JOURNAL OF MATHEMATICAL PHYSICS
VOLUME 44, NUMBER 11
NOVEMBER 2003
Existence of suitable weak solutions of complex
Ginzburg–Landau equations and properties
of the set of singular points
Xiaofeng Liua)
The Department of Mathematics, East China Normal University, Shanghai, 200062,
People’s Republic of China
Houyu Jiab)
The Department of Mathematics (Xixi Campus), Zhejiang University,
Hangzhou, 310028, People’s Republic of China
共Received 25 March 2003; accepted 12 August 2003兲
In this paper, we consider the supercritical complex Ginzburg–Landau equation.
We discuss the existence of suitable weak solution in ⍀, where ⍀ is a bounded
domain in Rn or the whole space. We also discuss the properties of the set of the
singular points of the suitable weak solution in Rn , which means that the possible
singular points are located in a bounded ball for any given time and there is no
singular point on the whole space after limited time. © 2003 American Institute of
Physics. 关DOI: 10.1063/1.1618360兴
I. INTRODUCTION AND RESULTS
In this paper we consider the following supercritical complex Ginzburg–Landau 共CGL兲 equations:
u t ⫽ 共 1⫹i ␮ 兲 ⌬u⫺ 共 1⫹i ␯ 兲 兩 u 兩 2 ␴ u⫹Ru
u 共 x,0兲 ⫽u 0 共 x 兲 ,
u 共 x,t 兲 ⫽0,
x苸 ⳵ ⍀,
in ⍀⫻ 关 0,T 兲 ,
共1兲
x苸⍀,
0⬍t⬍T,
where u(x,t) is a complex-valued field, the initial data u 0 苸L 2 , R⬎0, ␮ , ␯ 苸R and 2/n⬍ ␴ ⬍(n
⫹4)/2n , ⍀ is a bounded domain in Rn or the whole space. When ⍀⫽Rn , the third line of 共1兲
should be replaced by u(x,t)→0 as 兩 x 兩 →⬁ for 0⬍t⬍T. This equation, most often considered
with a cubic nonlinearity and n⫽3, has a long history in physics as a generic amplitude equation
near the onset of instabilities in fluid mechanical systems, as well as in the theory of phase
transitions and superconductivity. In this paper, we concentrate on mathematical questions related
to the regularity of weak solutions. In the case ␴ ⫽1, solutions of 共1兲 have similar scaling properties as solutions of the Navier–Stokes equations. It might be a good model problem in connection with regularity questions regarding the Navier–Stokes equations.
For Eq. 共1兲 in the periodic case, the existence of weak solutions in all cases was obtained in
Ref. 1. It is pointed out in Ref. 1 that in the subcritical case ( ␴ n⬍2), all weak solutions are
regular. When ␴ n⫽2, it is known that singularities cannot develop from sufficiently regular initial
data 共see Ref. 2 for case ␴ ⫽2, n⫽2 and Ref. 1 for the general case兲. When ␴ n⬎2, regularity
results are available only for special values of ␮ and ␯ 共see Ref. 1兲.
For Eq. 共1兲, the existence of a weak solution can be obtained by the Galerkin approximation.
The full regularity for weak solutions of 共1兲 in the subcritical and critical cases can be obtained by
a兲
Electronic mail: lxfn2@263.net
Electronic mail: mjhy@zju.edu.cn
b兲
0022-2488/2003/44(11)/5185/9/$20.00
5185
© 2003 American Institute of Physics
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5186
J. Math. Phys., Vol. 44, No. 11, November 2003
X. Liu and H. Jia
the usual bootstrap argument and a refined version of the standard bootstrap argument, respectively. In the supercritical case ( ␴ n⬎2) it is not known whether weak solutions are regular for
general values of ␮ and ␯.
In Ref. 3, the following version of CGL was considered:
u t ⫽ 共 1⫹i ␮ 兲 ⌬u⫺i ␯ 兩 u 兩 2 ␴ u⫹Ru
u 共 x,0兲 ⫽u 0 共 x 兲 ,
u 共 x,t 兲 ⫽0,
in ⍀⫻ 关 0,T 兲 ,
共2兲
x苸⍀,
x苸 ⳵ ⍀,
0⬍t⬍T.
The partial regularity of the suitable weak solution was discussed and the existence of the suitable
weak solution was mentioned but without proof in Ref. 3. In this paper we prove the existence of
the suitable weak solution of 共1兲. Actually, it also works for 共2兲. For more details on 共1兲 and 共2兲,
see Refs. 3, 1, and 2.
Now we give some notations. Let D⫽⍀⫻(0,T), B r be the ball in Rn with radius r,
Q r ⫽Q r (x,t)⫽ 兵 (y,s): 兩 y⫺x 兩 ⭐r,t⫺r 2 ⭐s⬍t 其 . Let E 0 (u)⫽sup0⬍t⬍T 1/2 兰 ⍀ 兩 u 兩 2 dx, E 1 (u)
⫽ 兰 T0 兰 ⍀ 兩 ⵜu 兩 2 dx dt, and E⫽E 0 ⫹E 1 . Let H 1 (⍀)⫽ 兵 u苸L 2 (⍀):ⵜu苸L 2 (⍀) 其 and H 10 (⍀) be the
completion of C⬁0 (⍀) in H 1 (⍀) and H ⫺1 (⍀) be the dual of H 10 (⍀). We omit ⍀ if there is no
confusion.
The definition of suitable weak solution is the following.
Definition 1.1: We say u is a suitable weak solution of (1) on D, if the following conditions are
satisfied:
(i) u苸L ⬁ (0,T;L 2 (⍀))艚L 2 (0,T;H 1 (⍀)).
(ii) u satisfies (1) in the sense of distributions on D.
(iii) (Generalized energy inequality) For each real-valued ␾ 苸C ⬁0 (D) with ␾ ⭓0, the following
inequality holds:
2
冕冕
⭐
兩 ⵜu 兩 2 ␾ dx dt⫹
冕冕
冕冕
兩 u 兩 2 ␴ ⫹2 ␾ dx dt
再
兩 u 兩 2 共 ␾ ⬘ ⫹2⌬ ␾ 兲 dx dt⫹2R 共 1⫹i ␮ 兲
冕冕
冎 冕冕
uⵜūⵜ ␾ dx dt ⫹2R
兩 u 兩 2 ␾ dx dt. 共3兲
We have the following existence result.
Theorem 1.1: Let u 0 苸L 2 (⍀). Then there exists a weak solution u of (1) on D satisfying
u苸L 2 共 0,T;H 10 兲 艚L ⬁ 共 0,T;L 2 兲 ,
共4兲
u 共 t 兲 →u 0 weakly in L 2 as t→0,
共5兲
and if ␾ 苸C ⬁ (D̄), ␾ ⭓0, and ␾ ⫽0 near ⳵ ⍀⫻(0,T), then for 0⬍t⬍T,
冕
⍀⫻ 兵 t 其
⭐
兩 u 兩 2 ␾ ⫹2
冕
⍀
冕冕
t
0
⍀
兩 ⵜu 兩 2 ␾ ⫹
兩 u 0 兩 2 ␾ 共 x,0兲 ⫹
冕冕
t
0
⍀
冕冕
t
0
⍀
兩 u 兩 2 ␴ ⫹2 ␾
再
兩 u 兩 2 共 ␾ ⬘ ⫹2⌬ ␾ 兲 ⫹2R 共 1⫹i ␮ 兲
冕冕
t
0
⍀
冎 冕冕
uⵜūⵜ ␾ ⫹2R
t
0
⍀
兩u兩2␾ .
共6兲
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J. Math. Phys., Vol. 44, No. 11, November 2003
Weak solutions of Ginzburg–Landau equations
5187
By the motivation of Ref. 4, we consider the property of the set of singular points of the
suitable weak solution with more assumptions on the initial data. We consider 共2兲 for simplicity
and have the following theorem.
Theorem 1.2: Suppose u 0 苸L 2 (Rn ), and
1
2
冕
Rn
共7兲
兩 u 0 兩 2 兩 x 兩 dx⫽G⬍⬁.
Then there exists a weak solution of the initial value problem of (2) which is regular in the region
兵 (x,t): 兩 x 兩 2 t n⫺ 2/␴ ⬎K 1 其 , where K 1 ⫽K 1 (E,G) is a constant depending on E and G.
Remark 1.1: The above theorem means that the set of singular points is restricted in a
bounded domain for every time. Actually, there is no singular point in Rn after a limited time, see
Lemma 3.1.
The following interpolation inequality is useful in our following proofs.
Lemma 1.1: If u苸H 1 (B r ), then
冕
Br
兩 u 兩 q dx⭐C
冉冕
Br
冊 冉冕 冊
a
兩 ⵜu 兩 2 dx
Br
兩u兩2
共 q/2兲 ⫺a
⫹
C
r 2a
冉冕 冊
Br
兩u兩2
q/2
,
共8兲
where a⫽ (n/4) (q⫺2), 2⭐q⭐ 2n/(n⫺2) , C independent of r. Moreover, if u has mean zero on
B r or if B r is replaced by all of Rn , then the second term on the right-hand side in (8) may be
omitted.
In the remainder of this paper we prove these two theorems, respectively.
II. PROOF OF THE THEOREM 1.1
We begin with some lemmas concerning the relevant linear systems.
Lemma 2.1: Suppose f 苸L 2 (0,T;H ⫺1 ), u苸L 2 (0,T;H 10 ), and
u t ⫽ 共 1⫹i ␮ 兲 ⌬u⫹ f
共9兲
in the sense of distributions on D. Then u t 苸L 2 (0,T;H ⫺1 ),
d
dt
冕
⍀
兩 u 兩 2 dx⫽2R
冕
⍀
u t ū dx
共10兲
in the sense of distributions on (0,T), and u苸C( 关 0,T 兴 ;L 2 ) after modification on a set of measure
zero. Solution of (9) is unique in L 2 (0,T;H 10 ) for given initial data u 0 苸L 2 .
Proof: The proof is standard, see Refs. 4 and 5.
Lemma 2.2: Let u 0 苸L 2 and w苸C ⬁ (D̄;R) with w⭓0. Then there exists unique function u
苸C( 关 0,T 兴 ;L 2 )艚L 2 (0,T;H 10 ) such that
u t ⫽ 共 1⫹i ␮ 兲 ⌬u⫺ 共 1⫹i ␯ 兲 wu⫹Ru
共11兲
in the sense of distributions on D and u(0)⫽u 0 .
Proof: The proof is similar to Ref. 4, and for the proof of uniqueness of solution we need the
Grownwall inequality.
Lemma 2.3: Let u 0 苸L 2 , w苸C ⬁ (D̄;R) with w⭓0 and u be a solution of (11). Then for any
␾ 苸C⬁0 (D̄) with ␾ ⫽0 near ⳵ ⍀⫻(0,T), for all t苸(0,T),
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5188
J. Math. Phys., Vol. 44, No. 11, November 2003
冕
⍀⫻ 兵 t 其
⫽
兩 u 兩 2 ␾ ⫹2
冕
⍀
冕冕
t
0
⍀
兩 ⵜu 兩 2 ␾ ⫹
兩 u 0 兩 2 ␾ 共 x,0兲 ⫹
⫹2R
冕冕
t
⍀
0
冕冕
t
⍀
0
冕冕
t
⍀
0
X. Liu and H. Jia
w兩u兩2␾
再
兩 u 兩 2 共 ␾ ⬘ ⫹2⌬ ␾ 兲 ⫹2R 共 1⫹i ␮ 兲
冕冕
t
0
⍀
uⵜūⵜ ␾
冎
兩u兩2␾ .
共12兲
Proof: Suppose for the moment that ␾ also vanishes near t⫽0; choose ⍀ 1 so that ⍀̄ 1 傺⍀ and
supp ␾ 傺⍀ 1 ⫻(0,T). Writing F⫽⫺(1⫹i ␯ )wu⫹Ru苸L 2 (D), we have
u t ⫽ 共 1⫹i ␮ 兲 ⌬u⫹F.
共13兲
Mollifying 共in Rn⫹1 ) each term of 共13兲, we obtain sequences of smooth functions 兵 u m 其 and 兵 F m 其
such that
d
u ⫽ 共 1⫹i ␮ 兲 ⌬u m ⫹F m ,
dt m
共14兲
with u m →u in L 2 (D), ⵜu m →ⵜu in L 2 (D), and F m →F in L 2 (D). Multiplying 共14兲 by ū m ␾ ,
integrating by part on D, and taking the real part, by Lemma 2.1, we have
⫺
冕冕
再
冋 冕冕
兩 u m 兩 2 ␾ ⬘ ⫽2R 共 1⫹i ␮ 兲 ⫺
⫹2R
冕冕
兩 ⵜu m 兩 2 ␾ ⫹
冕冕
u m ⵜū m ⵜ ␾ ⫹
冕冕
兩 u m兩 2⌬ ␾
冕冕
uⵜūⵜ ␾ ⫹2
冎 冕冕
兩u兩2⌬ ␾
册冎
F m ū m ␾ .
We pass to the limit as m→⬁ and obtain
⫺
冕冕
兩 u 兩 2 ␾ ⬘ ⫽⫺2
再
冕冕
冕冕
兩 ⵜu 兩 2 ␾ ⫹2R 共 1⫹i ␮ 兲
Fū ␾ .
⫹2R
On the other hand,
2R
冕冕
Fū ␾ ⫽2R
冉冕 冕
冊 冕冕
⫺ 共 1⫹i ␯ 兲 w 兩 u 兩 2 ␾ ⫹R 兩 u 兩 2 ␾ ⫽⫺2
w 兩 u 兩 2 ␾ ⫹2R
冕冕
兩u兩2␾ .
So we have
2
冕冕
兩 ⵜu 兩 2 ␾ ⫹
冕冕
w兩u兩2␾ ⫽
冕冕
兩 u 兩 2 共 ␾ ⬘ ⫹2⌬ ␾ 兲 ⫹2R
再
⫹2R 共 1⫹i ␮ 兲
冕冕
冕冕
兩u兩2␾
冎
uⵜūⵜ ␾ .
共15兲
For the general ␾ 苸C⬁ (D̄), ␾ ⭓0, by the argument in Ref. 4, we obtain 共12兲.
Utilize the ‘‘retard mollifier’’ ⌿ ␦ (u) as in Ref. 4, we have the following lemma.
Lemma 2.4: For any u苸L ⬁ (0,T;L 2 )艚L 2 (0,T;H 10 ),
sup
0⬍t⬍T
冕
⍀
兩 ⌿ ␦ 共 u 兲 兩 2 共 x,t 兲 dx⭐CE 0 共 u 兲 ,
共16兲
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J. Math. Phys., Vol. 44, No. 11, November 2003
冕冕
D
Weak solutions of Ginzburg–Landau equations
5189
兩 ⵜ⌿ ␦ 共 u 兲 兩 2 dx dt⭐CE 1 共 u 兲 ,
共17兲
where C denotes an universal constant.
Now we are ready to prove Theorem 1.1. For any large integer N, let ␦ ⫽T/N, and solve
d
u ⫽ 共 1⫹i ␮ 兲 ⵜu N ⫺ 共 1⫹i ␯ 兲 ⌿ ␦ 共 兩 ⌿ ␦ 共 u N 兲 兩 2 ␴ 兲 u N ⫹Ru N ,
dt N
共18兲
u N 苸L 2 共 0,T;H 10 兲 艚C 共关 0,T 兴 ;L 2 兲 ,
共19兲
u N 共 0 兲 ⫽u 0 .
共20兲
By induction, due to Lemma 2.2, such u N exist on each (m ␦ ,(m⫹1) ␦ ), 0⭐m⭐N⫺1. By 共10兲,
we have
冕
⍀⫻ 兵 t 其
兩 u N 兩 2 ⫹2
冕冕
t
0
⍀
兩 ⵜu N 兩 2 ⫹2
冕冕
t
0
⍀
⌿ ␦共 兩 ⌿ ␦共 u N 兲兩 2␴ 兲兩 u N兩 2⫽
冕
⍀
兩 u 0 兩 2 ⫹2R
冕冕
t
0
⍀
兩 u N兩 2,
then
冕
⍀⫻ 兵 t 其
兩 u N兩 2⭐
冕
⍀
兩 u 0 兩 2 ⫹2R
冕冕
t
⍀
0
兩 u N兩 2.
By the Grownwall inequality, for t苸(0,T),
冕
⍀⫻ 兵 t 其
兩 u N 兩 2 dx⭐exp共 2RT 兲
冕
⍀
兩 u 0兩 2,
then for t苸(0,T),
冕
⍀⫻ 兵 t 其
兩 u N 兩 2 ⫹2
冕冕
t
0
⍀
兩 ⵜu N 兩 2 ⫹2
冕冕
t
0
⍀
⌿ ␦ 共 兩 ⌿ ␦ 共 u N 兲 兩 2 ␴ 兲 兩 u N 兩 2 ⭐ 共 1⫹2RT exp共 2RT 兲兲
冕
⍀
兩 u 0兩 2,
so we have
兵 u N 其 is bounded in L ⬁ 共 0,T;L 2 兲 艚L 2 共 0,T;H 10 兲 .
共21兲
d
u 苸L p 共 0,T;H ⫺s 兲 ,
dt N
共22兲
We claim that
where s⬎n/2 for some 1⬍p⬍⬁.
By 共18兲 and 共21兲, it is sufficient to check that ⌿ ␦ ( 兩 ⌿ ␦ (u N ) 兩 2 ␴ )u N 苸L p (0,T;H ⫺s ). By Lemma
1.1, we have
冕
⍀
兩 u N 兩 2 ␴ ⫹1 ⭐C
冉冕
⍀
冊 冉冕 冊
a
兩 ⵜu N 兩 2
⍀
兩 u N兩 2
共 2 ␴ ⫹1 兲 /2 ⫺a
⫹
C
r 2a
冉冕 冊
⍀
兩 u N兩 2
共 2 ␴ ⫹1 兲 /2
,
where
a⫽
n 共 2 ␴ ⫺1 兲
.
4
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5190
J. Math. Phys., Vol. 44, No. 11, November 2003
X. Liu and H. Jia
For ␴ ⬍(4⫹n)/2n, let p⫽ 4/关 n(2 ␴ ⫺1) 兴 ⬎1, we have
冕 冉冕
T
0
⍀
兩 u N 兩 2 ␴ ⫹1 dx
冊
p
dt⭐C 共 E 共02 ␴ ⫹1 兲 /2 ⫺a 兲 p E 1 ⫹
CT 关 (2 ␴ ⫹1)p 兴 /2
E
⬍⬁,
r 2ap 0
so ⌿ ␦ ( 兩 ⌿ ␦ (u N ) 兩 2 ␴ )u N 苸L p (0,T;H ⫺s ).
When ⍀ is bounded in Rn , by 共21兲, 共22兲 and Theorem 2.1 in Chap. III in Ref. 5, 兵 u N 其 stays
in a compact set of L 2 (0,T;L 2 ), then there exists a strongly convergent subsequence of 兵 u N 其 , still
denoted by 兵 u N 其 , and u * with
u N →u * in L 2 共 0,T;L 2 兲 .
共23兲
By Lemma 1.1, let r 0 ⫽diam(⍀),
冕
⍀
兩 u N 兩 关 2 共 n⫹2 兲兴 /n dx⭐C
冉冕
⍀
兩 ⵜu N 兩 2 dx
冊冉 冕
⍀
兩 u N 兩 2 dx
冊
2/n
⫹
C
r 20
冉冕
⍀
兩 u N 兩 2 dx
冊
共 n⫹2 兲 /n
,
then integrate from 0 to T,
冕冕
T
0
⍀
兩 u N 兩 关 2 共 n⫹2 兲兴 /n dxdt⭐CE 1 E 2/n
0 ⫹
CT
r 20
E 共0n⫹2 兲 /n ⬍⬁,
so 兵 u N 其 is bounded in L 关 2(n⫹2) 兴 /n (D). By 共23兲, for 2⭐q⬍ 关 2(n⫹2) 兴 /n,
u N →u * in L q 共 0,T;L q 兲 .
共24兲
When ⍀⫽Rn , we can only obtain that there exists a subsequence, still denoted by 兵 u N 其 , and u *
2
satisfying u N →u * strongly in L 2 (0,T;L loc
), 兵 u N 其 is bounded in L 关 2(n⫹2) 兴 /n (D) and then u N
q
q
→u * strongly in L (0,T;L loc) for 2⭐q⬍ 关 2(n⫹2) 兴 /n, which is enough to get our results.
By 共21兲,
u N →u * weakly in L 2 共 0,T;H 10 兲 ,
共25兲
u N →u * weak-star in L ⬁ 共 0,T;L 2 兲 .
共26兲
and
By the definition of ⌿ ␦ , for some r⬍(n⫹2)/n ␴ ,
⌿ ␦ 共 兩 ⌿ ␦ 共 u N 兲 兩 2 ␴ 兲 → 兩 u * 兩 2 ␴ in L r 共 0,T;L r 兲 ,
共27兲
⌿ ␦ 共 u N 兲 →u * strongly in L q 共 D 兲 .
共28兲
and
So u * is the solution of 共18兲 in the sense of distributions. From now on, the proof is similar to that
in Ref. 4.
III. PROOF OF THE THEOREM 1.2
In this section, we consider the initial value problem of 共2兲 for simplicity. Let ⍀⫽Rn and u be
a suitable weak solution of 共2兲 on Rn ⫻(0,⬁) with initial data u 0 苸L 2 , and for ␾ 苸C⬁0 (Rn ⫻R)
with ␾ ⭓0,
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J. Math. Phys., Vol. 44, No. 11, November 2003
冕
⍀⫻ 兵 t 其
兩 u 兩 2 ␾ ⫹2
冕冕
t
0
⍀
冕
兩 ⵜu 兩 2 ␾ ⭐
⍀
Weak solutions of Ginzburg–Landau equations
兩 u 0 兩 2 ␾ 共 x,0兲 ⫹2R
再
⫹2R 共 1⫹i ␮ 兲
冕冕
冕冕
t
t
⍀
0
⍀
0
兩u兩2␾ ⫹
冎
冕冕
t
0
⍀
5191
兩 u 兩 2 共 ␾ ⬘ ⫹2⌬ ␾ 兲
uⵜūⵜ ␾ .
共29兲
Here we need a corollary in Ref. 3.
Proposition 3.1: Let ␣ ⬎2, 2 ␴ ⫹1⬍ ␣ ⬍ 关 2(n⫹2) 兴 /n, there exists ⑀ 0 ⬎0, if u is a suitable
weak solution of (2) on Q r , and moreover,
1
r
n⫹2⫺ ␣ / ␴
冕冕
Qr
兩u兩␣⭐ ⑀ 0 ,
共30兲
then u is C ␣ 0 in Q r/4 for some ␣ 0 苸(0,1/2).
By the same idea in Ref. 3, we let v ⫽e ⫺Rt u, if u is a suitable weak solution of 共2兲, then v is
a suitable weak solution of the following problem:
v t ⫽ 共 1⫹i ␮ 兲 ⌬ v ⫺i ␯ e2 ␴ Rt 兩 v 兩 2 ␴ v
v共 x,0兲 ⫽ v 0 共 x 兲 ,
v共 x,t 兲 ⫽0,
in ⍀⫻ 共 0,T 兲 ,
共31兲
x苸⍀,
x苸 ⳵ ⍀, t苸 共 0,T 兲 ,
and the corresponding generalized energy inequality
冕
⍀⫻ 兵 t 其
兩 v 兩 2 ␾ ⫹2
冕冕
t
⍀
0
兩ⵜ v兩2␾ ⭐
冕
⍀
冕冕
␮ 冕 冕
t
兩 v 0 兩 2 ␾ 共 x,0兲 ⫹
再
⍀
0
t
⫹2R 共 1⫹i 兲
0
⍀
兩 v 兩 2 共 ␾ ⬘ ⫹2⌬ ␾ 兲
v ⵜ v̄ ⵜ ␾
冎
共32兲
holds for any real-valued ␾ 苸C⬁ (D) with ␾ ⭓0 and ␾ (t)苸C⬁0 (Rn ) for each t苸 关 0,T 兴 .
To prove Theorem 1.2, we need the following lemmas.
Lemma 3.1: There exists an absolute constant C such that if t⭓CE 2 ␴ /(n ␴ ⫺2) , then u is regular
at (x,t).
Proof: By Lemma 1.1,
冕
R
兩 u 兩 ␣ dx⭐C
n
冉冕
R
兩 u 兩 2 dx
n
冊
共 ␣ /2兲 (1⫺n(1/2 ⫺ 1/␣ ))
冉冕
R
兩 ⵜu 兩 2 dx
n
冊
共 n ␣ /2兲 (1/2 ⫺ 1/␣ )
,
then
冕冕
T
0
R
冉 冕
兩 u 兩 ␣ dx dt⭐C supt
n
R
兩 u 兩 2 dx
n
冊
共 ␣ /2兲 (1⫺n(1/2 ⫺ 1/␣ ))
冕 冉冕
T
0
R
兩 ⵜu 兩 2 dx
n
冊
共 n ␣ /2兲 (1/2 ⫺ 1/␣ )
dt
⭐CE 共0␣ /2兲 (1⫺n(1/2 ⫺ 1/␣ )) E 共1n ␣ /2兲 (1/2 ⫺ 1/␣ ) T 1⫺ 共 n ␣ /2兲 (1/2 ⫺ 1/␣ )
⭐CE ␣ /2T 1⫺ 共 n ␣ /2兲 (1/2 ⫺ 1/␣ ) .
When E⭐( ⑀ 0 /C) 2/␣ T n/2 ⫺ 1/␴ , i.e., T⭓C( ⑀ 0 , ␣ )E 2 ␴ /(n ␴ ⫺2) ,
冕冕
T
0
Rn
兩 u 兩 ␣ dx dt⭐ ⑀ 0 T 共 n⫹2 兲 /2 ⫺ ␣ /2␴ .
By Proposition 3.1, if
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5192
J. Math. Phys., Vol. 44, No. 11, November 2003
冕冕
Qr
X. Liu and H. Jia
兩 u 兩 ␣ dx dt⭐ ⑀ 0 r n⫹2⫺ ␣ / ␴ ,
then u is regular in Q r/4 , so Lemma holds.
Lemma 3.2: Let ␶ ⫽ ⑀ ⫹ 兩 x 兩 , ⑀ ⭓0. If
(i) r⭓2, ␥ ⫹ n/r⬎0, ␣ ⫹ n/2⬎0, ␤ ⫹ n/2⬎0, 1/2 ⭐a⭐1,
(ii) ␥ ⫹ n/r ⫽a( ␣ ⫹ (n⫺2)/2)⫹(1⫺a)( ␤ ⫹ n/2),
(iii) a( ␣ ⫺1)⫹(1⫺a) ␤ ⭐ ␥ ⭐a ␣ ⫹(1⫺a) ␤ ,
then
兩 ␶ ␥ u 兩 L 2 ⭐C 兩 ␶ ␣ 兩 ⵜu 兩兩 L 2 兩 ␶ ␤ u 兩 L 2 .
a
1⫺a
共33兲
For n⫽3, the proof of the lemma has been given in Ref. 4; for the general case, the proof is
similar.
Lemma 3.3: Let u be a suitable weak solution of (2) with (29) holds and G⬍⬁. Then for a.e.
t⬎0,
1
2
冕
R
兩 u 兩 2 兩 x 兩 dx⫹
n
冕冕
t
Rn
0
兩 ⵜu 兩 2 兩 x 兩 dx dt⭐A 共 t 兲 ,
共34兲
where A(t)⫽G⫹Ct 1/2E.
Proof: By Lemma 3.2,
冕
兩u兩2
dx⭐C
Rn 兩 x 兩
冉冕
冊 冉冕
1/2
R
兩 u 兩 2 dx
n
R
兩 ⵜu 兩 2 dx
n
冊
1/2
,
then
冕冕
t
0
兩u兩2
1/2 1/2
1/2
dx dt⭐CE 1/2
0 E 1 t ⭐CEt .
Rn 兩 x 兩
Let ␹ (t) be C⬁ on s⭓0 with 0⭐ ␹ ⭐1, ␹ ⫽1 for s⭐1 and ␹ ⫽0 for s⭓2. For constants 1
⬎␭Ⰷ ⑀ ⬎0, we use the test function ␾ (x)⫽ 1/2 (␭ 2 ⫺ 兩 x 兩 2 ) 1/2␹ ( ⑀ /␭ 兩 x 兩 ) in the energy inequality
共32兲. It is easy to verify that 兩 ⵜ ␾ 兩 ⭐C, 兩 ⌬ ␾ 兩 ⭐C 兩 x 兩 ⫺1 , with C independent of ␭ and ⑀. Then
冕
Rn ⫻ 兵 t 其
兩 u 兩 2 ␾ dx⫹2
冕冕
t
0
R
兩 ⵜu 兩 2 ␾ dx dt⭐
n
冕
R
兩 u 0 兩 2 兩 x 兩 dx⫹C
n
冕冕 再
t
0
Rn
冎
兩u兩2
⫹ 兩 u 兩兩 ⵜu 兩 dx dt.
兩x兩
Let ⑀ →0 and then ␭→0,
冕
Rn ⫻ 兵 t 其
兩 u 兩 2 兩 x 兩 dx⫹2
冕冕
t
0
Rn
兩 ⵜu 兩 2 兩 x 兩 dx dt⭐G⫹CEt 1/2.
Now we prove Theorem 1.2. Let S be the set of singular points of u. By Lemma 3.1, if
(x,t)苸S, then
t⬍CE 2 ␴ / 共 n ␴ ⫺2 兲 .
共35兲
Let r 2 ⫽5/4 t, Q⫽Q r (x,r 2 ), by Proposition 3.1,
冕冕
Q
兩 u 兩 ␣ dx dt⬎C ⑀ 0 t n⫹2/2 ⫺ ␣ /2␴ .
共36兲
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J. Math. Phys., Vol. 44, No. 11, November 2003
Weak solutions of Ginzburg–Landau equations
5193
Let R⫽ 兩 x 兩 and suppose for the moment that R⭓2r. Take ␥ ⫽ ␣ ⫽ ␤ ⫽ 1/2 , a⫽ n/(n⫹2), and
p⫽ 关 2(n⫹1) 兴 /n, by Lemma 3.2,
p
2
4/n
兩兩 x 兩 1/2兩 u 兩兩 L p ⭐C 兩兩 x 兩 1/2兩 ⵜu 兩兩 L 2 兩兩 x 兩 1/2兩 u 兩兩 L 2 .
共37兲
Integrate from 0 to t,
冕冕
t
0
Rn
兩 x 兩 共 n⫹2 兲 /n 兩 u 兩 关 2 共 n⫹2 兲兴 /n dx dt⭐CA 共 t 兲 2/n
冕冕
t
0
Rn
兩 x 兩兩 ⵜu 兩 2 dx dt⭐CA 共 t 兲 共 2⫹n 兲 /n .
By 共36兲 and the Hölder inequality,
C ⑀ 0 t n⫹2/2⫺ ␣ /2␴ ⭐
冕冕
Q
兩 u 兩 ␣ dx dt⭐r n⫹2⫺ ␣ n/2
冉冕 冕
Q
兩 u 兩 关 2 共 n⫹2 兲兴 /n dx dt
冊
␣ n/ 关 2 共 n⫹2 兲兴
⭐Cr n⫹2⫺ ␣ n/2R ⫺ ␣ /2A 共 t 兲 ␣ /2⭐K 共 E,G 兲 t n⫹2/2 ⫺ ␣ n/4R ⫺ ␣ /2,
so
R 2 t n⫺ 2/␴ ⭐K 0 共 E,G, ␣ 兲 .
For R⭐2r,
R 2 t n⫺ 2/␴ ⭐5t n⫹1⫺2 ␴ ⭐C 共 E 2 ␴ / 共 n ␴ ⫺2 兲 兲 n⫹1⫺ 2/␴ .
Therefore, Theorem 1.2 holds with K 1 ⫽C max(K0 ,E2␴/n␴⫺2(n⫹1⫺ 2/␴ ) ).
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X. Yan, Commun. Partial Differ. Eqns. 24, 2263 共1999兲.
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