We start with the local existence of strong solutions whose proof can be found in Theorem 1.1 of [16].
Lemma 3.1
Under the assumptions of Theorem
1.1, a positive time
T
exists so that (1.2) has a unique local solution
\(a \in {\mathcal{C}_{b}}([0, T]; B^{3/2}_{2,1}({\mathbb{R}}^{3}))\); \(u,B\in{\mathcal{C}_{b}}([0, T]; \dot{B}^{1/2}_{2, 1}({\mathbb{R}}^{3})) \cap L^{1}([0,T]; \dot{B}^{5/2}_{2, 1}({\mathbb{R}}^{3}))\)
and, for
\(T\geq1\), we have
$$ \bigl\Vert (u,B)\bigr\Vert _{\widetilde{L}^{\infty}([0, T]; \dot{B}^{1/2}_{2, 1})}+ \int_{0}^{T} \bigl\Vert (u,B) (\tau)\bigr\Vert _{\dot{B}^{5/2}_{2, 1}}\delta\tau\leq C\bigl(\Vert u_{0}\Vert _{\dot{B}^{1/2}_{2, 1}}+\Vert B_{0}\Vert _{\dot{B}^{1/2}_{2, 1}}\bigr). $$
(3.1)
Similar to Theorem 2 in [8], we can obtain a higher-order regularity of the local solution to (1.2) as follows.
Proposition 3.1
Under the assumptions of Theorem
1.1, for any
\(t_{0}>0\), we have
$$\begin{aligned}& \bigl\Vert (u,B)\bigr\Vert _{\widetilde{L}^{\infty}([t_{0}, T]; \dot{B}^{3/2}_{2,1})} +\bigl\Vert (u,B) \bigr\Vert _{L^{1}([t_{0}, T]; \dot{B}^{7/2}_{2,1})}+\Vert \nabla \Pi \Vert _{L^{1}([t_{0}, T]; \dot{B}^{3/2}_{2,1})} \\& \quad \leq C\bigl(\Vert a_{0}\Vert _{{B}^{3/2}_{2,1}}\bigr) \bigl( \Vert u_{0}\Vert _{\dot{B}^{1/2}_{2,1}} +\Vert B_{0}\Vert _{\dot{B}^{1/2}_{2,1}}\bigr) (1+1/\sqrt{t_{0}})\exp \bigl\{ C\bigl(\Vert u_{0}\Vert _{\dot{B}^{1/2}_{2,1}} +\Vert B_{0}\Vert _{\dot{B}^{1/2}_{2,1}}\bigr) \bigr\} . \end{aligned}$$
(3.2)
Remark 3.1
Thanks to (3.1) and (3.2), there exists \(t_{1}\in(0, 1)\) so that \(u(t_{1}),B(t_{1}) \in\dot{B}^{1/2}_{2, 1}({\mathbb{R}}^{3}) \cap \dot{B}^{7/2}_{2, 1}({\mathbb{R}}^{3})\) and it satisfies (1.6).
We are in a position to prove Theorem 1.1.
Proof of Theorem 1.1
Thanks to Lemma 3.1, we conclude that: given \(a_{0}\in B^{3/2}_{2, 1}({\mathbb{R}}^{3})\), \(u_{0}, B_{0}\in\dot{B}^{1/2}_{2, 1}({\mathbb{R}}^{3})\) with \(\|u_{0}\|_{\dot{B}^{1/2}_{2, 1}}+\|B_{0}\|_{\dot{B}^{1/2}_{2, 1}}\) sufficiently small, (1.2) has a unique local solution \((a, u)\) satisfying \(a\in{\mathcal{C}}([0,{T}^{\ast}); B^{3/2}_{2, 1}({\mathbb{R}}^{3}))\), and \(u,B\in{\mathcal{C}}([0,{T}^{\ast}); \dot{B}^{1/2}_{2, 1}({\mathbb{R}}^{3})) \cap L^{1}_{\mathrm{loc}}((0,{T}^{\ast}); \dot{B}^{5/2}_{2, 1}({\mathbb{R}}^{3}))\) for some \(T^{\ast}>1\). Our aim o is to prove that \({T}^{\ast}=\infty\).
As \(\|u(t_{1})\|_{\dot{B}_{2, 1}^{1/2}\cap\dot{B}_{2, 1}^{7/2}}\), \(\|B(t_{1})\|_{\dot{B}_{2, 1}^{1/2}\cap\dot{B}_{2, 1}^{7/2}}\) is very small provided \(\|u_{0}\|_{\dot{B}_{2, 1}^{1/2}}\), \(\|B_{0}\|_{\dot{B}_{2, 1}^{1/2}}\) is sufficiently small. Let v solve the classical Navier-Stokes system (1.7). As \(u(t_{1})\) is sufficient small in \(\dot{B}_{2, 1}^{1/2}({\mathbb{R}}^{3})\), it follows from the classical theory of Navier-Stokes equations [15] that (1.7) has a unique global solution \(v \in\mathcal{C}([t_{1}, +\infty); \dot{B}_{2, 1}^{1/2}) \cap L^{1}([t_{1}, +\infty); \dot{B}_{2, 1}^{5/2})\) satisfying
$$ \begin{aligned} &\Vert v\Vert _{\widetilde{L}^{\infty}([t_{1}, +\infty); \dot{B}_{2, 1}^{1/2})}+ \Vert v\Vert _{L^{1}([t_{1}, +\infty); \dot{B}_{2, 1}^{5/2})}+\Vert \nabla\Pi_{v}\Vert _{L^{1}([t_{1}, +\infty); \dot{B}_{2, 1}^{1/2})} \leq C\bigl\Vert u(t_{1})\bigr\Vert _{\dot{B}_{2, 1}^{1/2}}, \\ &\Vert \partial_{t} v\Vert _{L^{1}([t_{1}, +\infty); \dot{B}_{2, 1}^{1/2})}\leq C \bigl\Vert u(t_{1})\bigr\Vert _{\dot{B}_{2, 1}^{1/2}}+\bigl\Vert \operatorname{div}(v \otimes v)\bigr\Vert _{ L^{1}([t_{1}, +\infty); \dot{B}_{2, 1}^{1/2})}\leq C\bigl\Vert u(t_{1})\bigr\Vert _{\dot{B}_{2, 1}^{1/2}}. \end{aligned} $$
(3.3)
With v thus obtained, we denote \(w \stackrel{\mathrm{def}}{=}u-v\). Then, thanks to (1.3) and (1.7), w, B solves (1.8). Then the proof of Theorem 1.1 reduces to proving the global well-posedness of (1.8). For simplicity, we just present the a priori estimates for smooth enough solutions of (1.8) on \([0, T^{\ast})\).
3.1 The higher regularities of v
Proposition 3.2
([8])
Let
\((v,\Pi_{v})\)
be the unique global solution of (1.7) which satisfies (3.3). Then, for
\(s_{1}\in[\frac{3}{2},\frac{7}{2}]\)
and
\(s_{2}\in[\frac{1}{2},\frac{3}{2}]\), we have
$$\begin{aligned}& \Vert v\Vert _{\widetilde{L}^{\infty}([t_{1}, +\infty); \dot{B}_{2, 1}^{s_{1}})}+ \bigl\Vert (\Delta v, \nabla \Pi_{v})\bigr\Vert _{L^{1}([t_{1}, +\infty); \dot{B}_{2, 1}^{s_{1}})}\leq C \Vert u_{0} \Vert _{\dot{B}_{2, 1}^{1/2}}, \end{aligned}$$
(3.4)
$$\begin{aligned}& \Vert \partial_{t} v\Vert _{\widetilde{L}^{\infty}([t_{1}, +\infty); \dot{B}_{2, 1}^{s_{2}})}+ \bigl\Vert (\partial_{t} \Delta v, \partial_{t} \nabla \Pi_{v})\bigr\Vert _{L^{1}([t_{1}, +\infty); \dot{B}_{2, 1}^{s_{2}})}\leq C \Vert u_{0} \Vert _{\dot{B}_{2, 1}^{1/2}}. \end{aligned}$$
(3.5)
Corollary 3.1
([8])
Under the assumptions of Proposition
3.2, one has
$$ \|\nabla v\|_{L^{2}([t_{1}, +\infty); L^{\infty})}+\|\Delta v-\nabla \Pi_{v} \|_{L^{2}([t_{1}, +\infty); L^{\infty})} \leq C \|u_{0}\|_{\dot{B}_{2, 1}^{1/2}}. $$
(3.6)
3.2 The estimate of \((w,B)\)
Lemma 3.2
(\(L^{2}\) estimate of \((w, B)\))
We have for
\(t_{1}< t< T^{\ast}\),
$$\begin{aligned}& \|w\|_{L^{\infty}([t_{1},t]; L^{2})}+\|\nabla w\|_{L^{2}([t_{1},t]; L^{2})}+\|B \|_{L^{\infty}([t_{1},t]; L^{2})}+\|\nabla B\|_{L^{2}([t_{1},t]; L^{2})} \\& \quad \leq C \bigl(\Vert u_{0}\Vert _{\dot{B}_{2, 1}^{1/2}}+\|B_{0}\|_{\dot{B}_{2, 1}^{1/2}} \bigr) \end{aligned}$$
(3.7)
with
C
being independent of
t.
Proof
First of all, thanks to (1.4), one deduces from the transport equation of (1.8) that
$$ \bigl(1+\Vert a_{0}\Vert _{\dot{B}^{3/2}_{2,1}} \bigr)^{-1} \leq\rho(t,x)\leq \underline{b}^{-1}, $$
(3.8)
from which, with \(1-\rho=\rho a\), we get by using the standard energy estimate to the w, B equation of (1.8)
$$\begin{aligned}& \frac{1}{2}\frac{d}{dt}\Vert \sqrt{\rho} w\Vert _{L^{2}}^{2} +\|\nabla w\|_{L^{2}}^{2} = \int_{\mathbb{R}^{3}} \bigl((1-\rho) (\partial_{t} v+v\cdot \nabla v)+\rho w\cdot\nabla v+B\cdot\nabla B \bigr) \cdot w \, dx, \\& \frac{1}{2}\frac{d}{dt}\|B\|_{L^{2}}^{2} +\|\nabla B\|_{L^{2}}^{2} = \int_{\mathbb{R}^{3}} ( B\cdot\nabla v\cdot B+ B\cdot\nabla w\cdot B )\, dx; \end{aligned}$$
thanks to \(\operatorname{div}B=0\), one has
$$\int_{\mathbb{R}^{3}}B\cdot\nabla w\cdot B \, dx+ \int_{\mathbb {R}^{3}}B\cdot\nabla B\cdot w \, dx=0. $$
Therefore, we get
$$\begin{aligned}& \frac{1}{2}\frac{d}{dt}\bigl(\Vert \sqrt{\rho} w \Vert _{L^{2}}^{2}+\Vert B\Vert _{L^{2}}^{2} \bigr) +\Vert \nabla w\Vert _{L^{2}}^{2}+\Vert \nabla B \Vert _{L^{2}}^{2} \\& \quad = \int_{\mathbb{R}^{3}} \bigl((1-\rho) (\partial_{t} v+v\cdot \nabla v)\cdot w+\rho w\cdot\nabla v\cdot w+B\cdot\nabla v\cdot B \bigr) \,dx \\& \quad \leq C \bigl(\Vert \sqrt{\rho} w\Vert _{L^{2}}\Vert a\Vert _{L^{2}}\Vert \partial _{t}v+v\cdot\nabla v\Vert _{L^{\infty}}+\Vert \nabla v\Vert _{L^{\infty}}\bigl(\Vert \sqrt { \rho} w\Vert _{L^{2}}^{2}+\Vert B\Vert _{L^{2}}^{2}\bigr) \bigr), \end{aligned}$$
(3.9)
from which we get
$$\begin{aligned}& \frac{1}{2}\frac{d}{dt}\bigl(\Vert \sqrt{\rho} w\Vert _{L^{2}}^{2}+\Vert B\Vert _{L^{2}}^{2} \bigr) \\& \quad \leq C \bigl(\Vert \sqrt{\rho} w\Vert _{L^{2}}\Vert a\Vert _{L^{2}}\Vert \partial _{t}v+v\cdot\nabla v\Vert _{L^{\infty}}+\Vert \nabla v\Vert _{L^{\infty}}\bigl(\Vert \sqrt { \rho} w\Vert _{L^{2}}^{2}+\Vert B\Vert _{L^{2}}^{2}\bigr) \bigr) \\& \quad \leq C \bigl(\bigl(\Vert \sqrt{\rho} w\Vert _{L^{2}}+\Vert B \Vert _{L^{2}}\bigr)\Vert a\Vert _{L^{2}}\Vert \partial_{t}v+v\cdot\nabla v\Vert _{L^{\infty}}+\Vert \nabla v \Vert _{L^{\infty}}\bigl(\Vert \sqrt{\rho} w\Vert _{L^{2}}^{2}+ \Vert B\Vert _{L^{2}}^{2}\bigr) \bigr), \end{aligned}$$
from which we infer for \(t\in(t_{1}, T^{\ast})\) that
$$\begin{aligned}& \frac{d}{dt} \bigl(e^{-2\int_{t_{1}}^{t}\Vert \nabla v(\tau)\Vert _{L^{\infty}} \,d\tau}\bigl(\bigl\Vert \sqrt{\rho} w (t)\bigr\Vert _{L^{2}}^{2}+\bigl\Vert B(t)\bigr\Vert _{L^{2}}^{2}\bigr) \bigr) \\& \quad \leq C \Vert a_{0}\Vert _{L^{2}} e^{-2\int_{t_{1}}^{t}\Vert \nabla v(\tau)\Vert _{L^{\infty}} \,d\tau} \bigl(\Vert \sqrt{\rho} w\Vert _{L^{2}}+\bigl\Vert B(t)\bigr\Vert _{L^{2}}\bigr) \Vert \Delta v-\nabla\Pi_{v}\Vert _{\dot{B}_{2, 1}^{3/2}}. \end{aligned}$$
This along with (1.6) and (3.4) implies
$$\begin{aligned} \Vert \sqrt{\rho} w\Vert _{L^{\infty}([t_{1},t];L^{2})}^{2}+\bigl\Vert B(t)\bigr\Vert _{L^{\infty}([t_{1},t];L^{2})}^{2} &\leq C e^{\int_{t_{1}}^{t}\Vert \nabla v(\tau)\Vert _{L^{\infty}} \,d\tau} \Vert \Delta v-\nabla \Pi_{v}\Vert _{L^{1}([t_{1}, t], \dot{B}_{2, 1}^{3/2})} \\ &\leq C \bigl\Vert u(t_{1})\bigr\Vert _{\dot{B}_{2, 1}^{3/2}}\exp \bigl\{ C\bigl\Vert u(t_{0})\bigr\Vert _{\dot{B}_{2, 1}^{1/2}} \bigr\} \\ &\leq C \bigl(\Vert u_{0}\Vert _{\dot{B}_{2, 1}^{1/2}}+\Vert B_{0}\Vert _{\dot{B}_{2, 1}^{1/2}}\bigr). \end{aligned}$$
Plugging the above estimate into (3.9) gives rise to
$$ \|\nabla w\|_{L^{2}([t_{1},t]; L^{2})} +\|\nabla B\|_{L^{2}([t_{1},t]; L^{2})}\leq C \bigl(\Vert u_{0}\Vert _{\dot{B}_{2, 1}^{1/2}}+\|B_{0}\|_{\dot{B}_{2, 1}^{1/2}} \bigr). $$
This completes the proof of Lemma 3.2. □
Lemma 3.3
(\(H^{1}\) estimate of \((w,B)\))
There exist two positive constants
\(c_{1}\)
and
\(c_{2}\)
such that for
\(t\in[t_{1}, T^{\ast})\),
$$\begin{aligned}& \bigl\Vert (\nabla w,\nabla B)\bigr\Vert ^{2}_{L^{\infty}([t_{1},t],L^{2})}+ \int_{t_{1}}^{t} \bigl(c_{1}\Vert \partial_{t} w\Vert _{ L^{2}}^{2} +c_{2} \bigl\Vert \bigl(\nabla ^{2}B, \nabla ^{2} w\bigr)\bigr\Vert _{ L^{2}}^{2}+\Vert \nabla \Pi_{w}\Vert _{ L^{2}}^{2} \bigr) \,dt' \\& \quad \leq C \Vert u_{0}\Vert _{\dot{B}_{2, 1}^{1/2}}^{2} \end{aligned}$$
(3.10)
with
C
being independent of
t.
Proof
Taking the \(L^{2}\) inner product of the w equation of (1.8) with \(\frac{1}{\rho}\Delta w \) and using (3.8), we obtain
$$\begin{aligned} \frac{1}{2}\frac{d}{dt}\Vert \nabla w\Vert _{L^{2}}^{2}+\biggl\Vert \frac{1}{\sqrt{\rho}} \Delta w\biggr\Vert _{L^{2}}^{2} \leq& C \biggl\Vert \frac{1}{\sqrt{\rho}} \Delta w\biggr\Vert _{L^{2}} \bigl\{ \Vert \nabla w\Vert _{L^{2}}\Vert v\Vert _{L^{\infty}}+ \Vert \nabla w\Vert _{L^{6}}\Vert w\Vert _{L^{3}} \\ &{}+ \Vert w\Vert _{L^{2}}\Vert \nabla v\Vert _{L^{\infty}}+\Vert \nabla \Pi_{w}\Vert _{L^{2}} \\ &{}+ \Vert a \Vert _{L^{2}} \Vert \partial_{t} v+ v\cdot\nabla v\Vert _{L^{\infty}}+ \Vert \nabla B\Vert _{L^{6}}\Vert B\Vert _{L^{3}} \bigr\} , \end{aligned}$$
which implies
$$\begin{aligned} \frac{d}{dt}\Vert \nabla w\Vert _{L^{2}}^{2}+\biggl\Vert \frac{1}{\sqrt{\rho}} \Delta w\biggr\Vert _{L^{2}}^{2} \leq& C\Vert \nabla w\Vert _{L^{2}}^{2} \Vert v\Vert _{L^{\infty}}^{2}+ \Vert \Delta w\Vert _{L^{2}}^{2} \Vert w\Vert _{L^{3}}^{2}+ \Vert w\Vert _{L^{2}}^{2}\Vert \nabla v\Vert _{L^{\infty}}^{2} \\ &{} +\Vert \nabla \Pi_{w}\Vert _{L^{2}}^{2}+ \Vert \Delta v-\nabla \Pi_{v}\Vert _{L^{\infty}}^{2}+ \Vert \Delta B\Vert _{L^{2}}^{2}\Vert B\Vert _{L^{3}}^{2}. \end{aligned}$$
Again thanks to the w equation of (1.8) and \(\operatorname{div}w=0\), one has
$$\begin{aligned} \Vert \Delta w\Vert _{L^{2}}^{2}+\Vert \nabla \Pi_{w}\Vert _{L^{2}}^{2} =& \Vert \Delta w- \nabla \Pi _{w}\Vert _{L^{2}}^{2} \\ \leq& C \bigl\{ \bigl\Vert \sqrt{\rho} \partial _{t} w(t)\bigr\Vert _{L^{2}}^{2}+\bigl\Vert w(t)\bigr\Vert _{L^{3}}^{2}\bigl\Vert \Delta w(t)\bigr\Vert _{L^{2}}^{2}+\Vert v \Vert _{L^{\infty}}^{2} \Vert \nabla w\Vert _{L^{2}}^{2} \\ &{} + \Vert \Delta v-\nabla\Pi_{v}\Vert _{L^{\infty}}^{2}+ \Vert w\Vert _{L^{2}}^{2}\Vert \nabla v\Vert _{L^{\infty}}^{2}+\bigl\Vert B(t)\bigr\Vert _{L^{3}}^{2} \bigl\Vert \Delta B(t)\bigr\Vert _{L^{2}}^{2} \bigr\} . \end{aligned}$$
As a consequence, we obtain for some positive constant \(c_{1}\),
$$\begin{aligned} \frac{d}{dt}\Vert \nabla w\Vert _{L^{2}}^{2}+c_{1} \bigl\Vert \nabla ^{2} w\bigr\Vert _{L^{2}}^{2} \leq& C \bigl\{ \Vert w\Vert _{L^{3}}^{2}\bigl\Vert \nabla ^{2} w\bigr\Vert _{L^{2}}^{2}+ \Vert \sqrt{\rho} \partial _{t} w\Vert _{L^{2}}^{2} \\ &{}+\Vert \nabla w \Vert _{L^{2}}^{2} \Vert v\Vert _{L^{\infty}}^{2}+ \Vert w\Vert _{L^{2}}^{2}\Vert \nabla v\Vert _{L^{\infty}}^{2} \\ &{}+ \Vert \Delta v-\nabla\Pi_{v}\Vert _{L^{\infty}}^{2}+\Vert \Delta B\Vert _{L^{2}}^{2} \Vert B\Vert _{L^{3}}^{2} \bigr\} . \end{aligned}$$
(3.11)
Taking the \(L^{2}\) inner product of the B equation of (1.8) with ΔB,
$$\begin{aligned} \frac{d}{dt}\Vert \nabla B\Vert _{L^{2}}^{2}+ \bigl\Vert \nabla ^{2} B\bigr\Vert _{L^{2}}^{2} \leq& C \bigl\{ \Vert B\Vert _{L^{3}}^{2}\bigl\Vert \nabla ^{2} w\bigr\Vert _{L^{2}}^{2}+\Vert w\Vert _{L^{3}}^{2}\bigl\Vert \nabla ^{2} B\bigr\Vert _{L^{2}}^{2} \\ &{}+\Vert \nabla B\Vert _{L^{2}}^{2} \Vert v\Vert _{L^{\infty}}^{2}+ \Vert B\Vert _{L^{2}}^{2}\Vert \nabla v\Vert _{L^{\infty}}^{2} \bigr\} . \end{aligned}$$
(3.12)
Along the same lines, we get by taking the \(L^{2}\) inner product of the w equation of (1.8) with \(\partial _{t} w\),
$$\begin{aligned} \begin{aligned}[b] \frac{d}{dt}\Vert \nabla w\Vert _{L^{2}}^{2}+ \Vert \sqrt{\rho}\partial_{t} w\Vert _{L^{2}}^{2} \leq{}&C \bigl(\Vert w\Vert _{L^{3}}^{2}\bigl\Vert \nabla^{2} w\bigr\Vert _{L^{2}}^{2}+\Vert \nabla w \Vert _{L^{2}}^{2} \Vert v\Vert _{L^{\infty}}^{2}+ \Vert w\Vert _{L^{2}}^{2}\Vert \nabla v\Vert _{L^{\infty}}^{2} \\ &{}+ \Vert \Delta v-\nabla\Pi_{v}\Vert _{L^{\infty}}^{2}+\Vert \Delta B\Vert _{L^{2}}^{2} \Vert B\Vert _{L^{3}}^{2} \bigr). \end{aligned} \end{aligned}$$
(3.13)
Thanks to (3.11), (3.12), and (3.13), there is a positive constant \(c_{2}\) so that
$$\begin{aligned}& \frac{d}{dt}\bigl(\Vert \nabla w\Vert _{L^{2}}^{2}+\Vert \nabla B\Vert _{L^{2}}^{2} \bigr)+c_{2}\Vert \partial _{t} w\Vert _{L^{2}}^{2} \\& \qquad {}+ \bigl( C_{3}-C_{2}\bigl( \Vert w\Vert _{L^{3}}^{2}+ \Vert B\Vert _{L^{3}}^{2}\bigr) \bigr) \bigl(\bigl\Vert \nabla ^{2} w\bigr\Vert _{L^{2}}^{2}+\bigl\Vert \nabla ^{2} B\bigr\Vert _{L^{2}}^{2}\bigr) \\& \quad \leq C_{4} \bigl(\Vert \nabla w\Vert _{L^{2}}^{2} \Vert v\Vert _{L^{\infty}}^{2}+ \Vert w\Vert _{L^{2}}^{2}\Vert \nabla v\Vert _{L^{\infty}}^{2}+ \Vert \Delta v-\nabla\Pi_{v}\Vert _{L^{\infty}}^{2} \\& \qquad {}+ \Vert \nabla B\Vert _{L^{2}}^{2} \Vert v\Vert _{L^{\infty}}^{2}+ \Vert B\Vert _{L^{2}}^{2} \Vert \nabla v\Vert _{L^{\infty}}^{2} \bigr) \\& \quad \leq C_{4} \bigl(\bigl(\Vert \nabla w\Vert _{L^{2}}^{2}+\Vert \nabla B\Vert _{L^{2}}^{2} \bigr) \Vert v\Vert _{L^{\infty}}^{2} \\& \qquad {}+ \bigl(\Vert w\Vert _{L^{2}}^{2}+\Vert B\Vert _{L^{2}}^{2} \bigr)\Vert \nabla v\Vert _{L^{\infty}}^{2}+ \Vert \Delta v- \nabla\Pi_{v}\Vert _{L^{\infty}}^{2} \bigr). \end{aligned}$$
(3.14)
Now let \(\tau^{\ast}\) be determined by
$$ \tau^{*}\stackrel{\mathrm{def}}{=}\sup \biggl\{ t\geq t_{1}, \bigl\Vert w(t)\bigr\Vert _{L^{3}}^{2} + \bigl\Vert B(t)\bigr\Vert _{L^{3}}^{2}\leq \frac{C_{3}}{2C_{2}} \biggr\} . $$
(3.15)
We claim that \(\tau^{\ast}=T^{\ast}\) provided that \(\|u_{0}\|_{\dot{B}^{1/2}_{2, 1}}+\|B_{0}\|_{\dot{B}^{1/2}_{2, 1}}\) is sufficiently small. Otherwise for \(t\in[t_{1},\tau^{\ast})\), it follows from (3.14) that
$$\begin{aligned}& \frac{d}{dt}\bigl(\Vert \nabla w\Vert _{L^{2}}^{2}+\Vert \nabla B\Vert _{L^{2}}^{2} \bigr)+c_{2}\Vert \partial _{t} w\Vert _{L^{2}}^{2}+\frac{C_{3}}{2}\bigl(\bigl\Vert \nabla ^{2} w\bigr\Vert _{L^{2}}^{2}+\bigl\Vert \nabla ^{2} B\bigr\Vert _{L^{2}}^{2}\bigr) \\& \quad \leq C_{4} \bigl(\bigl(\Vert \nabla w\Vert _{L^{2}}^{2}+\Vert \nabla B\Vert _{L^{2}}^{2} \bigr) \Vert v\Vert _{L^{\infty}}^{2} \\& \qquad {}+ \bigl(\Vert w\Vert _{L^{2}}^{2}+\Vert B\Vert _{L^{2}}^{2} \bigr)\Vert \nabla v\Vert _{L^{\infty}}^{2}+ \Vert \Delta v- \nabla\Pi_{v}\Vert _{L^{\infty}}^{2} \bigr). \end{aligned}$$
(3.16)
Applying Gronwall’s inequality to (3.16) and using (3.7), (3.6), (1.6), and (3.3) together with the interpolation inequality yield
$$\begin{aligned}& \Vert \nabla w\Vert _{L^{2}}^{2}+\Vert \nabla B\Vert _{L^{2}}^{2} \\& \quad \leq C_{4}\exp \biggl\{ C_{4} \int_{t_{1}}^{t}\Vert v\Vert _{L^{\infty}}^{2} \,dt' \biggr\} \biggl[\bigl\Vert \nabla B(t_{1})\bigr\Vert _{L^{2}}^{2}+ \int_{t_{1}}^{t} \bigl( \Vert \nabla v\Vert _{L^{\infty}}^{2}+ \Vert \Delta v-\nabla \Pi_{v} \Vert _{L^{\infty}}^{2} \bigr) \,dt' \biggr] \\& \quad \leq C_{5}\bigl(\Vert u_{0}\Vert _{\dot{B}^{1/2}_{2, 1}}^{2}+\Vert B_{0}\Vert _{\dot{B}^{1/2}_{2, 1}}^{2}\bigr). \end{aligned}$$
(3.17)
However, notice from (3.7) and (3.17) that
$$\begin{aligned} \bigl\Vert w(t)\bigr\Vert _{L^{3}}^{2}+\bigl\Vert B(t) \bigr\Vert _{L^{3}}^{2}&\leq C\bigl(\bigl\Vert w(t)\bigr\Vert _{L^{2}}\bigl\Vert \nabla w(t)\bigr\Vert _{L^{2}}+ \bigl\Vert B(t)\bigr\Vert _{L^{2}}\bigl\Vert \nabla B(t)\bigr\Vert _{L^{2}}\bigr) \\ &\leq C_{6}\bigl(\Vert u_{0}\Vert _{\dot{B}^{1/2}_{2, 1}}^{2}+ \Vert B_{0}\Vert _{\dot{B}^{1/2}_{2, 1}}^{2}\bigr)\leq \frac{C_{3}}{4C_{2}}\quad \mbox{for } t\in \bigl[t_{1},\tau^{\ast}\bigr) \end{aligned}$$
provided that \(\|u_{0}\|_{\dot{B}^{1/2}_{2, 1}}^{2}+\|B_{0}\|_{\dot{B}^{1/2}_{2, 1}}^{2}\leq \frac{C_{3}}{4C_{2}C_{6}}\), which contradicts with (3.15). This in turn shows that \(\tau^{\ast}=T^{\ast}\). Then integrating (3.16) and using (3.6) leads to (3.10). This completes the proof of the lemma. □
Lemma 3.4
(\(H^{2}\) estimate of \((w,B)\))
There exists a time independent constant
C
so that for
\(t\in[t_{1}, T^{\ast})\),
$$ \bigl\Vert \bigl(\nabla ^{2} w,\nabla ^{2} B \bigr)\bigr\Vert _{L^{\infty}([t_{1},t],L^{2})}+\bigl\Vert (\nabla w_{t},\nabla B_{t})\bigr\Vert _{L^{2}([t_{1},t],L^{2})}+\bigl\Vert \bigl(\nabla ^{2} w, \nabla ^{2} B\bigr)\bigr\Vert _{L^{2}([t_{1},t],L^{6})} \leq C. $$
(3.18)
Proof
Step 1. \(L^{2}\) estimate of \((\sqrt{\rho}w_{t},B_{t})\).
We get by first acting \(\partial_{t}\) to the w equation of (1.8) and then taking the \(L^{2}\) inner product of the resulting equation with \(\partial _{t} w\),
$$\begin{aligned}& \frac{1}{2}\frac{d}{dt}\|\sqrt{\rho} w_{t} \|_{L^{2}}^{2}+\|\nabla w_{t}\|_{L^{2}}^{2} \\& \quad = \int_{{\mathbb{R}}^{3}}(1-\rho)w_{t}\cdot\partial_{t} ( \Delta v-\nabla\Pi_{v}) \,dx \\& \qquad {}- \int_{{\mathbb{R}}^{3}}\rho_{t}w_{t} \cdot \bigl(w_{t}+ (w+v) \cdot \nabla w +w \cdot\nabla v+ (\Delta v-\nabla \Pi_{v}) \bigr) \,dx \\& \qquad {}- \int_{{\mathbb{R}}^{3}}\rho w_{t}\cdot \bigl((v+w)_{t} \cdot\nabla w + w_{t} \cdot\nabla v + w\cdot\nabla v_{t} \bigr) \,dx \\& \qquad {}+ \int_{{\mathbb{R}}^{3}}\partial _{t}(B\cdot \nabla B) w_{t} \,dx \\& \quad \stackrel{\mathrm{def}}{=} I+\mathit{II}+\mathit{III}+\mathit{IV}. \end{aligned}$$
(3.19)
The estimates of I, II, and III are similar to [8],
$$\begin{aligned}& |I | \leq C \Vert a_{0}\Vert _{\dot{B}^{3/2}_{2,1}} \Vert \sqrt{\rho}w_{t}\Vert _{L^{2}}\bigl\Vert \partial_{t} (\Delta v-\nabla \Pi_{v})\bigr\Vert _{L^{2}}, \end{aligned}$$
(3.20)
$$\begin{aligned}& |\mathit{II} | \leq \frac{1}{4} \Vert \nabla w_{t}\Vert _{L^{2}}^{2}+C \bigl\{ \Vert v\Vert _{L^{\infty}}^{4}+ \Vert \nabla v\Vert _{L^{6}}^{2}+ \Vert \Delta w\Vert _{L^{2}}^{2}+\Vert \Delta v-\nabla \Pi_{v}\Vert _{L^{4}}^{2} \\& \hphantom{ |\mathit{II} | \leq{}}{}+\bigl\Vert \nabla^{2} v\bigr\Vert _{L^{6}}^{2}+\Vert \sqrt{\rho}w_{t}\Vert _{L^{2}}^{2} \bigl(\Vert v\Vert _{L^{\infty}}^{2}+\Vert \nabla w\Vert _{L^{2}}\bigl\Vert \nabla^{2} w\bigr\Vert _{L^{2}} \bigr) \\& \hphantom{ |\mathit{II} | \leq{}}{}+ \Vert \sqrt{\rho}w_{t}\Vert _{L^{2}}\bigl\Vert \nabla(\Delta v-\nabla \Pi_{v})\bigr\Vert _{L^{4}} \bigr\} , \end{aligned}$$
(3.21)
$$\begin{aligned}& |\mathit{III} | \leq\frac{1}{16}\Vert \nabla w_{t}\Vert _{L^{2}}^{2} +C \bigl\{ \Vert \sqrt{ \rho} w_{t}\Vert _{L^{2}}^{2}\bigl(\Vert \nabla w \Vert _{L^{2}}\bigl\Vert \nabla^{2} w\bigr\Vert _{L^{2}}+\Vert \nabla v\Vert _{L^{\infty}}\bigr) \\& \hphantom{ |\mathit{III} | \leq{}}{}+\Vert \sqrt{\rho} w_{t}\Vert _{L^{2}} \bigl(\Vert v_{t}\Vert _{L^{\infty}}+\Vert \nabla v_{t}\Vert _{L^{4}}\bigr) \bigr\} , \end{aligned}$$
(3.22)
which, thanks to (3.10), yields
$$\begin{aligned} |\mathit{IV} | =&\biggl\vert \int_{{\mathbb{R}}^{3}}B_{t}\cdot\nabla B \cdot w_{t} \,dx+ \int_{{\mathbb{R}}^{3}}B\cdot\nabla B_{t}\cdot w_{t} \,dx\biggr\vert \\ \leq& \varepsilon \|\nabla w_{t} \|_{L^{2}}^{2}+C_{\varepsilon} \|B_{t} \|_{L^{2}}^{2}\| \nabla B\|_{L^{2}}\| \nabla^{2} B\|_{L^{2}}. \end{aligned}$$
This together with (3.19), (3.20), (3.21), and (3.22) yields
$$\begin{aligned}& \frac{d}{dt}\Vert \sqrt{\rho} w_{t}\Vert _{L^{2}}^{2}+\Vert \nabla w_{t}\Vert _{L^{2}}^{2} \\& \quad \leq C\Vert \sqrt{\rho} w_{t}\Vert _{L^{2}} \bigl[ \bigl\Vert \nabla(\Delta v-\nabla \Pi_{v})\bigr\Vert _{L^{4}} \\& \qquad {}+\bigl\Vert \partial_{t}(\Delta v-\nabla \Pi_{v})\bigr\Vert _{L^{2}}+\Vert v_{t}\Vert _{L^{\infty}}+\Vert \nabla v_{t}\Vert _{L^{4}} \bigr] \\& \qquad {} +C\Vert \sqrt{\rho} w_{t}\Vert _{L^{2}}^{2} \bigl[\Vert v\Vert _{L^{\infty}}^{2}+\Vert \nabla v\Vert _{L^{\infty}}+\Vert \nabla w\Vert _{L^{2}}\bigl\Vert \nabla^{2} w\bigr\Vert _{L^{2}} \bigr] \\& \qquad {} +C \bigl[\Vert v\Vert _{L^{\infty}}^{4}+ \Vert \nabla v\Vert _{L^{6}}^{2}+\Vert \Delta w\Vert _{L^{2}}^{2} +\Vert \Delta v-\nabla\Pi_{v}\Vert _{L^{4}}^{2}+\bigl\Vert \nabla^{2} v\bigr\Vert _{L^{6}}^{2} \bigr] \\& \qquad {} +C_{\varepsilon} \Vert B_{t} \Vert _{L^{2}}^{2} \bigl\Vert \nabla^{2} B\bigr\Vert _{L^{2}}\Vert \nabla B \Vert _{L^{2}}. \end{aligned}$$
(3.23)
On the other hand, acting by \(\partial _{t}\) to the B equation of (1.8) and then taking the \(L^{2}\) inner product of the resulting equation with \(\partial _{t} B\) we see that
$$\begin{aligned}& \frac{1}{2}\frac{d}{dt}\Vert B_{t}\Vert _{L^{2}}^{2}+\Vert \nabla B_{t}\Vert _{L^{2}}^{2} \\& \quad = \int_{{\mathbb{R}}^{3}}B_{t}\cdot\nabla(v+w)\cdot B_{t}+B\cdot\nabla (v+w)_{t}\cdot B_{t}-(v+w)_{t} \cdot\nabla B\cdot B_{t} \,dx \\& \quad \leq \varepsilon\bigl(\Vert \nabla w_{t}\Vert _{L^{2}}^{2}+\Vert \nabla B_{t}\Vert _{L^{2}}^{2}\bigr)+C\Vert B_{t}\Vert _{L^{2}}^{2}\bigl(\Vert \nabla w\Vert _{L^{2}}\bigl\Vert \nabla^{2} w\bigr\Vert _{L^{2}} \\& \qquad {} +\Vert \nabla B\Vert _{L^{2}}\bigl\Vert \nabla^{2} B\bigr\Vert _{L^{2}}+\Vert \nabla v\Vert _{L^{\infty}}\bigr)+C \Vert B_{t}\Vert _{L^{2}}\bigl(\Vert \nabla v_{t}\Vert _{L^{4}}+\Vert v_{t}\Vert _{L^{\infty}}\bigr). \end{aligned}$$
(3.24)
Thanks to (3.23) and (3.24), we have
$$\begin{aligned}& \frac{1}{2}\frac{d}{dt}\bigl(\Vert \sqrt{\rho} w_{t} \Vert _{L^{2}}^{2}+\Vert B_{t}\Vert _{L^{2}}^{2}\bigr)+\Vert \nabla B_{t}\Vert _{L^{2}}^{2}+\Vert \nabla w_{t}\Vert _{L^{2}}^{2} \\& \quad \leq C\bigl(\Vert \sqrt{\rho} w_{t}\Vert _{L^{2}}+ \Vert B_{t}\Vert _{L^{2}}\bigr) \\& \qquad {}\times\bigl[\bigl\Vert \nabla ( \Delta v-\nabla \Pi_{v})\bigr\Vert _{L^{4}}+\bigl\Vert \partial_{t}(\Delta v-\nabla \Pi_{v})\bigr\Vert _{L^{4}}+\Vert v_{t}\Vert _{L^{\infty}}+\Vert \nabla v_{t}\Vert _{L^{4}} \bigr] \\& \qquad {} +C\bigl(\Vert \sqrt{\rho} w_{t}\Vert _{L^{2}}^{2}+\Vert B_{t}\Vert _{L^{2}}^{2}\bigr) \\& \qquad {}\times \bigl[\Vert v\Vert _{L^{\infty}}^{2}+ \Vert \nabla v\Vert _{L^{\infty}}+\Vert \nabla w\Vert _{L^{2}} \bigl\Vert \nabla^{2} w\bigr\Vert _{L^{2}}+\Vert \nabla B \Vert _{L^{2}}\bigl\Vert \nabla^{2} B\bigr\Vert _{L^{2}} \bigr] \\& \qquad {} +C \bigl[\Vert v\Vert _{L^{\infty}}^{4}+ \Vert \nabla v\Vert _{L^{6}}^{2}+\Vert \Delta w\Vert _{L^{2}}^{2} +\Vert \Delta v-\nabla\Pi_{v}\Vert _{L^{4}}^{2}+\bigl\Vert \nabla^{2} v\bigr\Vert _{L^{6}}^{2} \bigr] \\& \quad \stackrel{\mathrm{def}}{=}C\bigl(\Vert \sqrt{\rho} w_{t} \Vert _{L^{2}}+\Vert B_{t}\Vert _{L^{2}} \bigr)f_{3}(t)+C\bigl(\Vert \sqrt {\rho} w_{t}\Vert _{L^{2}}^{2}+\Vert B_{t}\Vert _{L^{2}}^{2}\bigr)f_{1}(t) +f_{2}(t) \\& \quad \leq C\bigl(f_{3}(t)+f_{1}(t)\bigr) \bigl(\Vert \sqrt{\rho} w_{t}\Vert _{L^{2}}^{2}+\Vert B_{t}\Vert _{L^{2}}^{2}\bigr) +f_{2}(t)+f_{3}(t). \end{aligned}$$
We use
$$\begin{aligned}& f_{1}(t)\stackrel{\mathrm{def}}{=}\Vert v\Vert _{L^{\infty}}^{2}+\Vert \nabla v\Vert _{L^{\infty}}+\Vert \nabla w\Vert _{L^{2}}\bigl\Vert \nabla^{2} w\bigr\Vert _{L^{2}}+\Vert \nabla B\Vert _{L^{2}}\bigl\Vert \nabla^{2} B\bigr\Vert _{L^{2}}, \\& f_{2}(t) \stackrel{\mathrm{def}}{=}\Vert v\Vert _{L^{\infty}}^{4}+ \Vert \nabla v\Vert _{L^{6}}^{2}+ \Vert \Delta w\Vert _{L^{2}}^{2} +\Vert \Delta v-\nabla \Pi_{v}\Vert _{L^{4}}^{2}+\bigl\Vert \nabla^{2} v\bigr\Vert _{L^{6}}^{2}, \\& f_{3}(t) \stackrel{\mathrm{def}}{=}\bigl\Vert \nabla(\Delta v- \nabla \Pi_{v})\bigr\Vert _{L^{4}}+\bigl\Vert \partial_{t}(\Delta v-\nabla \Pi_{v})\bigr\Vert _{L^{2}}+\Vert v_{t}\Vert _{L^{\infty}}+\Vert \nabla v_{t}\Vert _{L^{4}}. \end{aligned}$$
Applying Gronwall’s inequality to (3.23) yields for \(t\in(t_{1},T^{\ast})\),
$$\begin{aligned}& \bigl\Vert \sqrt{\rho}w_{t}(t)\bigr\Vert _{L^{2}}^{2}+\bigl\Vert B_{t}(t)\bigr\Vert _{L^{2}}^{2}+ \int_{t_{1}}^{t}\bigl\Vert \nabla w_{t} \bigl(t'\bigr)\bigr\Vert _{L^{2}}^{2}+\bigl\Vert \nabla B_{t}\bigl(t'\bigr)\bigr\Vert _{L^{2}}^{2} \,dt' \\& \quad \leq C\exp \biggl\{ C \int_{t_{1}}^{t}\bigl(f_{1} \bigl(t'\bigr)+f_{3}\bigl(t'\bigr)\bigr) \,dt' \biggr\} \\& \qquad {}\times\biggl(\bigl\Vert (\sqrt{\rho}w_{t}) (t_{1})\bigr\Vert _{L^{2}}^{2}+\bigl\Vert B_{t}(t_{1})\bigr\Vert _{L^{2}}^{2} + \int_{t_{1}}^{t}\bigl(f_{2} \bigl(t'\bigr)+f_{3}\bigl(t'\bigr)\bigr) \,dt' \biggr). \end{aligned}$$
(3.25)
However, notice that \(\dot{B}^{3/2}_{2, 1}\hookrightarrow L^{\infty}\), \(\dot{B}^{1}_{2, 1}\hookrightarrow L^{6}\), \(\dot{B}^{3/4}_{2, 1}\hookrightarrow L^{4}\), and we deduce from (3.3), (3.4), (3.5), (3.7), and (3.10) that
$$ \int_{t_{1}}^{t}\bigl(f_{1} \bigl(t'\bigr)+f_{2}\bigl(t' \bigr)+f_{3}\bigl(t'\bigr)\bigr) \,dt'\leq C $$
with C being independent of t. Taking the \(L^{2}\) inner product of the w, B equation of (1.8) with \(w_{t}\), \(B_{t}\) at \(t=t_{1}\) and using (3.2) give rise to
$$\begin{aligned}& \bigl\Vert (\sqrt{\rho}w_{t}) (t_{1})\bigr\Vert _{L^{2}} \leq C\bigl(\bigl\Vert a(t_{1})\bigr\Vert _{L^{2}} \bigl\Vert (\partial_{t} v+v\cdot\nabla v) (t_{1})\bigr\Vert _{\dot{B}^{3/2}_{2, 1}} + \bigl\Vert B\cdot\nabla B(t_{1})\bigr\Vert _{L^{2}}\bigr) \\& \hphantom{\bigl\Vert (\sqrt{\rho}w_{t}) (t_{1})\bigr\Vert _{L^{2}}} \leq C\bigl(\bigl\Vert v(t_{1})\bigr\Vert _{\dot{B}^{7/2}_{2, 1}}+\bigl\Vert v(t_{1})\bigr\Vert _{\dot{B}^{3/2}_{2, 1}} \bigl\Vert v(t_{1})\bigr\Vert _{\dot{B}^{5/2}_{2, 1}}+\bigl\Vert B(t_{1})\bigr\Vert _{\dot{B}^{1/2}_{2, 1}}^{\frac{5}{4}}\bigl\Vert B(t_{1})\bigr\Vert _{\dot{B}^{5/2}_{2, 1}}^{\frac{3}{4}}\bigr) \\& \hphantom{\bigl\Vert (\sqrt{\rho}w_{t}) (t_{1})\bigr\Vert _{L^{2}}} \leq C, \\& \bigl\Vert B_{t}(t_{1})\bigr\Vert _{L^{2}} \leq \bigl( C \bigl\Vert v\cdot\nabla B(t_{1})\bigr\Vert _{L^{2}} + \bigl\Vert \Delta B(t_{1})\bigr\Vert _{L^{2}}+\bigl\Vert B\cdot\nabla v(t_{1})\bigr\Vert _{L^{2}}\bigr) \\& \hphantom{\bigl\Vert B_{t}(t_{1})\bigr\Vert _{L^{2}}} \leq C\bigl(\bigl\Vert v(t_{1})\bigr\Vert _{\dot{B}^{1/2}_{2, 1}}\bigl\Vert B(t_{1})\bigr\Vert _{\dot{B}^{1/2}_{2, 1}}^{\frac{1}{4}} \bigl\Vert B(t_{1})\bigr\Vert _{\dot{B}^{5/2}_{2, 1}}^{\frac{3}{4}} \\& \hphantom{\bigl\Vert B_{t}(t_{1})\bigr\Vert _{L^{2}}\leq{}}{}+\bigl\Vert B(t_{1})\bigr\Vert _{\dot{B}^{1/2}_{2, 1}}\bigl\Vert v(t_{1})\bigr\Vert _{\dot{B}^{1/2}_{2, 1}}^{\frac{1}{4}}\bigl\Vert v(t_{1})\bigr\Vert _{\dot{B}^{5/2}_{2, 1}}^{\frac{3}{4}} +\bigl\Vert B(t_{1})\bigr\Vert _{\dot{B}^{2}_{2, 1}} \bigr) \\& \hphantom{\bigl\Vert B_{t}(t_{1})\bigr\Vert _{L^{2}}} \leq C. \end{aligned}$$
As a consequence, we deduce from (3.25) that
$$ \sup_{ t\in[t_{1}, T^{\ast})}\bigl(\bigl\Vert \sqrt{ \rho}w_{t}(t)\bigr\Vert _{L^{2}}^{2}+\bigl\Vert B_{t}(t)\bigr\Vert _{L^{2}}^{2}\bigr)+ \int _{t_{1}}^{t}\bigl\Vert \nabla w_{t} \bigl(t'\bigr)\bigr\Vert _{L^{2}}^{2}+\bigl\Vert \nabla B_{t}\bigl(t'\bigr)\bigr\Vert _{L^{2}}^{2} \,dt'\leq C. $$
(3.26)
Step 2. The estimate of \((\nabla ^{2}w,\nabla ^{2}B)\).
We first observe from the w and B equation of (1.8) that
$$\begin{aligned}& \bigl\Vert \nabla ^{2} w\bigr\Vert _{L^{2}}+ \Vert \nabla \Pi_{w}\Vert _{L^{2}} \leq \varepsilon\bigl( \bigl\Vert \nabla ^{2} w\bigr\Vert _{L^{2}}+\bigl\Vert \nabla ^{2} B\bigr\Vert _{L^{2}}\bigr)+ C \bigl\{ \Vert \sqrt{\rho}w_{t} \Vert _{L^{2}}+\Vert \nabla w\Vert _{L^{2}}^{3}+\Vert \nabla B\Vert _{L^{2}}^{3} \\& \hphantom{\bigl\Vert \nabla ^{2} w\bigr\Vert _{L^{2}}+ \Vert \nabla \Pi_{w}\Vert _{L^{2}} \leq{}}{} +\Vert v\Vert _{L^{\infty}} \Vert \nabla w\Vert _{L^{2}} +\Vert \nabla v\Vert _{L^{\infty}} \Vert w\Vert _{L^{2}} +\Vert v\Vert _{\dot{B}^{7/2}_{2, 1}}+\Vert v\Vert _{\dot{B}^{3/2}_{2, 1}}\Vert v\Vert _{\dot{B}^{5/2}_{2, 1}} \bigr\} , \\& \bigl\Vert \nabla ^{2} B\bigr\Vert _{L^{2}} \leq \Vert B_{t} \Vert _{L^{2}}+\bigl\Vert (v+w)\cdot\nabla B\bigr\Vert _{L^{2}}+\bigl\Vert \nabla (v+w)\cdot B\bigr\Vert _{L^{2}} \\& \hphantom{\bigl\Vert \nabla ^{2} B\bigr\Vert _{L^{2}}}\leq\varepsilon\bigl(\bigl\Vert \nabla ^{2} w\bigr\Vert _{L^{2}}+\bigl\Vert \nabla ^{2} w\bigr\Vert _{L^{2}}\bigr)+C \bigl\{ \Vert \nabla w\Vert _{L^{2}}^{2} \Vert \nabla B\Vert _{L^{2}}+\Vert \nabla B\Vert _{L^{2}}^{2}\Vert \nabla w\Vert _{L^{2}} \\& \hphantom{\bigl\Vert \nabla ^{2} B\bigr\Vert _{L^{2}} \leq{}}{}+\Vert B_{t} \Vert _{L^{2}} +\Vert v\Vert _{L^{\infty}} \Vert \nabla B\Vert _{L^{2}}+\Vert \nabla v\Vert _{L^{\infty }}\Vert B\Vert _{L^{2}} \bigr\} , \end{aligned}$$
which along with (3.4), (3.5), (3.7), (3.10), and (3.26) ensures that
$$ \sup_{ t\in[t_{1}, T^{\ast})} \bigl(\bigl\Vert \nabla ^{2} w(t)\bigr\Vert _{L^{2}}+\bigl\Vert \nabla ^{2} B(t)\bigr\Vert _{L^{2}}+\bigl\Vert \nabla \Pi_{w}(t)\bigr\Vert _{L^{2}} \bigr)\leq C . $$
(3.27)
On the other hand, let \((v,q)\) solve
$$-\Delta v+\nabla q=f,\qquad \operatorname{div}v=0. $$
Then one has \(\nabla q=-\nabla (-\Delta )^{-1}\operatorname{div}f\) and, for any \(r\in (1,\infty)\),
$$\|\nabla q\|_{L^{r}}\leq C\|f\|_{L^{r}} \quad \mbox{and}\quad \| \Delta v \|_{L^{r}}\leq C\|f\|_{L^{r}}; $$
from this and the w equation of (1.8), we infer
$$\begin{aligned} \begin{aligned} \bigl\Vert \nabla ^{2} w\bigr\Vert _{L^{6}}+\Vert \nabla \Pi_{w}\Vert _{L^{6}} \leq{}& C \bigl( \Vert w_{t}\Vert _{L^{6}}+\Vert w \cdot\nabla w\Vert _{L^{6}}+\Vert v \cdot\nabla w\Vert _{L^{6}}+\Vert w \cdot \nabla v\Vert _{L^{6}} \\ &{} +\bigl\Vert (1-\rho)\bigr\Vert _{L^{6}}\Vert \partial_{t} v+v\cdot\nabla v\Vert _{L^{\infty}}+\Vert B \cdot \nabla B\Vert _{L^{6}} \bigr), \end{aligned} \end{aligned}$$
which along with (3.10) implies
$$\begin{aligned}& \bigl\Vert \nabla ^{2} w\bigr\Vert _{L^{6}}+\Vert \nabla \Pi_{w}\Vert _{L^{6}} \\& \quad \leq C \bigl( \Vert \nabla w_{t}\Vert _{L^{2}}+ \Vert \nabla w\Vert _{L^{2}}\bigl\Vert \nabla^{2} w\bigr\Vert _{L^{2}}^{\frac{1}{2}}\bigl\Vert \nabla^{2} w\bigr\Vert _{L^{6}}^{\frac{1}{2}}+\bigl\Vert \nabla ^{2} w\bigr\Vert _{L^{2}}\Vert v\Vert _{L^{\infty}}+\Vert \nabla w\Vert _{L^{2}}\Vert \nabla v\Vert _{L^{\infty}} \\& \qquad {} +\Vert v\Vert _{\dot{B}^{7/2}_{2, 1}}+\Vert v\Vert _{\dot{B}^{3/2}_{2, 1}} \Vert v\Vert _{\dot{B}^{5/2}_{2, 1}}+\Vert \nabla B\Vert _{L^{2}}\bigl\Vert \nabla^{2} B\bigr\Vert _{L^{2}}^{\frac{1}{2}}\bigl\Vert \nabla^{2} B\bigr\Vert _{L^{6}}^{\frac{1}{2}} \bigr) \\& \quad \leq \varepsilon\bigl(\bigl\Vert \nabla ^{2} w\bigr\Vert _{L^{6}}+\bigl\Vert \nabla ^{2} B\bigr\Vert _{L^{6}}\bigr)+C \bigl( \Vert \nabla w_{t}\Vert _{L^{2}}+\bigl\Vert \nabla^{2} w\bigr\Vert _{L^{2}}+\bigl\Vert \nabla^{2} B\bigr\Vert _{L^{2}}+\Vert v\Vert _{\dot{B}^{7/2}_{2, 1}}+\Vert v\Vert _{\dot{B}^{5/2}_{2,1}} \bigr). \end{aligned}$$
On the other hand, following the B equation of (1.8), we infer
$$ \bigl\Vert \nabla ^{2} B\bigr\Vert _{L^{6}} \leq \varepsilon\bigl(\bigl\Vert \nabla ^{2} w\bigr\Vert _{L^{6}}+\bigl\Vert \nabla ^{2} B\bigr\Vert _{L^{6}}\bigr)+C \bigl( \Vert \nabla B_{t}\Vert _{L^{2}}+\bigl\Vert \nabla^{2} w\bigr\Vert _{L^{2}}+\bigl\Vert \nabla^{2} B\bigr\Vert _{L^{2}}+\Vert v\Vert _{\dot{B}^{5/2}_{2,1}} \bigr). $$
Therefore, we get
$$\begin{aligned}& \bigl\Vert \nabla ^{2} w\bigr\Vert _{L^{6}}+\bigl\Vert \nabla ^{2} B\bigr\Vert _{L^{6}}+\Vert \nabla \Pi_{w}\Vert _{L^{6}} \\& \quad \leq C \bigl( \Vert \nabla w_{t}\Vert _{L^{2}}+ \Vert \nabla B_{t}\Vert _{L^{2}}+\bigl\Vert \nabla^{2} w\bigr\Vert _{L^{2}}+\bigl\Vert \nabla^{2} B\bigr\Vert _{L^{2}}+\Vert v\Vert _{\dot{B}^{7/2}_{2, 1}}+\Vert v\Vert _{\dot{B}^{5/2}_{2,1}} \bigr). \end{aligned}$$
Therefore thanks to (3.4), (3.26), and (3.10), we obtain
$$\begin{aligned}& \bigl\Vert \nabla ^{2} w\bigr\Vert _{L^{2}([t_{1},t]; L^{6})}^{2}+ \bigl\Vert \nabla ^{2} B\bigr\Vert _{L^{2}([t_{1},t]; L^{6})}^{2}+ \Vert \nabla \Pi_{w}\Vert _{L^{2}([t_{1},t]; L^{6})}^{2} \\& \quad \leq C \bigl\{ \Vert \nabla w_{t}\Vert _{L^{2}([t_{1},t]; L^{2})}^{2}+ \Vert \nabla B_{t}\Vert _{L^{2}([t_{1},t]; L^{2})}^{2} +\bigl\Vert \nabla ^{2} w\bigr\Vert _{L^{2}([t_{1},t]; L^{2})}^{2}+\bigl\Vert \nabla ^{2} B\bigr\Vert _{L^{2}([t_{1},t]; L^{2})}^{2} \\& \qquad {} +\Vert v\Vert _{L^{2}([t_{1},t]; \dot{B}^{7/2}_{2, 1})}^{2}+\Vert v\Vert _{L^{2}([t_{1},t]; \dot{B}^{5/2}_{2, 1})}^{2} \bigr\} \leq C. \end{aligned}$$
This completes the proof of the lemma. □
3.3 Proof of Theorem 1.1
We first rewrite the momentum equation in (1.3) as
$$ \partial_{t} u + u \cdot\nabla u -\Delta u+\nabla\Pi =(1-\rho) ( \partial_{t} u +u \cdot\nabla u)+B \cdot\nabla B. $$
Then it follows from the classical theory of the homogeneous Navier-Stokes equations (see [15] for instance) that with \(t\in [t_{1}, T^{\ast})\),
$$\begin{aligned}& \Vert u\Vert _{\widetilde{L}^{\infty}([t_{1}, t]; \dot{B}^{1/2}_{2, 1})} +\Vert u\Vert _{{L}^{1}([t_{1}, t]; \dot{B}^{5/2}_{2, 1})} +\Vert \nabla \Pi \Vert _{ L^{1}([t_{1}, t]; \dot{B}^{1/2}_{2, 1})} \\& \quad \leq C\bigl\Vert u(t_{1})\bigr\Vert _{\dot{B}^{1/2}_{2, 1}}+ \int_{t_{1}}^{t}\Vert \nabla u\Vert _{L^{\infty}} \Vert u\Vert _{\dot{B}^{1/2}_{2, 1}}\,dt'+\bigl\Vert (1-\rho) (\partial_{t} u+u \cdot\nabla u)\bigr\Vert _{L^{1}([t_{1}, t]; \dot{B}^{1/2}_{2, 1})} \\& \qquad {} +\Vert B \cdot\nabla B\Vert _{L^{1}([t_{1}, t]; \dot{B}^{1/2}_{2, 1})}. \end{aligned}$$
(3.28)
Applying the product law in Besov spaces gives
$$ \bigl\Vert (1-\rho) (\partial_{t} u+u \cdot\nabla u)\bigr\Vert _{L^{1}([t_{1}, t]; \dot{B}^{1/2}_{2, 1})} \leq C \|1-\rho\|_{L^{\infty}([t_{1}, t]; \dot{B}^{3/2}_{2, 1})} \|\partial_{t} u+u \cdot\nabla u\|_{L^{1}([t_{1}, t]; \dot{B}^{1/2}_{2, 1})}. $$
Yet thanks to Lemma 2.1 and (3.18), one has
$$\begin{aligned}& \Vert \partial_{t} u\Vert _{L^{1}([t_{1}, t]; \dot{B}^{1/2}_{2, 1})} \leq C t^{1/2} \Vert \partial_{t} w\Vert _{L^{2}([t_{1}, t]; H^{1})}+\bigl\Vert u(t_{1})\bigr\Vert _{\dot{B}^{1/2}_{2, 1}} \leq C\bigl(t^{1/2}+1 \bigr), \\& \Vert u \cdot\nabla u\Vert _{L^{1}([t_{1}, t]; \dot{B}^{1/2}_{2, 1})} \leq C \int_{t_{1}}^{t} \bigl(\Vert \nabla w \Vert _{L^{2}}\Vert \Delta w \Vert _{L^{2}}+\Vert v\Vert _{\dot{B}^{3/2}_{2, 1}}^{2} \bigr) \,dt'\leq C, \\& \Vert B \cdot\nabla B\Vert _{L^{1}([t_{1}, t]; \dot{B}^{1/2}_{2, 1})}\leq C \int_{t_{1}}^{t}\Vert B \Vert _{\dot{H}^{1}} \Vert \nabla B \Vert _{\dot{H}^{1}} \,dt'\leq C \int_{t_{1}}^{t}\Vert \nabla B \Vert _{L^{2}}\Vert \Delta B \Vert _{L^{2}} \,dt'\leq C. \end{aligned}$$
Thanks to Theorem 2.87 in [13] and Proposition 1 in [8], we have
$$\begin{aligned}& \Vert 1-\rho \Vert _{\widetilde{L}^{\infty}([t_{1}, t]; \dot{B}^{3/2}_{2, 1})} \\& \quad \leq C \Vert a\Vert _{\widetilde{L}^{\infty}([t_{1}, t]; \dot{B}^{3/2}_{2, 1})} \leq C\bigl\Vert a(t_{1})\bigr\Vert _{\dot{B}^{3/2}_{2, 1}}\exp \biggl\{ C \int_{t_{1}}^{t}\bigl\Vert u\bigl(t' \bigr)\bigr\Vert _{\dot{B}^{3/2}_{6, 1}} \,dt' \biggr\} . \end{aligned}$$
(3.29)
However, applying Lemma 2.1 leads to
$$ \Vert u\Vert _{\dot{B}^{3/2}_{6, 1}} \leq C\Vert \nabla w\Vert _{L^{6}}^{1/2} \bigl\Vert \nabla ^{2} w\bigr\Vert _{L^{6}}^{1/2}+\Vert v\Vert _{\dot{B}^{5/2}_{2, 1}}, $$
which along with (3.10) and (3.18) implies
$$\begin{aligned}& \Vert \nabla u\Vert _{L^{1}([t_{1}, t]; L^{\infty})}+\Vert u\Vert _{L^{1}([t_{1}, t]; \dot{B}^{3/2}_{6, 1})} \\& \quad \leq C\Vert u\Vert _{L^{1}([t_{1}, t]; \dot{B}^{3/2}_{6, 1})}\leq C \bigl\{ \Vert v\Vert _{L^{1}([t_{1}, t]; \dot{B}^{5/2}_{2, 1})} +t^{1/2}\Vert \Delta w\Vert _{L^{2}([t_{1}, t]; L^{2})}^{1/2} \Vert \Delta w\Vert _{L^{2}([t_{1}, t]; L^{6})}^{1/2} \bigr\} \\& \quad \leq C \bigl(1+t^{1/2}\bigr). \end{aligned}$$
Therefore, we obtain
$$ \bigl\Vert (1-\rho) (\partial_{t} u+u \cdot\nabla u)\bigr\Vert _{L^{1}([t_{1}, t]; \dot{B}^{1/2}_{2, 1})}\le C\bigl\Vert a(t_{1})\bigr\Vert _{\dot {B}^{3/2}_{2,1}}\exp \bigl\{ Ct^{1/2} \bigr\} . $$
Then applying Gronwall’s inequality to (3.28) gives rise to
$$\begin{aligned}& \Vert u\Vert _{\widetilde{L}^{\infty}([t_{1}, t]; \dot{B}^{1/2}_{2, 1})} +\Vert u\Vert _{{L}^{1}([t_{1}, t]; \dot{B}^{5/2}_{2, 1})}+\Vert \nabla \Pi \Vert _{ L^{1}([t_{1}, t]; \dot{B}^{1/2}_{2, 1})} \\& \quad \leq C \exp \{C \sqrt{t} \} \bigl(\Vert u_{0}\Vert _{\dot{B}^{1/2}_{2, 1}}+\Vert a_{0}\Vert _{B^{3/2}_{2, 1}}+C \bigr). \end{aligned}$$
(3.30)
We first observe from the B equation of (1.8) that
$$ \partial_{t} B -\Delta B =u \cdot\nabla B+B \cdot\nabla u. $$
Similarly, we get
$$\begin{aligned}& \Vert B\Vert _{\widetilde{L}^{\infty}([t_{1}, t]; \dot{B}^{1/2}_{2, 1})} +\Vert B\Vert _{{L}^{1}([t_{1}, t]; \dot{B}^{5/2}_{2, 1})} \\& \quad \leq C\bigl\Vert B(t_{1})\bigr\Vert _{\dot{B}^{1/2}_{2, 1}}+\Vert u \cdot\nabla B\Vert _{L^{1}([t_{1}, t]; \dot{B}^{1/2}_{2, 1})}+\Vert B\cdot \nabla u\Vert _{L^{1}([t_{1}, t]; \dot{B}^{1/2}_{2, 1})} \\& \quad \leq C\Vert B_{0}\Vert _{\dot{B}^{1/2}_{2, 1}}+C. \end{aligned}$$
(3.31)
From (3.29), (3.30), and (3.31), we infer by a standard argument that \({T}^{\ast}=\infty\). Moreover, the global solution thus obtained, \((a,u,B,\Pi)\), satisfies (1.5). This completes the proof of Theorem 1.1. □