SOLVABILITY OF A CLASS OF COMPLEX GINZBURG-LANDAU EQUATIONS IN PERIODIC SOBOLEV SPACES

. This paper is concerned with the Cauchy problem for the complex Ginzburg-Landau type equation u t = ( δ 1 + iδ 2 )∆ u − iµ | u | 2 σ u in (0 , ∞ ) × R d , where δ 1 > 0, δ 2 ,µ ∈ R and d ∈ N . Existence and uniqueness of spatially periodic solutions to the problem are established in a space which corresponds to the Sobolev space on the d -dimensional torus when 0 < σ < ∞ ( d = 1 , 2) and 0 < σ < 1 / ( d − 2) ( d ≥ 3). The result improves the case p = 2 of the result in the space W 1 ,p given by Gao-Wang [2, Theorem 1] in which it is assumed that d < p and σ < p/d .

1. Introduction. In this paper we consider the Cauchy problem for a class of complex Ginzburg-Landau equations    ∂u ∂t = (δ 1 + iδ 2 )∆u − iµ|u| 2σ u, (t, x) ∈ (0, ∞) × R d , Since spatially periodic functions can be regarded as those on the d-dimensional torus T d , the problem (CGL) 0 can be also translated to the problem on T d . In this paper we shall use W m,p per (R d ) instead of W m,p (T d ) (Sobolev space on T d ) because our interest is solvability of (CGL) 0 on R d and our treatment is based on functions on R d .
Then there exists a unique local solution of (CGL) 0 .
(II) Assume further that Then there exists a unique global solution of (CGL) 0 .
We focus our eyes on the case p = 2. If p = 2, then d satisfying (1) is restricted to d = 1 and the combination of σ and d satisfying (2) exists only when σ = 1 and d = 1. In other words, Gao and Wang have not dealt with the case d ≥ 2 or σ = 1.
The purpose of this paper is to relax the conditions (1) and (2) when p = 2 and to extend the restriction from σ ∈ N to σ ∈ R + := (0, ∞). Here we define local and global solutions of (CGL) 0 as follows.
In particular, if T = ∞, then u is said to be a global solution of (CGL) 0 . Now we state local and global existence of solutions to (CGL) 0 in the following two theorems, respectively. Theorem 1.2 (Local existence). Let u 0 ∈ W 1,2 per (R d ) and δ 1 > 0. Assume that σ satisfies Then there exists a unique local solution on [0, T ) of (CGL) 0 for some T > 0. Also, let u and v be local solutions on Then where L and ω are positive constants depending only on δ 1 , δ 2 , µ, σ, d, T and M .   This theorem can be regarded as a limiting case of the results for (CGL) κ as κ ↓ 0. Indeed, in [3,7,9,10] global existence of solutions to (CGL) κ was established under the condition (a) or the following (b) κ : To prove Theorems 1.2 and 1.3 we prepare fundamental estimates for (CGL) 0 which are given in Section 2. In Section 3 we first construct a mild solution of (CGL) 0 and we next prove local existence of solutions (Theorem 1.2). Section 4 is devoted to the proof of global existence (Theorem 1.3). In Section 5 we give some remarks on the inviscid limit (as δ 1 ↓ 0) of solutions to (CGL) 0 .

2.
Preliminaries. For δ 1 > 0 and δ 2 ∈ R we define G t as follows: First we show that G t plays a fundamental role in solving The following three lemmas can be proved by the direct calculations.
Lemma 2.2. Let G t be as in (5) with δ 1 > 0. Then for every t > 0, Now we define * by the convolution with respect to spatial variables: In the next lemma we describe that (δ 1 + iδ 2 )∆ generates an analytic semigroup on W 0,2 per (R d ) which can be represented by the convolution operator. Lemma 2.4. Let G t be as in (5) with δ 1 > 0. Define T (t) as T (0) := I and Then T (t) is a uniformly bounded C 0 semigroup on W 0,2 per (R d ) and its infinitesimal generator is given by (δ 1 +iδ 2 )∆ with domain W 2,2 per (R d ). Moreover, T (t) can be extended to an analytic semigroup, and so Proof. First we show that T (t) is a uniformly bounded C 0 semigroup and T (t) is differentiable for t > 0. From (7) and (10) it follows that and hence T (t) is uniformly bounded; note that T (0) = I. Using the Fourier transform, we can verify that Next, in the same way as in the proof of [1, Theorems 7.1], we can show that for δ 1 > 0, (9) and (10) give Thus by Pazy [11, Theorem 2.5.2] we obtain the conclusion.
We can also obtain the following two lemmas for W 1,2 per (R d ).
. Moreover all the above injections are continuous.
where * denotes the convolution w.r.t. spatial variables, G t is defined by (5) and Lemma 3.2. Let u 0 ∈ W 1,2 per (R d ) and assume that Then there exists T > 0 and a unique mild solution on [0, T ] of (CGL) 0 .
On the other hand, we see from (7) and (10) that Combining these inequalities and using Lemma 2.5, we obtain In view of (13) and (14) As in the proof of (14), we see from Lemma 2.7 that Taking the supremum on [0, T ], we have (15). Therefore if we take T sufficiently small, then the mapping S is a contraction on B R . Consequently, the contraction mapping principle yields that there exists a unique solution of (11) in B R . Finally we show uniqueness of mild solutions to (CGL) 0 . Let u and v be two mild solutions on [0, T ] of (CGL) 0 . Then we can see from Lemma 2.7, (7) and (8) that where Combining (17) with (16), we have Hence by Gronwall's inequality we obtain that u(t) − v(t) 1,2 ≤ 0. Therefore we conclude that u = v on [0, T ].

Local solutions.
We show that the mild solution of (CGL) 0 is the local one of (CGL) 0 in the sense of Definition 1.1.
Proof of Theorem 1.
Assume further that the assumption in Theorem 1.3 is satisfied. Then there exists a constant L 0 > 0 depending only on δ 1 , δ 2 , µ and σ such that Proof. Multiplying the equation in (CGL) 0 by u(t), integrating it over (0, 1) d , taking its real part and using integration by parts, we obtain 1 2 Thus we obtain (23). To prove (24) we set Multiplying the equation in (CGL) 0 by −∆u(t) and |u(t)| 2σ u(t), in the same way as in the proof of (23), we have and where C σ := σ/ √ 2σ + 1 > 0. By virtue of the conditions on δ 1 , δ 2 , µ and σ in Theorem 1.3, we can choose k ≥ 0 satisfying In fact, we have (28) by taking Combining (28) where C 2 is a positive constant satisfying w 0,2σ+2 ≤ C 2 w 1,2 for w ∈ W 1,2 per (R d ). This is nothing but the desired inequality (24).
We are in a position to complete the proof of Theorem 1.3.
Therefore we see from the standard argument that u can be extended to the global solution of (CGL) 0 . We finish the proof of Theorem 1.3.
Letting δ 1 ↓ 0 in (CGL) 0 , we obtain the Cauchy problem for nonlinear Schrödinger equations Here we point out some remarks in order.
Consequently, we conclude that u is a global weak solution of (NLS).