Abstract

We study the Cauchy problem for a type of generalized Zakharov system. With the help of energy conservation and approximate argument, we obtain global existence and uniqueness in Sobolev spaces for this system. Particularly, this result implies the existence of classical solution for this generalized Zakharov system.

1. Introduction

In this paper, we study a type of generalized Zakharov system which is given by with initial data where , , and is a fixed constant. In the above system, is a fractional differential operator. With this definition, maps to with the Fourier transform of with respect to the variable . In particular, .

When , system (1) and (2) reduces to the usual Zakharov system which was first obtained by Zakharov [1]; here, is the slowly varying amplitude of high-frequency electric field and      is the disturbing quantity of ion from its equilibrium. This model turned out to be very useful in laser plasmas, and many contributions have been made both in the physical and mathematical literature. For the local or global existence and uniqueness of smooth solutions for system (4), we refer to [26]. Well-posedness of (4) in lower regularity spaces was obtained in [7]. Existence of global attractors for dissipative Zakharov system was studied in [811]. For related Zakharov system including magnetic effects, one can see [1215].

On the other hand, Laskin [16, 17] discovered that the path integral over the Lévy-like quantum mechanical paths allows developing the generalization of the quantum mechanics. That is, if the path integral over Brownian trajectories leads to the well-known Schrödinger equation, then the path integral over Lévy trajectories leads to the fractional Schrödinger equation. So fractional Schrödinger equation is fundamental in the fractional quantum mechanics, and its global well-posedness is studied in [18, 19]. Inspired by this, we then replace the Laplacian in the Schrödinger equation of (4) by the fractional differential operator , and this is the main motivation of the paper.

In this work, we study global existence and uniqueness of smooth solutions for system (1) and (2). The main result is stated in the following theorem.

Theorem 1. Let , let be an integer, , and . Then system (1)(3) has a unique solution satisfying

Theorem 1 will be proved by using energy conservation and approximate argument. To this end, in the next section, we present some notations and useful lemmas which will be used throughout the paper. In Section 3, we study a regularized system of (1) and (2). Finally, the proof of Theorem 1 is given in Section 4.

2. Preliminaries

Firstly, we set some notations. For , we use to denote the fractional homogeneous Sobolev space, consisting of all tempered distribution such that is finite, where is defined via the Fourier transform Similarly, one can define the inhomogeneous Sobolev space equipped with the norm In particular, we have for . Throughout the paper, the initial data (3) is given in the product space defined by We endow with the natural norm

Next, we introduce the following calculus inequality, the proof of which can be found, for example, in [2022].

Lemma 2. Let and (the class of Schwarz functions); then where , , , .

We end this section with the following lemma, which states two conserved quantities for the smooth solutions of (1)(3). Here, we say a solution is a smooth solution of system (1)(3) provided that with sufficiently large and (1)(3) hold in the classical sense.

Lemma 3. Suppose that is a smooth solution of system (1)(3); then there hold

Proof. Multiplying on both sides of (1) and then choosing the imaginary part after integration in , it is easy to obtain
Now, we give the proof of the second conserved quantity. On one hand, multiplying on both sides of (1) and choosing the real part after integration in , we then get On the other hand, taking inner product of (2) with , we then obtain Combining the above two equalities gives .

3. Global Existence and Uniqueness for a Regularized System

In order to prove Theorem 1, we firstly study a regularized system for (1)(3) in this section. For , let us consider the following regularized system: where the operator and is the solution of the equation with initial data , . It is easy to see that the operator satisfies the following properties:(1), , ;(2)(3); (4)Roughly speaking, the fourth property says that the operator commutes with the operator ; of course, the operator can be replaced by other differential operators such as .

From the semigroup theory we know that the linear equation generates a unitary group in , so the solution of (15) can be expressed by the following integral form:

A few words about the regularized system (15) or (17). If we study directly the integral equation of the original system (1)(3), that is, where solves (2), we will find that it is difficult to apply fixed point theorem for this integral equation because the regularity of and is not the same (note that ). In fact, when estimating the norm of , we have where we need . However, this is not correct since only belongs to . For this reason, we first study the regularized system (15) by introducing the operator , and we can see that if . Then the well-posedness result of the regularized system can be easily proved through the integral equation (17) (see Theorem 6). Based on the solution of (15) and (16), we have to take in the regularized system to obtain the desired result as stated in Theorem 1. This step requires some uniform estimates for the solution of the regularized system, and these a priori estimates will be given in Section 4.

The main aim in this section is to obtain the existence and uniqueness of global solution for the regularized system (15) and (16). Due to the “good” operator , the global well-posedness result for the regularized system can be proved more easily. Before stating Theorem 6, we need the following two lemmas.

Lemma 4 (conserved quantities). Suppose that is a smooth solution of the regularized system (15) and (16); then there hold

The proof of Lemma 4 is similar to Lemma 3; thus, it is omitted here.

Lemma 5. Assume that is a smooth solution of the regularized system (15) and (16); then there holds where the constant depends on , and . In particular, the above estimate implies that

Proof. From Lemma 4, we know that By Hölder’s inequality and the embedding , the last term in the above inequality can be estimated as follows: Applying the Gagliardo-Nirenberg inequality we have Using this inequality and Young’s inequality, there holds where is a constant depending on , . Combining the above arguments, one can see that We firstly choose small enough to make sure that is absorbed by the term ; for such fixed , we then choose small enough to make sure that is absorbed by the term . Thus, we get Since , the estimate (22) follows easily from the embedding (). The proof of Lemma 5 is complete.

Now, we state the main result of this section.

Theorem 6. Let be an integer, and assume ; then for arbitrary , the regularized system (15) and (16) has a unique solution .

Proof. The proof consists of two parts: the first part is to prove local existence of the solution for the regularized system by using the standard Banach’s fixed point theorem, and the second part is to extend this local solution to be a global one with the help of some a priori estimates.
Step  1. Firstly, we get the local existence by using the contracting mapping principle. In order to achieve this aim we define by As satisfies (16), there holds
For , we now define the space where . From (31), it is easy to see that where depends on . By the definition of , one also has Combining the above estimates, we have Hence, if we choose sufficiently small, then maps into itself and is contractive. From the contraction mapping principle, (15) admits a unique solution ; , which, by (31), gives . Moreover, from the above procedure, we know that if is the largest existence time of the solution, then or as .
Step  2. In order to get the global existence result, it suffices to prove that for all . To this end, applying the operator to (15), then multiplying the resulted equation by , and integrating the imaginary part, one can obtain By (31) and Lemma 5, one can see that which implies Using this estimate and the fact that , one gets from (36) and Lemma 5 that With similar arguments as above, one can deduce from (16) and Lemma 2 that
Finally, collecting the above two estimates and using Gronwall’s inequality, there holds This inequality together with Lemma 5 gives which implies that . The proof of Theorem 6 is complete.

4. Proof of Theorem 1

In this section, we will present the proof of Theorem 1. In this proof, the key step is to obtain uniform estimates for the approximate solution with respect to . Note that the constant in (42) depends on , so this estimate is not useful in proving our global existence result for system (1)(3).

For , we now choose the regularized initial data with sufficiently large satisfying Now, we are going to give the uniform estimates for . These uniform estimates are stated in the following propositions.

Proposition 7. Suppose that is the solution of the regularized system (15) and (16) with satisfying (43); then for sufficiently small , there holds where the constant depends on , and , but is independent of and . In particular, the above estimate implies that

Proposition 7 follows easily from Lemma 5 and (43).

Proposition 8. Under the same assumption as Proposition 7, there holds for sufficiently small , where the constant depends on , , and . In particular, this estimate implies that

Proof. Taking one derivative with respect to on both sides of (15), one gets Then multiplying this equation by and integrating the imaginary part, one gets
Next, we take the inner product of (16) with and obtain where we have used the following estimate:
Note that (15) implies that
Now, it follows from (49)(52) that By Gronwall’s inequality, we have which gives, by (52) and Proposition 7, that Since Gronwall’s inequality gives for all . This estimate together with (55) yields the desired estimate.

Proposition 9. Under the same assumption as Proposition 7, there holds where the constant depends on , , and .

Proof. Applying the operator on both sides of (15) and (16), we get
Differentiating (58) with respect to , we can get Multiplying on both sides of the above equation and integrating the imaginary part, we have By Cauchy-Schwarz inequality, Lemma 2, Proposition 8, and Sobolev interpolation inequality, it is easy to get
Taking inner product of (59) with and then using (45) and Lemma 2, one can obtain Moreover, from (58), Lemma 2, and Proposition 8, we know that
From (62)(64), we have Then Gronwall’s inequality gives This estimate and (64) yield the desired estimate.

Applying the above procedure step by step, we finally obtain the following proposition.

Proposition 10. Under the same assumption as Proposition 7, there holds where the constant depends on , , , and .

Now, we give the proof of Theorem 1.

Proof of Theorem 1. By Propositions 710 and (15) and (16), there exists a subsequence of which converges weakly to (for simplicity, we use the same notation for the subsequence); that is, Moreover, by Sobolev compact embedding theorem, we also have for all . Note that this convergence result implies that converges to in the a.e. sense.
Since one has Thus, by (69)(71) and the uniqueness of weak limit, there are Now, letting in the regularized equations (15) and (16) and using the above convergence results as well as (43), we know that is a solution to system (1)(3). Moreover, with similar arguments as in [12], the solution is a continuous flow with respect to time; namely, . The existence part of Theorem 1 is finished.
It remains for us to prove the uniqueness. Suppose that , are both the solutions of system (1)(3). We set , and then satisfies the following equations: From (73) and (74), we can obtain
Taking one derivative of (73) with respect to , we get Multiplying on both sides of the above equation and integrating the imaginary part, we can obtain the following estimate (attention: and ): Furthermore, it is easy to see that and by (73),
Finally, it follows from (76)(81) that where the constant depends on and , . By Gronwall’s inequality and the zero initial condition (75), we get and . Thus the solution of system (1)(3) is unique. This ends the proof of Theorem 1.

Conflict of Interests

The authors declare that they do not have any commercial or associative interest that represents a conflict of interest in connection with the work.

Acknowledgments

The authors would like to thank the anonymous referee for his/her valuable remarks and suggestions on the earlier version of the paper. Jingjun Zhang is supported by the National Science Foundation under Grant no. 11201185 and Zhejiang Provincial Natural Science Foundation of China under Grant no. LQ12A01013.