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 