Electronic Journal of Differential Equations, Vol. 2020 (2020), No. 17, pp. 1–6.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

EXISTENCE AND UNIQUENESS FOR A GINZBURG-LANDAU

SYSTEM FOR SUPERCONDUCTIVITY

JISHAN FAN, YONG ZHOU

Abstract. We prove the existence of a unique solution for a time-dependent

Ginzburg-Landau model in superconductivity under the Coulomb gauge. Also
we prove the uniform-in-ε well-posedness of the solution, where ε is the coeffi-

cient of the double-well potential energy.

1. Introduction

This article concerns the Ginzburg-Landau model in superconductivity,

η∂tψ + iηkφψ +
( i
k
∇+A

)2

ψ + εa(|ψ|2 − 1)ψ = 0, (1.1)

∂tA+∇φ+ curl2A+ Re
{( i

k
∇ψ + ψA

)
ψ
}

= 0 (1.2)

in QT := (0, T )× Ω, with boundary and initial conditions

∇ψ · ν = 0, A · ν = 0, curlA× ν = 0 on (0, T )× ∂Ω, (1.3)

(ψ,A)(x, 0) = (ψ0, A0)(x) in Ω. (1.4)

Here Ω ⊂ Rd is a bounded domain with smooth boundary ∂Ω, ν is the outward
normal to ∂Ω, and T is any given positive constant. The unknowns ψ,A, and φ are
C-valued, Rd-valued, and R-valued functions, respectively, and they stand for the
order parameter, the magnetic potential, and the electric potential, respectively. η
and k are Ginzburg-Landau positive constants. ψ denotes the complex conjugate
of ψ, Reψ := (ψ+ψ)/2, |ψ|2 := ψψ is the density of superconducting carriers, and
i :=
√
−1. ε is a positive constant. We will assume a = 1.

It is well known that the Ginzburg-Landau equations are gauge invariant, namely
if (ψ,A, φ) is a solution of (1.1)-(1.4), then for any real-valued smooth function
χ, (ψeikχ, A +∇χ, φ − ∂tχ) is also a solution of (1.1)-(1.4). So, in order to obtain
the well-posedness of the problem, we need to impose suitable gauge condition.
From the physical point of view, one usually has four types of the gauge conditions:

• Coulomb gauge: divA = 0 in Ω and
∫

Ω
φdx = 0.

• Lorentz gauge: φ = −divA in Ω.
• Lorenz gauge: ∂tφ = −divA in Ω.
• Temporal gauge(Weyl gauge): φ = 0 in Ω.

2010 Mathematics Subject Classification. 35Q35, 35K55.

Key words and phrases. Ginzburg-Landau model; superconductivity; Coulomb gauge.
c©2020 Texas State University.

Submitted December 6, 2019. Published February 11, 2020.

1



2 J. FAN, Y. ZHOU EJDE-2020/17

For the initial data ψ0 ∈ H1(Ω), |ψ0| ≤ 1, A0 ∈ H1(Ω), Chen, Elliott and Tang
[3], Chen, Hoffmann and Liang [4], Du [5] and Tang [11] proved the existence and
uniqueness of a global strong solution to (1.1)-(1.4), in the case of the Coulomb,
Lorentz, and temporal gauges. For the initial data ψ0 ∈ H1(Ω), A0 ∈ H1(Ω),
Tang and Wang [12] obtained the existence and uniqueness of global strong solu-
tions, while Fan and Jiang [8] showed the existence of global weak solutions when
ψ0, A0 ∈ L2. Fan and Ozawa [9] (2-D) and Fan, Gao and Guo [7, 6] (3-D) prove
the uniqueness of a weak solution for ψ0, A0 ∈ Ld with d = 2, 3, which is criti-
cal. This comes from a scaling argument for (1.1) and (1.2). Move precisely, if
(ψ(t, x), A(t, x), φ(t, x)) is a solution of (1.1) and (1.2) associated with the initial
data (ψ0(x), A0(x)) without linear lower order term ψ, then

(λψ(λ2t, λx), λA(λ2t, λx), λ2φ(λ2t, λx)) =: (ψλ, Aλ, φλ) (1.5)

is also a solution for any λ > 0.
A Banach space B of distributions on R×Rd is a critical space if its norm verifies

for any λ and any u ∈ B,

‖u‖B = ‖λu(λ2·, λ·)‖B.
If we choose B as Lr(0,∞;Lp(Rd)), then (r, p) should satisfy

2

r
+
d

p
= 1.

In this article, we will choose the Coulomb gauge. First, we will prove the
following theorem.

Theorem 1.1. Let d = 3 and 0 < ε < 1. Let ψ0 ∈ H1, |ψ0| ≤ 1 and A0 ∈ H1.
Then the solution (ψ,A, φ) satisfies

|ψ| ≤ 1, ‖ψ‖L∞(0,T ;H1) + ‖ψ‖L2(0,T ;H2) + ‖∂tψ‖L2(0,T ;L2) ≤ C,
‖A‖L∞(0,T ;H1) + ‖A‖L2(0,T ;H2) + ‖∂tA‖L2(0,T ;L2) ≤ C,

‖φ‖L2(0,T ;H1) ≤ C
(1.6)

for any 0 < T < ∞. Here and later C will denote a positive constant independent
of ε.

Theorem 1.2. Let d = 3 and 0 ≤ ε ≤ 1 and ψ0, A0 ∈ L3(Ω). Then the problem
(1.1)-(1.4) has a unique solution (ψ,A, φ) satisfying

‖ψ‖L∞(0,T ;L3) + ‖ψ‖L5(0,T ;L5) + ‖ψ‖L2(0,T ;H1) + ‖|ψ|3/2‖L2(0,T ;H1) ≤ C,
‖∂tψ‖L2(0,T ;H−1) ≤ C,

‖A‖L∞(0,T ;L3) + ‖A‖L5(0,T ;L5) + ‖A‖L2(0,T ;H1) + ‖|A|3/2‖L2(0,T ;H1) ≤ C,
‖∂tA‖L2(0,T ;H−1) ≤ C, ‖∇φ‖L5/3(0,T ;L5/3) ≤ C

(1.7)

for any T > 0.

Remark 1.3. When a = −1, we are unable to prove a similar result at present.
Our results also hold true with the choice of Lorentz gauge.

In our proofs, we will use the following lemmas.

Lemma 1.4 ([1, 10]). Let Ω be a smooth and bounded open set in R3. Then there
exists C > 0 such that

‖f‖Lp(∂Ω) ≤ C‖f‖
1− 1

p

Lp(Ω)‖f‖
1/p
W 1,p(Ω) (1.8)



EJDE-2020/17 GINZBURG-LANDAU SYSTEM FOR SUPERCONDUCTIVITY 3

for any 1 < p <∞ and f : Ω→ R3 be in W 1,p(Ω).

Lemma 1.5 ([2]). Let Ω be a regular bounded domain in R3, let f : Ω → R3 be a
smooth enough vector field, and let 1 < p <∞. Then

−
∫

Ω

∆f · f |f |p−2 dx

=

∫
Ω

|f |p−2|∇f |2 dx+
4(p− 2)

p2

∫
Ω

|∇|f |
p
2 |2 dx−

∫
∂Ω

|f |p−2(ν · ∇)f · f dS.

(1.9)

Lemma 1.6 ([8]). ∇φ ∈ L5/3(0, T ;L5/3) satisfies

−∆φ = div Re
{( i

k
∇ψ + ψA

)
ψ
}

in Ω× (0, T ), (1.10)

∇φ · ν = 0 on (0, T )× ∂Ω. (1.11)

2. Proof of Theorem 1.1

We only need to show the a priori estimates (1.6). It is easy to show that (see
[3, 4, 5, 11])

|ψ| ≤ 1 in Ω× (0, T ). (2.1)

Testing (1.1) by ψ and taking the real parts, we see that

η

2

d

dt

∫
|ψ|2 dx+

∫ ∣∣ i
k
∇ψ + ψA

∣∣2 dx+ ε

∫
|ψ|4 dx = ε

∫
|ψ|2 dx,

which gives ∫ T

0

∫ ∣∣ i
k
∇ψ + ψA

∣∣2 dxdt ≤ C. (2.2)

Testing (1.2) by ∂tA+ curl2A, using (2.1), (2.2) and (1.11), we find that

d

dt

∫
| curlA|2 dx+

∫
(|∂tA|2 + | curl2A|2) dx

≤
∫ ∣∣ i

k
∇ψ + ψA

∣∣ |∂tA+ curl2A|dx

≤ 1

2

∫
(|∂tA|2 + | curl2A|2) dx+ C

∫ ∣∣ i
k
∇ψ + ψA

∣∣2 dx,

which leads to

‖A‖L∞(0,T ;H1) + ‖A‖L2(0,T ;H2) + ‖∂tA‖L2(0,T ;L2) ≤ C, (2.3)

whence
‖φ‖L2(0,T ;H1) ≤ C. (2.4)

Multiplying (1.1) by −∆ψ, integrating by parts and taking the real part, using
(2.1), (2.3) and (2.4), we obtain

η

2

d

dt

∫
|∇ψ|2 dx+

1

k2

∫
|∆ψ|2 dx

≤
∣∣Re

∫
iηkφψ ·∆ψ dx

∣∣+ 2
∣∣Re

1

k

∫
iA∇ψ ·∆ψ dx

∣∣
+ Re

∫
A2ψ∆ψ dx+ εRe

∫
(|ψ|2 − 1)ψ ·∆ψ dx

≤ 1

2

1

k2

∫
|∆ψ|2 dx+ C

∫
|∇φ||∇ψ|dx



4 J. FAN, Y. ZHOU EJDE-2020/17

+ C‖A‖2L∞‖∇ψ‖2L2 + C‖A‖L∞‖∇A‖L2‖∇ψ‖L2 + C‖∇ψ‖2L2 ,

which yields

‖ψ‖L∞(0,T ;H1) + ‖ψ‖L2(0,T ;H2) ≤ C. (2.5)

Whence

‖∂tψ‖L2(0,T ;L2) ≤ C. (2.6)

This completes the proof.

3. Proof of Theorem 1.2

To prove the existence, we only need to prove (1.7). First, we still have (2.2).
Multiplying (1.1) by |ψ|ψ, integrating by parts, and then taking the real part, we
obtain

η

3

d

dt

∫
|ψ|3 dx+

∫ ∣∣ i
k
∇ψ + ψA

∣∣2|ψ|dx+ ε

∫
|ψ|5 dx = ε

∫
|ψ|3 dx,

which gives

sup
0≤t≤T

∫
|ψ|3 dx+

∫ T

0

∫ ∣∣ i
k
∇ψ + ψA

∣∣2|ψ|dxdt ≤ C. (3.1)

Using the diamagnetic inequality∣∣1
k
∇|ψ|

∣∣ ≤ ∣∣ i
k
∇ψ + ψA

∣∣, (3.2)

and the Gagliardo-Nirenberg inequality

‖w‖Lp ≤ C‖w‖
3
p−

1
2

L2 ‖∇w‖
3
2−

3
p

L2 + C‖w‖L2 (3.3)

for w := |ψ|3/2 and p := 10
3 , we find that

‖ψ‖L5(0,T ;L5) ≤ C. (3.4)

Testing (1.2) by A and using (3.1), we observe that

1

2

d

dt

∫
|A|2 dx+

∫
| curlA|2 dx

≤
∫ ∣∣ i

k
∇ψ + ψA

∣∣ |ψ| |A|dx
≤
∫ ∣∣ i

k
∇ψ + ψA

∣∣ |ψ|1/2 |ψ|1/2 |A|dx
≤
∥∥ ∣∣ i
k
∇ψ + ψA

∣∣ |ψ|1/2∥∥
L2‖ |ψ|1/2‖L6‖A‖L3

≤ 1

2
‖ curlA‖2L2 + C

∥∥ ∣∣ i
k
∇ψ + ψA

∣∣ |ψ|1/2∥∥2

L2 ,

which leads to

‖A‖L∞(0,T ;L2) + ‖A‖L2(0,T ;H1) ≤ C. (3.5)

Here we have used the estimate ‖A‖L3 ≤ C‖ curlA‖L2 . Since∫ T

0

∫
|ψA|2 dxdt ≤ ‖ψ‖2L3

∫ T

0

‖A‖2L6 dt ≤ C,

it follows from (2.2) that

‖ψ‖L2(0,T ;H1) ≤ C. (3.6)



EJDE-2020/17 GINZBURG-LANDAU SYSTEM FOR SUPERCONDUCTIVITY 5

Testing (1.2) by |A|A and letting u := |A|3/2, using (1.3), (1.8), (1.9), (1.10), (1.11),
(3.1) and the vector identities

(ν · ∇)A ·A = (A · ∇)A · ν + (curlA× ν) ·A, (3.7)

(A · ∇)A · ν = −(A · ∇)ν ·A, (3.8)

we arrive at

d

dt

∫
u2 dx+ C0

∫
|∇u|2 dx+ C0

∫
|A||∇A|2 dx

≤ C
∫ ∣∣ i

k
∇ψ + ψA

∣∣ |ψ|u4/3 dx+ C

∫
|∇φ|u4/3 dx+ C

∫
∂Ω

u2 dS

≤ C
∥∥ ∣∣ i
k
∇ψ + ψA

∣∣ |ψ|1/2∥∥
L2‖ |ψ|1/2‖L6‖u4/3‖L3

+ C‖∇φ‖L3/2‖u4/3‖L3 + C‖u‖L2(Ω)‖u‖H1(Ω)

≤ C
∥∥ ∣∣ i
k
∇ψ + ψA

∣∣ |ψ|1/2∥∥
L2‖ |ψ|1/2‖L6‖u4/3‖L3 + C‖u‖L2(Ω)‖u‖H1(Ω)

≤ C
∥∥ ∣∣ i
k
∇ψ + ψA

∣∣ |ψ|1/2∥∥
L2‖u‖

1/3
L2 ‖u‖H1 + C‖u‖L2‖u‖H1

≤ C0

2
‖∇u‖2L2 + C

∥∥ ∣∣ i
k
∇ψ + ψA

∣∣ |ψ|1/2∥∥2

L2‖u‖
2
3

L2 + C‖u‖2L2 ,

which implies

‖A‖L∞(0,T ;L3) + ‖|A|3/2‖L2(0,T ;H1) ≤ C. (3.9)

Here we have used the estimate

‖∇φ‖L3/2 ≤ C
∥∥( i
k
∇ψ + ψA

)
ψ
∥∥
L3/2

≤ C
∥∥ ∣∣ i
k
∇ψ + ψA

∣∣ |ψ|1/2∥∥
L2‖ |ψ|1/2‖L6

≤ C
∥∥ ∣∣ i
k
∇ψ + ψA

∣∣ |ψ|1/2∥∥
L2 .

(3.10)

Using (3.3) for w = |A|3/2 and p = 10/3 and (3.9), we have

‖A‖L5(0,T ;L5) ≤ C. (3.11)

On the other hand, using (1.1), (1.2), (2.2), (3.1) and (3.9), we easily deduce
that

‖∂tψ‖L2(0,T ;H−1) + ‖∂tA‖L2(0,T ;H−1) ≤ C. (3.12)

This completes the proof of (1.7).
To prove the uniqueness, we use the method considered in [7, 6]. Here we remark

the only new estimate: if (ψi, Ai, φi) (i = 1, 2) are two weak solutions to the problem
(1.1)-(1.4), then the following monotone property holds:

Re

∫
(|ψ1|2ψ1 − |ψ2|2ψ2)(ψ1 − ψ2) dx ≥ 0.

The rest of the proof follows as in [7, 6]. This completes the proof.

Acknowledgements. This work was partially supported by NSFC (No. 11971234).
The authors are indebted to the referee for the careful reading of the manuscript
and for the suggestions that improved this article.



6 J. FAN, Y. ZHOU EJDE-2020/17

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Jishan Fan

Department of Applied Mathematics, Nanjing Forestry University, Nanjing 210037,
China

Email address: fanjishan@njfu.edu.cn

Yong Zhou (corresponding author)

School of Mathematics (Zhuhai), Sun Yat-sen University, Zhuhai, Guangdong 519082,

China
Email address: zhouyong3@mail.sysu.edu.cn


	1. Introduction
	2. Proof of Theorem ??
	3. Proof of Theorem ??
	Acknowledgements

	References