On the Stability of the Defocusing Mass-Critical Nonlinear Schrödinger Equation ()
1. Introduction
In this short note, we consider the defocusing mass-critical nonlinear Schrödinger equation in the exterior domain
in
(
) with Dirichlet boundary conditions:
(1)
Here
and the initial data
will only be required to the
space.
This equation has Hamiltonian
(2)
As (2) is preserved by (1), we shall refer to it as the mass and often write
or M for
.
H. Brezis and T. Gallouet [1] considered that
in
,
, the nonlinear Schrödinger equation in
of a bounded domain or an exterior domain of
with Dirichlet boundary conditions. In [2] , N. Burq, P. Gérard and N. Tzvetkov described nonlinear Schrödinger equations in exterior domains. In [3] [4] , R. 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
of a smooth, compact, strictly convex obstacle in
with Dirichlet 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
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
is called a solution to (1) if it lies in the class
for any compact interval
, and it satisfies the Duhamel formula
(3)
for all
. The interval I is said to be maximal if the solution cannot be extended beyond I. We say u is a global solution if
.
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
for the initial-value problem associated to the Equation (1).
Theorem 2 (Stability theorem). Suppose
, I is a compact interval and let
be an approximate solution to
(4)
in the sense that
(5)
for some function e.
Assume that
(6)
(7)
for some positive constants M and L.
Let
and
obey
(8)
for some
. Moreover, assume the smallness conditions
(9)
(10)
for some
, where
is a small constant.
Then, there exists a solution u to
(11)
on
with initial data
at time
satisfying
(12)
(13)
. (14)
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.
2. 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
to signify that there is a constant
such that
. We use the notation
whenever
. If the constant C involved has some explicit dependency, we emphasize it by a subscript. Thus
means that
for some constant
depending on u. We write
for the nonlinearity in (1).
We define that for some
,
![]()
![]()
We also define
to be the space dual to
with appropriate norm.
With these notations, the Strichartz estimates read as follows:
Theorem 3 (Strichartz estimates [3] [6] ). Let
be a time interval and let
, then the solution
to
![]()
satisfies
![]()
Proposition 4 (Local well-posedness). Given
, there exists
such that if
and
![]()
on some interval
,
, then there exists a unique solution
of (1) satisfying
. Besides,
![]()
The quantities
defined in (2) are conserved on I.
3. Proof of Theorem 2
We need the following lemma to prove this theorem.
Lemma 1. Let I be a compact interval and let
be an approximate solution to
(15)
in the sense that
(16)
for some function e.
Assume that
(17)
for some positive constant M.
Let
and
be such that
(18)
for some
.
Assume also the smallness conditions
(19)
(20)
(21)
for some
, where
is a small constant.
Then, there exists a solution u to
(22)
on
with initial data
at
satisfying
(23)
(24)
(25)
. (26)
Proof of Lemma 1. By symmetry, we may assume
. Let
, then w satisfies the following problem
![]()
where
.
For
, we define
![]()
By (19),
(27)
On the other hand, by Strichartz, (20), (21), we get
(28)
Combining (27) and (28), we obtain
![]()
By bootstrapping, we see if
is taken sufficiently small,
![]()
which implies (26).
Using (26) and (28), we see (23).
Moreover, by Strichartz, (18), (21) and (26),
![]()
which establishes (24) for
sufficiently small.
To show (25), we use Strichartz, (17), (18), (26), (19),
![]()
Choosing
sufficiently small, this finishes the proof of the lemma. W
We now turn to the proof of stability theorem.
Proof of Theorem 2. We now subdivide I into
subintervals
,
, such that
![]()
where
as in the lemma.
We need to replace
by
as the mass of the difference
might grow slightly in time.
By choosing
sufficiently small depending on J, M and
, we can apply the lemma to obtain for each j and all
,
![]()
![]()
![]()
![]()
provided we can show that analogues of (8) and (9) hold with
replaced by
.
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
sufficiently small depending on J, M and
, we can guarantee that the hypotheses of the lemma continue to hold as j varies. W
4. Conclusion
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.
Acknowledgements
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.