u}\Vert _{L^{3}}\leqslant C \Vert \nabla \mathbf {u}\Vert _{L^{2}}^{1/3} \Vert \nabla \mathbf {u}\Vert _{L^{4}}^{2/3} \quad \text{and}\quad \Vert \mathbf {u}\Vert _{L^{\infty }}\leqslant C \Vert \mathbf {u}\Vert _{L^{6}}^{1/3} \Vert \nabla \mathbf {u}\Vert _{L^{4}}^{2/3}. $$
Combining (3.14), (3.18), and (3.20), then integrating (3.21) over \((0, T)\), and choosing \(\delta >0\) suitably small in (3.22), by virtue of the Gronwall’s inequality, we obtain (3.8).
To prove (3.9), applying \(\dot{u}^{j}[\partial _{t}+\operatorname{div}(\mathbf {u}\cdot )]\) to \(\text{(1.1)}_{2}^{j}\), summing with respect to j, and integrating the resulting equation over \(\mathbb{R}^{3}\), we obtain after integration by parts
$$ \begin{aligned} \biggl(\frac {1}{2} \int \rho \vert \dot{\mathbf {u}} \vert ^{2}\,dx \biggr)_{t} ={}&- \int \dot{u}^{j} \bigl[\partial _{j}P_{t}+\operatorname{div}(\mathbf {u}\partial _{j}P) \bigr]\,dx \\ &{}+ \mu \int \dot{u}^{j} \bigl[\Delta {u}^{j}_{t}+\operatorname{div} \bigl(\mathbf {u}\Delta u^{j} \bigr) \bigr]\,dx \\ &{} +(\lambda +\mu ) \int \dot{u}^{j} \bigl[\partial _{j}\partial _{t}(\operatorname{div} \mathbf {u})+\operatorname{div} \bigl(\mathbf {u}\partial _{j}(\operatorname{div} \mathbf {u}) \bigr) \bigr]\,dx \\ & {}+ \int \dot{u}^{j} \bigl[\partial _{t} \bigl(\mathbf {B}\cdot \nabla B^{j} \bigr)+\operatorname{div} \bigl(\mathbf {u}\mathbf {B}\cdot \nabla B^{j} \bigr) \bigr]\,dx \\ &{} -\frac {1}{2} \int \dot{u}^{j} \bigl[\partial _{t}\partial _{j} \bigl( \vert \mathbf {B}\vert ^{2} \bigr)+\operatorname{div} \bigl(\mathbf {u}\partial _{j} \bigl( \vert \mathbf {B}\vert ^{2} \bigr) \bigr) \bigr]\,dx \triangleq \sum _{i=1}^{5}J_{i}, \end{aligned} $$
(3.23)
where the first term on the right-hand side can be estimated as follows, based on integration by parts and (3.12):
$$ \begin{aligned} J_{1} &= \int \bigl(\partial _{j} \dot{u}^{j} P(\rho )_{t}+ \partial _{k} \dot{u}^{j} u^{k} \partial _{j}P(\rho ) \bigr)\,dx \\ &= \int \bigl(-\gamma P(\rho )\operatorname{div} \mathbf {u}\partial _{j} \dot{u}^{j}- \partial _{j}\dot{u}^{j} u^{k} \partial _{k}P(\rho )-\partial _{j} \bigl( \partial _{k}\dot{u}^{j} u^{k} \bigr)P(\rho ) \bigr)\,dx \\ &= \int \bigl(-\gamma P(\rho )\operatorname{div} \mathbf {u}\partial _{j} \dot{u}^{j}+ \partial _{j}\dot{u}^{j} \partial _{k} u^{k} P(\rho )-\partial _{k} \dot{u}^{j} \partial _{j} u^{k} P(\rho ) \bigr) \,dx \\ &\leqslant \frac {\mu }{8} \Vert \nabla \dot{\mathbf {u}} \Vert _{L^{2}}^{2}+C \bigl( \Vert \nabla \mathbf {u}\Vert _{L^{4}}^{4}+ \bigl\Vert P(\rho ) \bigr\Vert _{L^{4}}^{4} \bigr). \end{aligned} $$
(3.24)
Similarly,
$$ J_{2}=\mu \int \dot{u}^{j} \bigl[\Delta {u}^{j}_{t}+\operatorname{div} \bigl(\mathbf {u}\Delta u^{j} \bigr) \bigr]\,dx\leqslant - \frac {3\mu }{4} \Vert \nabla \dot{\mathbf {u}} \Vert _{L^{2}}^{2}+C \Vert \nabla \mathbf {u}\Vert _{L^{4}}^{4} $$
(3.25)
and
$$ J_{3}\leqslant -\frac {\lambda +\mu }{2} \Vert \operatorname{div}\dot{\mathbf {u}} \Vert _{L^{2}}^{2}+ \frac {\mu }{4} \Vert \nabla \dot{\mathbf {u}} \Vert _{L^{2}}^{2}+C \Vert \nabla \mathbf {u}\Vert _{L^{4}}^{4}. $$
(3.26)
Next, integrating by parts, one has (keeping in mind that \(\operatorname{div} \mathbf {B}=0\))
$$ \begin{aligned} J_{4} &= \int \dot{u}^{j} \bigl[\partial _{t} \bigl(\mathbf {B}\cdot \nabla B^{j} \bigr)+\operatorname{div} \bigl(\mathbf {u}\mathbf {B}\cdot \nabla B^{j} \bigr) \bigr]\,dx \\ &= \int \bigl(\dot{u}^{j} \bigl(B_{t}^{i} \partial _{i}B^{j}+B^{i}\partial _{i}B_{t}^{j} \bigr)- \partial _{k} \dot{u}^{j}u^{k}B^{i}\partial _{i}B^{j} \bigr)\,dx \\ &\leqslant - \int \bigl(B^{j}\partial _{i}\dot{u}^{j}B_{t}^{i}+B_{t}^{j} \partial _{i}\dot{u}^{j}B^{i}\,dx+ \partial _{k}\dot{u}^{j}u^{k}B^{i} \partial _{i}B^{j} \bigr)\,dx \\ &\leqslant \frac {\mu }{8} \Vert \nabla \dot{\mathbf {u}} \Vert _{L^{2}}^{2}+C \Vert \mathbf {B}\Vert _{L^{3}}^{2} \Vert \nabla \mathbf {B}_{t} \Vert _{L^{2}}^{2}+ C \Vert \mathbf {B}\Vert _{L^{3}}^{4} \bigl\Vert \nabla ^{2}\mathbf {B}\bigr\Vert _{L^{2}}^{4}+ \Vert \mathbf {u}\Vert _{L^{\infty }}^{4}. \end{aligned} $$
(3.27)
Similarly,
$$ J_{5}\leqslant \frac {\mu }{8} \Vert \nabla \dot{ \mathbf {u}} \Vert _{L^{2}}^{2}+C \Vert \mathbf {B}\Vert _{L^{3}}^{2} \Vert \nabla \mathbf {B}_{t} \Vert _{L^{2}}^{2}+ C \Vert \mathbf {B}\Vert _{L^{3}}^{4} \bigl\Vert \nabla ^{2}\mathbf {B}\bigr\Vert _{L^{2}}^{4}+ \Vert \mathbf {u}\Vert _{L^{\infty }}^{4}. $$
(3.28)
Substituting (3.24)–(3.28) into (3.23), we obtain
$$ \begin{aligned} & \bigl( \Vert \sqrt{\rho } \dot{ \mathbf {u}} \Vert _{L^{2}}^{2} \bigr)_{t}+ \Vert \nabla \dot{\mathbf {u}} \Vert _{L^{2}}^{2} \\ &\quad \leqslant C \bigl( \Vert \nabla \mathbf {u}\Vert _{L^{4}}^{4}+ \bigl\Vert P(\rho ) \bigr\Vert _{L^{4}}^{4} \bigr)+C \Vert \mathbf {B}\Vert _{L^{3}}^{2} \Vert \nabla \mathbf {B}_{t} \Vert _{L^{2}}^{2}+ C \Vert \mathbf {B}\Vert _{L^{3}}^{4} \bigl\Vert \nabla ^{2}\mathbf {B}\bigr\Vert _{L^{2}}^{4}+ \Vert \mathbf {u}\Vert _{L^{\infty }}^{4}. \end{aligned} $$
(3.29)
On the other hand, it follows from (1.1)3 that
$$ \mathbf {B}_{tt}-\nu \Delta \mathbf {B}_{t}= ( \mathbf {B}\cdot \nabla \mathbf {u}- \mathbf {u}\cdot \nabla \mathbf {B}-\mathbf {B}\operatorname{div} \mathbf {u})_{t}. $$
(3.30)
Multiplying (3.30) by \(\mathbf {B}_{t}\) and integrating over \(\mathbb{R}^{3}\) yields
$$ \begin{aligned} &\biggl(\frac {1}{2} \Vert \mathbf {B}_{t} \Vert _{L^{2}}^{2} \biggr)_{t}+ \nu \Vert \nabla \mathbf {B}_{t} \Vert _{L^{2}}^{2} \\ &\quad = \int ( \mathbf {B}_{t} \cdot \nabla \mathbf {u}-\mathbf {u}\cdot \nabla \mathbf {B}_{t}-\mathbf {B}_{t} \cdot \operatorname{div} \mathbf {u}) \cdot \mathbf {B}_{t}\,dx \\ &\qquad {} + \int \bigl(-\mathbf {B}\cdot \nabla (\mathbf {u}\cdot \nabla \mathbf {u})+( \mathbf {u}\cdot \nabla \mathbf {u})\cdot \nabla \mathbf {B}+\mathbf {B}\operatorname{div}( \mathbf {u}\cdot \nabla \mathbf {u}) \bigr)\cdot \mathbf {B}_{t}\,dx \\ &\qquad {} + \int ( \mathbf {B}\cdot \nabla \dot{\mathbf {u}}-\dot{\mathbf {u}}\cdot \nabla \mathbf {B}-\mathbf {B}\cdot \operatorname{div}\dot{\mathbf {u}} )\cdot \mathbf {B}_{t} \,dx\triangleq \sum_{i=1}^{3}N_{i}. \end{aligned} $$
(3.31)
Now, we estimate \(N_{i}\) as follows. By using (2.1), (2.2), and integrating by parts, we obtain
$$ \begin{aligned} N_{1} &= \int ( \mathbf {B}_{t} \cdot \nabla \mathbf {u}- \mathbf {u}\cdot \nabla \mathbf {B}_{t}-\mathbf {B}_{t} \cdot \operatorname{div} \mathbf {u})\cdot \mathbf {B}_{t}\,dx \\ &\leqslant C \Vert \nabla \mathbf {B}_{t} \Vert _{L^{2}} \Vert \mathbf {B}_{t} \Vert _{L^{2}} \bigl( \Vert \nabla \mathbf {u}\Vert _{L^{3}}+ \Vert \mathbf {u}\Vert _{L^{\infty }} \bigr) \\ & \leqslant \frac {\nu }{8} \Vert \nabla \mathbf {B}_{t} \Vert _{L^{2}}^{2}+C \Vert \mathbf {B}_{t} \Vert _{L^{2}}^{2} \Vert \nabla \mathbf {u}\Vert _{L^{2}} \Vert \nabla \mathbf {u}\Vert _{L^{6}}, \end{aligned} $$
(3.32)
and
$$ \begin{aligned} N_{2} &= \int ( \mathbf {B}\cdot \nabla \dot{\mathbf {u}}- \dot{\mathbf {u}}\cdot \nabla \mathbf {B}-\mathbf {B}\cdot \operatorname{div} \dot{\mathbf {u}} )\cdot \mathbf {B}_{t}\,dx\leqslant \frac {1}{4} \Vert \nabla \dot{\mathbf {u}} \Vert _{L^{2}}^{2}+C \Vert \mathbf {B}\Vert _{L^{3}}^{2} \Vert \nabla \mathbf {B}_{t} \Vert _{L^{2}}^{2}, \end{aligned} $$
(3.33)
and
$$ \begin{aligned} N_{3} &= \int \bigl(-\mathbf {B}\cdot \nabla (\mathbf {u}\cdot \nabla \mathbf {u})+(\mathbf {u}\cdot \nabla \mathbf {u})\cdot \nabla \mathbf {B}+ \mathbf {B}\operatorname{div}(\mathbf {u}\cdot \nabla \mathbf {u}) \bigr)\cdot \mathbf {B}_{t}\,dx \\ &= \int \bigl( u^{k} \partial _{k} u^{j} B^{i} \partial _{i}B_{t}^{j} + u^{k} \partial _{k} u^{i} \partial _{i}B^{j} B_{t}^{j}-u^{k} \partial _{k} u^{i} \partial _{i}B^{j} B_{t}^{j} -u^{k} \partial _{k} u^{i} B^{j}\partial _{i}B_{t}^{j} \bigr)\,dx \\ &= \int \bigl( u^{k} \partial _{k} u^{j} B^{i} \partial _{i}B_{t}^{j} -u^{k} \partial _{k} u^{i} B^{j} \partial _{i}B_{t}^{j} \bigr)\,dx \\ &\leqslant \frac {\nu }{8} \Vert \nabla \mathbf {B}_{t} \Vert _{L^{2}}^{2}+C \Vert \nabla \mathbf {B}\Vert _{L^{2}}^{2} \Vert \nabla \mathbf {u}\Vert _{L^{2}}^{2} \Vert \nabla \mathbf {u}\Vert _{L^{6}}^{2}. \end{aligned} $$
(3.34)
Thus, substituting (3.32)–(3.34) into (3.31), we infer that
$$ \begin{aligned} \bigl( \Vert \mathbf {B}_{t} \Vert _{L^{2}}^{2} \bigr)_{t}+ \Vert \nabla \mathbf {B}_{t} \Vert _{L^{2}}^{2}\leqslant{}& \frac {1}{4} \Vert \nabla \dot{\mathbf {u}} \Vert _{L^{2}}^{2}+C \Vert \mathbf {B}\Vert _{L^{3}}^{2} \Vert \nabla \mathbf {B}_{t} \Vert _{L^{2}}^{2} \\ &{} +C \bigl( \Vert \mathbf {B}_{t} \Vert _{L^{2}}^{2} \Vert \nabla \mathbf {u}\Vert _{L^{2}} \Vert \nabla \mathbf {u}\Vert _{L^{6}}+ \Vert \nabla \mathbf {B}\Vert _{L^{2}}^{2} \Vert \nabla \mathbf {u}\Vert _{L^{2}}^{2} \Vert \nabla \mathbf {u}\Vert _{L^{6}}^{2} \bigr). \end{aligned} $$
(3.35)
The combination of (3.29) and (3.35) gives
$$ \begin{aligned} & \bigl( \Vert \sqrt{\rho } \dot{ \mathbf {u}} \Vert _{L^{2}}^{2}+ \Vert \mathbf {B}_{t} \Vert _{L^{2}}^{2} \bigr)_{t}+ \Vert \nabla \dot{ \mathbf {u}} \Vert _{L^{2}}^{2}+ \Vert \nabla \mathbf {B}_{t} \Vert _{L^{2}}^{2} \\ &\quad \leqslant C \bigl( \Vert \nabla \mathbf {u}\Vert _{L^{4}}^{4}+ \bigl\Vert P(\rho ) \bigr\Vert _{L^{4}}^{4} \bigr)+C \Vert \mathbf {B}\Vert _{L^{3}}^{2} \Vert \nabla \mathbf {B}_{t} \Vert _{L^{2}}^{2}+ C \Vert \mathbf {B}\Vert _{L^{3}}^{4} \bigl\Vert \nabla ^{2}\mathbf {B}\bigr\Vert _{L^{2}}^{4}+ \Vert \mathbf {u}\Vert _{L^{\infty }}^{4} \\ &\qquad {} +C \bigl( \Vert \mathbf {B}_{t} \Vert _{L^{2}}^{2} \Vert \nabla \mathbf {u}\Vert _{L^{2}} \Vert \nabla \mathbf {u}\Vert _{L^{6}}+ \Vert \nabla \mathbf {B}\Vert _{L^{2}}^{2} \Vert \nabla \mathbf {u}\Vert _{L^{2}}^{2} \Vert \nabla \mathbf {u}\Vert _{L^{6}}^{2} \bigr). \end{aligned} $$
(3.36)
The right-hand side of (3.36) on \([0, T]\) can be estimated as follows. It holds by (3.1) that
$$ \begin{aligned} \int _{0}^{T} \Vert \mathbf {B}\Vert _{L^{3}}^{4} \bigl\Vert \nabla ^{2} \mathbf {B}\bigr\Vert _{L^{2}}^{4}\,dt &\leqslant C\sup_{t\in [0, T]} \bigl( \Vert \mathbf {B}\Vert _{L^{3}}^{4} \bigl\Vert \nabla ^{2}\mathbf {B}\bigr\Vert _{L^{2}}^{2} \bigr) \int _{0}^{T} \bigl\Vert \nabla ^{2} \mathbf {B}\bigr\Vert _{L^{2}}^{2}\,dt \\ &\leqslant CKA_{2}(T) \Bigl(\sup_{t\in [0, T]} \Vert \mathbf {B}\Vert _{L^{3}}^{4} \Bigr). \end{aligned} $$
(3.37)
An application of the \(L^{p}\)-theory for the elliptic equation (1.1)2 leads to
$$ \begin{aligned} \Vert \nabla \mathbf {u}\Vert _{L^{6}} & \leqslant C \bigl( \Vert \rho \dot{\mathbf {u}} \Vert _{L^{2}}+ \bigl\Vert P(\rho ) \bigr\Vert _{L^{6}}+ \Vert \mathbf {B}\Vert _{L^{\infty }} \Vert \nabla \mathbf {B}\Vert _{L^{2}} \bigr) \\ & \leqslant C \bigl(M_{0}^{1/2} \Vert \sqrt{\rho } \dot{ \mathbf {u}} \Vert _{L^{2}}+M_{0}^{5 \gamma /6} \bigl\Vert P(\rho ) \bigr\Vert _{L^{1}}^{1/6}+ \Vert \mathbf {B}\Vert _{L^{3}} \bigl\Vert \nabla ^{2} \mathbf {B}\bigr\Vert _{L^{2}} \bigr) \\ & \leqslant C \bigl(M_{0}^{1/2} \Vert \sqrt{\rho } \dot{ \mathbf {u}} \Vert _{L^{2}}+M_{0}^{5 \gamma /6}E_{0}^{1/6}+ \Vert \mathbf {B}\Vert _{L^{3}} \bigl\Vert \nabla ^{2} \mathbf {B}\bigr\Vert _{L^{2}} \bigr) \\ & \leqslant C \bigl(M_{0}^{1/2}K^{1/2}+M_{0}^{5\gamma /6}+ \Vert \mathbf {B}\Vert _{L^{3}}K^{1/2} \bigr), \end{aligned} $$
(3.38)
where we have used (3.1). Thus
$$ \begin{aligned} & \int _{0}^{T} \bigl( \Vert \mathbf {B}_{t} \Vert _{L^{2}}^{2} \Vert \nabla \mathbf {u}\Vert _{L^{2}} \Vert \nabla \mathbf {u}\Vert _{L^{6}}+ \Vert \nabla \mathbf {B}\Vert _{L^{2}}^{2} \Vert \nabla \mathbf {u}\Vert _{L^{2}}^{2} \Vert \nabla \mathbf {u}\Vert _{L^{6}}^{2} \bigr)\,dt \\ &\quad \leqslant C K^{1/2} \Bigl(M_{0}^{1/2}K^{1/2}+M_{0}^{5\gamma /6}+ \Bigl( \sup_{t\in [0, T]} \Vert \mathbf {B}\Vert _{L^{3}} \Bigr)K^{1/2} \Bigr) \int _{0}^{T} \Vert \mathbf {B}_{t} \Vert _{L^{2}}^{2}\,dt \\ & \qquad {}+ C \Bigl(\sup_{t\in [0, T]} \Vert \nabla \mathbf {u}\Vert _{L^{2}}^{2} \Bigr) \Bigl(M_{0}K+M_{0}^{5\gamma /3}+ \Bigl(\sup_{t\in [0, T]} \Vert \mathbf {B}\Vert _{L^{3}}^{2} \Bigr)K \Bigr) \int _{0}^{T} \Vert \nabla \mathbf {B}\Vert _{L^{2}}^{2}\,dt \\ & \quad \leqslant C \Bigl(M_{0}^{1/2}K+M_{0}^{5\gamma /6}K^{1/2}+ \Bigl( \sup_{t\in [0, T]} \Vert \mathbf {B}\Vert _{L^{3}} \Bigr)K \Bigr)A_{1}(T). \end{aligned} $$
(3.39)
In addition, it holds by (1.1)3, (2.1), (2.2), and (3.1) that
$$ \bigl\Vert \nabla ^{2}\mathbf {B}\bigr\Vert _{L^{2}}^{2} \leqslant C \bigl( \Vert \mathbf {B}_{t} \Vert _{L^{2}}^{2}+K \Vert \mathbf {B}\Vert _{L^{3}}^{2} \Vert \nabla \mathbf {u}\Vert _{L^{2}}^{2} \bigr), $$
which, together with (3.36)–(3.39), (3.22), and Gronwall’s inequality, gives rise to (3.9). □
We are now in a position of providing the concluding estimates of \(A_{1}(T)\) and \(A_{2}(T)\).
Lemma 3.4
Let \((\rho , \mathbf{ u}, \mathbf{ B})\) be a smooth solution of (1.1)–(1.5) on \(\mathbb{R}^{3}\times [0,T]\), satisfying \(0\leqslant \rho \leqslant 2M_{0}\). Then there exist positive numbers \(\varepsilon _{1}\) and K, depending only on μ, λ, ν, γ, A, \(E_{0}\), \(M_{1}\), and \(M_{2}\), such that
$$ A_{1}(T)+A_{2}(T)\leqslant K, $$
(3.40)
provided
$$ A_{1}(T)+A_{2}(T)\leqslant 2K\quad \textit{and}\quad \Vert \mathbf{ B}_{0} \Vert _{L^{3}}^{\frac {1}{2\gamma }} \leqslant M_{0}\leqslant \varepsilon _{1}. $$
(3.41)
Proof
Indeed, it follows from (2.8) that
$$ \int _{0}^{T} \Vert \nabla \mathbf {u}\Vert _{L^{4}}^{4}\,dt\leqslant C \int _{0}^{T} \bigl( \Vert F \Vert _{L^{4}}^{4}+ \bigl\Vert P(\rho ) \bigr\Vert _{L^{4}}^{4}+ \bigl\Vert |\mathbf {B}|^{2} \bigr\Vert _{L^{4}}^{4}+ \Vert \omega \Vert _{L^{4}}^{4} \bigr)\,dt. $$
(3.42)
To estimate \(\|P(\rho )\|_{L^{4}}\), we multiply (3.12) by \(3(P(\rho ))^{2}\), integrate it over \(\mathbb{R}^{3}\) and utilize (2.3) to get that
$$ \begin{aligned} & \biggl( \int \bigl(P(\rho ) \bigr)^{3}\,dx \biggr)_{t}+ \frac {3\gamma -1}{2 \mu +\lambda } \bigl\Vert P(\rho ) \bigr\Vert _{L^{4}}^{4}+ \frac {3\gamma -1}{2(2\mu + \lambda )} \int \vert \mathbf {B}\vert ^{2}P^{3}\,dx \\ &\quad =-\frac {3\gamma -1}{2\mu +\lambda } \int F\cdot P^{3}\,dx \leqslant \frac {3\gamma -1}{2(2\mu +\lambda )} \bigl\Vert P(\rho ) \bigr\Vert _{L^{4}}^{4}+C \Vert F \Vert _{L^{4}}^{4}, \end{aligned} $$
(3.43)
thus
$$ \int _{0}^{T} \bigl\Vert P(\rho ) \bigr\Vert _{L^{4}}^{4}\,dt\leqslant C \int _{0}^{T} \Vert F \Vert _{L^{4}}^{4}\,dt+CM_{0}^{2 \gamma }. $$
(3.44)
Using (2.1), (2.2), (2.5)–(2.7), and the fact that \(0\leqslant \rho \leqslant 2M_{0}\), we deduce
$$ \begin{aligned} & \int _{0}^{T} \bigl( \Vert F \Vert _{L^{4}}^{4}+ \bigl\Vert |\mathbf {B}|^{2} \bigr\Vert _{L^{4}}^{4}+ \Vert \omega \Vert _{L^{4}}^{4} \bigr)\,dt \\ &\quad \leqslant \int _{0}^{T} \bigl( \Vert F \Vert _{L^{2}}^{2}+ \Vert \omega \Vert _{L^{2}}^{2} \bigr) \bigl( \Vert F \Vert _{L^{\infty }}^{2}+ \Vert \omega \Vert _{L^{\infty }}^{2} \bigr)\,dt+ \int _{0}^{T} \Vert \mathbf {B}\Vert _{L^{8}}^{8}\,dt \\ &\quad \leqslant \int _{0}^{T} \bigl( \Vert \nabla \mathbf {u}\Vert _{L^{2}}^{2}+ \bigl\Vert P(\rho ) \bigr\Vert _{L^{2}}^{2}+ \bigl\Vert |\mathbf {B}|^{2} \bigr\Vert _{L^{2}}^{2} \bigr) \bigl( \Vert \rho \dot{\mathbf {u}} \Vert _{L^{2}}+ \Vert \nabla \mathbf {B}\cdot \mathbf {B}\Vert _{L^{2}} \bigr) \\ & \qquad {}\times \bigl( \Vert \rho \dot{\mathbf {u}} \Vert _{L^{6}}+ \Vert \nabla \mathbf {B}\cdot \mathbf {B}\Vert _{L^{6}} \bigr)\,dt + \int _{0}^{T} \Vert \mathbf {B}\Vert _{L^{8}}^{8}\,dt \end{aligned} $$
(3.45)
Now, we estimate the right-hand side of (3.45) as follows:
$$ \begin{aligned} & \int _{0}^{T} \bigl( \Vert \nabla \mathbf {u}\Vert _{L^{2}}^{2}+ \bigl\Vert P(\rho ) \bigr\Vert _{L^{2}}^{2} \bigr) \Vert \rho \dot{\mathbf {u}} \Vert _{L^{2}} \Vert \rho \dot{\mathbf {u}} \Vert _{L^{6}}\,dt \\ &\quad \leqslant CM_{0}^{3/2} \bigl(A_{1}(T)+M_{0}^{\gamma } \bigr) \biggl( \int _{0}^{T} \Vert \sqrt{ \rho } \dot{\mathbf {u}} \Vert _{L^{2}}^{2}\,dt \biggr)^{1/2} \biggl( \int _{0}^{T} \Vert \nabla \dot{\mathbf {u}} \Vert _{L^{2}}^{2}\,dt \biggr)^{1/2} \\ &\quad \leqslant CM_{0}^{3/2} \bigl(A_{1}(T)+M_{0}^{\gamma } \bigr) A_{1}^{1/2}(T)A_{2}^{1/2}(T). \end{aligned} $$
(3.46)
It follows from (2.1) and (2.2) that
$$ \Vert \nabla \mathbf {B}\cdot \mathbf {B}\Vert _{L^{6}} \leqslant C \bigl\Vert \nabla ( \nabla \mathbf {B}\cdot \mathbf {B}) \bigr\Vert _{L^{2}}\leqslant C \Vert \nabla \mathbf {B}\Vert _{L^{2}}^{1/2} \bigl\Vert \nabla ^{2} \mathbf {B}\bigr\Vert _{L^{2}}^{3/2}. $$
(3.47)
Hence
$$ \begin{aligned} & \int _{0}^{T} \bigl( \Vert \nabla \mathbf {u}\Vert _{L^{2}}^{2}+ \bigl\Vert P(\rho ) \bigr\Vert _{L^{2}}^{2} \bigr) \Vert \rho \dot{\mathbf {u}} \Vert _{L^{2}} \Vert \nabla \mathbf {B}\cdot \mathbf {B}\Vert _{L^{6}} \,dt \\ &\quad \leqslant CM_{0}^{1/2} \bigl(A_{1}(T)+M_{0}^{\gamma } \bigr)A_{2}^{1/2}(T) \biggl( \int _{0}^{T} \Vert \nabla \mathbf {B}\Vert _{L^{2}}^{2}\,dt \biggr)^{1/4} \biggl( \int _{0}^{T} \bigl\Vert \nabla ^{2} \mathbf {B}\bigr\Vert _{L^{2}}^{2}\,dt \biggr)^{3/4} \\ &\quad \leqslant CM_{0}^{1/2} \bigl(A_{1}(T)+M_{0}^{\gamma } \bigr) A_{1}^{3/4}(T)A_{2}^{1/2}(T), \end{aligned} $$
(3.48)
and
$$ \begin{aligned} & \int _{0}^{T} \bigl( \Vert \nabla \mathbf {u}\Vert _{L^{2}}^{2}+ \bigl\Vert P(\rho ) \bigr\Vert _{L^{2}}^{2} \bigr) \Vert \rho \dot{\mathbf {u}} \Vert _{L^{6}} \Vert \nabla \mathbf {B}\cdot \mathbf {B}\Vert _{L^{2}} \,dt \\ &\quad \leqslant CM_{0} \bigl(A_{1}(T)+M_{0}^{\gamma } \bigr) \int _{0}^{T} \bigl\Vert \nabla ^{2} \mathbf {B}\bigr\Vert _{L^{2}} \Vert \mathbf {B}\Vert _{L^{3}} \Vert \nabla \dot{\mathbf {u}} \Vert _{L^{2}} \\ &\quad \leqslant CM_{0} \Bigl(\sup_{t\in [0, T]} \Vert \mathbf {B}\Vert _{L^{3}} \Bigr) \bigl(A_{1}(T)+M_{0}^{\gamma } \bigr) \biggl( \int _{0}^{T} \bigl\Vert \nabla ^{2} \mathbf {B}\bigr\Vert _{L^{2}}^{2}\,dt \biggr)^{1/2} \\ &\qquad {}\times \biggl( \int _{0}^{T} \Vert \nabla \dot{\mathbf {u}} \Vert _{L^{2}}^{2}\,dt \biggr)^{1/2} \\ &\quad \leqslant CM_{0} \Bigl(\sup_{t\in [0, T]} \Vert \mathbf {B}\Vert _{L^{3}} \Bigr) \bigl(A_{1}(T)+M_{0}^{\gamma } \bigr) A_{1}^{1/2}(T)A_{2}^{1/2}(T). \end{aligned} $$
(3.49)
It follows from (3.47) that
$$ \begin{aligned} & \int _{0}^{T} \bigl( \Vert \nabla \mathbf {u}\Vert _{L^{2}}^{2}+ \bigl\Vert P(\rho ) \bigr\Vert _{L^{2}}^{2} \bigr) \Vert \nabla \mathbf {B}\cdot \mathbf {B}\Vert _{L^{6}} \Vert \nabla \mathbf {B}\cdot \mathbf {B}\Vert _{L^{2}}\,dt \\ &\quad \leqslant C \bigl(A_{1}(T)+M_{0}^{\gamma } \bigr) \int _{0}^{T} \bigl\Vert \nabla ^{2} \mathbf {B}\bigr\Vert _{L^{2}}^{5/2} \Vert \mathbf {B}\Vert _{L^{3}} \Vert \nabla \mathbf {B}\Vert _{L^{2}}^{1/2} \\ &\quad \leqslant C \Bigl(\sup_{t\in [0, T]} \Vert \mathbf {B}\Vert _{L^{3}} \Bigr) \bigl(A_{1}(T)+M_{0}^{\gamma } \bigr)A_{2}^{1/2} \biggl( \int _{0}^{T} \bigl\Vert \nabla ^{2} \mathbf {B}\bigr\Vert _{L^{2}}^{2}\,dt \biggr)^{3/4} \biggl( \int _{0}^{T} \Vert \nabla \mathbf {B}\Vert _{L^{2}}^{2}\,dt \biggr)^{1/4} \\ &\quad \leqslant C \Bigl(\sup_{t\in [0, T]} \Vert \mathbf {B}\Vert _{L^{3}} \Bigr) \bigl(A_{1}(T)+M_{0}^{\gamma } \bigr) A_{1}^{3/4}(T)A_{2}^{1/2}(T). \end{aligned} $$
(3.50)
Similarly,
$$ \begin{aligned} & \int _{0}^{T} \bigl\Vert \vert \mathbf {B}\vert ^{2} \bigr\Vert _{L^{2}}^{2} \Vert \rho \dot{ \mathbf {u}} \Vert _{L^{2}} \bigl( \Vert \rho \dot{\mathbf {u}} \Vert _{L^{6}}+ \Vert \nabla \mathbf {B}\cdot \mathbf {B}\Vert _{L^{6}} \bigr)\,dt \\ &\quad \leqslant C \Bigl(\sup_{t\in [0, T]} \Vert \mathbf {B}\Vert _{L^{3}}^{2} \Bigr) \bigl(M_{0}^{3/2}A_{1}(T)+M_{0}^{1/2}A_{1}^{5/4}(T) \bigr) A_{1}^{1/2}(T)A_{2}^{1/2}(T), \end{aligned} $$
(3.51)
and
$$ \begin{aligned} & \int _{0}^{T} \bigl[ \bigl\Vert \vert \mathbf {B}\vert ^{2} \bigr\Vert _{L^{2}}^{2} \Vert \nabla \mathbf {B}\cdot \mathbf {B}\Vert _{L^{2}} \bigl( \Vert \rho \dot{\mathbf {u}} \Vert _{L^{6}}+ \Vert \nabla \mathbf {B}\cdot \mathbf {B}\Vert _{L^{6}} \bigr)+ \Vert \mathbf {B}\Vert _{L^{8}}^{8} \bigr]\,dt \\ &\quad \leqslant C \Bigl(\sup_{t\in [0, T]} \Vert \mathbf {B}\Vert _{L^{3}}^{3} \Bigr) \bigl(M_{0}A_{1}(T)+A_{1}^{3/4}(T)A_{2}^{1/4}(T) \bigr) A_{1}^{1/2}(T)A_{2}^{1/2}(T). \end{aligned} $$
(3.52)
Substituting (3.46)–(3.52) into (3.45), by virtue of (3.42) and (3.44), one obtains
$$\begin{aligned}& \int _{0}^{T} \bigl( \Vert \nabla \mathbf {u}\Vert _{L^{4}}^{4}+ \bigl\Vert P(\rho ) \bigr\Vert _{L^{4}}^{4} \bigr)\,dt \\& \quad \leqslant CM_{0}^{2\gamma }+C \bigl(A_{1}(T)+M_{0}^{\gamma } \bigr) \bigl(M_{0}^{3/2}+ M_{0}^{1/2}A_{1}^{1/4} \bigr)A_{1}^{1/2}(T)A_{2}^{1/2}(T) \\& \qquad {} +C \bigl(A_{1}(T)+M_{0}^{\gamma } \bigr) \Bigl(\sup _{t\in [0, T]} \Vert \mathbf {B}\Vert _{L^{3}} \Bigr) \bigl(M_{0}+A_{1}^{1/4}(T) \bigr)A_{1}^{1/2}(T)A_{2}^{1/2}(T) \\& \qquad {} +C \Bigl(\sup_{t\in [0, T]} \Vert \mathbf {B}\Vert _{L^{3}}^{2} \Bigr) \bigl(M_{0}^{3/2}A_{1}(T)+M_{0}^{1/2}A_{1}^{5/4}(T) \bigr) A_{1}^{1/2}(T)A_{2}^{1/2}(T) \\& \qquad {} +C \Bigl(\sup_{t\in [0, T]} \Vert \mathbf {B}\Vert _{L^{3}}^{3} \Bigr) \bigl(M_{0}A_{1}(T)+A_{1}^{3/4}(T)A_{2}^{1/4}(T) \bigr)A_{1}^{1/2}(T)A_{2}^{1/2}(T). \end{aligned}$$
(3.53)
Collecting (3.6), (3.8), (3.9), and (3.53) together, and choosing \(K\geqslant 1\), \(M_{0}\leqslant 1/2\), we immediately obtain
$$ \begin{aligned} &A_{1}(T)+A_{2}(T) \\ &\quad \leqslant C \bigl(M_{0}^{1/2}K+M_{0}^{5\gamma /6}K^{1/2}+e^{CK}M_{0}^{2 \gamma +1}+e^{CK}M_{0}^{2\gamma } \bigr)A_{1}(T) \\ &\qquad {} +C \bigl(e^{2CK}M_{0}^{4\gamma }+e^{4CK}M_{0}^{8\gamma } \bigr)A_{2}(T)+C \bigl(1+M_{0}^{\gamma }+M_{0}^{2\gamma } \bigr) \\ &\qquad {} + C \bigl(M_{0}^{1/2}+e^{CK}M_{0}^{2\gamma } \bigr) \bigl(A_{1}(T)+M_{0}^{\gamma } \bigr) \bigl(M_{0}+ A_{1}^{1/4}(T) \bigr)A_{1}^{1/2}(T)A_{2}^{1/2}(T) \\ &\qquad {} +Ce^{2CK}M_{0}^{4\gamma } \bigl(M_{0}^{3/2}A_{1}(T)+M_{0}^{1/2}A_{1}^{5/4}(T) \bigr) A_{1}^{1/2}(T)A_{2}^{1/2}(T) \\ &\qquad {} +Ce^{3CK}M_{0}^{6\gamma } \bigl(M_{0}A_{1}(T)+A_{1}^{3/4}(T)A_{2}^{1/4}(T) \bigr)A_{1}^{1/2}(T)A_{2}^{1/2}(T) \\ &\quad \leqslant C_{1}e^{4C_{0}K}M_{0}^{1/2} \bigl[A_{1}(T)+A_{2}(T) \bigr] +C_{2} \bigl(1+M_{0}^{\gamma }+M_{0}^{2\gamma } \bigr) \\ &\qquad {} +C_{3}e^{3C_{0}K}M_{0}^{1/2} \bigl(A_{1}^{5/4}(T)+A_{2}^{5/8}(T) \bigr)A_{1}^{1/2}(T)A_{2}^{1/2}(T), \end{aligned} $$
(3.54)
where \(C_{i}\) \((i=0, 1, 2, 3)>0\) depend on μ, λ, ν, A, \(E_{0}\), \(M_{1}\), and \(M_{2}\), but not on \(M_{0}\) and T.
To continue, set
$$ K \triangleq \max \{1, 8C_{2}, M_{1}+M_{2}\}. $$
Thus, if it holds that
$$ M_{0}\leqslant \varepsilon _{1} \triangleq \min \biggl\{ \frac {1}{2}, \biggl(\frac {1}{2C_{1}e^{4C_{0}K}} \biggr)^{2}, \biggl( \frac {1}{16C_{3}e^{3C_{0}K^{5/4}}} \biggr)^{2} \biggr\} , $$
then one infers from (3.54) that
$$ \begin{aligned} A_{1}(T)+A_{2}(T) & \leqslant 4C_{2}+16C_{3}e^{3C_{0}K} M_{0}^{1/2}K^{5/4}A_{1}^{1/2}(T)A_{2}^{1/2}(T) \\ &\leqslant \frac {K}{2}+\frac {1}{2} \bigl[A_{1}(T)+A_{2}(T) \bigr], \end{aligned} $$
(3.55)
provided (3.41) holds. Thus, the desired estimate of (3.40) immediately follows from (3.55) by choosing K and \(\varepsilon _{1}\) as above. □
In order to derive a uniform upper bound of the density, we still need the following t-weighted estimate.
Lemma 3.5
Let the conditions of Lemma 3.4be in force. Then there exists a positive constant \(\varepsilon _{2}\), depending only on μ, λ, ν, γ, A, \(E_{0}\), \(M_{1}\), and \(M_{2}\), such that if \(\|\mathbf{ B}_{0}\|_{L^{3}}^{1/2\gamma }\leqslant M_{0}\leqslant \varepsilon _{2}\), then for any \(0\leqslant t_{1}< t_{2}\leqslant T\),
$$ \begin{aligned} &\sup_{t_{1}\leqslant t\leqslant t_{2}} \bigl[(t-t_{1}) \bigl( \Vert \sqrt{\rho }\dot{\mathbf{ u}} \Vert _{L^{2}}^{2}+ \Vert \mathbf{ B}_{t} \Vert _{L^{2}}^{2}+ \bigl\Vert \nabla ^{2} \mathbf{ B} \bigr\Vert _{L^{2}}^{2} \bigr) \bigr] \\ &\quad {} + \int _{t_{1}}^{t_{2}}(t-t_{1}) \bigl( \Vert \nabla \dot{\mathbf{ u}} \Vert _{L^{2}}^{2}+ \Vert \nabla \mathbf{ B}_{t} \Vert _{L^{2}}^{2} \bigr)\,dt \leqslant C \bigl[1+M_{0}^{\frac {5\gamma }{2}}(t_{2}-t_{1}) \bigr], \end{aligned} $$
(3.56)
where the constant \(C>0\) depends on μ, λ, ν, γ, A, \(E_{0}\), \(M_{1}\), \(M_{2}\), K, and the coefficients of the Gagliardo–Nirenberg–Sobolev inequality in Lemma 2.1, but is independent of \(M_{0}\).
Proof
Indeed, it follows from (3.6), (3.38), and (3.40) that
$$ \Vert \nabla \mathbf {u}\Vert _{L^{6}} \leqslant C \bigl(M_{0}^{1/2} \Vert \sqrt{ \rho } \dot{\mathbf {u}} \Vert _{L^{2}}+M_{0}^{5\gamma /6}+M_{0}^{2\gamma } \bigl\Vert \nabla ^{2} \mathbf {B}\bigr\Vert _{L^{2}} \bigr), $$
(3.57)
thus
$$ \begin{aligned} &\Vert \nabla \mathbf {u}\Vert _{L^{4}}^{4}+ \bigl\Vert P(\rho ) \bigr\Vert _{L^{4}}^{4} \\ &\quad \leqslant C \Vert \nabla \mathbf {u}\Vert _{L^{2}} \Vert \nabla \mathbf {u}\Vert _{L^{6}}^{3}+CM_{0}^{3 \gamma } \\ &\quad \leqslant C \bigl(M_{0}^{5\gamma /2}+M_{0}^{3/2} \Vert \nabla \mathbf {u}\Vert _{L^{2}} \Vert \sqrt{\rho } \dot{\mathbf {u}} \Vert _{L^{2}}^{3}+M_{0}^{6\gamma } \Vert \nabla \mathbf {u}\Vert _{L^{2}} \bigl\Vert \nabla ^{2} \mathbf {B}\bigr\Vert _{L^{2}}^{3} \bigr). \end{aligned} $$
(3.58)
Multiplying (3.36) by \((t-t_{1})\), integrating it over \((t_{1}, t)\) with \(t_{1}\leqslant t\leqslant t_{2}\), using (3.6) and (3.40), we get
$$ \begin{aligned} &\sup_{t_{1}\leqslant t \leqslant t_{2}} \bigl[(t-t_{1}) \bigl( \Vert \sqrt{\rho } \dot{\mathbf {u}} \Vert _{L^{2}}^{2}+ \Vert \mathbf {B}_{t} \Vert _{L^{2}}^{2} \bigr) \bigr]+ \int _{t_{1}}^{t_{2}}(t-t_{1}) \bigl( \Vert \nabla \dot{\mathbf {u}} \Vert _{L^{2}}^{2}+ \Vert \nabla \mathbf {B}_{t} \Vert _{L^{2}}^{2} \bigr)\,dt \\ &\quad \leqslant C+ \int _{t_{1}}^{t_{2}}(t-t_{1}) \bigl( \Vert \nabla \mathbf {u}\Vert _{L^{4}}^{4}+ \bigl\Vert P(\rho ) \bigr\Vert _{L^{4}}^{4}+M_{0}^{8\gamma } \bigl\Vert \nabla ^{2} \mathbf {B}\bigr\Vert _{L^{2}}^{4} \bigr)\,dt \\ &\qquad {} +C \int _{t_{1}}^{t_{2}}(t-t_{1}) \bigl( \Vert \mathbf {B}_{t} \Vert _{L^{2}}^{2} \Vert \nabla \mathbf {u}\Vert _{L^{2}} \Vert \nabla \mathbf {u}\Vert _{L^{6}} \bigr)\,dt \\ &\qquad {} +C \int _{t_{1}}^{t_{2}}(t-t_{1}) \bigl( \Vert \nabla \mathbf {B}\Vert _{L^{2}}^{2} \Vert \nabla \mathbf {u}\Vert _{L^{2}}^{2} \Vert \nabla \mathbf {u}\Vert _{L^{6}}^{2}+ \Vert \mathbf {u}\Vert _{L^{\infty }}^{4} \bigr)\,dt \\ &\quad \triangleq C+J_{1}+J_{2}+J_{3}, \end{aligned} $$
(3.59)
where the second term on the right-hand side can be estimated as follows based on (3.58):
$$ \begin{aligned} J_{1}\leqslant{}& C \int _{t_{1}}^{t_{2}}(t-t_{1}) \bigl(M_{0}^{5 \gamma /2}+M_{0}^{3/2} \Vert \nabla \mathbf {u}\Vert _{L^{2}} \Vert \sqrt{\rho } \dot{\mathbf {u}} \Vert _{L^{2}}^{3} +M_{0}^{6\gamma } \Vert \nabla \mathbf {u}\Vert _{L^{2}} \bigl\Vert \nabla ^{2} \mathbf {B}\bigr\Vert _{L^{2}}^{3} \\ &{} +M_{0}^{8\gamma } \bigl\Vert \nabla ^{2} \mathbf {B}\bigr\Vert _{L^{2}}^{4} \bigr)\,dt \\ \leqslant{}& CM_{0}^{3/2}\sup_{t_{1}\leqslant t\leqslant t_{2}} \bigl[(t-t_{1}) \Vert \sqrt{\rho } \dot{\mathbf {u}} \Vert _{L^{2}}^{2} \bigr] \biggl( \int _{t_{1}}^{t_{2}} \Vert \nabla \mathbf {u}\Vert _{L^{2}}^{2}\,dt \biggr)^{1/2} \biggl( \int _{t_{1}}^{t_{2}} \Vert \sqrt{\rho }\dot{\mathbf {u}} \Vert _{L^{2}}^{2}\,dt \biggr)^{1/2} \\ &{} +CM_{0}^{6\gamma }\sup_{t_{1}\leqslant t\leqslant t_{2}} \bigl[(t-t_{1}) \bigl\Vert \nabla ^{2} \mathbf {B}\bigr\Vert _{L^{2}}^{2} \bigr] \biggl( \int _{t_{1}}^{t_{2}} \Vert \nabla \mathbf {u}\Vert _{L^{2}}^{2}\,dt \biggr)^{1/2} \biggl( \int _{t_{1}}^{t_{2}} \bigl\Vert \nabla ^{2} \mathbf {B}\bigr\Vert _{L^{2}}^{2}\,dt \biggr)^{1/2} \\ &{} +CM_{0}^{8\gamma }\sup_{t_{1}\leqslant t\leqslant t_{2}} \bigl[(t-t_{1}) \bigl\Vert \nabla ^{2} \mathbf {B}\bigr\Vert _{L^{2}}^{2} \bigr] \int _{t_{1}}^{t_{2}} \bigl\Vert \nabla ^{2} \mathbf {B}\bigr\Vert _{L^{2}}^{2}\,dt+C M_{0}^{5\gamma /2}(t_{1}-t_{2})^{2} \\ \leqslant{}& C M_{0}^{3/2}\sup_{t_{1}\leqslant t\leqslant t_{2}} \bigl[(t-t_{1}) \Vert \sqrt{\rho } \dot{\mathbf {u}} \Vert _{L^{2}}^{2} \bigr] +CM_{0}^{6\gamma } \sup _{t_{1}\leqslant t\leqslant t_{2}} \bigl[(t-t_{1}) \bigl\Vert \nabla ^{2} \mathbf {B}\bigr\Vert _{L^{2}}^{2} \bigr] \\ &{} + C M_{0}^{5\gamma /2}(t_{1}-t_{2})^{2}. \end{aligned} $$
By virtue of (3.57) and Cauchy–Schwarz inequality, we also have
$$\begin{aligned} J_{2} \leqslant& C \int _{t_{1}}^{t_{2}}(t-t_{1}) \Vert \mathbf {B}_{t} \Vert _{L^{2}}^{2} \Vert \nabla \mathbf {u}\Vert _{L^{2}} \bigl(M_{0}^{5 \gamma /6}+M_{0}^{1/2} \Vert \sqrt{\rho } \dot{\mathbf {u}} \Vert _{L^{2}} +M_{0}^{2 \gamma } \bigl\Vert \nabla ^{2} \mathbf {B}\bigr\Vert _{L^{2}} \bigr) \,dt \\ \leqslant& CM_{0}^{1/2}\sup_{t_{1}\leqslant t\leqslant t_{2}} \bigl[(t-t_{1}) \Vert \mathbf {B}_{t} \Vert _{L^{2}}^{2} \bigr] \biggl( \int _{t_{1}}^{t_{2}} \Vert \nabla \mathbf {u}\Vert _{L^{2}}^{2}\,dt \biggr)^{1/2} \biggl( \int _{t_{1}}^{t_{2}} \Vert \sqrt{\rho }\dot{\mathbf {u}} \Vert _{L^{2}}^{2}\,dt \biggr)^{1/2} \\ &{} +CM_{0}^{5\gamma /6}\sup_{t_{1}\leqslant t\leqslant t_{2}} \bigl[(t-t_{1}) \Vert \mathbf {B}_{t} \Vert _{L^{2}} \bigr] \biggl( \int _{t_{1}}^{t_{2}} \Vert \nabla \mathbf {u}\Vert _{L^{2}}^{2}\,dt \biggr)^{1/2} \biggl( \int _{t_{1}}^{t_{2}} \Vert \mathbf {B}_{t} \Vert _{L^{2}}^{2}\,dt \biggr)^{1/2} \\ &{} +CM_{0}^{2\gamma }\sup_{t_{1}\leqslant t\leqslant t_{2}} \bigl[(t-t_{1}) \Vert \mathbf {B}_{t} \Vert _{L^{2}}^{2} \bigr] \biggl( \int _{t_{1}}^{t_{2}} \bigl\Vert \nabla ^{2} \mathbf {B}\bigr\Vert _{L^{2}}^{2}\,dt \biggr)^{1/2} \biggl( \int _{t_{1}}^{t_{2}} \Vert \nabla \mathbf {u}\Vert _{L^{2}}^{2}\,dt \biggr)^{1/2} \\ \leqslant& C \bigl(M_{0}^{1/2}+M_{0}^{5\gamma /12} \bigr)\sup_{t_{1}\leqslant t \leqslant t_{2}} \bigl[(t-t_{1}) \Vert \mathbf {B}_{t} \Vert _{L^{2}}^{2} \bigr]+ C M_{0}^{5\gamma /2}(t_{1}-t_{2})^{2}+C. \end{aligned}$$
Similarly, we infer from (3.40) that
$$ \begin{aligned} J_{3}\leqslant{}& C \int _{t_{1}}^{t_{2}}(t-t_{1}) \Vert \nabla \mathbf {u}\Vert _{L^{2}}^{2} \bigl(M_{0}^{5\gamma /3}+M_{0} \Vert \sqrt{ \rho } \dot{\mathbf {u}} \Vert _{L^{2}}^{2} +M_{0}^{4\gamma } \bigl\Vert \nabla ^{2} \mathbf {B}\bigr\Vert _{L^{2}}^{2} \bigr)\,dt \\ \leqslant{}& C M_{0}\sup_{t_{1}\leqslant t\leqslant t_{2}} \bigl[(t-t_{1}) \Vert \sqrt{\rho }\dot{\mathbf {u}} \Vert _{L^{2}}^{2} \bigr]+C M_{0}^{4\gamma } \sup _{t_{1}\leqslant t\leqslant t_{2}} \bigl[(t-t_{1}) \bigl\Vert \nabla ^{2} \mathbf {B}\bigr\Vert _{L^{2}}^{2} \bigr] \\ &{} + C M_{0}^{10\gamma /3}(t_{1}-t_{2})^{2}+C. \end{aligned} $$
In addition, it holds by (1.1)3 and (3.40) that
$$ \begin{aligned} \sup_{t_{1}\leqslant t\leqslant t_{2}} \bigl[(t-t_{1}) \bigl\Vert \nabla ^{2} \mathbf {B}\bigr\Vert _{L^{2}}^{2} \bigr] &\leqslant C \sup_{t_{1} \leqslant t\leqslant t_{2}} \bigl[(t-t_{1}) \bigl( \Vert \mathbf {B}_{t} \Vert _{L^{2}}^{2} + \Vert \mathbf {B}\Vert _{L^{3}}^{2} \Vert \nabla \mathbf {u}\Vert _{L^{2}}^{2} \bigr) \bigr] \\ &\leqslant C \sup_{t_{1}\leqslant t\leqslant t_{2}} \bigl[(t-t_{1}) \Vert \mathbf {B}_{t} \Vert _{L^{2}}^{2} \bigr] + C M_{0}^{8\gamma }(t_{1}-t_{2})^{2}+C. \end{aligned} $$
(3.60)
Substituting \(J_{1}\), \(J_{2}\), and \(J_{3}\) into (3.59) and using (3.60), we obtain
$$ \begin{aligned} &\sup_{t_{1}\leqslant t \leqslant t_{2}} \bigl[(t-t_{1}) \bigl( \Vert \sqrt{\rho } \dot{\mathbf {u}} \Vert _{L^{2}}^{2}+ \Vert \mathbf {B}_{t} \Vert _{L^{2}}^{2}+ \bigl\Vert \nabla ^{2} \mathbf {B}\bigr\Vert _{L^{2}}^{2} \bigr) \bigr] \\ &\qquad {}+ \int _{t_{1}}^{t_{2}}(t-t_{1}) \bigl( \Vert \nabla \dot{\mathbf {u}} \Vert _{L^{2}}^{2}+ \Vert \nabla \mathbf {B}_{t} \Vert _{L^{2}}^{2} \bigr)\,dt \\ & \quad \leqslant C_{4} M_{0}\sup_{t_{1}\leqslant t\leqslant t_{2}} \bigl[(t-t_{1}) \Vert \sqrt{\rho }\dot{\mathbf {u}} \Vert _{L^{2}}^{2} \bigr] \\ &\qquad {}+ C_{5} \bigl(M_{0}^{1/2}+M_{0}^{5 \gamma /12} \bigr)\sup_{t_{1}\leqslant t\leqslant t_{2}} \bigl[(t-t_{1}) \Vert \mathbf {B}_{t} \Vert _{L^{2}}^{2} \bigr] \\ &\qquad {} + C_{6} M_{0}^{4\gamma }\sup_{t_{1}\leqslant t\leqslant t_{2}} \bigl[(t-t_{1}) \bigl\Vert \nabla ^{2} \mathbf {B}\bigr\Vert _{L^{2}}^{2} \bigr] +C M_{0}^{5 \gamma /2}(t_{1}-t_{2})^{2}+C. \end{aligned} $$
(3.61)
Thus, if \(M_{0}\) is chosen to be suitably small such that
$$ M_{0}\leqslant \varepsilon _{2} \triangleq \min \biggl\{ \frac {1}{2C_{4}}, \biggl(\frac {1}{2C_{5}} \biggr)^{{\frac {12}{5\gamma +6}}}, \biggl( \frac {1}{2C_{6}} \biggr)^{{\frac {1}{4\gamma }}} \biggr\} , $$
then we immediately obtain (3.56) from (3.61). □
We are now in a position of estimating an upper bound of the density.
Lemma 3.6
Let the conditions of Lemma 3.4and 3.5be in force. Then there exists a positive constant ε, depending only on μ, λ, ν, γ, A, \(E_{0}\), \(M_{1}\), and \(M_{2}\), such that if \(\|\mathbf{ B}_{0}\|_{L^{3}}^{1/2\gamma }\leqslant M_{0}\leqslant \varepsilon \), then
$$ \rho (x, t)\leqslant \frac {7M_{0}}{4} \quad \textit{for all } (x, t) \in \mathbb{R}^{3}\times [0, T]. $$
(3.62)
Proof
In view of (2.3)1, we can rewrite (1.1)1 in the form:
$$ D_{t} \rho =g(\rho )+b'(t), $$
(3.63)
where
$$ g(\rho )\triangleq -\frac {A \rho }{2\mu +\lambda }\rho ^{\gamma }, \qquad b(t) \triangleq -\frac {1}{2\mu +\lambda } \int _{0}^{t} \biggl( \rho F+\frac {1}{2} \rho \vert \mathbf {B}\vert ^{2} \biggr)\,ds. $$
(3.64)
Obviously, it holds that \(g(\infty )=-\infty \). So, to apply Lemma 2.3, we still need to deal with \(b(t)\). To do this, we first utilize (2.2), (2.5), (2.6), and the fact that \(0\leqslant \rho \leqslant 2M_{0}\) to deduce that for any \(0\leqslant t_{1} < t_{2} \leqslant T\),
$$ \begin{aligned} \bigl\vert b(t_{2})-b(t_{1}) \bigr\vert \leqslant{}& CM_{0} \int _{t_{1}}^{t_{2}} \bigl( \Vert F \Vert _{L^{\infty }}+ \Vert \mathbf {B}\Vert _{L^{\infty }}^{2} \bigr) \,dt \\ \leqslant{}& CM_{0} \int _{t_{1}}^{t_{2}} \bigl( \Vert F \Vert _{L^{2}}^{1/4} \Vert \nabla F \Vert _{L^{6}}^{3/4}+ \Vert \mathbf {B}\Vert _{L^{3}}^{2/3} \Vert \nabla \mathbf {B}\Vert _{L^{6}}^{4/3} \bigr)\,dt \\ \leqslant{}& C M_{0} \int _{t_{1}}^{t_{2}} \bigl( \Vert \nabla \mathbf{ u} \Vert _{L^{2}}+M_{0}^{\gamma /2}+ \Vert \mathbf{ B} \Vert _{L^{4}}^{2} \bigr)^{1/4} \\ &{}\times \bigl(M_{0} \Vert \nabla \dot{\mathbf{ u}} \Vert _{L^{2}}+ \Vert \mathbf {B}\Vert _{L^{3}}^{1/3} \Vert \nabla \mathbf{ B} \Vert _{L^{6}}^{5/3} \bigr)^{3/4}\,dt \\ &{} +C M_{0} \int _{t_{1}}^{t_{2}} \Vert \mathbf {B}\Vert _{L^{3}}^{2/3} \Vert \nabla \mathbf {B}\Vert _{L^{6}}^{4/3}\,dt \triangleq \sum_{i=1}^{7}N_{i}. \end{aligned} $$
(3.65)
The right-hand side of (3.65) can be estimated as follows. We deduce from (3.4), (3.56), and Hölder inequality that
$$ \begin{aligned} N_{1} & = C M_{0}^{7/4} \int _{t_{1}}^{t_{2}}(t-t_{1})^{-3/8} \bigl( \Vert \nabla \mathbf{ u} \Vert _{L^{2}}^{2} \bigr)^{1/8} \bigl[(t-t_{1}) \Vert \nabla \dot{\mathbf{ u}} \Vert _{L^{2}}^{2} \bigr]^{3/8}\,dt \\ &\leqslant C M_{0}^{7/4} \biggl( \int _{t_{1}}^{t_{2}}(t-t_{1})^{-3/4} \,dt \biggr)^{1/2} \biggl( \int _{t_{1}}^{t_{2}} \Vert \nabla \mathbf{ u} \Vert _{L^{2}}^{2}\,dt \biggr)^{1/8} \biggl( \int _{t_{1}}^{t_{2}}(t-t_{1}) \Vert \nabla \dot{\mathbf {u}} \Vert _{L^{2}}^{2}\,dt \biggr)^{3/8} \\ &\leqslant C M_{0}^{7/4} (t_{2}-t_{1})^{1/8} \bigl[1+M_{0}^{15\gamma /16}(t_{2}-t_{1})^{3/4} \bigr] \\ &\leqslant \frac {AM_{0}^{\gamma +1}}{8(2\mu +\lambda )}(t_{2}-t_{1})+C \bigl(M_{0}^{\frac {13-\gamma }{7}}+M_{0}^{\frac {14+\gamma }{2}} \bigr), \end{aligned} $$
and
$$\begin{aligned} N_{2} =&C M_{0} \int _{t_{1}}^{t_{2}} \Vert \nabla \mathbf{ u} \Vert _{L^{2}}^{1/4} \Vert \mathbf {B}\Vert _{L^{3}}^{1/4} \bigl\Vert \nabla ^{2} \mathbf{ B} \bigr\Vert _{L^{2}}^{3/4} \bigl[(t-t_{1}) \bigl\Vert \nabla ^{2}\mathbf{ B} \bigr\Vert _{L^{2}}^{2} \bigr]^{1/4}(t-t_{1})^{-1/4} \,dt \\ \leqslant& C M_{0}^{(2+\gamma )/2} \Bigl[\sup_{t_{1}\leqslant t \leqslant t_{2}} \bigl((t-t_{1}) \bigl\Vert \nabla ^{2} \mathbf {B}\bigr\Vert _{L^{2}}^{2} \bigr) \Bigr]^{1/4} \biggl( \int _{t_{1}}^{t_{2}} \Vert \nabla \mathbf{ u} \Vert _{L^{2}}^{2}\,dt \biggr)^{1/8} \\ &{}\times\biggl( \int _{t_{1}}^{t_{2}} \Vert \nabla \mathbf{ B} \Vert _{L^{2}}^{2}\,dt \biggr)^{3/8} \biggl( \int _{t_{1}}^{t_{2}}(t-t_{1})^{-1/2} \,dt \biggr)^{1/2} \\ \leqslant& C M_{0}^{(2+\gamma )/2} (t_{2}-t_{1})^{1/4} \bigl[1+M_{0}^{5 \gamma /8}(t_{2}-t_{1})^{1/2} \bigr] \\ \leqslant& \frac {AM_{0}^{\gamma +1}}{8(2\mu +\lambda )}(t_{2}-t_{1})+C \bigl(M_{0}^{\frac {3+\gamma }{3}}+M_{0}^{\frac {2+3\gamma }{2}} \bigr). \end{aligned}$$
Similarly, using (3.56) and Hölder inequality, we have
$$ \begin{aligned} N_{3} & = C M_{0}^{(14+\gamma )/8} \int _{t_{1}}^{t_{2}}(t-t_{1})^{-3/8} \bigl[(t-t_{1}) \Vert \nabla \dot{\mathbf{ u}} \Vert _{L^{2}}^{2} \bigr]^{3/8}\,dt \\ &\leqslant C M_{0}^{(14+\gamma )/8} \biggl( \int _{t_{1}}^{t_{2}}(t-t_{1})^{-3/5} \,dt \biggr)^{5/8} \biggl( \int _{t_{1}}^{t_{2}}(t-t_{1}) \Vert \nabla \dot{\mathbf {u}} \Vert _{L^{2}}^{2}\,dt \biggr)^{3/8} \\ &\leqslant C M_{0}^{(14+\gamma )/8} (t_{2}-t_{1})^{1/4} \bigl[1+M_{0}^{15 \gamma /16}(t_{2}-t_{1})^{3/4} \bigr] \\ &\leqslant \frac {AM_{0}^{\gamma +1}}{2\mu +\lambda } \biggl(\frac {1}{8}+CM_{0}^{ \frac {12+\gamma }{16}} \biggr) (t_{2}-t_{1})+CM_{0}^{ \frac {12-\gamma }{6}}, \end{aligned} $$
and
$$ \begin{aligned} N_{4} ={}&C M_{0}^{(8+\gamma )/8} \int _{t_{1}}^{t_{2}} \Vert \mathbf {B}\Vert _{L^{3}}^{1/4} \bigl\Vert \nabla ^{2}\mathbf{ B} \bigr\Vert _{L^{2}}^{3/4}\,dt \\ \leqslant{} &C M_{0}^{(8+5\gamma )/8} \Bigl[\sup_{t_{1}\leqslant t \leqslant t_{2}} \bigl((t-t_{1}) \bigl\Vert \nabla ^{2} \mathbf {B}\bigr\Vert _{L^{2}}^{2} \bigr) \Bigr]^{1/4} \biggl( \int _{t_{1}}^{t_{2}} \bigl\Vert \nabla ^{2} \mathbf{ B} \bigr\Vert _{L^{2}}^{2}\,dt \biggr)^{3/8} \\ &{} \times \biggl( \int _{t_{1}}^{t_{2}}(t-t_{1})^{-2/5} \,dt \biggr)^{5/8} \\ \leqslant{} &C M_{0}^{(8+5\gamma )/8} (t_{2}-t_{1})^{3/8} \bigl[1+M_{0}^{5 \gamma /8}(t_{2}-t_{1})^{1/2} \bigr] \\ \leqslant{}& \frac {AM_{0}^{\gamma +1}}{8(2\mu +\lambda )}(t_{2}-t_{1})+C \bigl(M_{0}^{\frac {5+2\gamma }{5}}+M_{0}^{1+3\gamma } \bigr). \end{aligned} $$
By virtue of (3.6) and (3.40), we infer from Cauchy–Schwarz inequality that
$$ \begin{aligned} N_{5} &=C M_{0}^{7/4} \int _{t_{1}}^{t_{2}} \Vert \mathbf {B}\Vert _{L^{3}}^{1/4} \Vert \nabla \mathbf{ B} \Vert _{L^{2}}^{1/4} \Vert \nabla \dot{\mathbf {u}} \Vert _{L^{2}}^{3/4}\,dt \\ & \leqslant C M_{0}^{(7+2\gamma )/4} \biggl( \int _{t_{1}}^{t_{2}}(t-t_{1}) \Vert \nabla \dot{\mathbf {u}} \Vert _{L^{2}}^{2}\,dt \biggr)^{3/8} \biggl( \int _{t_{1}}^{t_{2}}(t-t_{1})^{-3/5} \,dt \biggr)^{5/8} \\ &\leqslant C M_{0}^{(7+2\gamma )/4} (t_{2}-t_{1})^{1/4} \bigl[1+M_{0}^{15 \gamma /16}(t_{2}-t_{1})^{3/4} \bigr] \\ &\leqslant \frac {AM_{0}^{\gamma +1}}{2\mu +\lambda } \biggl(\frac {1}{8}+CM_{0}^{ \frac {12+7\gamma }{16}} \biggr) (t_{2}-t_{1})+CM_{0}^{ \frac {6+\gamma }{3}}, \end{aligned} $$
and
$$\begin{aligned} N_{6} ={}&C M_{0} \int _{t_{1}}^{t_{2}} \Vert \mathbf {B}\Vert _{L^{3}}^{1/2} \Vert \nabla \mathbf{ B} \Vert _{L^{2}}^{1/4} \bigl\Vert \nabla ^{2} \mathbf {B}\bigr\Vert _{L^{2}}^{5/4}\,dt \\ \leqslant{}& C M_{0}^{1+\gamma } \Bigl[\sup_{t_{1}\leqslant t \leqslant t_{2}} \bigl((t-t_{1}) \bigl\Vert \nabla ^{2} \mathbf {B}\bigr\Vert _{L^{2}}^{2} \bigr) \Bigr]^{1/4} ( \int _{t_{1}}^{t_{2}} \bigl( \bigl\Vert \nabla ^{2} \mathbf {B}\bigr\Vert _{L^{2}}^{2}\,dt \bigr)^{3/8} \\ &{} \times \biggl( \int _{t_{1}}^{t_{2}}(t-t_{1})^{-2/5} \,dt \biggr)^{5/8} \\ \leqslant{}& C M_{0}^{(1+\gamma } (t_{2}-t_{1})^{3/8} \bigl[1+M_{0}^{5 \gamma /8}(t_{2}-t_{1})^{1/2} \bigr] \\ \leqslant{}& \frac {AM_{0}^{\gamma +1}}{8(2\mu +\lambda )}(t_{2}-t_{1})+C \bigl(M_{0}^{1+\gamma }+M_{0}^{1+2\gamma } \bigr), \end{aligned}$$
and
$$ \begin{aligned} N_{7} &=C M_{0} \int _{t_{1}}^{t_{2}} \Vert \mathbf {B}\Vert _{L^{3}}^{2/3} \bigl\Vert \nabla ^{2} \mathbf {B}\bigr\Vert _{L^{2}}^{4/3}\,dt \\ & \leqslant C M_{0}^{\frac {3+4\gamma }{3}}\biggl( \int _{t_{1}}^{t_{2}} \bigl( \bigl\Vert \nabla ^{2}\mathbf {B}\bigr\Vert _{L^{2}}^{2}\,dt \bigr)^{2/3} \biggl( \int _{t_{1}}^{t_{2}}\,dt \biggr)^{1/3} \\ &\leqslant C M_{0}^{\frac {3+4\gamma }{3}} (t_{2}-t_{1})^{1/3} \\ &\leqslant \frac {AM_{0}^{\gamma +1}}{8(2\mu +\lambda )}(t_{2}-t_{1})+C M_{0}^{\frac {2+3\gamma }{2}}. \end{aligned} $$
Due to \(1<\gamma <6\), it is obvious that
$$ \frac {13-\gamma }{7}< \frac {12-\gamma }{6} $$
and hence, putting \(N_{i}\) (\(i=1, 2, \dots , 7\)) into (3.65), we obtain
$$ \bigl\vert b(t_{2})-b(t_{1}) \bigr\vert \leqslant C_{7} \bigl(M_{0}^{\frac {13-\gamma }{7}}+M_{0}^{ \frac {3+\gamma }{3}} \bigr)+\frac {AM_{0}^{\gamma +1}}{2\mu +\lambda } \biggl(\frac {7}{8}+C_{8}M_{0}^{\frac {12+\gamma }{16}} \biggr) (t_{2}-t_{1}). $$
(3.66)
Now, in view of (2.10) and (2.11), we set
$$ N_{0} \triangleq C_{7} \bigl(M_{0}^{\frac {13-\gamma }{7}}+M_{0}^{ \frac {3+\gamma }{3}} \bigr),\qquad N_{1}\triangleq \frac {AM_{0}^{ \gamma +1}}{2\mu +\lambda } \biggl(\frac {7}{8}+C_{8}M_{0}^{ \frac {12+\gamma }{16}} \biggr), \quad \text{and}\quad \xi ^{*} \triangleq M_{0}. $$
Thus, if \(M_{0}\) is chosen to be small enough such that (noting that \(1<\gamma <6\))
$$ M_{0} \triangleq \biggl\{ \varepsilon _{2}, \biggl( \frac {1}{4C_{7}} \biggr)^{\frac {7}{6-\gamma }}, \biggl(\frac {1}{2C_{7}} \biggr)^{ \frac {3}{\gamma }}, \biggl(\frac {1}{8C_{8}} \biggr)^{ \frac {16}{12+\gamma }} \biggr\} , $$
then it is easy to check that
$$ g(\xi )=-\frac {A \xi ^{\gamma +1}}{2\mu +\lambda }\leqslant -\frac {A M_{0}^{ \gamma +1}}{2\mu +\lambda }\leqslant -N_{1}, \quad \forall \xi \geqslant \xi ^{*}=M_{0}. $$
As a result, we conclude from (3.66) and Lemma 2.3 that
$$ \sup_{0\leqslant t \leqslant T} \bigl\Vert \rho (t) \bigr\Vert _{L^{\infty }}\leqslant M_{0}+N_{0} \leqslant M_{0}+\frac {3M_{0}}{4}=\frac {7M_{0}}{4}. $$
The proof of Lemma 3.6 is therefore complete. □
Proof of Proposition 3.1
In view of (3.40) and (3.62), we obtain the desired estimate of (3.2) with K and ε being the same as in Lemmas 3.4 and 3.6. □
With the help of Proposition 3.1, the higher-order estimates of the solution \((\rho , \mathbf {u}, \mathbf {B})\) can be shown in a manner similar to that in [9] (see Lemmas 4.1–4.6). For completeness and for convenience, we collect these estimates in the following lemma without proofs.
Proposition 3.2
Let the conditions of Proposition 3.1be in force. Then for any \(T > 0\) and \(0 < \tau < T\),
$$\begin{aligned} &\sup_{t\in [0, T]} \bigl( \bigl\Vert (\nabla \rho , \nabla P) \bigr\Vert _{H^{1} \cap W^{1, q}}+ \bigl\Vert (\rho _{t}, P_{t}) \bigr\Vert _{H^{1}} \bigr)+ \int _{0}^{T} \bigl\Vert (\rho _{tt}, P_{tt}) \bigr\Vert _{L^{2}}^{2}\,dt\leqslant C(T),\quad q \in (3, 6), \\ &\sup_{t\in [0, T]} \bigl( \bigl\Vert (\sqrt{\rho } \mathbf{ u}_{t}, \mathbf{ B}_{t}) \bigr\Vert _{L^{2}}^{2}+ \bigl\Vert (\nabla \mathbf{ u}, \nabla \mathbf{ B}) \bigr\Vert _{H^{1}}^{2} \bigr)+ \int _{0}^{T} \bigl( \bigl\Vert (\nabla \mathbf{ u}_{t}, \nabla \mathbf{ B}_{t}) \bigr\Vert _{L^{2}}^{2}+ \bigl\Vert \nabla ^{2} \mathbf{ u} \bigr\Vert _{H^{1}}^{2} \bigr)\,dt \\ &\quad \leqslant C(T), \\ &\sup_{t\in [\tau , T]} \bigl( \bigl\Vert \nabla ^{2}\mathbf{ u} \bigr\Vert _{H^{1} \cap W^{1, q}}^{2}+ \bigl\Vert \nabla ^{2} \mathbf{ B} \bigr\Vert _{H^{2}}^{2}+ \bigl\Vert (\nabla \mathbf{ u}_{t}, \nabla \mathbf{ B}_{t} ) \bigr\Vert _{H^{1}}^{2} \bigr)+ \int _{\tau }^{T} \bigl\Vert (\nabla \mathbf{ u}_{tt}, \nabla \mathbf{ B}_{tt}) \bigr\Vert _{L^{2}}^{2}\,dt \\ &\quad \leqslant C(\tau , T), \end{aligned}$$
where \(C(\tau , T)\) denotes a positive constant depending on τ and T, in addition to μ, λ, ν, γ, A, \(E_{0}\), \(M_{1}\), \(M_{2}\), K, and the coefficients of the Gagliardo–Nirenberg–Sobolev inequality in Lemma 2.1.
With Propositions 3.1–3.2 at hand, we can extend the local solutions obtained in Lemma 2.4 to be a global one in a similar manner as that in [9].
Proof of Theorem 1.1
By Lemma 2.4, there exists a \(T_{*} > 0\) such that the problem (1.1)–(1.5) has a classical solution \((\rho , \mathbf {u}, \mathbf {B})\) on \((0, T_{*}]\). Noting that
$$ A_{1}(0)+A_{2}(0)=M_{1}+M_{2}\leqslant K, \quad 0\leqslant \rho _{0} \leqslant M_{0}, $$
and using the continuity arguments, one infers that there exists a \(T_{1}\in (0, T_{*}]\) such that (3.1) holds for \(T = T_{1}\). Next, let
$$ T^{*} \triangleq \sup \{T | \text{ (3.1) holds}\}. $$
(3.67)
Then \(T^{*}\geqslant T_{1}>0\). We claim that
$$ T^{*} =\infty . $$
(3.68)
Indeed, if we had \(T^{*}<\infty \), then it would follow from Proposition 3.1 that (3.2) holds for \(T = T^{*}\), provided \(\|\mathbf {B}_{0}\|_{L^{3}}^{1/2\gamma }\leqslant M_{0}\leqslant \varepsilon \) and \(1<\gamma <6\). It would also follow from Proposition 3.2 that for any \(0<\tau < T\leqslant T^{*}\),
$$ \textstyle\begin{cases} (\nabla \mathbf {u}_{t}, \nabla \mathbf {B}_{t}, \nabla ^{2} \mathbf {u}, \nabla ^{3} \mathbf {B})\in C([\tau , T]; L^{2} \cap L^{4}), \\ \nabla ^{3} \mathbf {u}\in C([\tau , T]; L^{p}) \quad \text{with } p \in [2, q), q\in (3, 6), \\ (\nabla \mathbf {u}, \nabla \mathbf {B}, \nabla ^{2} \mathbf {u}, \nabla ^{2} \mathbf {B})\in C([\tau , T]; L^{2} \cap C(\mathbb{R}^{3})), \end{cases} $$
(3.69)
where we have also used the following embedding:
$$ L^{\infty } \bigl(\tau , T; H^{1} \bigr) \cap H^{1} \bigl( \tau , T; H^{-1} \bigr) \hookrightarrow C \bigl(\tau , T; L^{\alpha } \bigr),\quad \forall \alpha \in [2, 6). $$
Moreover, for any \(0<\tau <T\leqslant T^{*}\), it then follows from (1.1)1, Propositions 3.1 and 3.2 that
$$ \int _{\tau }^{T} \int \bigl\vert \partial _{t} \bigl(\rho \vert \mathbf {u}_{t} \vert ^{2} \bigr) \bigr\vert \,dx\,dt+ \int _{\tau }^{T} \int \bigl\vert \partial _{t} \bigl(\rho \vert \mathbf {u}\nabla \mathbf {u}\vert ^{2} \bigr) \bigr\vert \,dx\,dt\leqslant C(\tau , T), $$
which, together Proposition 3.2, yields
$$ \sqrt{\rho }\mathbf {u}_{t}, \sqrt{\rho }\mathbf {u}\cdot \nabla \mathbf {u}\in C \bigl(\tau , T; L^{2} \bigr), $$
and, consequently,
$$ \sqrt{\rho }\dot{\mathbf {u}} \in C \bigl(\tau , T; L^{2} \bigr). $$
In a similar manner, we can also show using (3.69) that
$$ \nabla \dot{\mathbf {u}} \in C \bigl(\tau , T; L^{2} \bigr). $$
This particularly implies that \((\rho , \mathbf {u}, \mathbf {B})(x, T^{*})\) satisfies the compatibility condition (1.9) with \(g(x)\triangleq \dot{\mathbf {u}}(x, T^{*})\) at \(t=T^{*}\). Thus, using Lemma 2.4, (3.2), and the continuity arguments, we know that there exists a \(T^{**}>T^{*}\) such that (3.1) holds for \(T = T^{**}\), which contradicts (3.67). Hence, (3.68) holds. This, together with Proposition 3.2 again, shows that the solution \((\rho , \mathbf {u}, \mathbf {B})\) is in fact the unique classical solution on \(\mathbb{R}^{3} \times (0, T ]\) for any \(0< T< T^{*}=\infty \). The proof of Theorem 1.1 is therefore complete. □