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