Abstract

In this paper, we consider the initial boudary value problem for modified Zakharov system in 3 dimensions with small initial condition. By using the continuity lemma and the linear interpolation theory, together with the properties of Sobolev spaces and the Galerkin method, we obtain the existence and uniqueness of the global solution.

Keywords

Modified Zakharov System, Galerkin Method, A Prior Estimate

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1. Introduction

In this paper, we study the global existence and uniqueness of solutions for a modified Zakharov system with initial boundary value conditions as follows.

(1.1)

(1.2)

(1.3)

(1.4)

(1.5)

where represents the slowly varying envelope of the electric field, denotes deviation of the ion density from its mean, and H is the dimensionless quantum parameter given by the ratio of the ion plasmon and electron thermal energies.

The classical Zakharov system was derived by Zakharov to describe the propagation of Langmuir waves in a plasma [1] . In the past decade, a large amount of work has been devoted to the existence problem for the classical Zakharov system. For instance, the Fourier restriction norm method, applies to this problem, and under appropriate assumption on the data, several existence results have been established. The local well-posedness of this problem on in the energy space was first obtained by Bourgain and Colliander [2] , which was improved to by Ginibre, Tsutsumi and Velo [3] . Recently, Bejenaru, Herr, Holmer and Tataru have obtained local well-posedness in [4] . Another approach was initiated by Colliander, Keel, Stafflani, Takaoka, and Tao in [5] , called the I-method. This method was successfully applied by these authors to the Kdv and modified Kdv equation. In [6] , this method was also used to construct global solutions for the one dimensional Zakharov system.

However, some important effects, such as quantum effects, have been ignored in the classical Zakharov system. The importance of quantum effects in ultrasmall electronic devices, dense astrophysical plasma systems and laser plasmas has produced an increasing interest on the investigation of the quantum counterpart of some of plasma physics phenomena [7] . By using a quantum fluid approach, Garcia has obtained the following modified Zakharov system [8] :

Recently, S. You and B. Guo have considered the existence and uniqueness of the global solution to the initial boundary value problem for the above system in 1 dimension and 2 dimensions, respectively [9] [10] . Motivated by their result, we consider the global existence of solutions in 3 dimensions with small initial condition. As is standard, the problem is to obtain a prior estimate of higher derivatives of solutions in some suitable function spaces. To overcome higher dimensional difficulty, we use continuity lemma and the linear interpolation theory, together with the properties of Sobolev spaces to handle it.

Now we give some notations:

-For, we denote by or simply, the space of all q times integrable functions in equipped with norm or simply.

-Let be the usual Sobolev space of complex-valued functions with the norm

-We denote by C a positive constant which may change from one line to the next line.

2. A Prior Estimates

To study the smooth solution of the modified Zakharov system, we introduce function, and transform Equations (1.1)-(1.5) into the form:

(2.1)

(2.2)

(2.3)

with initial condition

(2.4)

and boundary condition

(2.5)

(2.6)

Lemma 2.1. Assume that, and is the solution to (2.1)-(2.6), then we have

Proof. Multiplying Equation (2.1) by, and integrating over, we have

(2.7)

Since

Taking the imaginary part of Equation (2.7), then we have

Lemma 2.2. [11] (Continuity lemma) Assume that is a nonnegative continuous function defined on and

If and also hold, then is bounded in.

Lemma 2.3. [11] (Gargliardo-Nirenberg inequality) Assume that then

where

Lemma 2.4. Assume that is the solution to (2.1)-(2.6), and then we have

Proof. Multiplying Equation (2.1) by, and integrating over, we have

(2.8)

Since

then taking the real part of Equation (2.8), we have

(2.9)

where we have used the fact

Similarly, multiplying Equation (2.2) by, and integrating over, we have

(2.10)

i.e.

(2.11)

Adding Equation (2.9) to Equation (2.11), we deduce

Set then we have

therefore, using Young’s inequality, we have

(2.12)

Choosing such that, and by Lemma 2.3, we have

Set then the above inequality can be simplified as where

then using Lemma 2.2 yields

if the initial condition is small enough. Substituting it into Equation (2.12), we have

Lemma 2.5. Assume that is the solution to (2.1)-(2.6), and we have

Proof. Differentiating Equation (2.1) with respect to t, and then multiplying it by, integrating over, we have

(2.13)

Since

Taking the imaginary part of Equation (2.13) yields

therefore, using Hölder’s inequality and Sobolev imbedding, we have

(2.14)

Differentiating Equation (2.2) with respect to t, and then multiplying it by, integrating over, we have

i.e.

therefore

(2.15)

Adding Equation (2.14) to Equation (2.15), we have

Using Gronwall’s inequality, we have

From Equation (2.1), Equation (2.2), and Equation (2.3), it easily get

3. Existence and Uniqueness of Global Solution

Theorem 3.1. Assume that then there exists a unique global solution of the initial boundary value problem (2.1)-(2.6),

Proof. We first give the proof of the uniqueness of the solution. Suppose and are two solutions of (2.1)-(2.6). Let

therefore

(3.1)

(3.2)

(3.3)

with initial condition

(3.4)

Multiplying Equation (3.1) by, and integrating over, we have

taking the imaginary part yields

therefore

(3.5)

Multiplying Equation (3.3) by, and integrating over, we have

Since

therefore

(3.6)

Adding Equation (3.5) to Equation (3.6), we have

Using Gronwall’s inequality and the initial condition Equation (3.4), we can obtain

Next we show the existence of the solution.

By using the Garlerkin method, choose basic functions such that, where are different corresponding eigenvalues. The approximate solution of problem (2.1)- (2.6) can be written as

where the undetermined coefficients need to satisfy the following initial value problem of the system of ordinary differential equations

with initial conditions

According to the basic theory of ordinary differential equations, the above equations have a unique local solution.

Similarly to the proof of Lemma 2.1 and Lemma 2.4, we have

By compactness argument, we can choose subsequences, still denoted by such that

in weakly star;

in weakly star;

in weakly star;

in weakly star;

in strongly and a.e.;

in strongly and a.e.;

in weakly star;

in weakly star.

Indeed

By using the density of in, we get the existence of a local generalized solution for the problem (2.1)-(2.6). By continuous extension principle and a prior estimate in Section 2, we can get the existence of the global solution.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (No. 11271141, No. 11426069 and No.61375006).

NOTES

^*Corresponding author.

Conflicts of Interest

The authors declare no conflicts of interest.

References