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We consider the defocusing mass-critical nonlinear Schr ?dinger equation in the exterior domain in ( ). By analyzing Strichartz estimate and utilizing the inductive hypothesis method, we prove the stability for all initial data in .

In this short note, we consider the defocusing mass-critical nonlinear Schrödinger equation in the exterior domain

Here

This equation has Hamiltonian

As (2) is preserved by (1), we shall refer to it as the mass and often write

H. Brezis and T. Gallouet [

In [

Throughout this paper, we restrict ourselves to the following notion of solution.

Definition 1 (solution). Let I be a time interval containing zero, a function

for all

In this formulation, the Dirichlet boundary condition is enforced through the appearance of the linear propagator associated to the Dirichlet Laplacian.

Our stability theorem concerns mass-critical stability in

Theorem 2 (Stability theorem). Suppose

in the sense that

for some function e.

Assume that

for some positive constants M and L.

Let

for some

for some

Then, there exists a solution u to

on

The rest of the paper is organized as follows. In Section 2, we introduce our notations and state some previous results. In Section 3, we finally prove Theorem 2, except for proving a lemma about approximate solutions.

In this section we summarize some our notations and collect some lemmas that are used in the rest of the paper.

We write

We define that for some

We also define

With these notations, the Strichartz estimates read as follows:

Theorem 3 (Strichartz estimates [

satisfies

Proposition 4 (Local well-posedness). Given

on some interval

The quantities

We need the following lemma to prove this theorem.

Lemma 1. Let I be a compact interval and let

in the sense that

for some function e.

Assume that

for some positive constant M.

Let

for some

Assume also the smallness conditions

for some

Then, there exists a solution u to

on

Proof of Lemma 1. By symmetry, we may assume

where

For

By (19),

On the other hand, by Strichartz, (20), (21), we get

Combining (27) and (28), we obtain

By bootstrapping, we see if

which implies (26).

Using (26) and (28), we see (23).

Moreover, by Strichartz, (18), (21) and (26),

which establishes (24) for

To show (25), we use Strichartz, (17), (18), (26), (19),

Choosing

We now turn to the proof of stability theorem.

Proof of Theorem 2. We now subdivide I into

where

We need to replace

By choosing

provided we can show that analogues of (8) and (9) hold with

In order to verify this, we use an inductive argument.

By Strichartz, (8), (10) and the inductive hypothesis,

Similarly, by Strichartz, (9), (10) and the inductive hypothesis, we see

so we see

Choosing

In this paper, we consider a mass-critical stability of the defocusing mass-critical nonlinear Schrödinger equation. Then we prove two different types of perturbation to show the stability of nonlinear Schrödinger equation.

The research of Guangqing Zhang has been partially supported by the NSF grant of China (No. 51509073) and also “The Fundamental Research Funds for the Central Universities” (No. 2014B14214). The author would like to thank his tutor Zhen Hu for helpful conversations. The author also thanks the referees for their time and comments.

Zhang, G.Q. (2016) On the Stability of the Defocusing Mass- Critical Nonlinear Schrödinger Equation. International Journal of Modern Nonlinear Theory and Application, 5, 115-121. http://dx.doi.org/10.4236/ijmnta.2016.53012