The Initial Boundary Value Problem for Modified Zakharov System

(1)

http://dx.doi.org/10.4236/apm.2015.55028

How to cite this paper: Li, L. and Fang, S.M. (2015) The Initial Boundary Value Problem for Modified Zakharov System. Ad-vances in Pure Mathematics, 5, 278-285. http://dx.doi.org/10.4236/apm.2015.55028

The Initial Boundary Value Problem for

Modified Zakharov System

Lang Li, Shaomei Fang*

Department of Mathematics, South China Agricultural University, Guangzhou, China Email: *[email protected]

Received 6 July 2014; accepted 10 April 2015; published 16 April 2015

This work is licensed under the Creative Commons Attribution International License (CC BY).

http://creativecommons.org/licenses/by/4.0/

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 interpola-tion 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

1. Introduction

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

2 2 _0,

t

iE + ∆ −E H ∆ E−nE= (1.1) 2

2 2

0,

tt

n − ∆ +n H ∆ − ∆n E = (1.2)

( )

3

0 0 1

0 , 0 , t 0 , ,

t t t

E ₌ =E x n ₌ =n x n ₌ =n x x∈ Ω ⊂ (1.3)

( )

0, 0,

E E

E t

ν ν

∂Ω

∂Ω _∂Ω

∂ ∇ ∂

= = = ≥

∂ ∂ (1.4)

( )

0, n 0,

n

ν ∂Ω

∂Ω ∂ ∇

= =

∂ (1.5)

(2)

where E:3 1+ →3 represents the slowly varying envelope of the electric field, n:3 1+ → denotes dev-iation 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 clas-sical Zakharov system. For instance, the Fourier restriction norm method, applies to this problem, and under ap-propriate assumption on the data, several existence results have been established. The local well-posedness of this problem on 2 in the energy space H1× ×L2 Hˆ−1 was first obtained by Bourgain and Colliander [2], which was improved to H1 2× ×L2 H−1 by Ginibre, Tsutsumi and Velo [3]. Recently, Bejenaru, Herr, Holmer and Tataru have obtained local well-posedness in L2×H−1 2×H−3 2 [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 sys-tem. 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]:

2

2 2

0,

0.

t xx xxxx

tt xx xxxx xx

iE E H E nE

n n H n E

+ − − =

− + − =

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 1≤ ≤ ∞q , we denote by Lq

( )

Ω or simply Lq, the space of all q times integrable functions in Ω equipped with norm ⋅_Lq_{( )}_Ω or simply ⋅_Lq.

( )

(

)

( )

(

( )

)

(

)

1

d , 1 , ess sup : , .

p

p p

L L

f _Ω f x p f ∞ _Ω f x x p

Ω

 

=_ _ ≤ < ∞ = ∈ Ω = ∞



∫



-Let Hs

( )

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

( )

1 2 2

d .

s _s

H

f _Ω _{α ≤} D fα x

Ω

 

=  



∑ ∫



-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 ϕ, and transform Equa-tions (1.1)-(1.5) into the form:

2 2 _0,

t

iE + ∆ −E H ∆ E−nE= (2.1) 0,

t

n − ∆ =ϕ (2.2)

2 2

0,

t n H n E

ϕ − + ∆ − = (2.3)

with initial condition

( )

0 0 0

0 , 0 , 0 , .

t t t

E ₌ =E x n ₌ =n x ϕ ₌ =ϕ x x∈ Ω (2.4)

(3)

( )

0, E E

E

ν ν

∂Ω

∂Ω ∂Ω

∂ ∇ ∂

= = =

∂ ∂ (2.5)

( )

0, n 0.

n

ν ∂Ω

∂Ω ∂ ∇

= =

∂ (2.6)

Lemma 2.1. Assume that E0∈L2, and

(

E n, ,ϕ

)

is the solution to (2.1)-(2.6), then we have

( )

2 2

0 _L .

L

E t = E

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

(

2 2

)

, 0.

t

iE + ∆ −E H ∆ E−nE E = (2.7)

Since

(

)

2 2

3

2 2

1 d

Im , , Re d ,

2 d

t L L

iE E E E E x E

t

=

_∫

∆ ⋅ = − ∇



2

3 3 3

2 2

2 2 2

d , d d .

L

H ∆ E E x⋅ =H ∆E nE E x⋅ = n E x

∫

  

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

2 2 1 d

0. 2 d E L

t =

Lemma 2.2. [11] (Continuity lemma) Assume that f t

( )

is a nonnegative continuous function defined on +

 and

( )

k

( )

, , 0, 1.

f t ≤ +a bf t a b> k>

If

(

)

1

1 1

k k

k

k a b

k − − _< −

and f

( )

0 ≤a also hold, then f t

( )

is bounded in +.

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

( )

,

u∈L Ω m r

( )

,

D u∈L Ω Ω ⊂n,

1≤q, r≤ ∞ ≤ ≤, 0 j m, then

1 ,

q

p r

j m

L

L L

D u ≤C D uα u −α

where 0 j 1, 1 j 1 m

(

1

)

1.

m α p n α r n α q

 

≤ ≤ ≤ = + _ − _+ −

 

Lemma 2.4. Assume that

(

E n, ,ϕ

)

is the solution to (2.1)-(2.6), and E0

( )

x ∈H02,n0∈H01,ϕ0∈H01, then we have

2 1 1

0 0 0

2 2 2

0

sup _H _H _H .

t T

E n ϕ C

≤ ≤

 ₊ ₊ _≤

 

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

(

2 2

)

, 0.

t t

iE + ∆ −E H ∆ E−nE E = (2.8)

Since

(

)

2 2

3

2

3 3 3

2 2

2 2 2

1 d

, , d ,

2 d

1 d 1

d , d d ,

2 d 2

t t t L t L

t L t t

iE E i E E E x E

t

H E E x H E nE E x n E x

t

= ∆ ⋅ = − ∇

∆ ⋅ = ∆ ⋅ =

∫



  

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2 2

3 3

2 ₂ 2 2 2

d

d d 0,

d E L H E L n E x n Et x

t   ∇ + ∆ + − =      

∫

 

∫

(2.9)

where we have used the fact

3 3 3

2 2 2

d

d d d .

d n E x n Et x n Et x

t_

∫

=_

∫

+_

∫

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

(

nt − ∆ϕ,n

)

=0, (2.10)

i.e.

(

) (

)

(

)

(

)

(

)

2 2 2 2

2 2 2

3

2 2

2 2 2 2

2

2 2 2 2

, , ,

1 d 1 d

,

2 d 2 d

1 d 1 d

,

2 d 2 d

1 d 1 d d

d 0.

2 d 2 d 2 d

t t t

L L

t

L L

t

L L L

n n n n H n E

n H n E

t t

n n H n E

t t

H

n n n E x

t t t

ϕ ϕ ϕ

ϕ ϕ ϕ ϕ − ∆ = − ∆ + ∆ − = + ∇ − ∆ ∆ − = + ∇ − ∆ − = + ∇ + ∇ +

_∫

=  (2.11)

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

2 2 2 2 2

3

2

2 2 2 2 2 2 2

d 1 1

d 0.

d L L 2 L 2 L 2 L

H

E H E n E x n n

t ϕ   ∇ + ∆ + + + ∇ + ∇ =      

∫



Set

( )

2 2 2 2 2

3

2

2 2 2 2_d 1 2 1 2 2 _,

2 2 2

L L L L L

H

t E H E n E x n ϕ n

Ψ = ∇ + ∆ +

_∫

+ + ∇ + ∇



then we have

( )

t

( )

0 ,

Ψ = Ψ

therefore, using Young’s inequality, we have

( )

2 2 2 2 2

2 4

3

2

2 2 2 2 2 2

2 2 4

1 1

2 2 2

1

(0) d 0 .

4

L L L L L

L L

H

E H E n n

n E x n E

ϕ

ε

∇ + ∆ + + ∇ + ∇

≤ Ψ +

_∫

≤ Ψ + +



(2.12)

Choosing ε such that 1

4 2

ε

≤ , and by Lemma 2.3, we have

( )

2 4 2 2

2 1 4 1 3

0 0 .

L L L L

E C E C E E

ε ε

∇ ≤ Ψ + ≤ Ψ + ∇

Set 2

2 ,

L

m= ∇E then the above inequality can be simplified as

3 2_,

m≤ +a bm where

( )

0 2

1

0 , _L ,

a b C E

ε

= Ψ = then using Lemma 2.2 yields

2 2 , L E C ∇ ≤

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

2 2 2 2 2

2 2 2 2 2 2

.

L L L L L

E H E n ϕ n C

∇ + ∆ + + ∇ + ∇ ≤

(5)

have

2 2 2 2 2

2

2 2 2 2 2 2

2

0

sup _t _t _t .

H L L L L

L t T

E n n ϕ E ϕ C

≤ ≤

_∆ ₊ ₊ _{+ ∆} ₊ ₊ _≤

 

 

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

(

2 2

)

, 0.

tt t t t t t

iE + ∆ −E H ∆ E −n E−nE E = (2.13)

Since

(

)

2 2

3

2

3 3 3

2 2

2 2 2

1 d

Im , , d ,

2 d

1

d , d d .

2

tt t t _L t t t _L

t t t L t t t t

iE E E E E x E

t

H E E x H E n E E x n E x

= ∆ ⋅ = − ∇

∆ ⋅ = ∆ ⋅ =

∫



  

Taking the imaginary part of Equation (2.13) yields

(

)

2 2 1 d

Im , 0,

2 d_t Et L − n E Et t =

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

(

)

2 2 2 2 2

2 2 2

d

. d Et L C nt L E L Et L C Et L nt L

t ≤ ∞ ≤ + (2.14)

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

(

n_tt− ∆ϕ_t,n_t

)

=0,

i.e.

(

)

(

₂ ₂ 2

)

, , 0,

tt t t

n n + −∆ +n H ∆ − ∆n E n =

therefore

(

)

(

)

(

) (

)

2 2

2

2 4

2 2

2 2 2 2

2 2

2 2 2 4

d d

2 , 1 .

t L L L

t t L L L L t L

n n H n

t

E n C n E ∞ E E C n

+ ∇ + ∆

≤ ∆ ≤ + ∆ + ∇ ≤ +

(2.15)

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

(

2 2 2 2

) (

2 2

)

2 2 2 2 2 2 2

d

1 .

d Et L nt L nL H nL C Et L nt L

t + + ∇ + ∆ ≤ + +

Using Gronwall’s inequality, we have

2 2

2 2 2 2

.

t L t L L L

E + n + ∇n + ∆n ≤C

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

2 2

2

2 2 2

2

, , _t .

L L

L

E C ϕ C ϕ C

∆ ≤ ∆ ≤ ≤

3. Existence and Uniqueness of Global Solution

Theorem 3.1. Assume that E0

( )

x ∈H04,n0∈H02,ϕ0∈H02, then there exists a unique global solution of the in-itial boundary value problem (2.1)-(2.6),

( )

(

)

( )

(

)

2 2 2

, 0, ; , t , 0, ; ,

E t x L∞ T L E t x L∞ T L

∆ ∈ ∈

( )

(

2

)

( )

(

2

)

0

, 0, ; , _t , 0, ; ,

(6)

( )

(

2

)

( )

(

2

)

, 0, ; , _t , 0, ; .

t x L T L t x L T L

ϕ ∞ ϕ ∞

∆ ∈ ∈

Proof. We first give the proof of the uniqueness of the solution. Suppose

(

E n1, 1,ϕ1

)

and

(

E n2, 2,ϕ2

)

are

two solutions of (2.1)-(2.6). Let

1 2, 1 2, 1 2,

E=E −E n= −n n ϕ ϕ ϕ= −

therefore

2 2

1 1 2 2 0,

t

iE + ∆ −E H ∆ E−n E +n E = (3.1) 0,

t

n − ∆ =ϕ (3.2)

2 2

2

1 2 0,

t n H n E E

ϕ − + ∆ − + = (3.3)

with initial condition

0 0

0 0, t 0, t 0. t

E ₌ = n₌ = ϕ ₌ = (3.4)

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

(

2 2

)

1 1 2 2, 0,

t

iE + ∆ −E H ∆ E−n E +n E E =

taking the imaginary part yields

(

)

2 2

1 1 2 2 1 d

Im , 0,

2 d_t E L + −n E +n E E =

therefore

(

2

)

(

2

)

2 2 2 2

2 2 2

1 2

d

.

d_t EL ≤C E L∞ nL + n L∞ EL E L ≤C EL + nL (3.5)

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

(

₂ 2 2

)

1 2 , 0.

t n H n E E

ϕ − + ∆ − + ϕ =

Since

(

)

2

(

)

(

2 2

)

2 2 2

1 d

, , , ,

2 d

t _L n C n_L _L

t

ϕ ϕ  = ϕ −ϕ ≤  + ϕ

(

)

(

)

(

)

2

2 2

2 2 2 d

, , , ,

2 d

t L

H

H n H n H n n n

t

ϕ ϕ

∆  = ∆ =   = 

(

)

(

)

(

)

(

)

(

)

2

2 2

1 2 1 2 1 2 1 2

1 2

2 2

, ,

,

L L L L L L

L

E E E E E E E E

C E E E E

C E

ϕ ϕ

ϕ

∞ ∞

− + = − + −

≤ +

 

  _

 _

therefore

(

)

2 2 2 2 2

2

2 2 2 2

1 d d

.

2 d L 2 d L L L L

H

n C n E

t ϕ + t  ≤  + ϕ +  (3.6)

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

(

2 2 2

) (

2 2 2

)

2 2

2 2 2 2

d

. d_t ϕ L + nL + E L ≤C nL + ϕ L + E L

 

   

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

0, 0, 0.

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Next we show the existence of the solution.

By using the Garlerkin method, choose basic functions

{ }

ω ⊂j H04 such that −∆ωj =λ ωj j,

(

j=1, 2,,v

)

, where λj

(

j=1, 2,,v

)

are different corresponding eigenvalues. The approximate solution of problem (2.1)- (2.6) can be written as

( ) ( )

1

,

v

sv s s

Eν α t ω x

= =

∑

( ) ( )

1

,

v

sv s s

nν β t ω x

= =

∑

( ) ( )

1

,

v

sv s s

t x

ν

ϕ γ ω

= =

∑

where the undetermined coefficients α_sv

( )

t ,β_sv

( )

t ,γ_sv

( ) (

t s=1, 2,, ;v v=1, 2,

)

need to satisfy the follow-ing initial value problem of the system of ordinary differential equations

(

) (

)

2

(

2

) (

)

, , , , 0,

v v v v v

t s s s s

i E ω + ∆E ω −H ∆ E ω − n E ω =

(

ntv,ωs

) (

− ∆ϕ ωv, s

)

=0,

(

) (

)

₂

(

)

(

2

)

, , , , 0,

v v v v

t s n s H n s E s

ϕ ω − ω + ∆ ω − ω =

with initial conditions

( ) ( )

(

0, ,

)

(

0

( ) ( )

,

)

,

v

s s

E x ω x = E x ω x

( ) ( )

(

0, ,

)

(

0

( ) ( )

,

)

,

v

s s

n x ω x = n x ω x

( ) ( )

(

0, ,

)

(

0

( ) ( )

,

)

.

v

s s

x x x x

ϕ ω = ϕ ω

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

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

(

1 1 1 2

)

0 0

2

0

sup _t _t .

H H H L

t T

Eν nν Eν nν C

≤ ≤ + + + ≤

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

{ } { }

Eν , nν such that

Eν →E in L

(

0, ;T H04

)

∞

weakly star;

t t

Eν →E in L∞

(

0, ;T L2

)

weakly star;

nν →n in L

(

0, ;T H01

)

∞

weakly star;

t t

nν →n in L∞

(

0, ;T L2

)

weakly star;

Eν →E in L2

(

0, ;T L2

)

strongly and a.e.;

nν →n in 2

(

2

)

0, ;

L T L strongly and a.e.;

n Eν ν →nE in L∞

(

0, ;T L2

)

weakly star;

2 2

Eν → E in L∞

(

0, ;T L2

)

weakly star. Indeed

0 0 0

sup sup sup

0,

v v v v v v t T t T t T

v v v

L L

n E nE n E n E n E nE

n ∞ E E E ∞ n n

≤ ≤ − ≤ ≤ ≤ − + ≤ ≤ −

(8)

(

)

2 2

0

sup v v v 0.

L L t T

E E E _∞ E ∞ E E

≤ ≤ − ≤ + − →

By using the density of ωj in

( )

2

L Ω , 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).

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