1. Introduction
In this paper, we study the stability theory of solutions to the nonlinear Schrödinger equation (NLS).
We consider the Cauchy problem for the nonlinear Schrödinger equation
(1.1)
where
,
the solution ![](https://www.scirp.org/html/2-2340075\014f5438-c2ff-4f9e-b50f-173cfae5cc02.jpg)
is a complex-valued function in ![](https://www.scirp.org/html/2-2340075\ae0c3871-1773-4dd6-a111-eda548324187.jpg)
The Equation (1.1) is called mass-critical or
critical if
, and it is called mass-supercritical and energy-subcritical when
.
The solutions to (1.1) have the invariant scaling
(1.2)
Definition 1.1 (Solution) Let
such that
. A function
is a strong solution to (1.1)
if and only if it belongs to
, and for all
satisfies the integral equation
(1.3)
A function
is a weak solution to (1.1)
if and only if
, and for all ![](https://www.scirp.org/html/2-2340075\a11dac11-7c4d-439c-bcef-9cfa1f697e8f.jpg)
satisfies the integral Equation (1.3).
The solutions to (1.1) have the mass
![](https://www.scirp.org/html/2-2340075\2cf39c51-ecf3-4ac9-94fb-3f15191bc35d.jpg)
where
![](https://www.scirp.org/html/2-2340075\1553bc3b-1dda-4ced-b2c3-8fb544b3557e.jpg)
Energy
where,
![](https://www.scirp.org/html/2-2340075\f2ff6bd1-49e4-48a7-9a67-23f2385d9b25.jpg)
Definition 1.2 The problem (1.1) is locally wellposed in
if for any
there exist a time
and an open ball
in
such that
, and a subset
of
, such that for each
there exists a unique solution
to the Equation (1.3), and furthermore, the map
is continuous from
. If
can be taken arbitrarily large
, the problem is globally wellposed.
Definition 1.3 A global solution
to (1.1) is scattering in
as
if there exists
such that
![](https://www.scirp.org/html/2-2340075\bf9519c1-c2b7-48bc-928a-cc4531c36b12.jpg)
Similarly, we can define scattering in
for
.
For more definition of critical case see [1-3].
In this paper we discuss stability theory of the mass critical, mass-supercritical and energy-subcritical of solution to the nonlinear Schrödinger equation. In section three we discuss the stability of the mass critical solutions and in section four mass-supercritical and energysub-critical solutions are discussed.
Theorem 1.1 Let
and
. Then there exists a unique maximal-lifespan solution
to (1.1) with
and initial data
. Moreover:
1) The interval
is an open subset of
.
2) For all
, we have
so, we define
.
3) If the solution
does not blow up forward in time, then
, and moreover
scatters forward in time to
for some
. Converselyif
then there exists a unique maximallifespan solution
which scatters forward in time to
.
4) If the solution u does not blow up backward in time, then
and moreover
scatters backward in time to
for some
. Conversely, if
then there exists a unique maximallifespan solution
which scatters backward in time to
.
5) If
where a constant
depending only on
then
.
In particular, no blowup occurs and we have global existence and scattering both ways.
6) For every
and
there exists
With property: if
is a solution (not necessarily maximal-lifespan) such that
and
,
are such that
, then there exists a solution
with
such that ![](https://www.scirp.org/html/2-2340075\05a36f39-e3f6-4290-8684-e9927cab56c2.jpg)
and
for all
.
For proof: See [4-6].
Now in the following we will discuss Standard local well-posedness theorem.
Theorem 1.2 Let
,
and let
Assume that
if
is not an even integer. Then there exists
such that if between
and
there is a compact interval containing zero such that
(1.4)
then there exists a unique solution u to (1.1) on
. Furthermore, we have the bounds
(1.5)
(1.6)
(1.7)
where
for the closure of all test functions under this norm.
2. Strichartz Estimate
In this section we discus some notation and Strichartz estimate.
2.1. Some Notation
We write
anywhere in this work whenever there exists a constant
independent of the parameters, so that
. The shortcut
denotes a finite linear gathering of terms that “look like” X, but possibly with some factors changed by their complex conjugates.
We start by the definition of space-time norms
![](https://www.scirp.org/html/2-2340075\f22ef946-1878-435f-96fd-12c4973ac725.jpg)
The inhomogeneous Sobolev norm
(when
is an integer) is defined by:
![](https://www.scirp.org/html/2-2340075\e7aad18a-17a7-4cee-b8b1-3dd6a0265f10.jpg)
When s is any real number as
![](https://www.scirp.org/html/2-2340075\f618cb4c-fc96-4bf2-a8b7-340817fb81dc.jpg)
The homogeneous Sobolev norm
defines as:
![](https://www.scirp.org/html/2-2340075\c7346c9f-0d72-44b8-950a-3b9d51d277a6.jpg)
For any space time slab
We use
to denote the Banach space of function
whose norm is
![](https://www.scirp.org/html/2-2340075\04ec9856-9439-4d20-bfd8-13916aa31d2f.jpg)
With the usual adjustments when
or
is equal to infinity. When
we abbreviate
as
.
A Gagliardo-Nirenberg type inequality for Schrö- dinger equation the generator of the pseudo conformal transformation
plays the role of partial differentiation.
2.2. Strichartz Estimate
Definition 2.1 The exponent pair
is says the Schrödinger-admissible if
, and ![](https://www.scirp.org/html/2-2340075\f1b04605-5726-441d-81fa-4f1c37aa7195.jpg)
![](https://www.scirp.org/html/2-2340075\a9f61876-b94b-4fa5-bbb5-9ad3316e4e81.jpg)
Definition 2.2 The exponent pair
is says the Schrödinger-acceptable if
![](https://www.scirp.org/html/2-2340075\d0a2ca64-40a4-4d4a-953f-ef716fa15c1d.jpg)
Let
be the free Schrödinger evolution. From the explicit formula
![](https://www.scirp.org/html/2-2340075\d5207192-71b0-4266-b8e8-e9706a1045d4.jpg)
we obtain the standard dispersive inequality
(2.1)
for all
.
In particular, as the free propagator conserves the
- norm,
(2.2)
For all ![](https://www.scirp.org/html/2-2340075\f4f38f00-cdab-4552-a1dc-bb507eede20d.jpg)
![](https://www.scirp.org/html/2-2340075\d5f5937e-17f4-44a0-a380-5aaee53ed1eb.jpg)
If
solves the inhomogeneous Equation (1.1) for some
and ![](https://www.scirp.org/html/2-2340075\0fb58248-6ae5-42c1-8e20-4e6d4b90b48b.jpg)
in the integral.
Duhamel (1.3). Then we have
(2.3)
for some constant
depending only on the dimension
.
For some constant
depending only on
we have the Holder inequality
![](https://www.scirp.org/html/2-2340075\32cec1b4-55e8-4a28-ac75-79f0da1e3355.jpg)
We now return to prove Theorem 1.2.
Proof Theorem 1.2 The theorem follows from a contraction mapping argument. More accurate, defined
![](https://www.scirp.org/html/2-2340075\fc61057e-5ee9-47d4-984e-03883c99635a.jpg)
using the Strichartz estimates, we will show that the map
is a contraction on the set
where
![](https://www.scirp.org/html/2-2340075\a3fd418e-6051-4583-aee3-08d3ba499a47.jpg)
![](https://www.scirp.org/html/2-2340075\62c4df47-e998-4be8-b482-3554d108066d.jpg)
under the metric given by
![](https://www.scirp.org/html/2-2340075\75c1c34a-dad9-477a-9a55-ba8761f9ee76.jpg)
Here
denotes a constant that changes from line to line. Note that the norm appearing in the metric scales like
. Note also that both
and
are closed (and hence complete) in this metric.
Using the Strichartz inequality and Sobolev embedding, we find that for ![](https://www.scirp.org/html/2-2340075\229b5b48-0fc2-47b4-a27f-eeda5dbe240e.jpg)
![](https://www.scirp.org/html/2-2340075\20a7562c-f124-45f5-8bfb-fd9ccdb9561e.jpg)
And similarly,
![](https://www.scirp.org/html/2-2340075\6fe72511-5a4c-435b-99ae-d5fc43a3391a.jpg)
Arguing as above and invoking (1.4), we obtain
![](https://www.scirp.org/html/2-2340075\664de217-768f-4dfd-a215-93a8b245028b.jpg)
Thus, choosing
suciently small, we see that for
, the functional
maps the set
back to itself. To see that
is a contraction, we repeat the above calculations to obtain
![](https://www.scirp.org/html/2-2340075\6cedcca4-2a02-4312-bef1-ec43d7541053.jpg)
Therefore, choosing
even smaller (if necessary), we can ensure that
is a contraction on the set
. By the contraction mapping theorem, it follows that
has a fixed point in
. Furthermore, noting that
maps into
(not just
). We now turn our attention to the uniqueness. Since uniqueness is a local property, it enough to study a neighbourhood of
By Definition of solution (and the Strichartz inequality), any solution to (1.1) belongs to
on some such neighbourhood. Uniqueness thus follows from uniqueness in the contraction mapping theorem.
The claims (1.6) and (1.7) follow from another application of the Strichartz inequality. □
Remark 2.1 By the Strichartz inequality, we know that
![](https://www.scirp.org/html/2-2340075\9880dc76-5f12-48d8-a7e0-c2e7001108d9.jpg)
Thus, (1.4) holds with
for initial data with suciently small norm instead that, by the monotone convergence theorem, (1.4) holds provided
is chosen suciently small. Note that by scaling, the length of the interval
depends on the fine properties of
, not only on its norm.
3. Stability of the Mass Critical
In this section we discuss the stability theory at mass critical case. Consider the initial-value problem (1.1)
with
.An important part of the local well-posedness theory is the study of how the strong solutions built in the past subsection depend upon the initial data. More accurate, we want to know if the small perturbation of the initial data gives small changes in solution. In general, we take care in developing a stability theory for nonlinear Schrödinger Equation (1.1). Even though stability is a local question, it plays an important role in all existing treatments of the global well-posedness problem for nonlinear Schrödinger equation at critical case, for more see [7]. It has also proved useful in the treatment of local and global questions for more exotic nonlinearities [8,9]. In this section, we will only discus the stability theory for the mass-critical NLS.
Lemma 3.1 Let
be a compact interval and let
be an approximate solution to (1.1) meaning that
![](https://www.scirp.org/html/2-2340075\a3eccfc1-3246-4293-a6ca-8f152e897511.jpg)
for some function
. Suppose that
(3.1)
for some positive constant
. Let
and let
be such that
(3.2)
for some
. Suppose also the smallness conditions
(3.3)
(3.4)
(3.5)
for some
where
is a small constant. Then, there exists a solution
to (1.1) on
with initial data
at time
satisfying
(3.6)
(3.7)
(3.8)
(3.9)
Proof: By symmetry, we may assume
. Let
. Then
satisfies the initial value problem
![](https://www.scirp.org/html/2-2340075\94e9d896-c11b-4d6d-9c31-3b948a6ec78e.jpg)
For
we define
![](https://www.scirp.org/html/2-2340075\8a9120a1-8c44-4876-95c1-cc3877068f7b.jpg)
By (3.3),
(3.10)
Furthermore, by Strichartz, (3.4), and (3.5), we get
(3.11)
Combining (3.10) and (3.11), we obtain
![](https://www.scirp.org/html/2-2340075\87c3f384-931b-4871-843d-3bd772d2f169.jpg)
A standard continuity argument then shows that if
is taken sufficiently small,
![](https://www.scirp.org/html/2-2340075\307d0fcf-f69c-4d5c-b85f-981c098d8eb5.jpg)
which implies (3.9). Using (3.9) and (3.11), we obtained (3.6). Furthermore, by Strichartz, (3.2), (3.5), and (3.9),
![](https://www.scirp.org/html/2-2340075\aa340516-52ad-4e31-be2a-ea48b2bfdc40.jpg)
which establishes (3.7) for
sufficiently small.
To prove (3.8), we use Strichartz, (3.1), (3.2), (3.9), and (3.3):
![](https://www.scirp.org/html/2-2340075\9dedf9c1-9c6f-47f2-9b5d-c1bd0a837e32.jpg)
Choosing
sufficiently small, this finishes the proof. □
Based on the previous result, we are now able to prove stability for the mass-critical NLS.
Theorem 3.2 Let
be a compact interval and let
be an approximate solution to (1.1) in the sense that
![](https://www.scirp.org/html/2-2340075\944e0eae-900e-4a51-86d0-bb6fea4ac792.jpg)
for some function
. Assume that condition (3.1) in Lemma 3.1 holds and
(3.12)
for some positive constant
. Let
and let
obey (3.2) for some
. Furthermore, suppose the smallness conditions (3.4), (3.5) in Lemma 3.1. For some
where
is a small constant. Then, there exists a solution
to (1.1) on
with initial data
at time
satisfying
(3.13)
(3.14)
(3.15)
Proof: Subdivide
into
subintervals
such that
![](https://www.scirp.org/html/2-2340075\5762483f-f60c-4181-ba33-4a88a1ce88ac.jpg)
where
is as in Lemma 3.1. We replaced
by
as the mass of the difference
might grow slightly in time. By choosing
sufficiently small depending on
and
, we can apply Lemma 3.1 to obtain for each
and all ![](https://www.scirp.org/html/2-2340075\4b8f149b-ad90-42d4-b210-05bd2f8e50c1.jpg)
![](https://www.scirp.org/html/2-2340075\e501b3e6-3d4b-407e-97b4-f497258b1ae4.jpg)
![](https://www.scirp.org/html/2-2340075\1a427fa5-b880-4712-b24d-404304dce9d0.jpg)
![](https://www.scirp.org/html/2-2340075\c2c87f64-a85f-42b5-9b62-b2f38320b43c.jpg)
![](https://www.scirp.org/html/2-2340075\b2469642-92b6-4bad-a3d6-76002a21ec81.jpg)
Provided, we can prove that their counterparts of (3.2) and (3.4) hold with replace
by
. To verify this, we use an inductive argument. By Strichartz, (3.2), (3.5), and the inductive hypothesis,
![](https://www.scirp.org/html/2-2340075\2c80fdb5-b7bd-450b-ab24-ec024853dbc8.jpg)
Similarly, by Strichartz, (3.4), (3.4), and the inductive hypothesis,
![](https://www.scirp.org/html/2-2340075\aed1d24d-89dc-4a8e-89c6-1dc9e122f669.jpg)
Choosing
sufficiently small depending on
and
, we can ensure that hypotheses of Lemma 3.1 continue to hold as j varies. □
Lemma 3.3 (Stability) Fix
and
. For every
and
there exists
with the property: if
is such that
and that
approximately solves (1.1) in the sense that
. (3.16)
And
,
are such that
![](https://www.scirp.org/html/2-2340075\90ae8852-c65e-4f68-b5b2-decfa6c0b661.jpg)
Then there exists a solution
to (1.1) with
such that
.
Note that, the masses of
and
do not appear immediately in this lemma, although it is necessary that these masses are finite. Similar stability results for the energy-critical NLS (in
) instead of
, of course) have appeared in [10-14]. The mass-critical case it is actually slightly simpler as one does not need to deal with the existence of a derivative in the regularity class. For more see [15].
Proof: (Sketch) First let prove the claim when
is suciently small depending on
. Let
be the maximal-lifespan solution with initial data
. Writing
on the interval
, we see that
![](https://www.scirp.org/html/2-2340075\852aecd8-39bb-4465-8a89-fce3d8da77f7.jpg)
and
.
Thus, if we set
![](https://www.scirp.org/html/2-2340075\ac169c51-2773-4f8c-a853-9afe7a9cb66a.jpg)
by the triangle inequality, (2.3), and (3.16), we have
![](https://www.scirp.org/html/2-2340075\38943ea3-8fc6-4b40-b7fb-573d55283069.jpg)
hence, by (2.4) and the hypothesis
,
![](https://www.scirp.org/html/2-2340075\ef118d37-c854-4449-94ac-15f8e6c5b625.jpg)
where
depends only on
. If
is suciently small depending on
, and
is suciently small depending on
and
, then standard continuity arguments give
as desired. To deal with the case when
is large, simply iterate the case when
is small (shrinking
,
repeatedly) after a subdivision of the time interval
.
4. Stability of the Mass-Supercritical and Energy-Subcritical
In this section we discuss the Stability theory of the mass-supercritical and energy-subcritical to the nonlinear Schrödinger equation. Consider the initial-value problem (1.1) with
and
we chose
.
In this case the initial-value problem
is locally well-posed in
. Now we rewrite (1.1) as
(1.1)*
We discuss the stability by the following proposition. Before beginning we need define the Kato inhomogeneous Strichartz estimate. See [16]
(4.1)
Proposition 4.1 For each
there exists
and
such that the following holds.
Let
and solve
.
Let
for all
and define
![](https://www.scirp.org/html/2-2340075\388c3d35-0073-4470-956c-632c86225611.jpg)
If
![](https://www.scirp.org/html/2-2340075\c4ef13fb-716f-4d2f-b227-de1e146e1fa4.jpg)
And
(4.2)
Then
![](https://www.scirp.org/html/2-2340075\9f2ccb5d-eb18-4e47-9419-c75412eac93e.jpg)
Proof: Let w be defined by
then
solves the equation
(4.3)
since
.Can be divided
into
in intervals
Such that for all
, the quantity
, is Appropriate small
(δ to be selected below).
Integration (4.3) with initial time
is
(4.4)
where
.
Applying the Kato Strichartz estimate (4.1) on
, to obtain
. (4.5)
Note that
.
Similarly,
![](https://www.scirp.org/html/2-2340075\565425c5-1df8-48c0-9cfb-aa8f7a44fc55.jpg)
and
![](https://www.scirp.org/html/2-2340075\e80a6c88-ba6d-4838-8cdf-5478f06853af.jpg)
Substituting the above estimates in (4.5), to get,
(4.6)
As long as
![](https://www.scirp.org/html/2-2340075\0ba47ae6-b886-4626-9ddf-0d458bd10337.jpg)
and
(4.7)
We obtain
(4.8)
Taken now
in (4.4) and apply
to both sides to obtain
(4.9)
Since the Duhamel integral is restricted to
by again applying the Kato estimate, similarly to (4.6) we obtain,
![](https://www.scirp.org/html/2-2340075\29c02e83-fb5a-40b7-a167-60a1d0bfc9a6.jpg)
By (4.8) and (4.9), we bound the Former of expression to obtain
![](https://www.scirp.org/html/2-2340075\32dd0d5b-9174-411b-8096-aa2d163473ba.jpg)
Start iterates with
, we obtain
![](https://www.scirp.org/html/2-2340075\1d5d9d82-6d34-47fd-a0f2-c1cdb0fa7363.jpg)
To absorb the second part of (4.7) for all intervals
we require
(4.10)
We review that the dependence of parameters δ is an absolute constant chosen to meet the first part of (4.7). The inequality (4.10) determines how the small
needs to be taken in terms of
(and thus, in terms of
). We were given
which then determined
□