On the Stability of the Defocusing Mass-Critical Nonlinear Schrödinger Equation

We consider the defocusing mass-critical nonlinear Schrödinger equation in the exterior domain Ω in d  ( 2 d ≥ ). By analyzing Strichartz estimate and utilizing the inductive hypothesis method, we prove the stability for all initial data in ( ) 2 L Ω .


Introduction
In this short note, we consider the defocusing mass-critical nonlinear Schrödinger equation in the exterior domain Ω in As ( 2) is preserved by (1), we shall refer to it as the mass and often write ( ) H. Brezis and T. Gallouet [1] considered that the nonlinear Schrödinger equation in Ω of a bounded domain or an exterior domain of 2   with Dirichlet boundary conditions.In [2], N. Burq, P. Gérard and N.
Killip, M. Visan and X. Zhang considered the defocusing energy-critical nonlinear Schrödinger equation and the focusing cubic nonlinear Schrödinger equation in the exterior domain Ω of a smooth, compact, strictly convex obstacle in 3  with Di- richlet boundary conditions, respectively.
In [5], T. Tao and M. Visan established stability of energy-critical nonlinear Schrödinger equations in ( ) . However, we established stability of mass-critical nonlinear Schrödinger equations in the exterior domain 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 t I ∈ .The interval I is said to be maximal if the solution cannot be extended beyond I.We say u is a global solution if I =  .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 ( )

2
L Ω for the initial-value problem associated to the Equation (1).
Theorem 2 (Stability theorem).Suppose 2 d ≥ , I is a compact interval and let u  be an approximate solution to in the sense that for some function e.
Assume that ( ) , for some positive constants M and L.
Let 0 t I ∈ and ( ) for some 0 M ′ > .Moreover, assume the smallness conditions for some is a small constant.
Then, there exists a solution u to , , , The rest of the paper is organized as follows.In Section 2, we introduce our notations and state some results.In Section 3, we finally prove Theorem 2, except for proving a lemma about approximate solutions.

Preliminaries and Notations
In this section we summarize some our notations and collect some lemmas that are used in the rest of the paper.
We write A B  to signify that there is a constant 0 C > such that A CB ≤ .We use the notation Ã B whenever A B A   .If the constant C involved has some explicit dependency, we emphasize it by a subscript.Thus for some constant ( ) C u depending on u.We write ( ) the nonlinearity in (1).
We define that for some 0 We also define M u defined in (2) are conserved on I.

Proof of Theorem 2
We need the following lemma to prove this theorem.
Lemma 1.Let I be a compact interval and let u  be an approximate solution to in the sense that for some function e.
Assume that ( ) for some positive constant M.
Proof of Theorem 2. We now subdivide I into ( ) , 0 , as in the lemma.
We need to replace M ′ by 2M ′ as the mass of the difference u u −  might grow slightly in time.

By choosing 1
 sufficiently small depending on J, M and M ′ , we can apply the lemma to obtain for each j and all , provided we can show that analogues of (8) and (9) hold with 0 t replaced by j t .
In order to verify this, we use an inductive argument.