Existence and Uniqueness of Solutions for a Type of Generalized Zakharov System

which was first obtained by Zakharov [1]; here, E : R+×R → C is the slowly varying amplitude of high-frequency electric field and n : R+ × R → R 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 andmathematical literature. For the local or global existence and uniqueness of smooth solutions for system (4), we refer to [2–6]. Well-posedness of (4) in lower regularity spaces was obtained in [7]. Existence of global attractors for dissipative Zakharov system was studied in [8– 11]. For related Zakharov system including magnetic effects, one can see [12–15]. On the other hand, Laskin [16, 17] discovered that the path integral over the Lévy-like quantummechanical 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 wellposedness is studied in [18, 19]. Inspired by this, we then replace the Laplacian in the Schrödinger equation of (4) by the fractional differential operator Λ, 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.


Introduction
In this paper, we study a type of generalized Zakharov system which is given by with initial data  (0, ) =  0 () , (0, ) =  0 () ,   (0, ) =  1 () , where  ∈ R,  > 0, and  ∈ (1/2, 1) is a fixed constant.In the above system, Λ := (− 2  ) 1/2 is a fractional differential operator.With this definition, Λ 2 maps  to Λ 2  := F −1  (|| 2 F  ) with F  the Fourier transform of (, ) with respect to the variable .In particular, Λ 2 = − 2   .When  = 1, system (1) and (2) reduces to the usual Zakharov system which was first obtained by Zakharov [1]; here,  : R + × R → C is the slowly varying amplitude of high-frequency electric field and  : R + × R → R 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 [2][3][4][5][6].Well-posedness of (4) in lower regularity spaces was obtained in [7].Existence of global attractors for dissipative Zakharov system was studied in [8][9][10][11].For related Zakharov system including magnetic effects, one can see [12][13][14][15].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 wellposedness is studied in [18,19].Inspired by this, we then replace the Laplacian in the Schrödinger equation of (4) by the fractional differential operator Λ 2 , 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.
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.

Preliminaries
Firstly, we set some notations.For  ∈ R, we use Ḣ to denote the fractional homogeneous Sobolev space, consisting of all tempered distribution  such that ‖‖ Ḣ is finite, where ‖‖ Ḣ is defined via the Fourier transform Similarly, one can define the inhomogeneous Sobolev space   equipped with the norm In particular, we have ‖‖   ∼ ‖‖  2 + ‖Λ  ‖  2 for  ≥ 0.
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 [20][21][22].) , (10) 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 R, 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 R, we then get On the other hand, taking inner product of (2) with Λ −2   , we then obtain Combining the above two equalities gives Ψ() = Ψ(0).

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  ∈ (0, 1), let us consider the following regularized system: where the operator B  := ( + Λ 4 ) −1 and   =   (  ) is the solution of the equation with initial data   (, 0) =   0 ,    (, 0) =   1 .It is easy to see that the operator B  satisfies the following properties: Roughly speaking, the fourth property says that the operator B  commutes with the operator Λ; of course, the operator Λ can be replaced by other differential operators such as Λ  .
From the semigroup theory we know that the linear equation   = −Λ 2  generates a unitary group () = exp(−Λ 2 ) in   (R), 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 (, ) ∈ (R + ;   (R)× (−2)+1 (R))).In fact, when estimating the  ∞   norm of , we have where we need  ∈ (R + ;   ).However, this is not correct since  only belongs to (R + ;  (−2)+1 ).For this reason, we first study the regularized system (15) by introducing the operator B  , and we can see that ).Then the wellposedness 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  → 0 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 B  , the global wellposedness 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 ‖  0 ‖   , ‖  0 ‖ Ḣ1− and ‖  1 ‖ Ḣ− .In particular, the above estimate implies that Proof.From Lemma 4, we know that  ( 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 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.

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 ( 0 ,  0 , for sufficiently small  > 0, where the constant  depends on 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) where the constant  depends on ‖ 0 ‖  3 , ‖ 0 ‖  1+ , ‖ 1 ‖   ∩ Ḣ− , and .