Analytical and Numerical Approaches for Complicated Nonlinear EquationsView this Special Issue
Research Article | Open Access
On the Analyticity for the Generalized Quadratic Derivative Complex Ginzburg-Landau Equation
We study the analytic property of the (generalized) quadratic derivative Ginzburg-Landau equation in any spatial dimension with rough initial data. For , we prove the analyticity of local solutions to the (generalized) quadratic derivative Ginzburg-Landau equation with large rough initial data in modulation spaces . For , we obtain the analytic regularity of global solutions to the fractional quadratic derivative Ginzburg-Landau equation with small initial data in . The strategy is to develop uniform and dyadic exponential decay estimates for the generalized Ginzburg-Landau semigroup to overcome the derivative in the nonlinear term.
In this paper, we are interested in the Cauchy problem of the following generalized quadratic derivative complex Ginzburg-Landau equation (GDGL): where is a complex valued function of , , . is the dissipative coefficient, . is a given complex vector in . is a given complex valued function of . and denotes the fractional Laplacian defined by . It is well known that (1) can be rewritten into an integral equation as follows: where
Complex Ginzburg-Landau type equation is one of the most-studied equations in physics. It describes a lot of phenomena including nonlinear waves and the evolution of amplitudes of unstable modes for any process exhibiting a Hopf bifurcation. GDGL (1) is also called derivative fractional Ginzburg-Landau equation. For details of physical backgrounds of the fractional Ginzburg-Landau equation (1), one can refer to [1–3]. Equation (1) is both dissipative and dispersive. If , (1) is the quadratic derivative Ginzburg-Landau equation (QDGL): If , , (1) reduces to the well-known quadratic derivative Schrödinger equation (DNLS):
For DNLS (5), Christ  proved that when space dimension , the flow map is not continuous in any Sobolev space with any exponent for any short time in the sense that but after an arbitrarily short time. For (5), Stefanov  established in one space dimension the existence of local solution in with small total disturbance in . Han et al.  showed that (4) and (5) are locally well-posed in modulation space under the small condition of norm and they obtained the inviscid limit behavior between the solutions of (4) and (5) with initial data in as the dissipative parameter . For general , to the knowledge of the author, there are few results on (1). In this paper, we will study the analyticity of solutions of (1) for with rough initial data in certain modulation space. In the case , we prove that the local solution of (1) is real analytic with initial data in ; in the case , we show that (1) is globally well-posed with small initial data in and moreover the global solution of (1) is real analytic for any .
We now briefly sketch the idea of the proof. The basic strategy is to choose the working space to be some time dependent type exponential modulation space, say with , and consider the map: then use the standard contraction mapping method to prove that there exists a unique solution in this space. Due to the nice property of , the solution is naturally analytic for any . However, the main obstacle comes from the derivative in the nonlinear term. To resolve this difficulty, our idea is to make full use of the strong dissipative property of GDGL (1) when . Motivated by the work in [7, 8], we prove two exponential decay estimates of the generalized Ginzburg-Landau semigroup combined with frequency uniform decomposition operator and frequency dyadic decomposition operator : for all . Then we gain derivative in space from (7) in suitable space time norm which is sufficient to balance the one order derivative in the nonlinear term. More precisely, when , we choose the resolution space as and establish some linear estimates of and in this space like And satisfies similar estimates. Since one can choose sufficiently small to make sure that and finally verify that is a contractive map on which ensures that (1) has a local solution satisfying . Moreover, we can prove that
which implies that and hence the local solution is analytic for any . However, when , it is impossible to choose satisfying (10). To make this bound valid, one needs to impose additional small condition on . Then we will only obtain the existence of local solution with small initial data which is not ideal. So we intend to seek for different approach in this critical situation. The preferred working space would be . But when we bound it is easy to see that to control the right-hand side of (12), should be involved in the working space. So, the natural working space would be where . However, the obstacle comes again. The following low frequency projection term could not be bounded in this working space: To overcome this difficulty, our idea is to make use of the property and bound (13) by Then the time dependent type Besov norm should be included in the working space. So, finally we choose the resolution space as The corresponding condition imposed on the initial data would be stronger, , and sufficiently small. In Section 4, we develop estimates in this resolution space and combine them with the contraction mapping argument; finally we show that there exists a unique global solution to (1) in which is naturally analytic for .
Now let us recall some notations and basic facts that will be used in the sequel. , will denote universal positive constants which can be different at different places. (for ) means that . For any , we write and . Now we introduce some spaces. We denote by the Lebesgue space on which the norm is written as . Let be a Banach space. For any , we define for and with usual modification for . If , we will write and simply denote if . Now let us recall the notation and definitions in Littlewood-Paley theory . Let be a smooth radial cutoff function satisfying Denote and we introduce the function sequence , . Then , , are said to be the homogeneous dyadic decomposition operators and satisfying the operator identity: . The low frequency projection operators are defined by . It is easy to see that as in the sense of distributions. With this decomposition, the norms in homogeneous Besov spaces are defined as follows: And the space time homogenous Besov norms are defined by with the usual modification for . Such a kind of space was first used in Chemin . It is easy to see by Minkowski inequality that
We now recall the definition of Modulation space which was first introduced by Feichtinger  in 1983 (see also Gröchenig ). Let be a smooth cutoff function with , , and Then the frequency uniform decomposition operator is defined as Using this decomposition operator, for any , , , we define with usual modification for . is said to be a modulation space and it has been successfully applied to study nonlinear evolutions in recent years [8, 13–16]. Let , , ; the exponential modulation space was introduced in  with the following norm: We remark that when , this space can be viewed as modulation space with analytic regularity and when , it reduces to normal modulation space .
Let , and . Recall that the Gevrey class is defined as follows:
It is proved that is the Gevrey -class and any function in this space is real analytic . One can easily check that for . Therefore, any function in is real analytic. There is a very nice relationship between Gevrey class and exponential modulation spaces which is shown in Huang and Wang .
Lemma 1. Let , . Then
Remark 2. From this property we easily see that if we can prove the solution in exponential modulation space with positive regularity, then it is naturally analytic.
Inspired by (19), we define the following space time exponential modulation norm: with usual modification if .
In the end, let us recall the definition of multiplier space [18, 19]. Let . If there exists a such that holds for all , then is called a Fourier multiplier on . The linear space of all multipliers on is denoted by and the norm on which is defined as . Concerning the multipliers, there holds the following famous inequality which is also called the multiplier theorem.
Proposition 3 (see , Nikol’skij’s inequality). Let be a compact set, . Denote and assume that . Then there exists a constant such that holds for all and . In particular, if , then (28) holds for all .
The remaining part of this paper is organized as follows. In Section 2, we develop two decay estimates (7) associated with the generalized Ginzburg-Landau semigroup. In Section 3, we prove the analytic regularity property of the solutions to (1) when . In Section 4, we deal with the analytic property of (1) in the critical case . Finally, a short conclusion is given in Section 5.
2. Decay Estimates for GCGL Semigroup
In this part, we will set up some decay estimates for the generalized Ginzburg-Landau semigroup together with the frequency uniform decomposition operator and the dyadic operators . As explained in the introduction, these estimates are crucial to the proof of the main theorems.
Proposition 4 (uniform decay estimate). Suppose that , , . Then there exists (say ) such that holds for all and .
Proof. First, we choose a smooth cutoff function satisfying for and for . It is easy to see that equals on the support of and has similar property as . Applying this fact, we deduce that In view of Nikol’skij’s inequality, Since for , applying Leibniz’s Rule, we infer that where Due to the support property supp , (30)–(33), we conclude that, for , Note that (34) also holds for . Hence, we complete the proof of (29).
Proposition 5 (dyadic decay estimate). Suppose that , , . Then there exists (say ) such that holds for all .
Proof. When , Ginzburg-Landau semigroup is strong dissipative. Using the exponential decay property of and Nikol’skij’s inequality, we deduce that, for , By the support property of dyadic decomposition operator , there holds the following identity: so This completes the proof as desired.
3. Analytic Regularity for GDGL:
The main results of this paper are the following theorems.
Theorem 6 (analyticity for QDGL). Let , , , and . Assume that . Then there exists a such that (1) has a unique solution , . Moreover, the solution enjoys the following properties.(i).(ii)If , we have .
Theorem 7 (analyticity for GDGL (I)). Let , , , . Suppose that . There exists a such that (1) has a unique solution , . Moreover, the solution satisfies the following properties.(i).(ii)If , then .
Theorems 6 and 7 tell us that when , (1) is locally well-posed with any initial data in and moreover the local solution is analytic. However, the method used for Theorems 6 and 7 does not work for the critical case: . We need to impose stronger conditions on the initial data, that is, and be sufficiently small. We remark that there is no inclusion between and since while .
Theorem 8 (analyticity for GDGL (II)). Let , , and . Assume that is sufficiently small; then there exists a unique global solution to (1) satisfying and with .
In this section, we unify the proof of Theorems 6 and 7 in one part. The proof of Theorem 8 is left to Section 4. For convenience, we denote, for any , We first build up some linear estimates for and .
Proposition 9. Let , , and . There exists a constant () such that, for and , holds for all .
Proof. For any , multiplying (29) by and taking norm, we get
Taking the sequence norm on set to inequality (42), we obtain the estimate in (40). For , from (29), we see that
taking norm on (43) which implies (41).
Now let . We consider the estimate of . Again using uniform decay estimate (29), we have It follows from (44) that for Taking norm on (45) and applying Young’s inequality, we get In view of , we deduce that
So, taking the sequence norm on both sides of (47), we have the following.
Proposition 10. Let , . There exists a constant () such that, for and , holds for all .
Now we recall a nonlinear mapping estimate in which was shown in .
Proposition 11. Let . Consider and , , and with . Then one has
Now let us consider the following map: Let , . For any , one can choose satisfying . By Proposition 9, On the other hand, one can choose satisfying so, Hence, in view of (52) and (54), It follows from Propositions 10 and 11 that We can assume that . Hence, collecting (55)-(56), we have Now we can fix verifying . Put We have if and By the standard contraction mapping argument, there exists a unique solution to (1) in satisfying .
Next, we prove that In order to show (60), we need the following.
Proposition 12. Let , . There exists a constant () such that, for and ,