Stability Solution of the Nonlinear Schrödinger Equation

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 u     t i u F u        e 0 with small in a suitable space and e 0 u u   small in c s x H and then there exists a veritable solution to nonlinear Schrödinger equation which remains very close to in critical norms. u u

small in a suitable space and e 0 u u   small in c s x

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 where For more definition of critical case see [1][2][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.
and 0 .Then there exists a unique maximal-lifespan solution to (1.1) with and initial data .Moreover: 3) If the solution does not blow up forward in time, then , and moreover u scatters forward in time to e for some .Conversely, then there exists a unique maximal- x lifespan solution u which scatters forward in time to .4) If the solution u does not blow up backward in time, then and moreover u scatters backward in time to for some .Conversely, if then there exists a unique maximallifespan solution which scatters backward in time to .
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 0 A  0   there exists 0.

 
With property: if is a solution (not necessarily maximal-lifespan) such that , then there exists a solution : For proof: See [4][5][6].Now in the following we will discuss Standard local well-posedness theorem.
Theorem 1.2 Let , and let if is not an p even integer.Then there exists and I there is a compact interval containing zero such that

Strichartz Estimate
In this section we discus some notation and Strichartz estimate.

Some Notation
We write X Y  anywhere in this work whenever there exists a constant c in ependent of the parameters, so that d X cY  . The shortcut   O X otes a finite linear gathering of terms that "look like" X, but possibly with some factors changed by their complex conjugates.den We start by the definition of space-time norms The inhomogeneous Sobolev norm

 
s d H  (when s is an integer) is defined by:

∶
When s is any real number as

∶
For any space time slab , With the usual adjustments when or is equal to infinity.When q r q r  we abbreviate q r t x L L as .
, A Gagliardo-Nirenberg type inequality for Schrödinger equation the generator of the pseudo conformal    plays the role of partial differentiation.

Strichartz Estimate Definition 2.1
The exponent pair   , q r is says the Schrödinger-admissible if

Let
be the free Schrödinger evolution.From the explicit formula we obtain the standard dispersive inequality for all .0 t  In particular, as the free propagator conserves the 2 For all 0 and 2 , If solves the inhomogeneous Equation (1.1) for some : for some constant depending only on the dimension .
For some constant d depending only on we have the Holder inequality .
We now return to prove Theorem 1.2.
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 , .
denotes a constant that changes from line to line.Note that the norm appearing in the metric scales like x L .Note also that both 1 and are closed (and hence complete) in this metric.
Using the Strichartz inequality and Sobolev embedding, we find that for Arguing as above and invoking (1.4), we obtain , ,  suciently small, we see that for 0 0     , the functional maps the set 1 2 back to itself.To see that is a contraction, we repeat the above calculations to obtain .
even smaller (if necessary), we can ensure that is a contraction on the set 1 .By the contraction mapping theorem, it follows that has a fixed point in 1 2 .Furthermore, noting that Φ maps into Thus, (1.4) holds with I   for initial data with suciently small norm instead that, by the monotone convergence theorem, (1.4) holds provided I is chosen suciently small.Note that by scaling, the length of the interval I depends on the fine properties of , not only on its norm.0 u

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 4 p d  .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 I be a compact interval and let be an approximate solution to (1.1) for some function .Suppose that e   for some positive constant M .Let and let for some 0 M   .Suppose also the smallness conditions Furthermore, by Strichartz, (3.4), and (3.5), we get which implies (3.9).Using (3.9) and (3.11), we obtained (3.6).Furthermore, by Strichartz, (3.2), (3.5), and (3.9), To prove (3.8), we use Strichartz, (3.1), (3.2), (3.9), and (3.3): sufficiently small, this finishes the proof.□ Based on the previous result, we are now able to prove stability for the mass-critical NLS.
 is a small constant.Then, there exists a solution u to (1.1) on with initial data is as in Lemma 3.1.We replaced 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 Lemma 3.1 to obtain for each j and all Provided, we can prove that their counterparts of (3.2) and (3.4) hold with replace 0 by t j t .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,  sufficiently small depending on , , J M and M  , we can ensure that hypotheses of Lemma 3.1 continue to hold as j varies.□ Lemma 3.3 (Stability) Fix u and .For every and and that approximately solves (1.1) in the sense that And , are such that Then there exists a solution to (1.1) with such that : Note that, the masses of u and 0 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 L  , of course) have appeared in [10][11][12][13][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 A is suciently small depending on .Let be the maximal-lifespan solution with initial data Thus, if we set by the triangle inequality, (2.3), and (3.16), we have (2.4) and the hypothesis   where d depends only on d .
C  If A is uciently small depending on d , and s  is suciently small depending on  and d , then s andard continuity arguments give

X t   as de
To deal with the case when sired.
A is large, simply iterate the case when A is small (shrinking  ,  repeat ly after a subdivision of the time interval ed ) I .

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.
We discuss the stability by the following proposition.Before beginning we need define the Kato inhomogeneous Strichartz estimate.See [16] . , d for all t and define def 2 .
.     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 0  needs to be taken in terms of A (and thus, in terms of ).We were given which then determined N N .
A □  for the closure of all test functions under this norm.

2 . 1
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 1 2 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 By the Strichartz inequality, we know that

Theorem 3 . 2
Let I be a compact interval and let be an approximate solution to (1.1) in the sense that u 


To absorb the second part of (4.7) for all intervals ,