Local Well-Posedness of Classical Solutions to the Time-Dependent Ginzburg–Landau Model for Superconductivity in ${\mathbb{R}}^{\mathit{n}}$

Abstract

In this paper, we prove the local well-posedness of classical solutions $(\psi ,A,\varphi )$ to the $nD(n\ge 3)$ time-dependent Ginzburg–Landau model in superconductivity with the choice of Coulomb gauge and the main assumptions ${\psi }_0,A_0\in H^s\left({\mathbb{R}}^n\right)$ with $\mathrm{div}{A}_0=0$ in ${\mathbb{R}}^n$ and $s>{\displaystyle \frac{n}{2}}$. This result can be used in the proof of regularity criterion and global-in-time well-posedness of the strong solution.

1. Introduction

In this paper, we consider the $nD(n\ge 3)$ time-dependent Ginzburg–Landau model in superconductivity with the Coulomb gauge:

$$\begin{array}{ccc}& & \eta {\partial }_t\psi +i\eta k\varphi \psi +{\left({\displaystyle \frac{i}{k}}\nabla +A\right)}^2\psi +{(|\psi |}^2-1)\psi =0,\hfill \end{array}$$

(1)

$$\begin{array}{ccc}& & {\partial }_{t}A+\nabla \varphi -\Delta A+\mathrm{Re}\left\{\left({\displaystyle \frac{i}{k}}\nabla \psi +\psi A\right)\overline{\psi }\right\}=0,\hfill \end{array}$$

(2)

$$\begin{array}{ccc}& & \mathrm{div}A=0\mathrm{in}{\mathbb{R}}^n\times (0,\infty ),\hfill \end{array}$$

(3)

$$\begin{array}{ccc}& & (\psi ,A)(·,0)=({\psi }_0,A_0)(·)\mathrm{in}{\mathbb{R}}^n,\hfill \end{array}$$

(4)

where $\psi ,A$, and $\varphi$ denote $\mathbb{C}$-valued, ${\mathbb{R}}^n$-valued, and $\mathbb{R}$-valued functions, respectively, and they stand for the order parameter, the magnetic potential, and the electric potential, respectively. Two positive constants $\eta$ and k are Ginzburg–Landau constants, $i:=\sqrt{-1}$. $\overline{\psi }$ denotes the complex conjugate of ${\displaystyle \psi ,\mathrm{Re}\psi :=\frac{\psi +\overline{\psi }}{2}}$ is the real part of $\psi$, and ${|\psi |}^2:=\psi \overline{\psi }$ is the density of superconductivity carriers. Here, we choose the Coulomb gauge which means that (3) holds true.

There are many studies of the case where $n=3$. For the given initial data ${\psi }_0\in H^1\cap L^{\infty },A_0\in H^1$, Guo-Yuan [1] proved the global well-posedness of strong solutions. The local well-posedness of mild solutions was proved in [2] by the method of Kato [3] when ${\psi }_0,A_0\in L^3(n=3)$. Fan-Jiang [4] showed the global existence of weak solutions when ${\psi }_0,A_0\in L^2$.

There are also many studies considering the problem in a bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^3$ [5,6,7,8,9,10,11,12,13]. The papers [5,6,7] showed the global well-posedness of strong solutions with the choice of Lorentz gauge, Coulomb gauge, or the temporal gauge, respectively. Tang [8] considered an inverse problem. Tang and Wang [9] studied the long-time behavior of the problem. Fan and Ozawa [10] showed the well-posedness of the weak solutions with the applied magnetic field. Fan, Samet, and Zhou [11] studied the uniform regularity. Fan, Gao and Guo [12,13] proved the uniqueness of the weak solutions in critical spaces.

Very recently, Gao–Fan–Nakamura [14] showed the weak–strong uniqueness. Fan and Zhou [15] established a regularity criterion, which in fact can be refined as

$$\nabla \psi \in L^2(0,T;L^n).$$

(5)

This was proved to be true in [16] when $3\le n\le 17$. The key estimate was to show that

$${\parallel \nabla \psi \parallel }_{L^4(0,T;L^4)}+{\parallel A\parallel }_{L^{\infty }(0,T;L^4)}\le C.$$

(6)

It should be pointed out that the local well-posedness of smooth solutions was used but not proven in [14,15,16].

The aim of this paper is to study the well-posedness of the problem in ${\mathbb{R}}^n$ for any $n\ge 3$. We will prove the following:

Theorem 1.

Let ${\psi }_0,A_0\in H^s\left({\mathbb{R}}^n\right)$ with $\mathrm{div}{A}_0=0$ in ${\mathbb{R}}^n$ and $s>{\displaystyle \frac{n}{2}}$. Let ${\varphi }_0$ satisfy

$$-\Delta {\varphi }_0=\mathrm{Rediv}\left\{\left({\displaystyle \frac{i}{k}}\nabla {\psi }_0+{\psi }_0{A}_0\right)\overline{{\psi }_0}\right\}in{\mathbb{R}}^n.$$

(7)

Then, the problem (1)–(4) has a unique smooth solution $(\psi ,A,\varphi )$ satisfying

$$\psi ,A,\varphi \in C([0,T];H^s)\cap L^2(0,T;H^{s+1})$$

(8)

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

Remark 1.

Our Theorems 1 and 2 can be used to study the regularity criterion and global well-posedness of the problem, see [14,15,16].

Theorem 2.

Let ${\psi }_0,A_0\in H^s\left({\mathbb{R}}^n\right)$ with $\mathrm{div}{A}_0=0$ in ${\mathbb{R}}^n$ and $s>{\displaystyle \frac{n}{2}}$ and (7) holds true. Then, (6) holds true for any $T>0$.

We will prove Theorem 1 by the Banach fixed point theorem. We denote the nonempty closed set

$$\mathcal{A}:=\{(\tilde{\psi },\tilde{A},\tilde{\varphi })\in \mathcal{A};(\tilde{\psi },\tilde{A},\tilde{\varphi })(·,0)=({\psi }_0,A_0,{\varphi }_0),\mathrm{div}\tilde{A}=0,\parallel (\tilde{\psi },\tilde{A},\tilde{\varphi }){\parallel }_{\mathcal{A}}\le R\}$$

with the norm

$$\parallel (\tilde{\psi },\tilde{A},\tilde{\varphi }){\parallel }_{\mathcal{A}}:=\parallel (\tilde{\psi },\tilde{A}){\parallel }_{C([0,T];H^s)}+{\parallel (\tilde{\psi },\tilde{A},\tilde{\varphi })\parallel }_{L^2(0,T;H^{s+1})}.$$

Given $(\tilde{\psi },\tilde{A},\tilde{\varphi })\in \mathcal{A}$, we consider the following linear problems:

$$\begin{array}{ccc}& & \eta {\partial }_t\psi -{\displaystyle \frac{1}{k^2}}\Delta \psi -\psi =f:=-{\displaystyle \frac{2i}{k}}\tilde{A}·\nabla \tilde{\psi }-|\tilde{A}{|}^2\tilde{\psi }-i\eta k\tilde{\varphi }\tilde{\psi }-{|\tilde{\psi }|}^2\tilde{\psi },\hfill \end{array}$$

(9)

$$\begin{array}{ccc}& & \psi (·,0)={\psi }_0\mathrm{in}{\mathbb{R}}^n.\hfill \end{array}$$

(10)

$$\begin{array}{ccc}& & -\Delta \varphi =\mathrm{Rediv}\left\{\left({\displaystyle \frac{i}{k}}\nabla \psi +\psi \tilde{A}\right)\overline{\psi }\right\}.\hfill \end{array}$$

(11)

$$\begin{array}{ccc}& & {\partial }_{t}A-\Delta A=-\nabla \varphi -\mathrm{Re}\left\{\left({\displaystyle \frac{i}{k}}\nabla \psi +\psi \tilde{A}\right)\overline{\psi }\right\},\hfill \end{array}$$

(12)

$$\begin{array}{ccc}& & A(·,0)=A_0\mathrm{and}\mathrm{div}{A}_0=0\mathrm{in}{\mathbb{R}}^n.\hfill \end{array}$$

(13)

Let $(\psi ,A,\varphi )$ be the unique smooth solution to the above problems; we define the fixed point map $F:(\tilde{\psi },\tilde{A},\tilde{\varphi })\in \mathcal{A}\to (\psi ,A,\varphi )\in \mathcal{A}$. We will prove that the map F maps $\mathcal{A}$ into $\mathcal{A}$ for suitable constant A and small T, and F is a contraction mapping, and thus, F has a unique fixed point in $\mathcal{A}$. This proves Theorem 1.

Our proof will use the following bilinear product estimate due to Kato–Ponce [17]:

$$\parallel {\Lambda }^s{\left(fg\right)\parallel }_{l^p}\lesssim {\parallel f\parallel }_{L^{p_1}}\parallel {\Lambda }^s{g\parallel }_{L^{q_1}}+\parallel {\Lambda }^s{f\parallel }_{L^{p_2}}{\parallel g\parallel }_{L^{q_2}}$$

(14)

with $s>0,\Lambda :={(-\Delta )}^{{\displaystyle \frac{1}{2}}}$ and ${\displaystyle \frac{1}{p}}={\displaystyle \frac{1}{p_1}}+{\displaystyle \frac{1}{q_1}}={\displaystyle \frac{1}{p_2}}+{\displaystyle \frac{1}{q_2}}$. Here, the operator $\Lambda$ is the fractional Laplacian, which has been studied in [18,19]. We will use (14) in the following calculations:

$${∥{\Lambda }^m\left[\left({\displaystyle \frac{i}{k}}\nabla \psi \right)\overline{\psi }\right]∥}_{L^2}\lesssim {\parallel \nabla \psi \parallel }_{L^n}{\parallel {\Lambda }^m\psi \parallel }_{L^{{\displaystyle \frac{2n}{n-2}}}}+{\parallel \psi \parallel }_{L^{\infty }}{\parallel {\Lambda }^m\nabla \psi \parallel }_{L^2}$$

(15)

in (26) and (34).

2. Local Well-Posedness Analysis

This section is devoted to the proof of Theorem 1.

Lemma 1.

Let $(\tilde{\psi },\tilde{A},\tilde{\varphi })\in \mathcal{A}$ be given. Then, the problem (9) and (10) has a unique solution ψ satisfying

$${\parallel \psi \parallel }_{C([0,T];H^s)}+{\parallel \psi \parallel }_{L^2(0,T;H^{s+1})}\le C_0$$

(16)

for some small $0<T\le 1$.

Here and later on, $C_0$ will denote a constant independent of R.

Proof. 

Since the Equation (9) is linear with regular $(\tilde{\psi },\tilde{A},\tilde{\varphi })$ and the existence and uniqueness are well known, we only need to prove the a priori estimates.

Testing (9) by $\overline{\psi }$ and taking the real parts, one has

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\eta }{2}}{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}{\int |\psi |}^2\mathrm{d}x+{\displaystyle \frac{1}{k^2}}\int {|\nabla \psi |}^2\mathrm{d}x& =& \int |\psi {|}^2\mathrm{d}x+\mathrm{Re}\int f\overline{\psi }\mathrm{d}x\hfill \\ & \le & \int |\psi {|}^2\mathrm{d}x+{\parallel f\parallel }_{L^2}{\parallel \psi \parallel }_{L^2}\hfill \end{array}$$

(17)

with

$$\begin{array}{ccc}\hfill {\parallel f\parallel }_{L^2}& \le & {\displaystyle \frac{2}{k}}{\parallel \tilde{A}\parallel }_{L^{\infty }}{\parallel \nabla \tilde{\psi }\parallel }_{L^2}+{\parallel \tilde{A}\parallel }_{L^{\infty }}^2{\parallel \tilde{\psi }\parallel }_{L^2}+{\eta k\parallel \tilde{\varphi }\parallel }_{L^2}{\parallel \tilde{\psi }\parallel }_{L^{\infty }}+{\parallel \tilde{\psi }\parallel }_{L^{\infty }}^2{\parallel \tilde{\psi }\parallel }_{L^2}\hfill \\ & \le & C{R}^2+C{R}^3+CR{\parallel \tilde{\varphi }\parallel }_{L^2}\hfill \end{array}$$

(18)

for some absolute constant C.

Inserting (18) into (17), we see that

$${\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}{\parallel \psi \parallel }_{L^2}\le C{\parallel \psi \parallel }_{L^2}+C{R}^2+C{R}^3+CR{\parallel \tilde{\varphi }\parallel }_{L^2},$$

which gives

$$\begin{array}{ccc}\hfill {\parallel \psi \parallel }_{L^2}& \le & \left[{\parallel {\psi }_0\parallel }_{L^2}+(C{R}^2+C{R}^3)T+CR{\int }_0^T{\parallel \tilde{\varphi }\parallel }_{L^2}\mathrm{d}t\right]e^{CT}\hfill \\ & \le & \left[{\parallel {\psi }_0\parallel }_{L^2}+(C{R}^2+C{R}^3)T+CR{\left({\int }_0^T{\parallel \tilde{\varphi }\parallel }_{L^2}^2\mathrm{d}t\right)}^{{\displaystyle \frac{1}{2}}}\sqrt{T}\right]e^{CT}\hfill \\ & \le & [{\parallel {\psi }_0\parallel }_{L^2}+(C{R}^2+C{R}^3)T+CR\sqrt{R}\sqrt{T}]e^{CT}\hfill \\ & \le & C_0\hfill \end{array}$$

(19)

if $R^{2}T+R^{3}T+R\sqrt{R}\sqrt{T}+T\le 1$.

Applying ${\Lambda }^s$ to (9), testing by ${\Lambda }^s\overline{\psi }$ and taking the real parts, we obtain

$$\begin{array}{ccc}& & {\displaystyle \frac{\eta }{2}}{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}\int |{\Lambda }^s{\psi |}^2\mathrm{d}x+{\displaystyle \frac{1}{k^2}}\int {|{\Lambda }^{s+1}\psi |}^2\mathrm{d}x\hfill \\ & \le & \int |{\Lambda }^s{\psi |}^2\mathrm{d}x+\parallel {\Lambda }^s{f\parallel }_{L^2}{\parallel {\Lambda }^s\psi \parallel }_{L^2},\hfill \end{array}$$

(20)

which gives

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}{\parallel {\Lambda }^s\psi \parallel }_{L^2}& \le & C\parallel {\Lambda }^s{\psi \parallel }_{L^2}+C{\parallel {\Lambda }^{s}f\parallel }_{L^2}\hfill \\ & \le & C\parallel {\Lambda }^s{\psi \parallel }_{L^2}+C{\parallel \tilde{A}\parallel }_{H^s}{\parallel \tilde{\psi }\parallel }_{H^{s+1}}+C{\parallel \tilde{A}\parallel }_{H^s}^2{\parallel \tilde{\psi }\parallel }_{H^s}\hfill \\ & & +C{\parallel \tilde{\varphi }\parallel }_{H^s}{\parallel \tilde{\psi }\parallel }_{H^s}+C{\parallel \tilde{\psi }\parallel }_{H^s}^3\hfill \\ & \le & C\parallel {\Lambda }^s{\psi \parallel }_{L^2}+CR{\parallel \tilde{\psi }\parallel }_{H^{s+1}}+C{R}^3+CR{\parallel \tilde{\varphi }\parallel }_{H^s}.\hfill \end{array}$$

(21)

Solving this inequality, one has

$$\begin{array}{ccc}\hfill \parallel {\Lambda }^s{\psi \parallel }_{L^2}& \le & \left({\parallel {\Lambda }^s{\psi }_0\parallel }_{L^2}+C{R}^{3}T+CR{\int }_0^T{\parallel \tilde{\psi }\parallel }_{H^{s+1}}\mathrm{d}t+CR{\int }_0^T{\parallel \tilde{\varphi }\parallel }_{H^s}\mathrm{d}t\right)e^{CT}\hfill \\ & \le & ({\parallel {\Lambda }^s{\psi }_0\parallel }_{L^2}+C{R}^{3}T+CR\sqrt{R}\sqrt{T})e^{CT}\le C_0\hfill \end{array}$$

(22)

if $R^{3}T+R\sqrt{R}\sqrt{T}+T\le 1$.

Inserting (22) into (19), we derive

$${\parallel \psi \parallel }_{L^2(0,T;H^{s+1})}\le C_0.$$

(23)

This completes the proof of Lemma 1. □

Lemma 2.

Let $\tilde{A}$ and ψ be given. Then, the problem (11) has a unique solution ϕ satisfying

$${\parallel \varphi \parallel }_{L^2(0,T;H^{s+1})}\le C_0,{\parallel \varphi \parallel }_{C([0,T];H^s)}\le C+CR.$$

(24)

for some small $0<T\le 1$.

Proof. 

Since Equation (11) is a Poisson equation with regular $(\psi ,\tilde{A})$, the existence and uniqueness are well known, and we only need to show the a priori estimates.

$$\begin{array}{ccc}\hfill {\parallel \varphi \parallel }_{L^2}& \le & {C\parallel \nabla \varphi \parallel }_{L^{{\displaystyle \frac{2n}{n+2}}}}\le C{∥\left({\displaystyle \frac{i}{k}}\nabla \psi +\psi \tilde{A}\right)\overline{\psi }∥}_{L^{{\displaystyle \frac{2n}{n+2}}}}\hfill \\ & \le & {C(\parallel \nabla \psi \parallel }_{L^2}+{\parallel \psi \parallel }_{L^{\infty }}{\parallel \tilde{A}\parallel }_{L^2}){\parallel \psi \parallel }_{L^n}\hfill \\ & \le & C+C{\parallel \tilde{A}\parallel }_{L^2},\hfill \end{array}$$

(25)

$$\begin{array}{ccc}\hfill \parallel {\Lambda }^{m+1}{\varphi \parallel }_{L^2}& \le & C{∥{\Lambda }^m\left\{\left({\displaystyle \frac{i}{k}}\nabla \psi +\psi \tilde{A}\right)\overline{\psi }\right\}∥}_{L^2}\hfill \\ & \le & {C(\parallel \psi \parallel }_{L^{\infty }}\parallel {\Lambda }^{m+1}{\psi \parallel }_{L^2}+{\parallel \nabla \psi \parallel }_{L^n}{\parallel {\Lambda }^m\psi \parallel }_{L^{{\displaystyle \frac{2n}{n-2}}}})+{C\parallel \psi \parallel }_{H^m}^2{\parallel \tilde{A}\parallel }_{H^m}\hfill \\ & \le & {C\parallel \psi \parallel }_{H^{m+1}}+C{\parallel \tilde{A}\parallel }_{H^m}\hfill \end{array}$$

(26)

Integrating (25) and (26), we obtain

$$\begin{array}{ccc}\hfill {\parallel \varphi \parallel }_{L^2(0,T;H^{s+1})}& \le & {C\parallel \psi \parallel }_{L^2(0,T;H^{s+1})}+C\sqrt{R}\sqrt{T}\hfill \\ & \le & C_0\hfill \end{array}$$

(27)

if $RT\le 1$ by taking $m:=s$.

Similarly, taking $m:=s-1$ in (26), one has

$$\begin{array}{c}\hfill {\parallel \varphi \parallel }_{C([0,T];H^s)}\le C+CR.\end{array}$$

(28)

This completes the proof of Lemma 2. □

Lemma 3.

Let $(\tilde{\psi },\tilde{A},\tilde{\varphi })\in \mathcal{A}$ be given and ϕ be given in Lemma 2. Then, the problem (12) and (13) has a unique smooth solution A satisfying

$$\begin{array}{ccc}& & \mathrm{div}A=0in{\mathbb{R}}^n\times (0,T),\hfill \end{array}$$

(29)

$$\begin{array}{ccc}& & {\parallel A\parallel }_{C([0,T];H^s)}+{\parallel A\parallel }_{L^2(0,T;H^{s+1})}\le C_0\hfill \end{array}$$

(30)

for some small $0<T\le 1$.

Proof. 

Since the Equation (12) is linear with regular $(\tilde{\psi },\tilde{A},\tilde{\varphi })$ and $\varphi$, the existence and uniqueness are well known, we only need to show (29) and (30).

Taking $\mathrm{div}$ to (29) and using (11), we have

$${\partial }_t\mathrm{div}A-\Delta \mathrm{div}A=0.$$

(31)

Using $\mathrm{div}{A}_0=0$ in ${\mathbb{R}}^n$, we obtain (29).

Testing (12) by A and using (29), we infer that

$$\begin{array}{ccc}\hfill {\displaystyle \frac{1}{2}}{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}{\int |A|}^2\mathrm{d}x+\int {|\nabla A|}^2\mathrm{d}x& \le & \int \left|{\displaystyle \frac{i}{k}}\nabla \psi +\psi \tilde{A}\right||\psi ||A|\mathrm{d}x\hfill \\ & \le & \left({∥{\displaystyle \frac{i}{k}}\nabla \psi ∥}_{L^2}+{\parallel \psi \parallel }_{L^{\infty }}{\parallel \tilde{A}\parallel }_{L^2}\right){\parallel \psi \parallel }_{L^{\infty }}{\parallel A\parallel }_{L^2}\hfill \\ & \le & (C+{C\parallel \tilde{A}\parallel }_{L^2}{)\parallel A\parallel }_{L^2}\hfill \\ & \le & {\parallel A\parallel }_{L^2}^2+C+C{R}^2,\hfill \end{array}$$

(32)

which implies

$${\parallel A\parallel }_{L^2}\le C_0$$

(33)

if $R^{2}T+T\le 1$.

Applying ${\Lambda }^s$ to (12), testing by ${\Lambda }^{s}A$ and using (29), we conclude that

$$\begin{array}{ccc}& & {\displaystyle \frac{1}{2}}{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}\int |{\Lambda }^s{A|}^2\mathrm{d}x+\int {|{\Lambda }^{s+1}A|}^2\mathrm{d}x\le \left|\int {\Lambda }^s\left[\left({\displaystyle \frac{i}{k}}\nabla \psi +\psi \tilde{A}\right)\overline{\psi }\right]·{\Lambda }^{s}A\mathrm{d}x\right|\hfill \\ & \le & {C[\parallel \psi \parallel }_{L^{\infty }}{\parallel {\Lambda }^{s+1}\psi \parallel }_{L^2}+{\parallel \nabla \psi \parallel }_{L^n}{\parallel {\Lambda }^s\psi \parallel }_{L^{{\displaystyle \frac{2n}{n-2}}}}+{\parallel \psi \parallel }_{H^s}^2{\parallel \tilde{A}\parallel }_{H^s}]{\parallel {\Lambda }^{s}A\parallel }_{L^2}\hfill \\ & \le & C({\parallel {\Lambda }^{s+1}\psi \parallel }_{L^2}+{\parallel \tilde{A}\parallel }_{H^s}){\parallel {\Lambda }^{s}A\parallel }_{L^2}\hfill \\ & \le & C(\parallel {\Lambda }^{s+1}{\psi \parallel }_{L^2}+R)\parallel {\Lambda }^s{A\parallel }_{L^2},\hfill \end{array}$$

(34)

which leads to (30) if

$$R^{2}T+T\le 1.$$

This completes the proof of Lemma 3. □

We take $R:=100C_0$; then, it is easy to deduce that F maps $\mathcal{A}$ into $\mathcal{A}$ if T is small enough.

We will prove:

Lemma 4.

Let $({\tilde{\psi }}_j,{\tilde{A}}_j,{\tilde{\varphi }}_j)\in \mathcal{A}$ and denote $({\psi }_j,A_j,{\varphi }_j):=F({\tilde{\psi }}_j,{\tilde{A}}_j,{\tilde{\varphi }}_j)(j=1,2)$. Then,

$$\begin{array}{ccc}& & \parallel {\psi }_2-{\psi }_1{\parallel }_{L^2}^2+{\parallel A_2-A_1\parallel }_{L^2}^2+{\int }_0^T{\parallel {\varphi }_2-{\varphi }_1\parallel }_{L^2}^2\mathrm{d}t\hfill \\ & \le & CT{e}^{CT}({\parallel {\tilde{\psi }}_2-{\tilde{\psi }}_1\parallel }_{L^2}^2+{\parallel {\tilde{A}}_2-{\tilde{A}}_1\parallel }_{L^2}^2+\parallel {\tilde{\varphi }}_2-{\tilde{\varphi }}_1{\parallel }_{L^2}^2)\hfill \end{array}$$

(35)

if T is small enough.

Proof. 

It is easy to see that

$$\begin{array}{ccc}& & \eta {\partial }_t({\psi }_2-{\psi }_1)-{\displaystyle \frac{1}{k^2}}\Delta ({\psi }_2-{\psi }_1)-({\psi }_2-{\psi }_1)\hfill \\ & =& -{\displaystyle \frac{2i}{k}}({\tilde{A}}_2\nabla {\tilde{\psi }}_2-{\tilde{A}}_1\nabla {\tilde{\psi }}_1)-(|{\tilde{A}}_2{|}^2{\tilde{\psi }}_2-|{\tilde{A}}_1{|}^2{\tilde{\psi }}_1)\hfill \\ & & -i\eta k({\tilde{\varphi }}_2{\tilde{\psi }}_2-{\tilde{\varphi }}_1{\tilde{\psi }}_1)-(|{\tilde{\psi }}_2{|}^2{\tilde{\psi }}_2-|{\tilde{\psi }}_1|{\tilde{\psi }}_1).\hfill \end{array}$$

(36)

Testing (36) by $\overline{{\psi }_2-{\psi }_1}$ and taking the real parts, we deduce

$$\begin{array}{ccc}& & {\displaystyle \frac{\eta }{2}}{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}\int |{\psi }_2-{\psi }_1{|}^2\mathrm{d}x+{\displaystyle \frac{1}{k^2}}\int {|\nabla ({\psi }_2-{\psi }_1)|}^2\mathrm{d}x\hfill \\ & \le & \int |{\psi }_2-{\psi }_1{|}^2\mathrm{d}x+\left|{\displaystyle \frac{2}{k}}\int ({\tilde{A}}_2{\tilde{\psi }}_2-{\tilde{A}}_1{\tilde{\psi }}_1)\nabla ({\psi }_2-{\psi }_1)\mathrm{d}x\right|\hfill \\ & & +C({\parallel {\tilde{A}}_2-{\tilde{A}}_1\parallel }_{L^2}+{\parallel {\tilde{\psi }}_2-{\tilde{\psi }}_1\parallel }_{L^2}){\parallel {\psi }_2-{\psi }_1\parallel }_{L^2}+C{\parallel {\tilde{\varphi }}_2-{\tilde{\varphi }}_1\parallel }_{L^2}{\parallel {\psi }_2-{\psi }_1\parallel }_{L^2}\hfill \\ & \le & {\displaystyle \frac{1}{2k^2}}{\parallel \nabla ({\psi }_2-{\psi }_1)\parallel }_{L^2}^2+C{\parallel {\psi }_2-{\psi }_1\parallel }_{L^2}^2\hfill \\ & & +C({\parallel {\tilde{\psi }}_2-{\tilde{\psi }}_1\parallel }_{L^2}^2+{\parallel {\tilde{A}}_2-{\tilde{A}}_1\parallel }_{L^2}^2+{\parallel {\tilde{\varphi }}_2-{\tilde{\varphi }}_1\parallel }_{L^2}^2),\hfill \end{array}$$

which implies

$$\parallel {\psi }_2-{\psi }_1{\parallel }_{L^2}^2+{\int }_0^T{\parallel \nabla ({\psi }_2-{\psi }_1)\parallel }_{L^2}^2\mathrm{d}t\le \mathrm{RHS}\mathrm{of}\left(35\right).$$

(37)

Similarly, it is easy to observe that

$$\begin{array}{ccc}& & {\partial }_t(A_2-A_1)+\nabla ({\varphi }_2-{\varphi }_1)-\Delta (A_2-A_1)\hfill \\ & =& -\mathrm{Re}\left\{\left({\displaystyle \frac{i}{k}}\nabla {\psi }_2+{\psi }_2{\tilde{A}}_2\right)\overline{{\psi }_2}-\left({\displaystyle \frac{i}{k}}\nabla {\psi }_1+{\psi }_1{\tilde{A}}_1\right)\overline{{\psi }_1}\right\}.\hfill \end{array}$$

(38)

Testing (38) by $A_2-A_1$ and using (29), we have

$$\begin{array}{ccc}& & {\displaystyle \frac{1}{2}}{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}\int |A_2-A_1{|}^2\mathrm{d}x+\int {|\nabla (A_2-A_1)|}^2\mathrm{d}x\hfill \\ & \le & C({\parallel \nabla ({\psi }_2-{\psi }_1)\parallel }_{L^2}{\parallel {\psi }_2\parallel }_{L^{\infty }}+{\parallel \nabla {\psi }_2\parallel }_{L^n}{\parallel {\psi }_2-{\psi }_1\parallel }_{L^{{\displaystyle \frac{2n}{n-2}}}}){\parallel A_2-A_1\parallel }_{L^2}\hfill \\ & & +C({\parallel {\psi }_2-{\psi }_1\parallel }_{L^2}+{\parallel {\tilde{A}}_2-{\tilde{A}}_1\parallel }_{L^2}){\parallel A_2-A_1\parallel }_{L^2}\hfill \\ & \le & C({\parallel \nabla ({\psi }_2-{\psi }_1)\parallel }_{L^2}+{\parallel {\psi }_2-{\psi }_1\parallel }_{L^2}+{\parallel {\tilde{A}}_2-{\tilde{A}}_1\parallel }_{L^2}){\parallel A_2-A_1\parallel }_{L^2},\hfill \end{array}$$

which implies

$$\begin{array}{ccc}& & \parallel A_2-A_1{\parallel }_{L^2}^2+{\int }_0^T{\parallel \nabla (A_2-A_1)\parallel }_{L^2}^2\mathrm{d}t\hfill \\ & \le & C\left(T\parallel {\tilde{A}}_2-{\tilde{A}}_1{\parallel }_{L^2}^2+{\int }_0^T{\parallel {\psi }_2-{\psi }_1\parallel }_{H^1}^2\mathrm{d}t\right)e^{CT}\hfill \\ & \le & \mathrm{RHS}\mathrm{of}\left(35\right).\hfill \end{array}$$

(39)

Taking $\mathrm{div}$ to (38), we arrive at

$$\begin{array}{ccc}\hfill \parallel {\varphi }_2-{\varphi }_1{\parallel }_{L^2}& \le & C\parallel \nabla ({\varphi }_2-{\varphi }_1){\parallel }_{L^{{\displaystyle \frac{2n}{n+2}}}}\hfill \\ & \le & C{∥\left({\displaystyle \frac{i}{k}}\nabla {\psi }_2+{\psi }_2{\tilde{A}}_2\right)\overline{{\psi }_2}-\left({\displaystyle \frac{i}{k}}\nabla {\psi }_1+{\psi }_1{\tilde{A}}_1\right)\overline{{\psi }_1}∥}_{L^{{\displaystyle \frac{2n}{n+2}}}}\hfill \\ & \le & C({\parallel \nabla ({\psi }_2-{\psi }_1)\parallel }_{L^2}+{\parallel {\psi }_2-{\psi }_1\parallel }_{L^2}+{\parallel {\tilde{A}}_2-{\tilde{A}}_1\parallel }_{L^2}).\hfill \end{array}$$

(40)

Integrating (40), one has

$${\int }_0^T{\parallel {\varphi }_2-{\varphi }_1\parallel }_{L^2}^2\mathrm{d}t\le \mathrm{RHS}\mathrm{of}\left(35\right).$$

(41)

This completes the proof of Lemma 4. □

Proof of Theorem 1.

By Lemmas 1–4 and the Banach fixed point theorem, it is easy to show that Theorem 1 holds true.

This completes the proof. □

3. A Regularity Criterion

A regularity criterion in Besov space to the Navier–Stokes system has been studied in [20]. This section is devoted to the proof of Theorem 2. We only need to show the a priori estimates.

Proof. 

First, following the method in [16], we have

$$\begin{array}{ccc}& & |\psi |\le {C,\parallel \psi \parallel }_{L^{\infty }(0,T;L^2)}+{\parallel \psi \parallel }_{L^2(0,T;H^1)}\le C,\hfill \end{array}$$

(42)

$$\begin{array}{ccc}& & {\parallel A\parallel }_{L^{\infty }(0,T;H^1)}+{\parallel A\parallel }_{L^2(0,T;H^2)}\le C,\hfill \end{array}$$

(43)

$$\begin{array}{ccc}& & {\parallel \varphi \parallel }_{L^2(0,T;H^1)}\le C.\hfill \end{array}$$

(44)

Testing (1) by $-\Delta \overline{\psi }$ and taking the real parts, using (42)–(44), we obtain

$$\begin{array}{ccc}& & {\displaystyle \frac{\eta }{2}}{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}\int |\nabla \psi {|}^2\mathrm{d}x+{\displaystyle \frac{1}{k^2}}\int |\Delta \psi {|}^2\mathrm{d}x+\int |A{|}^2|\nabla \psi {|}^2\mathrm{d}x\hfill \\ & =& \mathrm{Re}\int i\eta k\varphi \psi \Delta \overline{\psi }\mathrm{d}x-\mathrm{Re}\int {\displaystyle \frac{2i}{k}}A·\nabla \psi \Delta \overline{\psi }\mathrm{d}x\hfill \\ & & -\mathrm{Re}\int \nabla |A{|}^2·\psi \nabla \overline{\psi }\mathrm{d}x+\mathrm{Re}\int (|\psi {|}^2\psi -\psi )\Delta \overline{\psi }\mathrm{d}x\hfill \\ & \le & {C\parallel \varphi \parallel }_{L^2}{\parallel \Delta \psi \parallel }_{L^2}+{C\parallel A\parallel }_{L^4}{\parallel \nabla \psi \parallel }_{L^4}{\parallel \Delta \psi \parallel }_{L^2}\hfill \\ & & +{\displaystyle \frac{1}{4}}\int |A{|}^2|\nabla \psi {|}^2\mathrm{d}x+{C\parallel \nabla A\parallel }_{L^2}^2+{C\parallel \psi \parallel }_{L^2}{\parallel \Delta \psi \parallel }_{L^2}\hfill \\ & \le & {C\parallel \varphi \parallel }_{L^2}{\parallel \Delta \psi \parallel }_{L^2}+{C\parallel A\parallel }_{L^4}·{\parallel \psi \parallel }_{L^{\infty }}^{{\displaystyle \frac{1}{2}}}{\parallel \Delta \psi \parallel }_{L^2}^{{\displaystyle \frac{1}{2}}}·{\parallel \Delta \psi \parallel }_{L^2}\hfill \\ & & +{\displaystyle \frac{1}{4}}\int |A{|}^2|\nabla \psi {|}^2\mathrm{d}x+C+C{\parallel \Delta \psi \parallel }_{L^2}\hfill \\ & \le & {\displaystyle \frac{1}{4}}\int |A{|}^2|\nabla \psi {|}^2\mathrm{d}x+{\displaystyle \frac{1}{4k^2}}\int |\Delta \psi {|}^2\mathrm{d}x+{C\parallel \varphi \parallel }_{L^2}^2+C{\parallel A\parallel }_{L^4}^4+C.\hfill \end{array}$$

(45)

Here, we have used the Gagliardo–Nirenberg inequality [21]:

$${\parallel \nabla \psi \parallel }_{L^4}^2\lesssim {\parallel \psi \parallel }_{L^{\infty }}{\parallel \Delta \psi \parallel }_{L^2}.$$

(46)

Testing (2) by ${|A|}^{2}A$, using (42) and (46), we have

$$\begin{array}{ccc}& & {\displaystyle \frac{1}{4}}{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}\int |A{|}^4\mathrm{d}x-\int {|A|}^{2}A·\Delta A\mathrm{d}x\hfill \\ & =& -\int |A{|}^{2}A·\nabla \varphi \mathrm{d}x-\mathrm{Re}\int {|A|}^{2}A\left\{\left({\displaystyle \frac{i}{k}}\nabla \varphi +\psi A\right)\overline{\psi }\right\}\mathrm{d}x\hfill \\ & \le & {\parallel A\parallel }_{L^4}^3{\parallel \nabla \varphi \parallel }_{L^4}+{\parallel A\parallel }_{L^4}^3{∥{\displaystyle \frac{i}{k}}\nabla \psi +\psi A∥}_{L^4}{\parallel \psi \parallel }_{L^{\infty }}\hfill \\ & \le & {C\parallel A\parallel }_{L^4}^3{∥{\displaystyle \frac{i}{k}}\nabla \psi +\psi A∥}_{L^4}{\parallel \psi \parallel }_{L^{\infty }}\hfill \\ & \le & {C\parallel A\parallel }_{L^4}^3{(\parallel \nabla \psi \parallel }_{L^4}+{\parallel A\parallel }_{L^4})\hfill \\ & \le & {C\parallel A\parallel }_{L^4}^3{(\parallel \psi \parallel }_{L^{\infty }}^{{\displaystyle \frac{1}{2}}}{\parallel \Delta \psi \parallel }_{L^2}^{{\displaystyle \frac{1}{2}}}+{\parallel A\parallel }_{L^4})\hfill \\ & \le & {\displaystyle \frac{1}{4k^2}}{\parallel \Delta \psi \parallel }_{L^2}^2+C{\parallel A\parallel }_{L^4}^4.\hfill \end{array}$$

(47)

Here, we have used the estimate

$${\parallel \nabla \varphi \parallel }_{L^4}\lesssim {∥\left({\displaystyle \frac{i}{k}}\nabla \psi +\psi A\right)\overline{\psi }∥}_{L^4}.$$

(48)

Combining (45) and (47) and using

$$-\int |A{|}^{2}A·\Delta A\mathrm{d}x\ge 0,$$

(49)

and using the Gronwall inequality, we reach

$${\parallel \psi \parallel }_{L^2(0,T;H^2)}+{\parallel A\parallel }_{L^{\infty }(0,T;L^4)}\le C,$$

(50)

whence

$${\parallel \nabla \psi \parallel }_{L^4(0,T;L^4)}\le C.$$

(51)

This completes the proof. □

4. Conclusions

In this paper, we prove the local well-posedness of strong solutions and a regularity criterion to the $nD(n\ge 3)$ time-dependent Ginzburg–Landau system in superconductivity with the Coulomb gauge. In the future, we wish to study the global well-posedness of the problem.