Qiaoxin Li

(The Graduate School of China Academy of Engineering Physics, Beijing 100088,E-mail:liqiaoxin@126.com)

GLOBAL WEAK SOLUTION TO THE NONLINEAR SCHRöDINGER EQUATIONS WITH DERIVATIVE∗

Qiaoxin Li

(The Graduate School of China Academy of Engineering Physics, Beijing 100088,E-mail:liqiaoxin@126.com)

In this paper,we study the nonlinear Schrödinger equations with derivative.By using the Galërkin method and a priori estimates,we obtain the global existence of the weak solution.

Schrödinger equations;global weak solution;a priori estimates;derivative

2000 Mathematics Subject Classification 35Q55

Ann.of Appl.Math.

31:2(2015),165-174

1　Introduction

In this paper we consider the following nonlinear Schrödinger equations

with the initial condition and periodic boundary condition

The single equation of system(1.1)describes the propagation of circularly polarized Alfvén waves in the magnetized plasma with a constant magnetic field(see[1-3]).

In[4],the existence of a global weak solution was verified for the following equation

where α,β,σ are real constants with β≠0,σ∈[1,2].

In[5],Guo and Tan studied the following equation

they used the Galërkin method and a priori estimates to obtain the unique existence of smooth solutions under some conditions on the constants α,β,γ,the function g(·)and initial data u(x,0).In addition,they also discussed the decay behaviors of the smooth solutions as|x|→∞.Tan and Zhang[6]also discussed the unique existence of the weak solution to equation(1.4).

A large amount of work has been devoted to studying the local and global well posedness of the nonlinear Schrödinger equation with derivative.Takaoka[9]proved the local wellposedness for s≥1/2.Biagioni and Linares[10]got the ill-posedness for s＜1/2.For the global well-posedness,Colliander proved it for s＞1/2.In 2013,Wu[12]proved that it is globally well-posed in energy space,provided that the initial data u0∈H1(R)withFurther discussion can be found in[13-15].

System(1.1)is relevant in the theory of polarized Alfvén waves and the propagation of the ultra-short pulse.In this paper,we prove the existence of the global weak solution to system(1.1)with the periodic boundary value problem by using the Galërkin method and a priori estimates.

Before starting the main results,we review the notations and the calculus inequalities used in this paper.

Let Lm(Ω),1≤m≤∞be the classical Lebesgue space with the norm

The usual L2inner product is,wheredenotes the complex conjugate of v,and the norm of L2is

Denote H1(Ω)be the Sobolev space of complex-valued functions with the norm

C is a generic constant and may assume different values in different formulates.

The following auxiliary lemmas will be needed.

Lemma 1.1(The Gagliardo-Nirenberg inequality)Assume, 1≤q,r≤∞.Let p and α satisfy

Then

In particular

Lemma 1.2(The Gronwall inequality)Let c be a constant,and b(t),u(t)be nonnegative continuous functions in the interval[0,T]satisfying

Then u(t)satisfies the estimate

for t∈[0,T].

Further,the weak solutions to(1.1)-(1.3)are defined as follows:

Definition 1.1 We call functions u(x,t)and v(x,t)weak solutions to the initial problems(1.1)-(1.3)if they satisfy the following weak forms

for any functions

then system(1.1)-(1.3)has a global weak solution,which satisfies

Theorem 1.2 Let u0(x)∈H1(R),v0(x)∈H1(R)and one of the following conditions holds

then system(1.1)has a global weak solution,which satisfies

2　a priori Estimates

In this section,we give the demonstration of a priori estimates that guarantee the existence of the global weak solution to system(1.1)-(1.3).

Lemma 2.1 If u0∈L2(Ω),v0∈L2(Ω),then for the solution(u,v)to problem(1.1)-(1.3),we have

Proof Taking the inner product for the first equation of system(1.1)withand the second equation with,respectively,and integrating the resulting equations with respect to x on Ω,and then taking the imaginary part of the resulting equations,we obtain

we can obtain

which implies the equality(2.1).

The proof of Lemma 2.1 is completed.

then we can get

where C is a constant depending only on‖u0‖H1and‖v0‖H1.

Proof For system(1.1),we have the conservation of total energy

For the case 1≤p＜2,using the inequality(1.7),we have

Using Hölder’s inequality

Therefore

Using Sobolev’s embedding,we have

Therefore

Similarly

And so

Combining(2.6)-(2.9),we have

And combining(2.5)with(2.10),we get

For the case p≥2,using Young inequality,we get

Using Hölder’s inequality and Lemma 2.1,we have

Therefore

because of condition(ii),we obtain

where C is a constant depending only on‖u0‖H1,‖v0‖H1.

This completes the proof of Lemma 2.2.

3　Global Weak Solution

In this section,we prove the existence of the global weak solution to problem(1.1)-(1.3) by using Galerkin-Fourier method.We need the following lemmas.

Lemma 3.1 Let B0,B and B1be three Banach spaces.Assume that B0⊂B⊂B1and Bi,i=0,1 are reflective.Suppose also that B0is compactly embedded in B.Let

where T is finite and 1＜pi＜∞,i=0,1.W is equipped with the norm

Then W is compactly embedded in Lp0(0,T;B).

Lemma 3.2 Suppose that Q is a bounded domain inand‖gµ‖Lq(Q)≤C.Furthermore,suppose that

Then

Lemma 3.3X is a Banach space.Suppose that g∈Lp(0,T;X),∈Lp(0,T;X) (1≤p≤∞).Then g∈C([0,T],X)(after possibly being redefined on a set of measure zero).

In the following,we prove the existence of weak solution to problem(1.1)-(1.3).

Proof of Theorem 1.1 We prove Theorem 1.1 by the following three steps.

Step 1 Construct the approximate solutions by the Galerkin-Fourier method.

Let{ωj(x)}(j=1,2,···)be the periodic eigenvectors of the operator A=-△,which satisfies

For every integer m,we are looking for an approximate solution to system(1.1)of the form

where αjmand βjmsatisfy the following nonlinear equations

and the nonlinear equations(3.1)satisfy the following initial-value conditions

Then(3.1)becomes the system of nonlinear ODE subject to the initial condition(3.2). According to the standard existence theory for nonlinear ordinary differential equations, there exists a unique solution to(3.1)and(3.2)for a.e.0≤t≤tm.By a priori estimates we obtain that tm=T.

Step 2 A priori estimates.

As the proofs of Lemmas 2.1 and 2.2,we have

So

Using the Sobolev embedding theorem,we get

So by(3.5)and(3.6),we obtain

Therefore

Step 3 Passaging to the limit.

By applying(3.3)and(3.7),we deduce that there exist a subsequence uµfrom um,vkfrom vmsuch that

By(3.3),we have

By(3.7),we have

Define

We equip W with the norm:

By using(3.3),(3.12)and Lemma 3.2,we have

Fixing j,by(3.1),we get

By applying(3.8),(3.9),(3.13)and(3.14),we deduce that there exist a subsequence uµfrom um,vkfrom vmsuch that

we obtain

Then from(3.5),we have

the above equalities hold for any fixed j.By the density of the basis ωj(j∈Z),we have

Hence(u,v)satisfies(1.1)and(1.9).By(3.3),(3.7)and Lemma 3.3,we obtain that

Then

But from(3.2),we have

Therefore u(0)=u0,v(0)=v0.

Theorem 1.1 generalizes the result of the global existence of weak solution to the nonlinear Schrödinger equations in[17].

Remark By using the a priori estimates of the solution to system(1.1)-(1.3)for the period L,as in[18],we can derive the global weak solution as L→∞.So,Theorem 1.2 is obtained.

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(edited by Liangwei Huang)

∗Manuscript April 21,2015