﻿Stability Solution of the Nonlinear Schrödinger Equation

International Journal of Modern Nonlinear Theory and Application
Vol. 2  No. 2 (2013) , Article ID: 32976 , 8 pages DOI:10.4236/ijmnta.2013.22015

Stability Solution of the Nonlinear Schrödinger Equation

Mujahid Abd Elmjed M-Ali

Department of Mathematics, Faculty of Education, Kassala University, Kassala, Sudan

Email: mujahid@mail.ustc.edu.cn

Copyright © 2013 Mujahid Abd Elmjed M-Ali. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Received February 19, 2013; revised March 28, 2013; accepted April 30, 2013

Keywords: NLS; Wellposed

ABSTRACT

In this paper we discuss stability theory of the mass critical, mass-supercritical and energy-subcritical of solution to the nonlinear Schrödinger equation. In general, we take care in developing a stability theory for nonlinear Schrödinger equation. By stability, we discuss the property: the approximate solution to nonlinear Schrödinger equation obeying with small in a suitable space and small in and then there exists a veritable solution to nonlinear Schrödinger equation which remains very close to in critical norms.

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

is a complex-valued function in

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

satisfies the integral Equation (1.3).

The solutions to (1.1) have the mass

where

Energy where,

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

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

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

The inhomogeneous Sobolev norm (when is an integer) is defined by:

When s is any real number as

The homogeneous Sobolev norm defines as:

For any space time slabWe use to denote the Banach space of function whose norm is

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

Definition 2.2 The exponent pair is says the Schrödinger-acceptable if

Let be the free Schrödinger evolution. From the explicit formula

we obtain the standard dispersive inequality

(2.1)

for all.

In particular, as the free propagator conserves the - norm,

(2.2)

For all

If solves the inhomogeneous Equation (1.1) for some

and

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

Proof Theorem 1.2 The theorem follows from a contraction mapping argument. More accurate, defined

using the Strichartz estimates, we will show that the map is a contraction on the set where

under the metric given by

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

And similarly,

Arguing as above and invoking (1.4), we obtain

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

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

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

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

For we define

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

A standard continuity argument then shows that if is taken sufficiently small,

which implies (3.9). Using (3.9) and (3.11), we obtained (3.6). Furthermore, by Strichartz, (3.2), (3.5), and (3.9),

which establishes (3.7) for sufficiently small.

To prove (3.8), we use Strichartz, (3.1), (3.2), (3.9), and (3.3):

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

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

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

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,

Similarly, by Strichartz, (3.4), (3.4), and the inductive hypothesis,

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

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

and

.

Thus, if we set

by the triangle inequality, (2.3), and (3.16), we have

hence, by (2.4) and the hypothesis,

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

If

And

(4.2)

Then

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,

and

Substituting the above estimates in (4.5), to get,

(4.6)

As long as

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 toby again applying the Kato estimate, similarly to (4.6) we obtain,

By (4.8) and (4.9), we bound the Former of expression to obtain

Start iterates with, we obtain

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

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