ABSTRACT

This paper studies the global behavior defocusing nonlinear Schrödinger equation in dimension d = 2, and we will discuss the case This means that the solutions, and called critical solution. We show that u scatters forward and backward to a free solution and the solution is globally well posed.

1. Introduction

We consider the Cauchy problem for the nonlinear Schrö- dinger equation in dimension d = 2.

(1.1)

where, , and, When (1.1) is called defocusing when (1.1) is called focusing. In this paper we discuss the case when and (defocusing case).

If is a solution to (1.1) on a time interval then

(1.2)

is a solution to (1.1) on with.

This scaling saves the norm of u,

(1.3)

Thus (1.1) under previous hypotheses is called L²-critical or mass critical.

Proposition 1.1. Suppose that and

then, for any initial data, there exist such that there exists a unique solution

of the nonlinear Schrödinger Equation (1.1). If then for some non-increasing, and if is sufficiently small u exists globally.

In this paper we will discuss the case,

This means that a solution u

and called critical solution.

Definition 1.1. Let, is a solution to (1.1) if for any compact

and for all

(1.4)

The space caused from strichartz estimates. This norm is invariant under the scaling (1.2).

Definition 1.2. If there exist a solution u to (1.1) defined on blows up forward in time, such that

(1.5)

And u blows up backward in time, such that

(1.6)

Definition 1.3. If there exist we say that a solution u to (1.1) scatter forward in time such that,

(1.7)

A solution is said to scatter backward in time if there exist

Such that,

(1.8)

We note that the Equation (1.1) has preserved quantities, the mass

(1.9)

And energy

(1.10)

For more see [1].

Proposition 1.2. let p be the -critical exponent

, then the NLS (1.1) is locally well posed in

in the critical case. More precisely, given any, there exists such that whenever has norm at most R, and K is a time interval containing 0 such that

Then for any u₀ in the ball

there exists a unique strong solution to (1.1), and moreover the map, is Lipschitz from B to, where defined in Equation (2.5).

Proposition 1.3. let K be a time interval containing and let be two classical solutions to (1.1) with same initial datum u₀ for some fixed μ and p, assume also that we have the temperate decay hypothesis for q = 2, ∞. Then.

Proposition 1.4. Let, given there exists a maximal lifespan solution u to (1.1) define on, with. Furthermore1) k is an open neighborhood of.

2) We say u is a blow up in the contrast direction If or is finite.

3) If we have compact time intervals for bounded sets of initial data, then the map that takes initial data to the corresponding solution is uniformly continuous in these intervals.

4) We say that u scatters forward to a free solution, if and u does not blow up forward in time. And we say that u scatters backward to a free solution, if and u does not blow up backward in time.

To Proof: see [1-3].

2. Strichartz Estimates

In this section we discuss some notation and Strichartz estimates for critical NLS (1.1) and we turn to prove Propositions 1.1 and 1.3.

2.1. Some Notation

If X, Y are nonnegative quantities, we use or, to denote the estimate for some c and to denote the estimate

We defined the Fourier transform on by

We use to denote the Banach space for any space time slab of function with norm is

With the usual amendments when q or r is equal to infinity. When we cut short as.

Defined the fractional differentiation operators, by

where, specially, we will use to signify the spatial gradient and define the Sobolev norms as

Let be the free Schrödinger propagator; in terms of the Fourier transform, this is given by,

.

A Gagliardo-Nirenberg type inequality for Schrödinger equation the generator of the spurious conformal transformation plays the role of the partial differentiation.

2.2. Strichartz Estimates

Let be the free Schrödinger evolution, from the explicit formula

(2.1)

Specially, as the free propagator saves the -norm,

For all and, where

Proposition 2.1. There holds that

(2.2)

In fact, this follows directly from the formula (2.1).

Definition 2.1. Define an admissible pair to be pair

with, , With

Theorem 2.2. If solves the initial value problem

On an interval K, then

(2.3)

For all admissible pairs,. denotes the Lebesgue dual.

To prove: see [4,5].

Definition 2.2. Define the norm

(2.4)

(2.5)

We also define the space to be the space dual to with suitable norm. By theorem.2.2,

(2.6)

Theorem 2.3. If is small, then (1.1) is globally well posed, for more see [6,7].

Proof: by (2.3) and (2.6)

(2.7)

If is small enough and by the continuity method, then we have global well-posedness. Furthermore, for any there exist such that

Then

(2.8)

So by (2.6), when

(2.9)

Thus, the limit

(2.10)

Exists, and,

(2.11)

A conformable argument can be made for

indeed, if, then can be division into subintervals K with

on each subinterval. Using the Duhamel formula on each interval individually, we obtain global well-posedness and scattering.                               

Now we return to prove Proposition 1.1 and Proposition 1.3.

Proof proposition 1.1:

We suppose in what follows that. Let

and for some to be chosen, be such that

(2.12)

We deem the space

And the mapping,

(2.13)

We want to prove that the δ small adequate, is contraction. We use first Strichartz estimates, to compute that

where

Then, distinctly,

So that for is small enough, is settled under. In addition,

Again, decreasing may be, we get a contraction.

If, then, and from Strichartz estimateswe see that if is small enough, then (2.12) is satisfied for                             

Proof Proposition 1.3: By time translation symmetry we can take. By time reversal symmetry we may assume that K lies in the upper time axis. Let, and then, , and v obey the variance equation

Since v and lies in the we may calling Duhamel’s and conclude

for all. By Minkowski’s inequality, and the unitarity of, conclude that,

Since u and v are in, and the function is locally Lipcshitz, we have the bound

Apply Gronwall’s inequality to conclude that

for all and hence.   

3. Decay Estimates

Consider the defocusing nonlinear Schrödinger Equation (1.1), in, where and, for. We suppose that at

(3.1)

First we have the following result.

Theorem 3.1. Suppose that, if and let u be a solution to (1.1), identical to an initial data

such that. If d = 2let r be such that, , then there exists a constant c > 0 such that if R is the solution of, , with

, then

Furthermore, c depends only on d, p, r and,

.

The method made up in rescheduling, by the average of a time dependent rescheduling the equation, and to use the energy of the equation, to get by interpolation decay estimates in suitable norms. The asymptotically average, is normally obtained directly by using the pseudo conformal law, the above result was in fact partially proved in [8], under a bit different point of view: look for a time dependent change of coordinates, which maintain the Galilean invariance, and the construction directly a Lyapunov functional by a suitable ansatz. This Lyapunov functional is surely the energy of the rescaled equation. Our aim here is to study with further details the rescaled wave function and its energy. Found to be the method provides rates which are seems completely new in the limiting case of the logarithmic nonlinear Schrödinger equation. Because of the reversibility of the Schrödinger equation and standard results of scattering theory, one cannot foresee the convergence of the rescaled wave function to some a intuition given limiting wave function, but found to be some convexity properties of the energy can be used to state an asymptotically stabilization result. From the general theory of Schrödinger equations, it is well known that the Cauchy problems (1.1)-(3.1) is well posed for any initial data in when, and that the solution u belongs to

As usual for Schrödinger equations is critical when.

Let be such that

.

where and are positive derivable real functions of the time.

It is simple to check that with this change of coordinates, satisfies the following equation,

where, with the choice, which means that and u are linked by,

(3.2)

where and and has to satisfy the following time-dependent defocusing nonlinear Schrö- dinger equation,

(3.3)

We note that so that

for all.

Also we note that if and then

(3.4)

To extract the controlling impacts as we fix and R such that,

where

(3.5)

Because p is critical, this ansatz is actually the only one that sets to 1 at least three of the four coefficients in the equation for, with and solves the equation,

(3.6)

With the choice and, integration of (3.5) with respect to t gives and this is possible if, and only if, for all thus the function is globally defined on increasing, and as.

Supposing that, is an increasing positive function such that, , where if.

Consider now the energy functional linked to Equation (3.6)

(3.7)

where R has to be understood as a function of.

Lemma 3.2. Suppose that, if, and let u be a solution to (1.1), identical to an initial data,

such that.

With the above notations, E is a decreasing positive functional. Thus is bounded by, with the notations of Theorem 3.1.

Proof: The proof follows by a direct computation.

Because of (3.6), only the coefficients of, and contribute to the decay of the energy.

For more see [9].                              

Proof of Theorem 3.1: Suppose that p is critical. By Lemma 3.2 and pursuant to the time-dependent rescaling (3.2),

Thus

Is bounded by, the remainder of the proof follows the same lines that in Theorem 7.2.1 of [10], see also [11,12], using maintain the L²-norm and the Sobolev-Gagliardo-Nirenberg inequality.      

Proposition 3.3. Consider the two-dimensional defocusing cubic NLS (1.1), (is -critical). Let then there exists a global -well posed solution to (1.1), and moreover the norm of u₀ is finite.

Proof: By time reflection symmetry and adhesion arguments we may heed attention to the time interval Since u₀ lies in, it lies in. Apply the well posedness theory (Proposition 1.2) we can find an -well posed solution, on some time interval with depending on the profile of u₀.

Specially the norm of u is finite. Next we apply the pseudoconformal law to deduce that,

Since, we got a solution from t = 0 to t = T. To go to all the way to. We apply the pseudoconformal transformation at time t = T, obtaining an initial datum at time by the formula

From we see that v has finite energy:

And, the pseudoconformal transformation saves mass and hence

So we see that has a finite norm. Thus, we can use the global -well posedness theory, backwards in time to obtain an -well posed solution

, to the equation particularly,

We reverse the pseudoconformal transformation, which defines the original field u on the new slab

We see that the, and

norm of u are finite. This is sufficient to make u an -well posed solution to NLS on the time interval; for v classical. And for general, the claim follows by a limiting argument using the -well posedness theory. Adhesion together the two intervals and, we have obtained a global solution u to (1.1).

4. Some Lemma

Consider the defocusing case of the NLS (1.1) and if, the energy and mass together will control the norm of the solution:

Conversely, energy and mass are controlled by the norm (the Gagliardo-Nirenberg inequality showed that):

This bound and the energy conservation law and mass conservation law showed that for any -well posed solution, the norm of the solution at time t is bounded by a quantity depending only on the norm of the initial data.

Proposition 4.1. The cubic NLS (1.1) with, d = 2 is globally well posed in. Actually, for and any time interval, K the Cauchy problem (1.1) has a well posed solution

Lemma 4.2. If the following holds:

.

Proof: The proof depends on the noticing that;

.

With

Thus

By standard Gagliardo-Nirenberg inequality,

Lemma.4.3. Let. For any spacetime slab, , and for any

(4.1)

The estimate (4.1) is very helpful when u is high hesitancy and v is low hesitancy, as it moves abundance of derivatives onto the low hesitancy term. In particular, this estimate shows that there is little interaction between high and low hesitancy. This estimate is basically the repeated Strichartz estimate of Bourgain in [13]. We make the trivial remark that the norm of uv is the same as that of, , or, thus the above estimate also applies to expressions of the form

Proof: We fix, and permit our tacit constants to depend on. We begin by dealing with homogeneous case, with and, And consider the more general problem of proving,

(4.2)

where the scaling invariance of this estimate, first, our objective is to prove this for and.

May be recast (4.2) using duality and renormalization as

(4.3)

Since, we may restrict attention to the interactions with

In fact, in the residual case we can multiply by

to return to the condition under discussion. In fact, we may further restrict attention to the case where since, in the other case, we can move the frequencies between the two factors and reduce the case where, which can be dealt by Strichartz estimates when Next, we decompose dyadically and in dyadic multiples of the size of by rewriting the quantity to be controlled as (N, dyadic):

Note that subscripts on have been inserted to invoke the localizations to, , , consecutive. In the case, we have that and this expound, why g may be so localized. By renaming components, we may suppose that and.

Write We change variables by writing

And

We show that by calculation

Thus, upon changing variables in the inner two integrals, we encounter

where

Apply the Cauchy-Schwarz on the u, v integration and change back to the original variables to obtain

We recall that and use Cauchy-Schwarz in the integration, taking into consideration the localization, to get

Choose and. with to obtain

This summarizes to get the claimed homogeneous estimate. Now we discuss the inhomogeneous estimate (4.1). For simplicity we set, and . Then we use Duhamel’s formula to write

We obtain

The first term was treated in the first part of the proof. The second and the third are similar and so we consider I₂ only. By the Minkowski inequality,

And in this case the lemma follows from the homogeneous estimate proved above. Finally, again by Minkowski’s inequality we have

And the proof follows by inserting in the integral the homogeneous estimate above.                   

Lemma.4.4. Let is nearly periodic modulo G. Then there exist functions, and, and for every there exists such that we have the spatial concentration estimate,

(4.3)

And hesitancy concentration estimate,

(4.4)

For all

Remark 4.5. Informally, this lemma confirms that the mass is spatially concentrated in the ball

And is hesitancy concentrated in the ball

.

Note that we have presently no control about how, , vary in time; (for more see [14- 17]).

Proof: By hypothesis, lay in GI for some compact subset I in. For every, compactness argument shows that there exists, (depending on) such that

And hesitancy concentration estimate

For all. By inspecting what the symmetry group G does to the spatial and hesitancy distribution of the mass of a function, then the claim follows.         

Corollary 4.6. Fix and d, and assume that m₀ is finite. Then there exists a maximal-lifespan solution of mass precisely m₀ which blows up both forward and backward in time, and functions, , and, with property, for every (depending on, d, m₀) such that we have the concentration estimates (4.3), (4.4) For all

REFERENCES