Acta Mathematica Sinica, English Series Aug., 2010, Vol. 26, No. 8, pp. 1601–1612 Published online: July 15, 2010 DOI: 10.1007/s10114-010-7313-6 Http://www.ActaMath.com
Acta Mathematica Sinica, English Series Springer-Verlag Berlin Heidelberg & The Editorial Office of AMS 2010
On the Generalized 2-D Stochastic Ginzburg–Landau Equation De Sheng YANG School of Mathematical Sciences and Computing Technology, Central South University, Changsha 410083, P. R. China E-mail :
[email protected] Abstract This paper proves the existence and uniqueness of solutions in a Banach space for the generalized stochastic Ginzburg–Landau equation with a multiplicative noise in two spatial dimensions. The noise is white in time and correlated in spatial variables. The condition on the parameters is the same as in the deterministic case. The Banach contraction principle and stochastic estimates in Banach spaces are used as the main tool. Keywords Generalized stochastic Ginzburg–Landau equation, multiplicative noise, stochastic partial differential equation MR(2000) Subject Classification 60H15
1
Introduction
This paper is concerned with the existence and uniqueness of solutions to the the generalized two-dimensional (2-D) stochastic Ginzburg–Landau (G–L) equation with a multiplicative noise. This equation has the following It¨o form du = ((1 + iν)Δu + γu − (1 − iμ)|u|2σ u + F (u))dt + udW (t), u(t, x) = 0, t ∈ R+ , u(0, x) = u0 ,
(t, x) ∈ R+ × D,
x ∈ ∂D,
x ∈ D,
where D = (0, L1 ) × (0, L2 ), i =
(1.1)
√ −1, γ > 0, μ and ν are given real constants. The derivative
term F (u) = λ1 · ∇(|u|2 u) + (λ2 · ∇u)|u|2 with two complex constant vectors λ1 and λ2 . The random source W (t) is a Wiener process with the variance operator QQ∗ , and Q satisfies some regular conditions which will be specified later. The deterministic generalized G–L equation, i.e., Eq. (1.1) without the stochastic term, derived by Doelman [1], aries in various areas of physics and chemistry such as counterpropagating waves and the turbulent flow in chemical reaction. Many mathematical results on the existence and uniqueness, as well as the asymptotic behavior of solutions have been studied under various assumptions on the parameters (see, e.g., [2–8]). In particular, the paper [8] proved that the deterministic 2-D generalized G–L equation possesses a unique solution under the following assumptions on the parameters σ, ν and μ: (A) Either (i) σ > 2 or (ii) σ = 2, |λi |(i = 1, 2) are sufficiently small; √ 2σ + 1 |ν + μ|. (B) − 1 + νμ < σ Received June 19, 2007, accepted January 25, 2009
(1.2)
Yang D. S.
1602
In the present paper, we introduce a multiplicative noise with the form u dWdt(t) into the deterministic derivative G–L equation and prove that its stochastic version (1.1) has a unique mild solution under the same assumptions (1.2). The noise models small irregular fluctuations generated by microscopic effects in the flux of molecular collisions, and thus the stochastic model may be more realistic. We also note that the existence of the stochastic complex G–L equation without the derivative term is obtained in [9]. However, due to the derivative term and different assumptions on the parameters, we introduce the energy Jκ (u) =
1 κ 2(σ+1) |∇u|2 + |u| , 2 2(σ + 1) 2(σ+1)
which leads to the estimates different from those in [9]. Moreover, since the equation is restricted to the two-dimensional case, we have to work in a Banach space C([0, T ]; H01 ) ∩ L2 (0, T ; W 1,p0 ) with p0 = 2(σ + 1). Hence the stochastic estimates in this Banach space are necessary. The organization of this paper is as follows. In Section 2, we first introduce some notations, the known results, and then state the main theorem of this paper. Section 3 is devoted to the proof of the existence of a unique local solution to Eq. (1.1) by the Banach contraction principle. Finally, the energy estimate in Section 4 shows that the local solution is also global in time. 2
Notations and Main Result
Throughout the paper, we always denote by |·|X the norm in a vector space X and by Lp , W m,p and H0m,p the usual Sobolev spaces of complex-valued functions on the domain D. The inner product in L2 is denoted by (·, ·), i.e., u(x)¯ v(x)dx, (u, v) = Re D 2
for u, v ∈ L , where u ¯ stands for the conjugate of u. We also denote by XR a vector space of realvalued functions. Sometimes we use the abbreviations H = L2 , V = W01,2 , | · |p = | · |Lp , HR = L2R , HR1 = WR1,2 and · = | · |V , and it is understood that the integral is taken over D, unless specified otherwise. Let U and V be two Banach spaces. We denote by L (U ; V ) the Banach space of all bounded linear operators from U to V , endowed with the sup-norm. If U and V are two separable Hilbert spaces, we denote by L20 (U ; V ) the space of Hilbert–Schmidt operators from U into V with the norm 12 1 |Φei |2V , |Φ|L20 (U;V ) = (tr(Φ∗ Φ)) 2 = i∈N
where {ei }i∈N is any orthonormal basis of U . The Hilbert–Schmidt spaces L20 (HR ; HR ) and L20 (HR ; HR1 ) are simply written as L20 and L21 , respectively. We use the space R(U ; B) of γ-operators from a Hilbert space U to a Banach space B such that the image of the canonical Gaussian distribution on U extends to a Borel probability measure on B. It is a Banach space and the corresponding norm is endowed by 12 2 ˜ γj Rej |B , |R|R(U ;B) = E| j∈N
On the Generalized 2-D Stochastic Ginzburg–Landau Equation
1603
for any sequence {γj }j∈N of independent standard normal real-valued random variables on a ˜ F˜ , P˜ ) and any orthonormal basis {ej }j∈N of U . This series is convergent and probability (Ω, does not depend on {γj }j∈N , nor on {ej }j∈N . It is well known that L20 (U ; B) = R(U ; B) if B is also a Hilbert space. Moreover, as shown in [10], given R ∈ R(U ; B) and L ∈ L (B; V ), then LR ∈ R(U ; V ) and |LR|R(U ;V ) ≤ |L|L (B;V ) |R|R(U;B) . We denote by A the realization in H of the differential operator −(1 + iν)Δ with Dirichlet boundary conditions. Then the operator A is positive, self-adjoint and sectorial on the domain D(A) = H 2 ∩ V . By the spectral theory, we may define the fractional powers As of A with the domain D(As ) for any s ∈ [0, 1]. Note that the semigroup (S(t))t≥0 generated by the operator A is analytic on Lp for all 1 ≤ p ≤ ∞, and enjoys the following properties (11): H02β ⊂ D(Aβ ) ⊂ H 2β , S(t)Aβ = Aβ S(t), |A S(t)u|p ≤ Ct β
|S(t)u|q ≤ Ct
−β
1 1 q−p
β ∈ [0, 1],
β ≥ 0, |u|p ,
|u|p ,
β ≥ 0, t ≥ 0,
(2.1)
q ≥ p ≥ 1, t ≥ 0,
for some positive constant C. In the sequel, the constant C > 0 is generic and may take different values. In particular, C(R) denotes a positive constant dependent on the parameter R. Let us write f μ (u) = −(1 − iμ)|u|2σ u. Then the equation (1.1) can be rewritten as the following abstract form du + Audt = [γu + f μ (u) + F (u)]dt + udW, u(0) = u0 .
(2.2)
In what follows we fix any T > 0, p0 = 2(σ + 1) and work in the Banach space BT = C([0, T ]; V ) ∩ L2 (0, T ; W 1,p0 ) with the norm |u|BT = sup u(s) + |u|L2 (0,T ;W 1,p0 ) . 0≤s≤T
Given a stochastic basis (Ω, F , {Ft }0≤t≤T , P ), we let the process W (t) be a QQ∗ -Wiener process on the real Hilbert space HR given as the expansion W (t) =
∞
βj (t)Qej ,
j=1
where {ej }j∈N is an orthonormal basis of HR and {βj }j∈N a sequence of independent standard Brownian motions on the probability space (Ω, F , P ). We always assume the operator Q ∈ R(HR ; WR1,p0 +ε ) for some ε > 0. Clearly, we also have Q ∈ L21 ∩ R(HR ; L∞ ). We now state the following existence and uniqueness of the result. Theorem 2.1 If the assumption (1.2) holds, then for any initial datum u0 ∈ V ,there exists a unique mild solution u to Eq. (2.2) in L2 (Ω; BT ). That is, for almost each ω ∈ Ω, u has the
Yang D. S.
1604
Itˆ o’s integral form
t
S(t − s)[γu(s) − (1 − iμ)f (u(s)) + F (u(s))]ds u(t) = S(t)u0 + 0 t S(t − s)u(s)dW (s). +
(2.3)
0
Since nonlinear terms f (u) and F (u) are not Lipschitz-continuous, we will use a truncation argument, leading to a local existence result. Then by some a priori estimates we obtain that the solution is also global. 3
Local Existence
Let ρ ∈ C0∞ (R; [0, 1]) be a cut-off function such that ρ(r) = 1 for r ∈ [0, 1] and ρ(r) = 0 for u r ≥ 2. For any R > 0, u ∈ V and t ∈ [0, T ], put ρR (u)(t) = ρ RBt . The truncated equation corresponding to (2.2) is the following stochastic partial differential equation (SPDE) du + Audt = [γu + ρR (u)(f μ (u) + F (u))]dt + udW, u(0) = u0 , which has mild form
(3.1)
t
S(t − s)[γu(s) + fR (u(s)) + FR (u(s))]ds u(t) = S(t)u0 + 0 t S(t − s)u(s)dW (s), +
(3.2)
0
where for brevity we use the notation fR (u) = ρR (u)f μ (u),
FR (u) = ρR (u)F (u).
This section is devoted to the proof of the existence of local solutions to Eq. (2.2) by the above cut-off equation. That is, Theorem 3.1 If the initial value u0 ∈ V , then there is a unique mild solution u(t) to Eq. (2.2) on an interval [0, τ ∗ ) for some stopping time τ ∗ , which satisfies u ∈ L2 (Ω; Bτ ∗ ). Moreover, if τ ∗ < ∞, then P-a.s. lim sup |u|Bt = ∞. t→τ ∗
Proof We first prove the existence and uniqueness of the truncated equation and then define a stopping time τ ∗ such that Eq. (2.2) possesses a unique local solution on [0, τ ∗ ). With this aim, for T0 ∈ [0, T ], we denote by XT0 the space of all measurable and Ft -adapted processes u with values in BT0 such that u ∈ L2 (Ω; BT0 ). For u ∈ XT0 , we write 1 |u|XT0 = E|u|2BT0 2 . Let u0 ∈ V . For any t ∈ [0, T0 ] and u ∈ XT0 , we define the operator t S(t − s)[γu(s) + fR (u(s)) + FR (u(s))]ds JT0 ,R (u)(t) = S(t)u0 + 0
t
+ 0
S(t − s)u(s)dW (s).
On the Generalized 2-D Stochastic Ginzburg–Landau Equation
1605
The goal is to prove that the operator JT0 ,R maps XT0 into itself and is a contraction for small T0 . Fix T0 , R and write I1 (u)(t) = S(t)u0 , t I2 (u)(t) = S(t − s)[γu(s) + fR (u(s))]ds, 0 t S(t − s)FR (u(s))ds, I3 (u)(t) = 0 t S(t − s)u(s)dW (s). I4 (u)(t) = 0
4
We have JT0 ,R = i=1 Ii . Clearly, I1 is a constant mapping from XT0 into itself. The other terms Ii (i = 2, 3, 4) are easily proven to belong to XT0 for any XT0 by a similar argument to that below. To show that JT0 ,R is a contraction, let u1 , u2 ∈ XT0 . We put tR j = inf{t ≤ T0 : uj Bt ≥ 2R} R for j = 1, 2, and assume that tR older’s inequality and the Sobolev embedding 1 ≤ t2 . Using the H¨ 1 k inequality H → L (1 ≤ k < ∞), for any m ≥ 1, we have
u1 u2
ρR |fR (u1 ) − fR (u2 )|m ≤ (f (u1 ) − f (u2 ))ρR 2 2
m
u1 u2
ρR +
(ρR (u1 ) − ρR (u2 ))f (u2 )ρR 2 2 m 2σ ≤ C((|u1 |2σ 4mσ + |u2 |4mσ )|u1 − u2 |2m
+ R−1 |u2 |2σ+1 m(2σ+1) u1 − u2 )χ{u1 ≤4R} χ{u2 ≤4R} ≤ CR2σ u1 − u2 ,
(3.3)
where χX denotes the characteristic function of the set X. Thus the properties of the semigroup S(t) yields E sup I2 (u1 )(t) − I2 (u2 )(t)2 0≤t≤T0
≤ CE sup
0≤t≤T0
t 0
(t − s)
− 12
2 (|u1 (s) − u2 (s)|2 + |fR (u1 (s)) − fR (u2 (s))|2 )ds
≤ C(R2σ + 1)2 T0 |u1 − u2 |2XT0 ,
(3.4)
where the constant C is independent of T0 and R. Note that F (u) is locally Lipschitz-continuous from V into Lp1 for any p1 ∈ (1, 2). Indeed, for any u1 , u2 ∈ XT0 ,p , we have |FR (u1 ) − FR (u2 )|p1
u1 u2
ρR ≤ (F (u1 ) − F (u2 ))ρR 2 2 p
1
u1 u2
ρR +
(ρR (u1 ) − ρR (u2 ))F (u2 )ρR 2 2 p1
Yang D. S.
1606
≤ C( |u1 |2 ∇(u1 − u2 ) |p1 +| (|u1 | + |u2 |)|∇u2 ||u1 − u2 | p1
+ R−1 |u2 |2 |∇u2 | p1 u1 − u2 )χ{u1 ≤4R} χ{u2 ≤4R} ≤ C(u1 2 + u2 2 + R−1 u2 3 )u1 − u2 χ{u1 ≤4R} χ{u2 ≤4R} ≤ CR2 u1 − u2 ,
(3.5)
which gives E sup I3 (u1 )(t) − I3 (u2 )(t)2 0≤t≤T0
≤ CE sup
0≤t≤T0
t
0
≤ CE sup
0≤t≤T0
0
2(1− p1 )
≤ CR4 T0
t
1
2
1 2
|A S(t − s)(FR (u1 (s)) − FR (u2 (s)))|2 ds |(t − s)
− p1
1
2 |FR (u1 (s)) − FR (u2 (s))|p1 ds
|u1 − u2 |2XT0 .
(3.6)
To estimate the stochastic integral, define L : z → S(t − s)(u1 − u2 )z and write m0 = By the H¨ older’s inequality, we have Lz ≤ C(u1 − u2 )z ≤ C|z|W 1,p0 |u1 − u2 |
1
W 1,2(1+ σ )
|Lz|W 1,p0 ≤ C(t − s) ≤ C(t − s)
1 p0
− m1
1 p0
− 12 − p 1+ε 0
0
2(p0 +ε) 2+p0 +ε .
with t ≥ s;
|(u1 − u2 )z|W 1,m0 u1 − u2 |z|W 1,p0 +ε with t > s.
(3.7)
Consequently, applying the first inequality in (3.7), (2.1) and the maximal inequality of stochastic convolutions (see [12, Theorem 6.10]) gives E sup I4 (u1 )(t) − I4 (u2 )(t)2 0≤t≤T0
≤ CE
T0
0
|(u1 (s) − u2 (s))Q|2L 1 ds
≤ C|Q|2R(HR ;W 1,p0 ) ) E
≤ CE
T0
T0
0 σ 2(σ+1)
≤ CT0
0
|L|2L (W 1,p0 ;V ) ds
|u1 (s) − u2 (s)|2
1
ds σ+2
σ
u1 (s) − u2 (s) σ+1 |u(s)| σ+1 ds 1,2(1+ 2 ) E sup u1 (t) − u2 (t)
σ σ+1
0≤t≤T0
σ
σ 2(σ+1)
σ
W
≤ CT02(σ+1) E ≤ CT0
T0
W 1,2(1+ σ )
0
≤ CE
2
σ+2 σ+1 L2 (0,T0 ;W 1,p0 )
|u|
sup u1 (t) − u2 (t)2 + |u|2L2 (0,T0 ;W 1,p0 )
0≤t≤T0
|u1 − u2 |2XT0 .
(3.8)
Hence the estimates (3.4), (3.6) and (3.8) lead to E sup JT0 ,R (u1 (t)) − JT0 ,R (u2 (t)) 0≤t≤T0
1
1− p1
≤ C(R2σ T02 + R2 T0
1
1
+ T02
− p1
0
1
+ T02 )|u1 − u2 |XT0 .
(3.9)
On the Generalized 2-D Stochastic Ginzburg–Landau Equation
1607
We turn to estimating E|JT0 ,R (u1 ) − JT0 ,R (u2 )|2YT with YT0 = L2 (0, T0 ; W 1,p0 ). Using (3.3) 0 and (3.5) once again yields E|I2 (u1 ) − I2 (u2 )|2YT ≤ C(1 + R2σ )2 T02 |u1 − u2 |2XT , 0
(3.10)
0
and 2(1+ p1 − p1 )
E|I3 (u1 ) − I3 (u2 )|2YT ≤ CR4 T0
0
1
0
|u1 − u2 |2XT ,
(3.11)
0
for any p1 ∈ ( 2(σ+1) σ+2 , 2). Moreover, we apply the second inequality in (3.7) for the stochastic integral to obtain E|I4 (u1 ) − I4 (u2 )|2YT0
2 T0 t
≤E S(t − s)(u (s) − u (s))dW (s) 1 2
≤C
0
0
T0
E
0
W 1,p0
t
0
|S(t − s)(u1 (s) − u2 (s))Q|2R(HR ;W 1,p0 ) dsdt
≤ C|Q|2R(HR ;W 1,p0 +ε ) ≤C ≤
T0
t
E
dt
T0
E
0 1
t
|L|2L (W 1,p0 +ε ;W 1,p0 ) dsdt
0
1
1
|t − s|2( p0 − 2 − p0 +ε ) u1 (s) − u2 (s)2 dsdt
0 0 2( p1 − p 1+ε )+1 0 0 |u1 CT0
− u2 |2XT0 .
(3.12)
Thus we collect the inequalities (3.10), (3.11) and (3.12) to get E|JT0 ,R (u1 ) − JT0 ,R (u2 )|2YT
0
2(1+ p1 − p1 )
≤ C((1 + R2σ )2 T02 + R4 T0
0
1
2( p1 − p
1 )+1 0 +ε
0
+ T0
)|u1 − u2 |2XT0 .
(3.13)
Finally, the above estimate (3.13) along with (3.9) gives |JT0 ,R (u1 ) − JT0 ,R (u2 )|XT0 1− p1
1
≤ C((1 + R2σ )(T02 + T0 ) + R2 (T0 1 1 2 − p0
+ T0
1 p0
+ T0
− p 1+ε + 12 0
1
1+ p1 − p1
+ T0
)|u1 − u2 |XT0 ,
0
1
) (3.14)
where p1 ∈ ( 2(σ+1) σ+2 , 2). Hence JT0 ,R is a contraction mapping on XT0 for a sufficiently small T0 . The Banach contraction principle shows that JT0 ,R has a unique fixed point in XT0 , which is the unique local solution to Eq. (3.1) on the interval [0, T0 ]. Since T0 does not depend on the initial value u0 , this solution may be extended to the whole interval [0, T ]. We denote by uR this unique mild solution and let τR = inf{t ∈ [0, T ] : |uR |Bt ≥ R}, with the usual convention that inf ∅ = ∞. Since the sequence of stopping time τR is nondecreasing on R, we can put τ ∗ = supR→∞ τR . Moreover, we can define a local solution to Eq. (2.2) as u(t) = uR (t) on [0, τR ], which is well defined since uR1 (t) = uR2 (t) P-a.s. for any
Yang D. S.
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0 ≤ t ≤ τR1 ∧ τR2 . Actually, set τ = τR1 ∧ τR2 with R1 ≤ R2 . For t ∈ [0, τ ], we have t uR1 (t) − uR2 (t) = S(t − s)(γ(uR1 (s) − uR2 (s)) + FR1 (uR1 (s)) − FR2 (uR2 (s)))ds 0 t (fR1 (uR1 (s)) − fR2 (uR2 (s)))ds + 0 t S(t − s)(uR1 (s) − uR2 (s))dW (s). + 0
Proceeding as in the proof of (3.14), we can find a continuous increasing function c(t) with c(0) = 0 such that |uR1 (s) − uR2 (s)|Bt ≤ c(t)|uR1 (s) − uR2 (s)|Bt , which gives uR1 (t) = uR2 (t) for t sufficiently small. Repeating the same argument in the interval [t, 2t] and so on yields uR1 (t) = uR2 (t) in the whole interval [0, τ ]. In the end, if τ ∗ < ∞, the definition of u yields lim sup |u|Bt = ∞, t→τ ∗
which shows u is a unique local solution to Eq. (2.2) on the interval [0, τ ∗ ), and thus we complete the proof. 4
Global Existence
This section is devoted to the proof of Theorem 2.1. We will exploit an energy inequality and thus prove that the solution u to Eq. (2.2) satisfies E|u|2BT < ∞. To derive the energy estimate, we define 1 κ Jκ (u) = |∇u|22 + |u|pp00 , p0 = 2(σ + 1), 2 p0 for any κ > 0. Then we have the following Lemma 4.1 (Energy inequality) there exists κ0 > 0 such that E sup Jκ0 (u(t)) + E 0≤t≤τ
Let τ = inf{τ ∗ , T }. If the assumption (1.2) is fulfilled, then
τ 0
Proof We first note that
|Δu(t)|22 dt ≤ C(u0 , |Q|R(HR ;W 1,p0 +ε ) , T ). R
|u|2σ u ¯Δu u)2 (∇u)2 ] = −Re(1 + iα) |u|2(σ−1) [(σ + 1)|u|2 |∇u|2 + σ(¯
−(Δu, f α (u)) = Re(1 + iα)
2 ¯ 1 u∂j u 2(σ−1) =− |u| , (¯ u∂j u, u∂j u ¯)M (α, σ) u ¯∂j u 2 j=1 √ σ+1 (1−iα)σ 2σ+1 with the notation M (α, σ) = (1−iα)σ . If |α| < , then matrix M (α, σ) is defiσ σ+1 nitely positive and thus the small eigenvalue λα is positive. This gives (4.1) −(Δu, f α (u)) + λα |u|2σ |∇u|2 ≤ 0.
On the Generalized 2-D Stochastic Ginzburg–Landau Equation
1609
We now apply the It¨o’s formula with 12 |∇u(t)|22 and obtain t 1 1 1 t |∇u(t)|22 = |∇u0 |2 − (|Δu(s)|22 − γ|∇u(s)|22 )ds + |u(s)Q|2L 1 ds 2 2 2 2 0 0 t t (Δu(s), f μ (u(s)) + F (u(s)))ds − (Δu(s), u(s)dW (s)). − 0
(4.2)
0
Applying once again It¨ o’s formula to p10 |u(t)|pp00 yields t 1 1 2(p −1) p0 p0 |u(t)|p0 = |u0 |p0 + (γ|u(s)|pp00 − |u(s)|2(p00 −1) )ds p0 p0 0 t t (Δu(s), f ν (u(s)))ds + (|u(s)|2σ u(s), F (u(s)))ds − 0 0 t 1 |u(s)|u(s)|σ Q|2L 0 + σ||u(s)|σ+1 Q|2L 0 ds + N (t), (4.3) + 2 2 2 0 t where N (t) represents the stochastic integral 0 (|u(s)|2σ u(s), u(s)dW (s)). For any κ > 0, use (4.2) and (4.3) to get t (|∇u(s)|22 + κ|u(s)|pp00 )ds Jκ (u(t)) ≤ Jκ (u0 ) + γ 0 t t 2(p −1) (|Δu(s)|22 + κ|u(s)|2(p00 −1) )ds − (Δu(s), f μ (u(s)) + κf ν (u(s)))ds − 0 0 t 1 t (κ|u(s)|2σ u(s) − Δu(s), F (u(s)))ds + |u(s)Q|2L 1 ds + 2 2 0 0 t 1 ||u(s)|σ u(s)Q|2L 0 + σ||u(s)|σ+1 Q|2L 0 ds +κ 2 2 2 0 t − (Δu(s), u(s)dW (s)) + κN (t). (4.4) 0
We multiply (4.1) by 1 + κ and plug this resultant into (4.4) to obtain t t (|∇u(s)|22 + κ|u(s)|pp00 )ds + G(α, κ, r)(s)ds Jκ (u(t)) ≤ Jκ (u0 ) + γ 0 0 t t 2(p −1) (|Δu(s)|22 + κ|u(s)|2(p00 −1) )ds − (1 + κ)λα ||u(s)|σ ∇u(s)|22 ds − (1 − r) 0 0 t 1 t (κ|u(s)|2σ u(s) − Δu(s), F (u(s)))ds + |u(s)Q|2L 1 ds + κN (t) + 2 2 0 0 t 1 ||u(s)|σ u(s)Q|2L 0 + σ||u(s)|σ+1 Q|2L 0 ds +κ 2 2 2 0 t − (Δu(s), u(s)dW (s)), (4.5) 0
where r ∈ (0, 1), κ > 0 and |α| <
√ 2σ+1 σ
are three constants to be determined later, and the
function G is 2(p −1)
G(α, κ, r) = −r(|Δu|22 + κ|u|2(p00 −1) ) − (Δu, f μ (u) + κf ν (u) − (1 + κ)f α (u)), which is non-positive under the assumption (1.2) (B) for some suitable parameters r, κ and α.
Yang D. S.
1610
Indeed, we can write G as
1 Δ¯ u 2σ , G = − (Δu, |u| u)M1 (α, κ, r) 2 ¯ |u|2σ u
with M1 (α, κ, r) =
2r i(μ + κν − α(1 + κ)) −i(μ + κν − α(1 + κ)) 2rκ
Like the deterministic case (see [8]), if −1+μν < and |α0 | <
√ 2σ+1 σ
.
√ 2σ+1 |ν+μ|, σ
then there exist r0 ∈ (0, 1), κ0 > 0 such that the matrix M1 is non-negative and thus we have G(α0 , κ0 , r0 ) ≤ 0.
We continue to estimate the other terms in (4.5). Firstly, the H¨ older’s inequality and Young’s inequality yield |u|2 |∇u||Δu| + κ |u|2σ+3 |∇u| |(−Δu + κ|u|2σ u, F (u))| ≤ (3|λ1 | + |λ2 |) ≤
1−r 1+κ 2(p −1) (|Δu|22 + κ|u|2(p00 −1) ) + (3|λ1 | + |λ2 |)2 ||u|2 ∇u|22 . (4.6) 4 1−r
If the assumption (1.2) (A) holds, then the second term on the right-hand side of (4.6) can be bounded by (1 + κ)λα ||u|σ ∇u|22 + C|∇u|22 . 2 Since the Hilbert–Schmidt operator Q ∈ L21 , we have ∞ 1 σ 2 σ+1 2 ||u| uQ|L 0 + σ||u| |u|2(σ+1) |Qej |2 Q|L 0 ≤ C 2 2 2 j=1 0 ≤ C|u|p2(p 0 −1)
∞
|Qej |22(p0 −1) σ
j=1
1 − r 2(p0 −1) |u|2(p0 −1) + C, 4 and by the H¨ older inequality and the embedding inequality H 1 → L4 , 0 ≤ C|u|p2(p |Q|2L 1 ≤ 0 −1) 2
|uQ|2L 1 = 2
∞
(4.7)
uQej 2 ≤ |Δu|2 |Q|L20 .
j=1
Finally collecting these estimates (4.5) to (4.7), we obtain t λα (1 + κ0 ) t Jκ0 (u(s))ds − 0 ||u(s)|σ ∇u(s)|22 ds Jκ0 (u(t)) ≤Jκ0 (u0 ) + C 2 0 0 1 − r0 t 2(p −1) (|Δu(s)|22 + κ0 |u(s)|2(p00 −1) )ds + κ0 N (t) + M (t) + C1 , (4.8) − 2 0 t where M (t) = − 0 (Δu(s), u(s)dW (s)). Applying the Burkholder–Davis–Gundy inequality to E sup0≤t≤τ |M (t)| yields
E sup |M (t)| ≤ CE 0≤t≤τ
τ
0
≤ C|Q|L20 E
|∇u(s)|22 |u(s)Q|2L 1 ds 2 0
τ
12
|∇u(s)|22 |Δu|2 ds
12
On the Generalized 2-D Stochastic Ginzburg–Landau Equation
1 2 κ0
≤ CE sup J (u(t)) 0≤t≤τ
1611 τ
|Δu(s)|22 ds
0
1 1 − r0 E ≤ E sup Jκ0 (u(t)) + 4 0≤t≤τ 4
τ 0
14
|Δu(s)|22 ds + C.
(4.9)
We apply again the Burkholder–Davis–Gundy inequality to E sup0≤t≤τ |N (t)|, giving κ0 E sup |N (t)| ≤ CE 0≤t≤τ
τ
0
||u(s)|
p0
Q|22 ds
≤ C|Q|R(HR ;L∞ ) E 1 2 κ0
0
0 |u(s)|2p p0 ds
≤ CE sup J (u(t)) 0≤t≤τ
τ
12 12 12
τ
Jκ0 (u(s))ds
0
1 ≤ E sup Jκ0 (u(t)) + CE 4 0≤t≤τ
0
τ
Jκ0 (u(s))ds.
(4.10)
Substituting (4.9) and (4.10) into (4.8) and applying the Gronwall lemma yield 1 − r0 τ |Δu(s)|22 ds ≤ C(u0 , |Q|R(HR ;W 1,p0 +ε ) , T ), E sup Jκ0 (u(t)) + R 2 0≤t≤τ 0
which completes the proof. To complete the proof of Theorem 2.1, it suffices to show P (τ applying the Sobolev inequality W 2,2 → W 1,p0 yields
∗
= ∞) = 1. Actually,
E sup |u(t)|2Bτ ≤ C(u0 , |Q|R(HR ;W 1,p0 +ε ) , T ). R
By the Chebyshev’s inequality, Lemma 4.1 and the definition of u, we have P (τ ∗ < ∞) = lim P (τ ∗ ≤ T ) = lim P (τ = τ ∗ ) T →∞
T →∞
= lim lim P (τR ≤ τ ) ≤ lim lim P (|u|Bτ ≥ |u|BτR ) T →∞ R→∞
T →∞ R→∞
E|u(t)|2Bτ = 0, T →∞ R→∞ R2
≤ lim lim P (|u(t)|Bτ ≥ R) ≤ lim lim T →∞ R→∞
∗
which gives P-a.s. τ = ∞. References [1] Doering, C. R.: On the Nonlinear Evolution of Patterns (Modulation Equations and Their solutions), Ph. D. thesis, University of Utrecht, the Netherlands, 1990 [2] Doering, C. R., Gibbon, J. D., Holm, D., et al.: Low-dimensional behavior in the complex Ginzburg–Landau equations. Nonlinearity, 1, 279–309 (1988) [3] Doering, C. R., Gibbon, J. D., Levermore, C. D.: Weak and strong solutions of the complex Ginzburg– Landau equation. Physica D, 71, 285–318 (1994) [4] Duan, J., Holmes, P.: On the Cauchy problem of a generalized Ginzburg–Landau equation. Nonlinear Anal., 22, 1033–1040 (1994) [5] Duan, J., Holmes, P., Titi, E. S.: Global existence theory for a generalized Ginzburg–Landau equation. Nonlinearity, 6, 915–933 (1993) [6] Gao, H., Duan, J.: On the initial value problem for the generalied 2D Ginzburg–Landau equation. J. Math. Anal. Appl., 216, 536–548 (1997) [7] Guo, B., Wang, B.: Finite dimensional behaviour for the derivative Ginzburg–Landau equation in two spatial dimensions. Physica D, 89, 83–99 (1995) [8] Li, Y., Guo, B.: Global existence of solutions to the 2D Ginzburg–Landau equation. J. Math. Anal. Appl., 249, 412–432 (2000)
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Yang D. S.
[9] Barton-Smith, M.: Global solution for a stochastic Ginzburg–Landau equation with multiplicative noise. Stochastic Analysis and Application, 22(1), 1–18 (2004) [10] de Bouard, A., Debussche, A.: A stochastic nonlinear Schr¨ odinger equation with multiplicative noise. Comm. Math. Phys., 205, 161–181 (1999) [11] Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations, SpringerVerlag, Berlin/New York, 1983 [12] Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992
