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Article

Local Well-Posedness of Classical Solutions to the Time-Dependent Ginzburg–Landau Model for Superconductivity in Rn

1
Department of Applied Mathematics, Nanjing Forestry University, Nanjing 210037, China
2
Department of Mathematics, Wenzhou University, Wenzhou 325035, China
3
Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China
*
Author to whom correspondence should be addressed.
Mathematics 2025, 13(11), 1697; https://doi.org/10.3390/math13111697
Submission received: 28 April 2025 / Revised: 17 May 2025 / Accepted: 20 May 2025 / Published: 22 May 2025
(This article belongs to the Section C1: Difference and Differential Equations)

Abstract

In this paper, we prove the local well-posedness of classical solutions ( ψ , A , ϕ ) to the n D ( n 3 ) time-dependent Ginzburg–Landau model in superconductivity with the choice of Coulomb gauge and the main assumptions ψ 0 , A 0 H s ( R n ) with div A 0 = 0 in R n and s > n 2 . This result can be used in the proof of regularity criterion and global-in-time well-posedness of the strong solution.
MSC:
35A05; 35A40; 35K55; 82D55

1. Introduction

In this paper, we consider the n D ( n 3 ) time-dependent Ginzburg–Landau model in superconductivity with the Coulomb gauge:
η t ψ + i η k ϕ ψ + i k + A 2 ψ + ( | ψ | 2 1 ) ψ = 0 ,
t A + ϕ Δ A + Re i k ψ + ψ A ψ ¯ = 0 ,
div A = 0 in R n × ( 0 , ) ,
( ψ , A ) ( · , 0 ) = ( ψ 0 , A 0 ) ( · ) in R n ,
where ψ , A , and ϕ denote C -valued, R n -valued, and R -valued functions, respectively, and they stand for the order parameter, the magnetic potential, and the electric potential, respectively. Two positive constants η and k are Ginzburg–Landau constants, i : = 1 . ψ ¯ denotes the complex conjugate of ψ , Re ψ : = ψ + ψ ¯ 2 is the real part of ψ , and | ψ | 2 : = ψ ψ ¯ 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 ψ 0 H 1 L , A 0 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 ψ 0 , A 0 L 3 ( n = 3 ) . Fan-Jiang [4] showed the global existence of weak solutions when ψ 0 , A 0 L 2 .
There are also many studies considering the problem in a bounded domain Ω 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
ψ L 2 ( 0 , T ; L n ) .
This was proved to be true in [16] when 3 n 17 . The key estimate was to show that
ψ L 4 ( 0 , T ; L 4 ) + A L ( 0 , T ; L 4 ) C .
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 R n for any n 3 . We will prove the following:
Theorem 1.
Let ψ 0 , A 0 H s ( R n ) with div A 0 = 0 in R n and s > n 2 . Let ϕ 0 satisfy
Δ ϕ 0 = Rediv i k ψ 0 + ψ 0 A 0 ψ 0 ¯ i n R n .
Then, the problem (1)–(4) has a unique smooth solution ( ψ , A , ϕ ) satisfying
ψ , A , ϕ C ( [ 0 , T ] ; H s ) L 2 ( 0 , T ; H s + 1 )
for some 0 < T .
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 ψ 0 , A 0 H s ( R n ) with div A 0 = 0 in R n and s > 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
A : = { ( ψ ˜ , A ˜ , ϕ ˜ ) A ; ( ψ ˜ , A ˜ , ϕ ˜ ) ( · , 0 ) = ( ψ 0 , A 0 , ϕ 0 ) , div A ˜ = 0 , ( ψ ˜ , A ˜ , ϕ ˜ ) A R }
with the norm
( ψ ˜ , A ˜ , ϕ ˜ ) A : = ( ψ ˜ , A ˜ ) C ( [ 0 , T ] ; H s ) + ( ψ ˜ , A ˜ , ϕ ˜ ) L 2 ( 0 , T ; H s + 1 ) .
Given ( ψ ˜ , A ˜ , ϕ ˜ ) A , we consider the following linear problems:
η t ψ 1 k 2 Δ ψ ψ = f : = 2 i k A ˜ · ψ ˜ | A ˜ | 2 ψ ˜ i η k ϕ ˜ ψ ˜ | ψ ˜ | 2 ψ ˜ ,
ψ ( · , 0 ) = ψ 0 in R n .
Δ ϕ = Rediv i k ψ + ψ A ˜ ψ ¯ .
t A Δ A = ϕ Re i k ψ + ψ A ˜ ψ ¯ ,
A ( · , 0 ) = A 0 and div A 0 = 0 in R n .
Let ( ψ , A , ϕ ) be the unique smooth solution to the above problems; we define the fixed point map F : ( ψ ˜ , A ˜ , ϕ ˜ ) A ( ψ , A , ϕ ) A . We will prove that the map F maps A into A for suitable constant A and small T, and F is a contraction mapping, and thus, F has a unique fixed point in A . This proves Theorem 1.
Our proof will use the following bilinear product estimate due to Kato–Ponce [17]:
Λ s ( f g ) l p f L p 1 Λ s g L q 1 + Λ s f L p 2 g L q 2
with s > 0 , Λ : = ( Δ ) 1 2 and 1 p = 1 p 1 + 1 q 1 = 1 p 2 + 1 q 2 . Here, the operator Λ is the fractional Laplacian, which has been studied in [18,19]. We will use (14) in the following calculations:
Λ m i k ψ ψ ¯ L 2 ψ L n Λ m ψ L 2 n n 2 + ψ L Λ m ψ L 2
in (26) and (34).

2. Local Well-Posedness Analysis

This section is devoted to the proof of Theorem 1.
Lemma 1.
Let ( ψ ˜ , A ˜ , ϕ ˜ ) A be given. Then, the problem (9) and (10) has a unique solution ψ satisfying
ψ C ( [ 0 , T ] ; H s ) + ψ L 2 ( 0 , T ; H s + 1 ) C 0
for some small 0 < T 1 .
Here and later on, C 0 will denote a constant independent of R.
Proof. 
Since the Equation (9) is linear with regular ( ψ ˜ , A ˜ , ϕ ˜ ) and the existence and uniqueness are well known, we only need to prove the a priori estimates.
Testing (9) by ψ ¯ and taking the real parts, one has
η 2 d d t | ψ | 2 d x + 1 k 2 | ψ | 2 d x = | ψ | 2 d x + Re f ψ ¯ d x | ψ | 2 d x + f L 2 ψ L 2
with
f L 2 2 k A ˜ L ψ ˜ L 2 + A ˜ L 2 ψ ˜ L 2 + η k ϕ ˜ L 2 ψ ˜ L + ψ ˜ L 2 ψ ˜ L 2 C R 2 + C R 3 + C R ϕ ˜ L 2
for some absolute constant C.
Inserting (18) into (17), we see that
d d t ψ L 2 C ψ L 2 + C R 2 + C R 3 + C R ϕ ˜ L 2 ,
which gives
ψ L 2 ψ 0 L 2 + ( C R 2 + C R 3 ) T + C R 0 T ϕ ˜ L 2 d t e C T ψ 0 L 2 + ( C R 2 + C R 3 ) T + C R 0 T ϕ ˜ L 2 2 d t 1 2 T e C T [ ψ 0 L 2 + ( C R 2 + C R 3 ) T + C R R T ] e C T C 0
if R 2 T + R 3 T + R R T + T 1 .
Applying Λ s to (9), testing by Λ s ψ ¯ and taking the real parts, we obtain
η 2 d d t | Λ s ψ | 2 d x + 1 k 2 | Λ s + 1 ψ | 2 d x | Λ s ψ | 2 d x + Λ s f L 2 Λ s ψ L 2 ,
which gives
d d t Λ s ψ L 2 C Λ s ψ L 2 + C Λ s f L 2 C Λ s ψ L 2 + C A ˜ H s ψ ˜ H s + 1 + C A ˜ H s 2 ψ ˜ H s + C ϕ ˜ H s ψ ˜ H s + C ψ ˜ H s 3 C Λ s ψ L 2 + C R ψ ˜ H s + 1 + C R 3 + C R ϕ ˜ H s .
Solving this inequality, one has
Λ s ψ L 2 Λ s ψ 0 L 2 + C R 3 T + C R 0 T ψ ˜ H s + 1 d t + C R 0 T ϕ ˜ H s d t e C T ( Λ s ψ 0 L 2 + C R 3 T + C R R T ) e C T C 0
if R 3 T + R R T + T 1 .
Inserting (22) into (19), we derive
ψ L 2 ( 0 , T ; H s + 1 ) C 0 .
This completes the proof of Lemma 1. □
Lemma 2.
Let A ˜ and ψ be given. Then, the problem (11) has a unique solution ϕ satisfying
ϕ L 2 ( 0 , T ; H s + 1 ) C 0 , ϕ C ( [ 0 , T ] ; H s ) C + C R .
for some small 0 < T 1 .
Proof. 
Since Equation (11) is a Poisson equation with regular ( ψ , A ˜ ) , the existence and uniqueness are well known, and we only need to show the a priori estimates.
ϕ L 2 C ϕ L 2 n n + 2 C i k ψ + ψ A ˜ ψ ¯ L 2 n n + 2 C ( ψ L 2 + ψ L A ˜ L 2 ) ψ L n C + C A ˜ L 2 ,
Λ m + 1 ϕ L 2 C Λ m i k ψ + ψ A ˜ ψ ¯ L 2 C ( ψ L Λ m + 1 ψ L 2 + ψ L n Λ m ψ L 2 n n 2 ) + C ψ H m 2 A ˜ H m C ψ H m + 1 + C A ˜ H m
Integrating (25) and (26), we obtain
ϕ L 2 ( 0 , T ; H s + 1 ) C ψ L 2 ( 0 , T ; H s + 1 ) + C R T C 0
if R T 1 by taking m : = s .
Similarly, taking m : = s 1 in (26), one has
ϕ C ( [ 0 , T ] ; H s ) C + C R .
This completes the proof of Lemma 2. □
Lemma 3.
Let ( ψ ˜ , A ˜ , ϕ ˜ ) A be given and ϕ be given in Lemma 2. Then, the problem (12) and (13) has a unique smooth solution A satisfying
div A = 0 i n R n × ( 0 , T ) ,
A C ( [ 0 , T ] ; H s ) + A L 2 ( 0 , T ; H s + 1 ) C 0
for some small 0 < T 1 .
Proof. 
Since the Equation (12) is linear with regular ( ψ ˜ , A ˜ , ϕ ˜ ) and ϕ , the existence and uniqueness are well known, we only need to show (29) and (30).
Taking div to (29) and using (11), we have
t div A Δ div A = 0 .
Using div A 0 = 0 in R n , we obtain (29).
Testing (12) by A and using (29), we infer that
1 2 d d t | A | 2 d x + | A | 2 d x i k ψ + ψ A ˜ | ψ | | A | d x i k ψ L 2 + ψ L A ˜ L 2 ψ L A L 2 ( C + C A ˜ L 2 ) A L 2 A L 2 2 + C + C R 2 ,
which implies
A L 2 C 0
if R 2 T + T 1 .
Applying Λ s to (12), testing by Λ s A and using (29), we conclude that
1 2 d d t | Λ s A | 2 d x + | Λ s + 1 A | 2 d x Λ s i k ψ + ψ A ˜ ψ ¯ · Λ s A d x C [ ψ L Λ s + 1 ψ L 2 + ψ L n Λ s ψ L 2 n n 2 + ψ H s 2 A ˜ H s ] Λ s A L 2 C ( Λ s + 1 ψ L 2 + A ˜ H s ) Λ s A L 2 C ( Λ s + 1 ψ L 2 + R ) Λ s A L 2 ,
which leads to (30) if
R 2 T + T 1 .
This completes the proof of Lemma 3. □
We take R : = 100 C 0 ; then, it is easy to deduce that F maps A into A if T is small enough.
We will prove:
Lemma 4.
Let ( ψ ˜ j , A ˜ j , ϕ ˜ j ) A and denote ( ψ j , A j , ϕ j ) : = F ( ψ ˜ j , A ˜ j , ϕ ˜ j ) ( j = 1 , 2 ) . Then,
ψ 2 ψ 1 L 2 2 + A 2 A 1 L 2 2 + 0 T ϕ 2 ϕ 1 L 2 2 d t C T e C T ( ψ ˜ 2 ψ ˜ 1 L 2 2 + A ˜ 2 A ˜ 1 L 2 2 + ϕ ˜ 2 ϕ ˜ 1 L 2 2 )
if T is small enough.
Proof. 
It is easy to see that
η t ( ψ 2 ψ 1 ) 1 k 2 Δ ( ψ 2 ψ 1 ) ( ψ 2 ψ 1 ) = 2 i k ( A ˜ 2 ψ ˜ 2 A ˜ 1 ψ ˜ 1 ) ( | A ˜ 2 | 2 ψ ˜ 2 | A ˜ 1 | 2 ψ ˜ 1 ) i η k ( ϕ ˜ 2 ψ ˜ 2 ϕ ˜ 1 ψ ˜ 1 ) ( | ψ ˜ 2 | 2 ψ ˜ 2 | ψ ˜ 1 | ψ ˜ 1 ) .
Testing (36) by ψ 2 ψ 1 ¯ and taking the real parts, we deduce
η 2 d d t | ψ 2 ψ 1 | 2 d x + 1 k 2 | ( ψ 2 ψ 1 ) | 2 d x | ψ 2 ψ 1 | 2 d x + 2 k ( A ˜ 2 ψ ˜ 2 A ˜ 1 ψ ˜ 1 ) ( ψ 2 ψ 1 ) d x + C ( A ˜ 2 A ˜ 1 L 2 + ψ ˜ 2 ψ ˜ 1 L 2 ) ψ 2 ψ 1 L 2 + C ϕ ˜ 2 ϕ ˜ 1 L 2 ψ 2 ψ 1 L 2 1 2 k 2 ( ψ 2 ψ 1 ) L 2 2 + C ψ 2 ψ 1 L 2 2 + C ( ψ ˜ 2 ψ ˜ 1 L 2 2 + A ˜ 2 A ˜ 1 L 2 2 + ϕ ˜ 2 ϕ ˜ 1 L 2 2 ) ,
which implies
ψ 2 ψ 1 L 2 2 + 0 T ( ψ 2 ψ 1 ) L 2 2 d t RHS of ( 35 ) .
Similarly, it is easy to observe that
t ( A 2 A 1 ) + ( ϕ 2 ϕ 1 ) Δ ( A 2 A 1 ) = Re i k ψ 2 + ψ 2 A ˜ 2 ψ 2 ¯ i k ψ 1 + ψ 1 A ˜ 1 ψ 1 ¯ .
Testing (38) by A 2 A 1 and using (29), we have
1 2 d d t | A 2 A 1 | 2 d x + | ( A 2 A 1 ) | 2 d x C ( ( ψ 2 ψ 1 ) L 2 ψ 2 L + ψ 2 L n ψ 2 ψ 1 L 2 n n 2 ) A 2 A 1 L 2 + C ( ψ 2 ψ 1 L 2 + A ˜ 2 A ˜ 1 L 2 ) A 2 A 1 L 2 C ( ( ψ 2 ψ 1 ) L 2 + ψ 2 ψ 1 L 2 + A ˜ 2 A ˜ 1 L 2 ) A 2 A 1 L 2 ,
which implies
A 2 A 1 L 2 2 + 0 T ( A 2 A 1 ) L 2 2 d t C T A ˜ 2 A ˜ 1 L 2 2 + 0 T ψ 2 ψ 1 H 1 2 d t e C T RHS of ( 35 ) .
Taking div to (38), we arrive at
ϕ 2 ϕ 1 L 2 C ( ϕ 2 ϕ 1 ) L 2 n n + 2 C i k ψ 2 + ψ 2 A ˜ 2 ψ 2 ¯ i k ψ 1 + ψ 1 A ˜ 1 ψ 1 ¯ L 2 n n + 2 C ( ( ψ 2 ψ 1 ) L 2 + ψ 2 ψ 1 L 2 + A ˜ 2 A ˜ 1 L 2 ) .
Integrating (40), one has
0 T ϕ 2 ϕ 1 L 2 2 d t RHS of ( 35 ) .
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
| ψ | C , ψ L ( 0 , T ; L 2 ) + ψ L 2 ( 0 , T ; H 1 ) C ,
A L ( 0 , T ; H 1 ) + A L 2 ( 0 , T ; H 2 ) C ,
ϕ L 2 ( 0 , T ; H 1 ) C .
Testing (1) by Δ ψ ¯ and taking the real parts, using (42)–(44), we obtain
η 2 d d t | ψ | 2 d x + 1 k 2 | Δ ψ | 2 d x + | A | 2 | ψ | 2 d x = Re i η k ϕ ψ Δ ψ ¯ d x Re 2 i k A · ψ Δ ψ ¯ d x Re | A | 2 · ψ ψ ¯ d x + Re ( | ψ | 2 ψ ψ ) Δ ψ ¯ d x C ϕ L 2 Δ ψ L 2 + C A L 4 ψ L 4 Δ ψ L 2 + 1 4 | A | 2 | ψ | 2 d x + C A L 2 2 + C ψ L 2 Δ ψ L 2 C ϕ L 2 Δ ψ L 2 + C A L 4 · ψ L 1 2 Δ ψ L 2 1 2 · Δ ψ L 2 + 1 4 | A | 2 | ψ | 2 d x + C + C Δ ψ L 2 1 4 | A | 2 | ψ | 2 d x + 1 4 k 2 | Δ ψ | 2 d x + C ϕ L 2 2 + C A L 4 4 + C .
Here, we have used the Gagliardo–Nirenberg inequality [21]:
ψ L 4 2 ψ L Δ ψ L 2 .
Testing (2) by | A | 2 A , using (42) and (46), we have
1 4 d d t | A | 4 d x | A | 2 A · Δ A d x = | A | 2 A · ϕ d x Re | A | 2 A i k ϕ + ψ A ψ ¯ d x A L 4 3 ϕ L 4 + A L 4 3 i k ψ + ψ A L 4 ψ L C A L 4 3 i k ψ + ψ A L 4 ψ L C A L 4 3 ( ψ L 4 + A L 4 ) C A L 4 3 ( ψ L 1 2 Δ ψ L 2 1 2 + A L 4 ) 1 4 k 2 Δ ψ L 2 2 + C A L 4 4 .
Here, we have used the estimate
ϕ L 4 i k ψ + ψ A ψ ¯ L 4 .
Combining (45) and (47) and using
| A | 2 A · Δ A d x 0 ,
and using the Gronwall inequality, we reach
ψ L 2 ( 0 , T ; H 2 ) + A L ( 0 , T ; L 4 ) C ,
whence
ψ L 4 ( 0 , T ; L 4 ) C .
This completes the proof. □

4. Conclusions

In this paper, we prove the local well-posedness of strong solutions and a regularity criterion to the n D ( n 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.

Author Contributions

Investigation, J.F. and Y.Z.; Writing—original draft, J.F.; Writing—review & editing, Y.Z.; Project administration, J.F. All authors have read and agreed to the published version of the manuscript.

Funding

This work is partially supported by NSFC (No. 11971234).

Data Availability Statement

The original contributions presented in this study are included in the article. Further inquiries can be directed to the corresponding author.

Acknowledgments

The authors are indebted to the referees for useful suggestions.

Conflicts of Interest

The authors declare no conflict of interest.

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Fan, J.; Zhou, Y. Local Well-Posedness of Classical Solutions to the Time-Dependent Ginzburg–Landau Model for Superconductivity in Rn. Mathematics 2025, 13, 1697. https://doi.org/10.3390/math13111697

AMA Style

Fan J, Zhou Y. Local Well-Posedness of Classical Solutions to the Time-Dependent Ginzburg–Landau Model for Superconductivity in Rn. Mathematics. 2025; 13(11):1697. https://doi.org/10.3390/math13111697

Chicago/Turabian Style

Fan, Jishan, and Yong Zhou. 2025. "Local Well-Posedness of Classical Solutions to the Time-Dependent Ginzburg–Landau Model for Superconductivity in Rn" Mathematics 13, no. 11: 1697. https://doi.org/10.3390/math13111697

APA Style

Fan, J., & Zhou, Y. (2025). Local Well-Posedness of Classical Solutions to the Time-Dependent Ginzburg–Landau Model for Superconductivity in Rn. Mathematics, 13(11), 1697. https://doi.org/10.3390/math13111697

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