Abstract and Applied Analysis

Volume 2014 (2014), Article ID 607028, 11 pages

http://dx.doi.org/10.1155/2014/607028

Research Article

On the Analyticity for the Generalized Quadratic Derivative Complex Ginzburg-Landau Equation

Chunyan Huang

School of Statistics and Mathematics , Central University of Finance and Economics, Beijing 100081, China

Received 10 December 2013; Accepted 15 February 2014; Published 16 April 2014

Academic Editor: Hossein Jafari

Copyright © 2014 Chunyan Huang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

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.

1. Introduction

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 [4] 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 [5] established in one space dimension the existence of local solution in with small total disturbance in . Han et al. [6] 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 [9]. 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 [10]. It is easy to see by Minkowski inequality that

We now recall the definition of Modulation space which was first introduced by Feichtinger [11] in 1983 (see also Gröchenig [12]). 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 [8] 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 [17]. 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 [7].

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 [9], 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 .

Theorem 8 states that (1) is globally well-posed with small initial data in and the solution is actually real analytic for any .

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 .

For the proof of (49), one only needs to take in (44) and then norm together with Young’s inequality will imply the result.

Now we recall a nonlinear mapping estimate in which was shown in [7].

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 ,

hold for all and .

Proof. In view of (29), we immediately have (61).

Let us apply Proposition 12:

which implies (60).

We now extend the solution from to for some . Consider the mapping

and the resolution space

where will be chosen below. Taking in (60), one has that

For any , in view of (65), we can choose a sufficiently large such that

Hence, in view of Proposition 9,

and one can choose verifying , so

In view of (67) and (68),

It follows from Propositions 10 and 11

Hence, collecting (69) and (70), we have

Now we can choose satisfying

It follows that for any . So, we have extended the solution from to . Noticing that

we easily see that the solution can be extended to and finally find a satisfying the conclusions in Theorems 6 and 7.

Remark 13. As explained in the introduction, the method used here does not work for the critical case and we need to find other ways to deal with this situation in the next section.

4. Analytic Regularity for GDGL:

Recall that, in Section 2, we verify the following frequency uniform exponential decay estimate for the generalized Ginzburg-Landau semigroup :

and the frequency dyadic decay estimate:

Based on (74) and (75), we will establish some estimates for and in suitable space time exponential modulation and Besov norms, respectively. First, let us consider the linear estimates for in space time exponential modulation norm.

Proposition 14. Assume that , , ; then there hold

Proof. For , from (74)

Taking norm to (77),

For , (78) also holds. Hence taking a sequence norm on both sides of (78) implies the first estimate. For the second estimate, one only needs to take norm to (74) and then a sequence norm.

More generally, using the method in Proposition 14, one can verify that, for the infinite time interval , there holds the following.

Proposition 15. Let , , ; then for any ,

Next, we proceed with the estimates of in space time exponential modulation norm.

Proposition 16. Let , , . Then there exists a constant such that, for , , and ,

holds for all .

Proof. By the decay estimate (74), we easily see that

Multiplying (81) by ,

When , applying Young’s inequality to (82) with ,

So, taking the sequence norm on both sides of (83) implied the conclusion.

In view of and taking special index in Proposition 16, we obtain the following.

Corollary 17. Under the same conditions as above, one has

We now consider the linear estimates of and in space time homogeneous Besov norms based on dyadic decay estimate (75).

Proposition 18. Assume that , ; then

Proof. Let us take norm on both sides of dyadic decay estimate (75):

Then multiplying the above inequality by and taking a sequence norm, we obtain the result as desired.

Proposition 19. Let , , . Then

In particular,

Proof. Using dyadic decay estimate (75), we see that

Taking norm on both sides of the above estimate and with Young’s inequality, we get

Arguing as before, we multiply (90) by and take a sequence norm and then finally get the conclusion as desired.

With the preparation above, we are now ready to start the proof of Theorem 8. In the following let . We choose the following resolution space:

with metric , where and . Consider the map

We will first carry out the computation of . Set . By (74), for ,

Denote . Taking norm and then sequence norm on to (93), we get

For , there holds

From (94) and (95), it is easy to see that

Similarly,

Hence

By Corollary 17 and Proposition 11,

In view of , we observe that

To bound the right-hand side of (100), let us recall the paraproduct decomposition [10, 20]: for any two tempered distributions , can be decomposed as the summation of two parts:

From the support property in frequency space, there holds

Thus

With this tool, we now estimate .

By taking norm, we deduce that

Multiplying (105) by and taking a sequence norm, we get

where in the last inequality, we use Young’s inequality. From (100) and (106), we obtain

Collecting (98), (99), and (107), we see that

On the other hand, by Propositions 18 and 19, (100), and (106), similar reasoning yields that

Thus from (108) and (109), we have

Suppose that

By the standard contraction mapping principle, we see that there exists a unique solution to in space . In particular, there hold

So for ,

We now extend the regularity of solutions. Consider the integral equation

Let . Denote

Choose the resolution space to be

with metric . is to be determined later. Similar to (109), we have

To estimate , by Proposition 15

For the remaining part concerning , we use and Proposition 18 to bound

Arguing as before, with Corollary 17 and Proposition 11, we get

Similar as in (107), we have

Then

Choose satisfying and ; then by the contraction mapping argument, we see that has a unique solution in which completes the proof of Theorem 8.

Remark 20. The same reasoning can show that if is sufficiently small, then there exists a unique global solution to (1) satisfying and with . We omit the details here.

5. Conclusion

In this paper, we investigate and prove some well-posedness results and analytic regularity property of the solutions to the (fractional) quadratic derivative Ginzburg-Landau equation with rough initial data in certain critical modulation spaces and Besov type spaces. The basic strategy is via contraction mapping principle. The main obstacle that comes from the one order derivative in the nonlinearity is finally resolved by developing suitable exponential decay estimates for the (fractional) Ginzburg-Landau semigroup. The suggested approach can be used for more general fractional nonlinear partial differential equations.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The author would like to express great thanks to the editor and the referees for their helpful suggestions and comments. The author is supported in part by the National Science Foundation of China, Grant 11201498.

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