Regularity Criteria for a Ginzburg–Landau–Navier–Stokes in a Bounded Domain

Abstract

In this work, we prove some regularity criteria for a Ginzburg–Landau–Navier–Stokes system with the Coulomb gauge in a bounded domain $\Omega \subset {\mathbb {R}}^3\,$.

1 Introduction

In this work, we consider the following Ginzburg–Landau–Navier–Stokes system with the Coulomb gauge:

$$\begin{aligned}&{\mathrm {div}}\,u=0, \end{aligned}$$

(1.1)

$$\begin{aligned}&\partial _tu+u\cdot \nabla u+\nabla \pi -\Delta u=|\psi |^2\nabla h,\end{aligned}$$

(1.2)

$$\begin{aligned}&\eta \partial _t\psi +i\eta k\phi \psi +u\cdot \nabla \psi + \left( \frac{i}{k}\nabla +A\right) ^2\psi +(|\psi |^2-1)\psi =0,\end{aligned}$$

(1.3)

$$\begin{aligned}&\partial _tA+\nabla \phi -\Delta A+\text{ Re }\left\{ \left( \frac{i}{k}\nabla \psi +\psi A\right) {{\overline{\psi }}}\right\} =0,\end{aligned}$$

(1.4)

$$\begin{aligned}&{\mathrm {div}}\,A=0\ \ \text{ in }\ \ \Omega \times (0,\infty ),\end{aligned}$$

(1.5)

$$\begin{aligned}&u=0,\frac{\partial \psi }{\partial n}=0,A\cdot n=0,{\mathrm {rot}}\,A\times n=0\ \ \text{ on }\ \ \partial \Omega \times (0,\infty ),\end{aligned}$$

(1.6)

$$\begin{aligned}&(u,\psi ,A)(\cdot ,0)=(u_0,\psi _0,A_0)(\cdot )\ \ \text{ in }\ \ \Omega , \end{aligned}$$

(1.7)

where u is the velocity, $\pi$ is the pressure, $\psi$ is complex the order parameter, A is the vector potential, and $\phi$ is the electric potential, respectively. $\eta$ and k are the positive Ginzburg–Landau constants. ${{\overline{\psi }}}$ is the complex conjugate of $\psi ,\ \text{ Re }\psi :=\displaystyle \frac{\psi +{{\overline{\psi }}}}{2}$ is the real part of $\psi$, $|\psi |^2:=\psi {{\overline{\psi }}}$ is the density of superconductivity carriers and $i:=\sqrt{-1}$. The function $h:=h(x)$ denotes a potential function; we will assume that h is a smooth function. $\Omega$ is a bounded domain with smooth boundary $\partial \Omega$, and n is the unit outward normal vector to $\partial \Omega$.

When h is a constant, system (1.1) and (1.2) reduces to the well-known Navier–Stokes. Papers [1, 2] showed the following regularity criteria:

$$\begin{aligned} \int _0^T\frac{\Vert u(t)\Vert _{L_\mathrm{w}^p}^\frac{2p}{p-3}}{\log (e+\Vert u(t) \Vert _{L_\mathrm{w}^p})}\mathrm {d}t<\infty \ \ \text{ with }\ \ 3<p\le \infty , \end{aligned}$$

(1.8)

or

$$\begin{aligned} u\in L^2(0,T;\hbox {BMO}), \end{aligned}$$

(1.9)

or

$$\begin{aligned} \int _0^T\frac{\Vert \nabla u(t)\Vert _{L_\mathrm{w}^q}^\frac{2q}{2q-3}}{\log (e+\Vert \nabla u(t)\Vert _{L_\mathrm{w}^q})}\mathrm {d}t<\infty \ \ \text{ with }\ \ \frac{3}{2}<q\le \infty , \end{aligned}$$

(1.10)

or

$$\begin{aligned} \nabla u\in L^1(0,T;\hbox {BMO}). \end{aligned}$$

(1.11)

Here $L_\mathrm{w}^q$ is the usual weak $L^q$ space (see Definition 1.1 for details), and BMO is the space of bounded mean oscillation whose norm is defined by

$$\begin{aligned} \Vert f\Vert _\mathrm{BMO}:=\Vert f\Vert _{L^2}+[f]_\mathrm{BMO}, \end{aligned}$$

with

$$\begin{aligned}{}[f]_\mathrm{BMO}:= & {} \sup \limits _{\begin{array}{c} x\in \Omega \\ r\in (0,d) \end{array}} \frac{1}{|\Omega _r(x)|}\int _{\Omega _r(x)}|f(y)-f_{\Omega _r(x)}|\mathrm {d}y,\\ f_{\Omega _r(x)}:= & {} \frac{1}{|\Omega _r(x)|}\int _{\Omega _r(x)}f(y)\mathrm {d}y, \end{aligned}$$

$\Omega _r(x):=B_r(x)\cap \Omega$, $B_r(x)$ is the ball with center x and radius r, and d is the diameter of $\Omega$. $|\Omega _r(x)|$ denotes the Lebesgue measure of $\Omega _r(x)$.

On the other hand, when $u=0$, system (1.3), (1.4) and (1.5) reduces to the time-dependent Ginzburg–Landau, which has received many studies [3,4,5,6,7,8,9,10,11,12]. Paper [4] showed the existence of global weak solutions. Paper [10, 12] proved the uniqueness of weak solutions.

The aim of this paper is to prove some regularity criteria of the problem in a bounded domain. We will prove

Theorem 1.1

Let $u_0\in H_0^1\cap H^2,\psi _0,A_0\in H^1$ with $|\psi _0|\le 1$, ${\mathrm {div}}\,u_0={\mathrm {div}}\,A_0=0$ in $\Omega$. Let $(u,\pi ,\psi ,A,\phi )$ be a local strong solution to the problem (1.1)–(1.7). If (1.8) or (1.9) holds true with $0<T<\infty$, then the solution $(u,\pi ,\psi ,A,\phi )$ can be extended beyond $T>0$.

Theorem 1.2

Let $u_0\in H_0^1\cap H^3,\psi _0,A_0\in H^1$ with $|\psi _0|\le 1$, ${\mathrm {div}}\,u_0={\mathrm {div}}\,A_0=0$ in $\Omega$. Let $(u,\pi ,\psi ,A,\phi )$ be a local strong solution to the problem (1.1)–(1.7). If (1.10) or (1.11) holds true with $0<T<\infty$, then the solution $(u,\pi ,\psi ,A,\phi )$ can be extended beyond $T>0$.

Remark 1.1

We can prove similar results under the Lorentz gauge.

Definition 1.1

Let $f\in L^{p,q}$ be such that

$$\begin{aligned} \left( \frac{p}{q}\int _0^\infty [t^\frac{1}{p} f^*(t)]^q\frac{\mathrm {d}t}{t}\right) ^\frac{1}{q}<\infty , \end{aligned}$$

where $f^*(t)$ is the nonincreasing function equimeasurable with |f| on $(0,\infty )$. We say that f belongs to the Lorentz space $L^{p,\infty }\equiv L_\mathrm{w}^p$ if

$$\begin{aligned} \mathrm {mes}\{x\in \Omega :|f(x)|>\alpha \}\le A\alpha ^{-p}\ \ \mathrm {for}\ \ \mathrm {all}\ \ \alpha >0. \end{aligned}$$

In the following proofs, we will use the following Gagliardo–Nirenberg inequality [13]:

$$\begin{aligned} \Vert u\Vert _{L^{\frac{2r}{r-2},2}}\le C\Vert u\Vert _{L^2}^{1-\frac{3}{r}}\Vert u\Vert _{H^1}^\frac{3}{r}\ \ \mathrm {with}\ \ 3<r<\infty , \end{aligned}$$

(1.12)

and the generalized Hölder inequality [14]:

$$\begin{aligned} \Vert uv\Vert _{L^{p,q}}\le C\Vert u\Vert _{L^{p_1,q_1}}\Vert v\Vert _{L^{p_2,q_2}} \end{aligned}$$

(1.13)

with $\displaystyle \frac{1}{p}=\frac{1}{p_1}+\frac{1}{p_2}$ and $\displaystyle \frac{1}{q}=\frac{1}{q_1}+\frac{1}{q_2}$.

In the following proofs, we will also use the following three lemmas:

Lemma 1.1

We have

$$\begin{aligned} \Vert f\Vert _{L^\infty (\Omega )}\le C(1+\Vert f\Vert _{\mathrm{BMO}(\Omega )}\log ^\frac{1}{2}(e+\Vert f\Vert _{W^{1,m}(\Omega )})) \end{aligned}$$

(1.14)

for all $f\in W_0^{1,m}(\Omega )$ with $3<m<\infty$.

Proof

When $\Omega :={\mathbb {R}}^3\,$, (1.14) is proved by Ogawa [15]. For a bounded domain $\Omega$ in ${\mathbb {R}}^3\,$, we define

$$\begin{aligned} {\tilde{f}}:=\left\{ \begin{array}{lll} f&{}\quad \mathrm {in}&{} \quad \Omega ,\\ 0&{}\quad \mathrm {in}&{} \quad \Omega ^c:={\mathbb {R}}^3\,\setminus \Omega . \end{array} \right. \end{aligned}$$

Then we have [16, p.71]:

$$\begin{aligned} \Vert {\tilde{f}}\Vert _{W^{1,m}({\mathbb {R}}^3\,)}=\Vert f\Vert _{W^{1,m}(\Omega )}, \end{aligned}$$

and it is obvious that

$$\begin{aligned} \Vert {\tilde{f}}\Vert _{L^\infty ({\mathbb {R}}^3\,)}=\Vert f\Vert _{L^\infty (\Omega )},\Vert {\tilde{f}} \Vert _{\mathrm{BMO}({\mathbb {R}}^3\,)}\le C\Vert f\Vert _{\mathrm{BMO}(\Omega )}. \end{aligned}$$

Thus (1.14) is proved. $\square$

Lemma 1.2

([17]). We have

$$\begin{aligned} \Vert u\Vert _{L^4(\Omega )}^2\le C\Vert u\Vert _{L^2(\Omega )}\Vert u\Vert _{\mathrm{BMO}(\Omega )}. \end{aligned}$$

(1.15)

Lemma 1.3

([18]). There holds the following logarithmic Sobolev inequality:

$$\begin{aligned} \Vert \nabla f\Vert _{L^\infty }\le C(1+\Vert \nabla f\Vert _{\mathrm{BMO}}\log (e+\Vert f\Vert _{W^{s,p}})\ \ \text{ with }\ \ s>1+\frac{3}{p} \end{aligned}$$

(1.16)

for any $f\in W^{s,p}(\Omega )$ and $\Omega \subset {\mathbb {R}}^3\,$.

Applying ${\mathrm {div}}\,$ to (1.3) and using (1.5), we see that

$$\begin{aligned} -\Delta \phi ={\mathrm {div}}\,\text{ Re }\left\{ \frac{i}{k}{{\overline{\psi }}}\nabla \psi +| \psi |^2A\right\} . \end{aligned}$$

(1.17)

2 Proof of Theorem 1.1

This section is devoted to the proof of Theorem 1.1. It is easy to show the local well-posedness of strong solutions; we only need to establish a priori estimates.

First, we set

$$\begin{aligned} f:=|\psi |,\psi :=fe^{i\theta },\ \ \text{ and }\ \ V:=-A+\nabla \theta . \end{aligned}$$

Then we have

$$\begin{aligned} \eta \partial _tf+u\cdot \nabla f=\frac{1}{k^2}\Delta f-f(f^2-1+V^2). \end{aligned}$$

(2.1)

Testing (2.1) by $(f-1)_+$ and using (1.1), we see that

$$\begin{aligned}&\frac{\eta }{2}\frac{\mathrm {d}}{\mathrm {d}t}\int (f-1)_+^2\mathrm {d}x +\frac{1}{k^2}\int |\nabla (f-1)_+|^2\mathrm {d}x \\&\quad =-\int f(f^2-1+V^2)(f-1)_+\mathrm {d}x\le 0, \end{aligned}$$

which gives

$$\begin{aligned} (f-1)_+=0, \end{aligned}$$

and thus

$$\begin{aligned} |\psi |\le 1. \end{aligned}$$

(2.2)

Testing (1.3) by ${{\overline{\psi }}}$, taking the real parts and using (1.1), we get

$$\begin{aligned} \frac{\eta }{2}\frac{\mathrm {d}}{\mathrm {d}t}\int |\psi |^2\mathrm {d}x+ \int \left| \frac{i}{k}\nabla \psi +\psi A\right| ^2\mathrm {d}x+\int |\psi |^4\mathrm {d}x=\int |\psi |^2\mathrm {d}x, \end{aligned}$$

which leads to

$$\begin{aligned} \int |\psi |^2\mathrm {d}x+\int _0^T\int \left| \frac{i}{k}\nabla \psi +\psi A\right| ^2\mathrm {d}x\mathrm {d}t\le C. \end{aligned}$$

(2.3)

Testing (1.4) by A, using (1.5), (2.2) and (2.3), we find that

$$\begin{aligned} \frac{1}{2}\frac{\mathrm {d}}{\mathrm {d}t}\int |A|^2\mathrm {d}x+\int |{\mathrm {rot}}\,A|^2\mathrm {d}x= & {} -\text{ Re }\int \left( \frac{i}{k}\nabla \psi +\psi A\right) {{\overline{\psi }}} A\mathrm {d}x\\\le & {} \left\| \frac{i}{k}\nabla \psi +\psi A\right\| _{L^2}\Vert \psi \Vert _{L^\infty }\Vert A\Vert _{L^2}\\\le & {} \left\| \frac{i}{k}\nabla \psi +\psi A\right\| _{L^2}\Vert A\Vert _{L^2}, \end{aligned}$$

which implies

$$\begin{aligned} \Vert A\Vert _{L^\infty (0,T;L^2)}+\Vert A\Vert _{L^2(0,T;H^1)}\le C. \end{aligned}$$

(2.4)

It follows from (2.2), (2.3) and (2.4) that

$$\begin{aligned} \int _0^T\int |\psi A|^2\mathrm {d}x\mathrm {d}t\le \Vert \psi \Vert _{L^\infty (0,T;L^\infty )}\int _0^T\int |A|^2\mathrm {d}x\mathrm {d}t\le C, \end{aligned}$$

whence

$$\begin{aligned} \Vert \psi \Vert _{L^2(0,T;H^1)}\le C. \end{aligned}$$

(2.5)

Testing (1.4) by $-\Delta A$, using (1.5), (1.6), (2.2) and (2.3), we get

$$\begin{aligned}&\frac{1}{2}\frac{\mathrm {d}}{\mathrm {d}t}\int |{\mathrm {rot}}\,A|^2\mathrm {d}x+\int |\Delta A|^2\mathrm {d}x\\&\quad =\int \text{ Re }\left[ \left( \frac{i}{k}\nabla \psi +\psi A\right) {{\overline{\psi }}}\right] \Delta A\mathrm {d}x\\&\quad \le \left\| \frac{i}{k}\nabla \psi +\psi A\right\| _{L^2}\Vert \psi \Vert _{L^\infty }\Vert \Delta A\Vert _{L^2}\\&\quad \le \frac{1}{2}\Vert \Delta A\Vert _{L^2}^2+C\left\| \frac{i}{k}\nabla \psi +\psi A\right\| _{L^2}^2, \end{aligned}$$

which implies

$$\begin{aligned} \Vert A\Vert _{L^\infty (0,T;H^1)}+\Vert A\Vert _{L^2(0,T;H^2)}\le C. \end{aligned}$$

(2.6)

Here we used the well-known facts

$$\begin{aligned} \Vert A\Vert _{H^1}\le C(\Vert A\Vert _{L^2}+\Vert {\mathrm {rot}}\,A\Vert _{L^2}), \end{aligned}$$

(2.7)

and

$$\begin{aligned} \Vert A\Vert _{H^2}\le C(\Vert A\Vert _{L^2}+\Vert \Delta A\Vert _{L^2}) \end{aligned}$$

(2.8)

due to ${\mathrm {div}}\,A=0$ in $\Omega$ and $A\cdot n=0,{\mathrm {rot}}\,A\times n=0$ on $\partial \Omega$.

$$\begin{aligned} \Vert \nabla \phi \Vert _{L^2(0,T;L^2)}\le C\left\| \frac{i}{k}\nabla \psi +\psi A\right\| _{L^2(0,T;L^2)}\Vert \psi \Vert _{L^\infty (0,T;L^\infty )}\le C. \end{aligned}$$

(2.9)

Testing (1.2) by u and using (1.1) and (2.2), we infer that

$$\begin{aligned} \frac{1}{2}\frac{\mathrm {d}}{\mathrm {d}t}\int |u|^2\mathrm {d}x+\int |\nabla u|^2\mathrm {d}x= & {} \int |\psi |^2\nabla h\cdot u\mathrm {d}x\le \Vert \psi \Vert _{L^\infty }^2\Vert \nabla h\Vert _{L^2}\Vert u\Vert _{L^2}\\\le & {} C\Vert u\Vert _{L^2}, \end{aligned}$$

which yields

$$\begin{aligned} \Vert u\Vert _{L^\infty (0,T;L^2)}+\Vert u\Vert _{L^2(0,T;H^1)}\le C. \end{aligned}$$

(2.10)

(I) Let (1.8) hold true.

Testing (1.2) by $-\Delta u+\nabla \pi$, using (1.1), (2.2), (1.12) and (1.13), we compute

$$\begin{aligned}&\frac{1}{2}\frac{\mathrm {d}}{\mathrm {d}t}\int |\nabla u|^2\mathrm {d}x+\int |\nabla \pi -\Delta u|^2\mathrm {d}x\nonumber \\&\quad = -\int u\cdot \nabla u\cdot (\nabla \pi -\Delta u)\mathrm {d}x+\int |\psi |^2\nabla h(\nabla \pi -\Delta u)\mathrm {d}x\nonumber \\&\quad \le \Vert u\Vert _{L_\mathrm{w}^p}\Vert \nabla u\Vert _{L^{\frac{2p}{p-2},2}}\Vert \nabla \pi -\Delta u\Vert _{L^2}+\Vert \psi \Vert _{L^\infty }^2\Vert \nabla h\Vert _{L^2}\Vert \nabla \pi -\Delta u\Vert _{L^2}\nonumber \\&\quad \le \Vert u\Vert _{L_w^p}\Vert \nabla u\Vert _{L^2}^{1-\frac{3}{p}}\Vert \nabla \pi -\Delta u\Vert _{L^2}^{1+\frac{3}{p}}+C\Vert \nabla \pi -\Delta u\Vert _{L^2}\nonumber \\&\quad \le \frac{1}{2}\Vert \nabla \pi -\Delta u\Vert _{L^2}^2+C\Vert u\Vert _{L_\mathrm{w}^p}^\frac{2p}{p-3}\Vert \nabla u\Vert _{L^2}^2+C\nonumber \\&\quad \le \frac{1}{2}\Vert \nabla \pi -\Delta u\Vert _{L^2}^2+\frac{C\Vert u\Vert _{L_\mathrm{w}^p}^\frac{2p}{p-3}}{\log (e+\Vert u\Vert _{L_\mathrm{w}^p})}\Vert \nabla u\Vert _{L^2}^2\log (e+\Vert u\Vert _{L_\mathrm{w}^p})+C\nonumber \\&\quad \le \frac{1}{2}\Vert \nabla \pi -\Delta u\Vert _{L^2}^2+\frac{C\Vert u\Vert _{L_\mathrm{w}^p}^\frac{2p}{p-2}}{\log (e+\Vert u\Vert _{L_\mathrm{w}^p})}\Vert \nabla u\Vert _{L^2}^2\log (e+y)+C, \end{aligned}$$

(2.11)

which gives

$$\begin{aligned} \int |\nabla u|^2\mathrm {d}x+\int _0^t\int |\Delta u|^2\mathrm {d}x\mathrm {d}s\le C(e+y)^{C_0\epsilon } \end{aligned}$$

(2.12)

with

$$\begin{aligned} y(t):=\sup \limits _{[t_0,t]}\Vert u(\cdot ,s)\Vert _{W^{1,m}}\ \ \text{ and }\ \ 3<m\le 6, \end{aligned}$$

for any $0<t_0\le t\le T$, where $C_0$ is an absolute constant, provided that

$$\begin{aligned} \int _{t_0}^T\frac{\Vert u(t)\Vert _{L_\mathrm{w}^p}^\frac{2p}{p-3}}{\log (e+ \Vert u\Vert _{L_\mathrm{w}^p})}\mathrm {d}t\le \epsilon<<1. \end{aligned}$$

(2.13)

Here we have used the well-known $H^2$-estimate of Stokes system:

$$\begin{aligned} \Vert u\Vert _{H^2}\le C\Vert \nabla \pi -\Delta u\Vert _{L^2}. \end{aligned}$$

(2.14)

Integrating (2.11) over $(t_0,t)$ and using (2.12) and (2.13), we obtain

$$\begin{aligned} \int _{t_0}^t\int |\partial _tu|^2\mathrm {d}x\mathrm {d}s\le C(e+y)^{C_0\epsilon }. \end{aligned}$$

(2.15)

Equation (1.3) can be rewritten as

$$\begin{aligned}&\eta \partial _t\psi +u\cdot \nabla \psi +i\eta k\phi \psi -\frac{1}{k^2}\Delta \psi + \frac{2i}{k}A\cdot \nabla \psi +|A|^2\psi +|\psi |^2\psi -\psi \nonumber \\&\quad =0. \end{aligned}$$

(2.16)

Testing (2.16) by $-\Delta {{\overline{\psi }}}$ and taking the real parts, using (2.2), (2.6), (2.11), (1.14) and (1.13), we have

$$\begin{aligned}&\frac{\eta }{2}\frac{\mathrm {d}}{\mathrm {d}t}\int |\nabla \psi |^2\mathrm {d}x +\frac{1}{k^2}\int |\Delta \psi |^2\mathrm {d}x\nonumber \\&\quad = \text{ Re }\int u\cdot \nabla \psi \cdot \Delta {{\overline{\psi }}}\mathrm {d}x\nonumber \\&\qquad +\text{ Re }\int i\eta k\phi \psi \Delta {{\overline{\psi }}}\mathrm {d}x-\text{ Re } \int \frac{2i}{k}A\cdot \nabla \psi \cdot \Delta {{\overline{\psi }}}\mathrm {d}x\nonumber \\&\qquad +\text{ Re }\int |A|^2\psi \Delta {{\overline{\psi }}}\mathrm {d}x +\text{ Re }\int |\psi |^2\psi \Delta {{\overline{\psi }}}\mathrm {d}x+\int |\nabla \psi |^2\mathrm {d}x \nonumber \\&\quad \le \Vert u\Vert _{L_\mathrm{w}^p}\Vert \nabla \psi \Vert _{L^{\frac{2p}{p-2},2}}\Vert \Delta \psi \Vert _{L^2}+C\Vert \phi \Vert _{L^2}\Vert \Delta \psi \Vert _{L^2} \nonumber \\&\qquad +C\Vert A\Vert _{L^4}\Vert \nabla \psi \Vert _{L^4}\Vert \Delta \psi \Vert _{L^2}\nonumber \\&\qquad +C\Vert A\Vert _{L^4}^2\Vert \Delta \psi \Vert _{L^2}+C\Vert \psi \Vert _{L^2}\Vert \Delta \psi \Vert _{L^2}+\Vert \nabla \psi \Vert _{L^2}^2\nonumber \\&\quad \le C\Vert u\Vert _{L_\mathrm{w}^p}\Vert \nabla \psi \Vert _{L^2}^{1-\frac{3}{p}}\Vert \Delta \psi \Vert _{L^2}^{1+\frac{3}{p}}+C\Vert \phi \Vert _{L^2}\Vert \Delta \psi \Vert _{L^2}\nonumber \\&\qquad +C\Vert \nabla \psi \Vert _{L^4}\Vert \Delta \psi \Vert _{L^2}+C\Vert \Delta \psi \Vert _{L^2}\nonumber \\&\quad \le \frac{1}{32k^2}\Vert \Delta \psi \Vert _{L^2}^2+C\Vert u\Vert _{L_\mathrm{w}^p}^\frac{2p}{p-3} \Vert \nabla \psi \Vert _{L^2}^2+C\Vert \phi \Vert _{L^2}^2+C\nonumber \\&\quad \le \frac{1}{32k^2}\Vert \Delta \psi \Vert _{L^2}^2+C\frac{\Vert u\Vert _{L_\mathrm{w}^p}^\frac{2p}{p-3}}{\log (e+\Vert u\Vert _{L_\mathrm{w}^p})}\log (e+y)\Vert \nabla \psi \Vert _{L^2}^2\nonumber \\&\qquad +C\Vert \phi \Vert _{L^2}^2+C, \end{aligned}$$

(2.17)

which implies

$$\begin{aligned} \int |\nabla \psi |^2\mathrm {d}x+\int _{t_0}^t\int |\Delta \psi |^2 \mathrm {d}x\mathrm {d}s\le C(e+y)^{C_0\epsilon }. \end{aligned}$$

(2.18)

Here we have used the Gagliardo–Nirenberg inequalities

$$\begin{aligned}&\Vert \nabla \psi \Vert _{L^4}^2\le C\Vert \psi \Vert _{L^\infty }\Vert \Delta \psi \Vert _{L^2}, \end{aligned}$$

(2.19)

$$\begin{aligned}&\Vert \nabla \psi \Vert _{L^2}^2\le C\Vert \psi \Vert _{L^2}\Vert \Delta \psi \Vert _{L^2}, \end{aligned}$$

(2.20)

and the fact

$$\begin{aligned} \Vert \psi \Vert _{H^2}\le C\Vert \Delta \psi \Vert _{L^2}. \end{aligned}$$

(2.21)

Similarly, testing (2.16) by $\partial _t{{\overline{\psi }}}$ and taking the real parts, we obtain

$$\begin{aligned} \int _{t_0}^t\int |\partial _t\psi |^2\mathrm {d}x\mathrm {d}s\le C(e+y)^{C_0\epsilon }. \end{aligned}$$

(2.22)

Taking $\partial _t$ to (1.2), testing by $\partial _tu$, using (1.1), (2.2), (2.12), (2.15) and (2.22), we obtain

$$\begin{aligned}&\frac{1}{2}\frac{\mathrm {d}}{\mathrm {d}t}\int |\partial _tu|^2\mathrm {d}x +\int |\nabla \partial _tu|^2\mathrm {d}x \nonumber \\&\quad =-\int \partial _tu\cdot \nabla u\cdot \partial _tu\mathrm {d}x+\int \partial _t|\psi |^2\nabla h\partial _tu\mathrm {d}x\nonumber \\&\quad \le \Vert \nabla u\Vert _{L^2}\Vert \partial _tu\Vert _{L^4}^2+ C\Vert \psi \Vert _{L^\infty }\Vert \partial _t\psi \Vert _{L^2}\Vert \nabla h\Vert _{L^\infty }\Vert \partial _tu\Vert _{L^2}\nonumber \\&\quad \le C\Vert \nabla u\Vert _{L^2}\Vert \partial _tu\Vert _{L^2}^\frac{1}{2} \Vert \nabla \partial _tu\Vert _{L^2}^\frac{3}{2}+C\Vert \partial _t\psi \Vert _{L^2}^2 +C\Vert \partial _tu\Vert _{L^2}^2\nonumber \\&\quad \le \frac{1}{2}\Vert \nabla \partial _tu\Vert _{L^2}^2+C\Vert \nabla u\Vert _{L^2}^4\Vert \partial _tu\Vert _{L^2}^2+C\Vert \partial _t\psi \Vert _{L^2}^2 +C\Vert \partial _tu\Vert _{L^2}^2 \end{aligned}$$

(2.23)

which leads to

$$\begin{aligned} \int |\partial _tu|^2\mathrm {d}x+\int _{t_0}^t\int |\nabla \partial _tu|^2 \mathrm {d}x\mathrm {d}s\le C(e+y)^{C_0\epsilon }. \end{aligned}$$

(2.24)

On the other hand, Eq. (1.2) can be rewritten as

$$\begin{aligned} -\Delta u+\nabla \pi =f:=|\psi |^2\nabla h-\partial _tu-u\cdot \nabla u. \end{aligned}$$

(2.25)

Then we have

$$\begin{aligned} \Vert u\Vert _{H^2}\le & {} C\Vert f\Vert _{L^2}=C\Vert |\psi |^2\nabla h-\partial _tu-u\cdot \nabla u\Vert _{L^2}\\\le & {} C+C\Vert \partial _tu\Vert _{L^2}+C\Vert u\Vert _{L^6}\Vert \nabla u\Vert _{L^3}\\\le & {} C+C\Vert \partial _tu\Vert _{L^2}+C\Vert \nabla u\Vert _{L^2}\cdot \Vert \nabla u\Vert _{L^2}^\frac{1}{2}\Vert u\Vert _{H^2}^\frac{1}{2}, \end{aligned}$$

whence

$$\begin{aligned} \Vert u\Vert _{H^2}\le & {} C+C\Vert \partial _tu\Vert _{L^2}+C\Vert \nabla u\Vert _{L^2}^3\nonumber \\\le & {} C(e+y)^{C_0\epsilon }, \end{aligned}$$

(2.26)

which implies

$$\begin{aligned} \Vert u\Vert _{L^\infty (0,T;H^2)}\le C. \end{aligned}$$

(2.27)

(2.18) and (2.27) give

$$\begin{aligned} \Vert \psi \Vert _{L^\infty (0,T;H^1)}+\Vert \psi \Vert _{L^2(0,T;H^2)}\le C. \end{aligned}$$

(2.28)

(1.17), (2.6) and (2.28) lead to

$$\begin{aligned} \Vert \phi \Vert _{L^\infty (0,T;H^1)}+\Vert \phi \Vert _{L^2(0,T;H^2)}\le C. \end{aligned}$$

(2.29)

(II) Let (1.9) hold true.

We still have (2.9) (with $p=\infty$) and using Lemma 1.1, we get (2.12). Provided that

$$\begin{aligned} \int _{t_0}^T\Vert u(t)\Vert _{\mathrm{BMO}}^2\mathrm {d}t\le \epsilon<<1. \end{aligned}$$

Similar to (2.17), we have

$$\begin{aligned}&\frac{\eta }{2}\frac{\mathrm {d}}{\mathrm {d}t}\int |\nabla \psi |^2 \mathrm {d}x +\frac{1}{k^2}\int |\Delta \psi |^2\mathrm {d}x \nonumber \\&\quad =\text{ Re }\int u\cdot \nabla \psi \cdot \Delta \psi \mathrm {d}x+\text{ the } \text{ same } \text{ other } \text{ terms }. \end{aligned}$$

(2.30)

Now we bound the first term of RHS of (2.30) as follows:

$$\begin{aligned}&\text{ Re }\int u\cdot \nabla \psi \cdot \Delta \psi \mathrm {d}x\le \Vert u \Vert _{L^4}\Vert \nabla \psi \Vert _{L^4}\Vert \Delta \psi \Vert _{L^2}\nonumber \\&\quad \le \Vert u\Vert _{L^2}^\frac{1}{2}\Vert u\Vert _{\mathrm{BMO}}^\frac{1}{2}\cdot \Vert \psi \Vert _{L^\infty }^\frac{1}{2}\Vert \Delta \psi \Vert _{L^2}^\frac{1}{2}\cdot \Vert \Delta \psi \Vert _{L^2}\nonumber \\&\quad \le C\Vert u\Vert _{\mathrm{BMO}}^\frac{1}{2}\Vert \Delta \psi \Vert _{L^2}^\frac{3}{2}\le \frac{1}{32k^2}\Vert \Delta \psi \Vert _{L^2}^2+C\Vert u\Vert _{\mathrm{BMO}}^2. \end{aligned}$$

(2.31)

And thus we have

$$\begin{aligned} \int |\nabla \psi |^2\mathrm {d}x+\int _0^t\int (|\Delta \psi |^2+ |\partial _t\psi |^2)\mathrm {d}x\mathrm {d}s\le C. \end{aligned}$$

(2.32)

Then we still have (2.27) and (2.29).

This completes the proof. $\square$

3 Proof of Theorem 1.2

This section is devoted to the proof of Theorem 1.2. We only need to establish some a priori estimates.

We still have (2.1), (2.2), (2.3), (2.4), (2.5), (2.6), (2.7) and (2.10).

(I). Let (1.10) hold true.

We still have (2.17). We bound the first term of the RHS of (2.17) as follows:

$$\begin{aligned} \text{ Re }\int u\cdot \nabla \psi \cdot \Delta {{\overline{\psi }}}\mathrm {d}x= & {} \sum \limits _{i,j}\text{ Re }\int u_i\partial _i\psi \partial _j^2{{\overline{\psi }}}\mathrm {d}x =-\sum \limits _{i,j}\text{ Re }\int \partial _ju_i\partial _i\psi \partial _j{{\overline{\psi }}}\mathrm {d}x\nonumber \\\le & {} C\Vert \nabla u\Vert _{L_\mathrm{w}^p}\Vert \nabla \psi \Vert _{L^{\frac{2q}{q-1},2}}^2\le C\Vert \nabla u\Vert _{L_\mathrm{w}^q}\Vert \nabla \psi \Vert _{L^2}^{2-\frac{3}{q}}\Vert \Delta \psi \Vert _{L^2}^\frac{3}{q}\nonumber \\\le & {} \frac{1}{32k^2}\Vert \Delta \psi \Vert _{L^2}^2+C\Vert \nabla u\Vert _{L_\mathrm{w}^q}^\frac{2q}{2q-3}\Vert \nabla \psi \Vert _{L^2}^2 \le \frac{1}{32k^2}\Vert \Delta \psi \Vert _{L^2}^2 \nonumber \\&+\,C\frac{\Vert \nabla u\Vert _{L_\mathrm{w}^q}^\frac{2q}{2q-3}}{\log (e+\Vert \nabla u\Vert _{L_\mathrm{w}^q})}\log (e+z)\Vert \nabla \psi \Vert _{L^2}^2 \end{aligned}$$

(3.1)

with

$$\begin{aligned} z(t):=\sup \limits _{[t_0,t]}\Vert u(\cdot ,s)\Vert _{H^3}. \end{aligned}$$

The other terms can be bounded as before.

Then we have

$$\begin{aligned} \int |\nabla \psi |^2\mathrm {d}x+\int _{t_0}^t\int (|\Delta \psi |^2+ |\partial _t\psi |^2)\mathrm {d}x\mathrm {d}s\le C(e+z)^{C_0\epsilon } \end{aligned}$$

(3.2)

provided that

$$\begin{aligned} \int _{t_0}^T\frac{\Vert \nabla u(t)\Vert _{L_\mathrm{w}^q}^\frac{2q}{2q-3}}{\log (e+\Vert \nabla u\Vert _{L_\mathrm{w}^q})}\mathrm {d}t\le \epsilon<<1. \end{aligned}$$

(3.3)

Similarly to (2.23), we have

$$\begin{aligned}&\frac{1}{2}\frac{\mathrm {d}}{\mathrm {d}t}\int |\partial _tu|^2\mathrm {d}x +\int |\nabla \partial _tu|^2\mathrm {d}x \\&\quad =-\int \partial _tu\cdot \nabla u\cdot \partial _tu\mathrm {d}x+\int \partial _t|\psi |^2\cdot \nabla h\cdot \partial _tu\mathrm {d}x\\&\quad \le C\Vert \nabla u\Vert _{L_\mathrm{w}^q}\Vert \partial _tu\Vert _{L^{\frac{2q}{q-1},2}}^2 +C\Vert \partial _t\psi \Vert _{L^2}\Vert \partial _tu\Vert _{L^2}\\&\quad \le C\Vert \nabla u\Vert _{L_\mathrm{w}^q}\Vert \partial _tu\Vert _{L^2}^{2-\frac{3}{q}}\Vert \nabla \partial _tu\Vert _{L^2}^\frac{3}{q}+C\Vert \partial _t\psi \Vert _{L^2}\Vert \partial _tu\Vert _{L^2}\\&\quad \le \frac{1}{2}\Vert \nabla \partial _tu\Vert _{L^2}^2+C\Vert \nabla u \Vert _{L_\mathrm{w}^q}^\frac{2q}{2q-3}\Vert \partial _tu\Vert _{L^2}^2+C\Vert \partial _t\psi \Vert _{L^2}\Vert \partial _tu\Vert _{L^2}\\&\quad \le \frac{1}{2}\Vert \nabla \partial _tu\Vert _{L^2}^2+C\frac{\Vert \nabla u \Vert _{L_\mathrm{w}^q}^\frac{2q}{2q-3}}{\log (e+\Vert \nabla u\Vert _{L_\mathrm{w}^q})}\log (e+z)\Vert \partial _tu\Vert _{L^2}^2 \\&\qquad +C\Vert \partial _t\psi \Vert _{L^2}^2 +C\Vert \partial _tu\Vert _{L^2}^2, \end{aligned}$$

which gives

$$\begin{aligned} \int |\partial _tu|^2\mathrm {d}x+\int _0^t\int |\nabla \partial _tu|^2 \mathrm {d}x\mathrm {d}s\le C(e+z)^{C_0\epsilon }. \end{aligned}$$

(3.4)

Testing (1.2) by $\partial _tu$, using (1.1), (2.10) and (3.4), we have

$$\begin{aligned} \frac{1}{2}\frac{\mathrm {d}}{\mathrm {d}t}\int |\nabla u|^2\mathrm {d}x +\int |\partial _tu|^2\mathrm {d}x= & {} -\int u\cdot \nabla u\cdot \partial _tu\mathrm {d}x+\int |\psi |^2\nabla h\cdot \partial _tu\mathrm {d}x\\\le & {} \Vert u\Vert _{L^6}\Vert \nabla u\Vert _{L^2}\Vert \partial _tu\Vert _{L^3}+C\Vert \partial _tu\Vert _{L^2}\\\le & {} C\Vert \nabla u\Vert _{L^2}^4+C\Vert \nabla \partial _tu\Vert _{L^2}^2+C, \end{aligned}$$

which gives

$$\begin{aligned} \int |\nabla u|^2\mathrm {d}x\le C(e+z)^{C_0\epsilon }. \end{aligned}$$

(3.5)

Applying $\partial _t$ to (1.2), testing by $-\Delta \partial _tu+\nabla \partial _t\pi$, using (1.1), (2.2), (3.2), (3.4) and (3.5), we have

$$\begin{aligned}&\frac{1}{2}\frac{\mathrm {d}}{\mathrm {d}t}\int |\nabla \partial _tu|^2\mathrm {d}x +\int |\nabla \partial _t\pi -\Delta \partial _tu|^2\mathrm {d}x\\&\quad =-\int (\partial _tu\cdot \nabla u+u\cdot \nabla \partial _tu)(\nabla \partial _t \pi -\Delta \partial _tu)\mathrm {d}x \\&\qquad +\int \partial _t|\psi |^2\cdot \nabla h(\nabla \partial _t\pi -\Delta \partial _tu)\mathrm {d}x\\&\quad \le (\Vert \partial _tu\Vert _{L^\infty }\Vert \nabla u\Vert _{L^2}+\Vert u\Vert _{L^6}\Vert \nabla \partial _tu\Vert _{L^3})\Vert \nabla \partial _t\pi -\Delta \partial _tu\Vert _{L^2} \\&\qquad +C\Vert \partial _t\psi \Vert _{L^2}\Vert \nabla \partial _t\pi - \Delta \partial _tu\Vert _{L^2}\\&\quad \le C\Vert \nabla \partial _tu\Vert _{L^2}^\frac{1}{2}\Vert \Delta \partial _tu\Vert _{L^2}^\frac{1}{2} \cdot \Vert \nabla u\Vert _{L^2}\Vert \nabla \partial _t\pi - \Delta \partial _tu\Vert _{L^2} \\&\qquad + C\Vert \partial _t\psi \Vert _{L^2}\Vert \nabla \partial _t\pi - \Delta \partial _tu\Vert _{L^2}\\&\quad \le \frac{1}{2}\Vert \nabla \partial _t\pi -\Delta \partial _tu\Vert _{L^2}^2+C\Vert \nabla u\Vert _{L^2}^4\Vert \nabla \partial _tu\Vert _{L^2}^2+C\Vert \partial _t\psi \Vert _{L^2}^2, \end{aligned}$$

which implies

$$\begin{aligned} \int |\nabla \partial _tu|^2\mathrm {d}x+\int _{t_0}^t\int |\nabla \partial _t \pi -\Delta \partial _tu|^2\mathrm {d}x\mathrm {d}s\le C(e+z)^{C_0\epsilon }. \end{aligned}$$

(3.6)

Here we have used the fact

$$\begin{aligned} \Vert \partial _tu\Vert _{H^2}\le C\Vert \nabla \partial _t\pi -\Delta \partial _tu\Vert _{L^2}. \end{aligned}$$

(3.7)

From (2.25), (2.2), (3.2), (3.6) and (3.5), we have

$$\begin{aligned} \Vert u\Vert _{H^3}\le & {} C(\Vert \nabla f\Vert _{L^2}+\Vert u\Vert _{L^2})\\\le & {} C\Vert \nabla (|\psi |^2\nabla h)\Vert _{L^2}+C\Vert \nabla \partial _tu\Vert _{L^2}+C\Vert \nabla (u\cdot \nabla u)\Vert _{L^2}+C\\\le & {} C\Vert \nabla \psi \Vert _{L^2}+C\Vert \nabla \partial _tu\Vert _{L^2}+C\Vert u \Vert _{L^6}\Vert \nabla ^2u\Vert _{L^3}+C\Vert \nabla u\Vert _{L^4}^2+C\\\le & {} C\Vert \nabla \psi \Vert _{L^2}+C\Vert \nabla \partial _tu\Vert _{L^2}+C\Vert u\Vert _{L^6}\Vert \nabla u\Vert _{L^2}^\frac{1}{4}\Vert u\Vert _{H^3}^\frac{3}{4} \\&+C\Vert \nabla u\Vert _{L^2}^\frac{5}{4}\Vert u\Vert _{H^3}^\frac{3}{4}+C, \end{aligned}$$

and therefore

$$\begin{aligned} \Vert u\Vert _{H^3}\le & {} C\Vert \nabla \psi \Vert _{L^2}+C\Vert \nabla \partial _tu\Vert _{L^2}+ C\Vert \nabla u\Vert _{L^2}^5+C\\\le & {} C(e+z)^{C_0\epsilon }, \end{aligned}$$

which implies

$$\begin{aligned} \Vert u\Vert _{L^\infty (0,T;H^3)}\le C. \end{aligned}$$

(3.8)

We still have (2.29).

(II) Let (1.11) hold true.

Similarly to (3.1) for $q=\infty$ and using Lemma 1.2, we still have (3.2), provided that

$$\begin{aligned} \int _{t_0}^t\Vert \nabla u\Vert _{\mathrm{BMO}}\mathrm {d}t\le \epsilon<<1. \end{aligned}$$

(3.9)

We still have (3.4) (for $q=\infty$ and Lemma 1.2) and (3.5).

We still have (3.6), (3.8), (2.29) and (2.32).

This completes the proof. $\square$

Appendix

In this appendix, we will prove the local well-posedness of strong solutions to the problem (1.1)–(1.7).

First, we give the definition of strong solutions to the problem.

Definition 4.1

(strong solutions). $(u,\pi ,\psi ,A,\phi )$ is called a strong solution to the problem (1.1)–(1.7) in $\Omega \times (0,T)$ under the Coulomb gauge if

$$\begin{aligned} \begin{aligned}&(u,\psi ,A)(\cdot ,0)=(u_0,\psi _0,A_0)(\cdot )\in H_0^1\cap H^2\times H^1\times H^1\ \ \text{ with }\\&|\psi _0|\le 1,{\mathrm {div}}\,u_0={\mathrm {div}}\,A_0=0\ \ \text{ in }\ \ \Omega , \end{aligned} \end{aligned}$$

(4.1)

and

$$\begin{aligned} \begin{aligned}&u\in L^\infty (0,T;H_0^1\cap H^2)\cap L^2(0,T;H^3),\\&\pi \in L^\infty (0,T;H^1)\cap L^2(0,T;H^2),\\&\psi \in L^\infty (0,T;H^1)\cap L^2(0,T;H^2),|\psi |\le 1\ \ \text{ in }\ \ \Omega \times (0,T),\\&A\in L^\infty (0,T;H^1)\cap L^2(0,T;H^2),\\&\phi \in L^\infty (0,T;H^1)\cap L^2(0,T;H^2),\\&\partial _tu\in L^\infty (0,T;L^2)\cap L^2(0,T;H^1),\\&\partial _t\psi ,\partial _tA\in L^2(0,T;L^2), \end{aligned} \end{aligned}$$

(4.2)

and the equations

$$\begin{aligned} (1.1)-(1.5)\ \ holds\ \ almost\ \ everywhere\ \ on\ \ \Omega \times (0,T). \end{aligned}$$

(4.3)

In this appendix, we will prove

Theorem 4.1

Let (4.1) hold true. Then the problem (1.1)–(1.7) has a unique strong solution $(u,\pi ,\psi ,A,\phi )$ satisfying (4.2) for some $0<T\le \infty$.

The proof of the uniqueness part is standard with regularity (4.2), and thus we omit the details here. We will use the Galerkin method to show the existence part; the key step of the Galerkin method is to show the a priori estimates. Thus we only need to show the a priori estimates.

Proof of Theorem 4.1

We still have (2.2), (2.3), (2.4), (2.5), (2.6), (2.9) and (2.10).

Similar to (2.11), we have

$$\begin{aligned}&\frac{1}{2}\frac{\mathrm {d}}{\mathrm {d}t}\int |\nabla u|^2\mathrm {d}x+ \int |\nabla \pi -\Delta u|^2\mathrm {d}x\\&\quad =-\int u\cdot \nabla u\cdot (\nabla \pi -\Delta u)\mathrm {d}x+ \int |\psi |^2\nabla h\cdot (\nabla \pi -\Delta u)\mathrm {d}x\\&\quad \le \Vert u\Vert _{L^6}\Vert \nabla u\Vert _{L^3}\Vert \nabla \pi -\Delta u\Vert _{L^2} +\Vert \psi \Vert _{L^\infty }^2\Vert \nabla h\Vert _{L^2}\Vert \nabla \pi -\Delta u\Vert _{L^2}\\&\quad \le C\Vert \nabla u\Vert _{L^2}\cdot \Vert \nabla u\Vert _{L^2}^\frac{1}{2}\Vert \nabla \pi -\Delta u\Vert _{L^2}^\frac{3}{2}+C\Vert \nabla \pi -\Delta u\Vert _{L^2}\\&\quad \le \frac{1}{2}\Vert \nabla \pi -\Delta u\Vert _{L^2}^2+C\Vert \nabla u\Vert _{L^2}^6+C, \end{aligned}$$

which gives

$$\begin{aligned} \Vert u\Vert _{L^\infty (0,T;H^1)}+\Vert u\Vert _{L^2(0,T;H^2)}\le C \end{aligned}$$

(4.4)

for some $0<T\le \infty$.

Similar to (2.17), we get

$$\begin{aligned}&\frac{\eta }{2}\frac{\mathrm {d}}{\mathrm {d}t}\int |\nabla \psi |^2\mathrm {d}x +\frac{1}{k^2}\int |\Delta \psi |^2\mathrm {d}x\\&\quad = \text{ Re }\int u\cdot \nabla \psi \cdot \Delta {{\overline{\psi }}}\mathrm {d}x +\text{ Re }\int i\eta k\phi \psi \Delta {{\overline{\psi }}}\mathrm {d}x -\text{ Re }\int \frac{2i}{k}A\cdot \nabla \psi \cdot \Delta {{\overline{\psi }}}\mathrm {d}x\\&\qquad +\text{ Re }\int |A|^2\psi \Delta {{\overline{\psi }}}\mathrm {d}x+\text{ Re } \int |\psi |^2\psi \Delta {{\overline{\psi }}}\mathrm {d}x+\int |\nabla \psi |^2\mathrm {d}x\\&\quad \le C\Vert u\Vert _{L^4}\Vert \nabla \psi \Vert _{L^4}\Vert \Delta \psi \Vert _{L^2}+C\Vert \phi \Vert _{L^2}\Vert \psi \Vert _{L^\infty }\Vert \Delta \psi \Vert _{L^2}\\&\qquad +C\Vert A\Vert _{L^4}\Vert \nabla \psi \Vert _{L^4}\Vert \Delta \psi \Vert _{L^2}+C\Vert A\Vert _{L^4}^2\Vert \psi \Vert _{L^\infty }\Vert \Delta \psi \Vert _{L^2}\\&\qquad +C\Vert \psi \Vert _{L^\infty }^2\Vert \psi \Vert _{L^2}\Vert \Delta \psi \Vert _{L^2}+\Vert \nabla \psi \Vert _{L^2}^2\\&\quad \le C\Vert \psi \Vert _{L^\infty }^\frac{1}{2}\Vert \Delta \psi \Vert _{L^2}^\frac{3}{2}+C\Vert \phi \Vert _{L^2}\Vert \Delta \psi \Vert _{L^2}+C\Vert \Delta \psi \Vert _{L^2}+\Vert \nabla \psi \Vert _{L^2}^2\\&\quad \le \frac{1}{2k^2}\Vert \Delta \psi \Vert _{L^2}^2+C+C\Vert \phi \Vert _{L^2}^2+C\Vert \nabla \psi \Vert _{L^2}^2, \end{aligned}$$

which leads to

$$\begin{aligned} \Vert \psi \Vert _{L^\infty (0,T;H^1)}+\Vert \psi \Vert _{L^2(0,T;H^2)}\le C \end{aligned}$$

(4.5)

for some $0<T\le \infty$.

Similar to (2.22), we have

$$\begin{aligned} \Vert \partial _t\psi \Vert _{L^2(0,T;L^2)}\le C \end{aligned}$$

(4.6)

for some $0<T\le \infty$.

Similar to (2.23), we observe that

$$\begin{aligned}&\frac{1}{2}\frac{\mathrm {d}}{\mathrm {d}t}\int |\partial _tu|^2\mathrm {d}x +\int |\nabla \partial _tu|^2\mathrm {d}x \\&\quad =-\int \partial _tu\cdot \nabla u\cdot \partial _tu\mathrm {d}x+\int \partial _t|\psi |^2\nabla h\cdot \partial _tu\mathrm {d}x\\&\quad \le \Vert \nabla u\Vert _{L^2}\Vert \partial _tu\Vert _{L^4}^2+C\Vert \psi \Vert _{L^\infty } \Vert \partial _t\psi \Vert _{L^2}\Vert \nabla h\Vert _{L^\infty }\Vert \partial _tu\Vert _{L^2}\\&\quad \le C\Vert \partial _tu\Vert _{L^2}^\frac{1}{2}\Vert \nabla \partial _tu\Vert _{L^2}^\frac{3}{2} +C\Vert \partial _t\psi \Vert _{L^2}\Vert \partial _tu\Vert _{L^2}\\&\quad \le \frac{1}{2}\Vert \nabla \partial _tu\Vert _{L^2}^2+C\Vert \partial _tu\Vert _{L^2}^2+ C\Vert \partial _t\psi \Vert _{L^2}^2, \end{aligned}$$

which implies

$$\begin{aligned} \Vert \partial _tu\Vert _{L^\infty (0,T;L^2)}+\Vert \partial _tu\Vert _{L^2(0,T;H^1)}\le C \end{aligned}$$

(4.7)

for some $0<T\le \infty$.

We still have (2.27), (2.28) and (2.29).

This completes the proof. $\square$