The Mass-Critical for the Nonlinear Schrödinger Equation in d = 2 Mujahid

This paper studies the global behavior defocusing nonlinear Schrödinger equation in dimension d = 2, and we will discuss the case 4 , 2 p d d   . This means that the solutions         2 4 2 , 0, , 0, t x t x u C T L L T     , and called critical solution. We show that u scatters forward and backward to a free solution and the solution is globally well posed.


Introduction
(1.1) is called focusing.In this paper we discuss the case when and 2 p  1    (defocusing case).
If   , u t x is a solution to (1.1) on a time interval   0,T , then is a solution to (1.1) on with 2 0, T This scaling saves the norm of u, Thus (1.1) under previous hypotheses is called L 2 -critical or mass critical.(1.7) A solution is said to scatter backward in time if there exist We note that the Equation (1.1) has preserved quantities, the mass And energy 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 Proposition 1.3.let K be a time interval containing 0 and let be two classical solutions to (1.1) with same initial datum u 0 for some fixed μ and p, assume also that we have the temperate decay hypothesis  there exists a maximal lifespan solution u to (1.1) define on , with .Furthermore, inf k 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

 
sup k   and u does not blow up forward in time.And we say that u scatters backward to a free solution, if   inf k   and u does not blow up backward in time.

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.

Some Notation
If X, Y are nonnegative quantities, we use We defined the Fourier transform on by 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 q r  we cut short as .
q r t x L L , Defined the fractional differentiation operators , specially, we will use  to signify the spatial gradient x  and define the Sobolev norms as : : Let e it 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 def 2 J x it    plays the role of the partial differentiation.

Strichartz Estimates
Let e it be the free Schrödinger evolution, from the explicit formula Specially, as the free propagator saves the 2 x L -norm, and 1 , where In fact, this follows directly from the formula (2.1).Definition 2.1.Define an admissible pair to be pair For all admissible pairs , .

 
the Lebesgue dual .p  To prove: see [4,5].Definition 2.2.Define the norm , admissible su : p We also define the space to be the space dual to bally well posed, for more see [6,7].
Proof: by (2.3) and (2.6) , is small enough and by the continuity method, then we have global well-posedness.Furthermore, for any 0 Thus, the limit Exists, and, A conformable argument can be made for .
, then can be divi- 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 4 and for some 0 We deem the space , And the mapping, We want to prove that the δ small adequate, , , 0 2 .
Again, decreasing may be .By time reversal symmetry we may assume that K lies in the upper time axis we may calling Duhamel's and conclude for all.By Minkowski's inequality, and the unitarity of , conclude that, , and the function for all t K  and hence u u  .□

Decay Estimates
Consider the defocusing nonlinear Schrödinger Equation (1.1), in 2     , where and 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   let r be such that, 2 r    , then there exists a constant c > 0 such that if R is the solution of, 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 z is locally Lipcshitz, we have the bound well known that the Cauchy problems (1.1)-(3.1) is well posed for any initial data in when 0 H  p    , and that the solution u belongs to As usual for Schrödinger equations is critical when 4 p d  .
Let  be such that , e , 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, and u are linked by, where and and  has to satisfy the following time-dependent defocusing nonlinear Schrödinger equation, To extract the controlling impacts as we fix , t    and R such that, 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 this is possible if, and only if, for all thus the function Consider now the energy functional linked to Equation 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,   With the above notations, E is a decreasing positive functional.Thus   E  is bounded by   0 0 E E  , with the notations of Theorem 3.1.
Proof: The proof follows by a direct computation.Because of (3.6),only the coefficients of Fo more see [9].□ r Proof of Theorem 3.1: Suppose that p is critical.By Lemma 3.2 and pursuant to the time-dependent rescaling (3.2), To go to all the way to t   .We apply the pseu , we got a solu doconformal transformation at time t = T, obtaining an , .
And, the pseudoconformal transformation saves mass and hence H -well posed solution We reverse the p doconformal transformation, which defines the original field u on the new , the claim follows by a limiting argument using th 1 x L -well posedness theory.Adhesion together the tw intervals o

Some Lemma
Consider the defocusing ase of the NLS (1.1) and if an ss are controlled by the Conversely, energy d ma 1 x H norm (the Gagliardo-Nirenberg inequality showed that): This bound and the energy conservation law and mass conservation law showed that for any H -well posed solution, the  , 0 t K  , and for any 0.
The estimate elpful hen 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 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 sa between    , we may restrict attention to the interactions with 1 2 .
   In fact, in the residual case we can multiply by  , .

   
We change variables by writing Write pon changing grals, we encounter Thus, u variables in the inner two inte Apply the Cauchy-Schwarz on the u, v integration and change back to the original variables to obtain We recall that J N  and use Cauchy-Schwarz in the integration, taking into consideration the localization 2 ~ΛN


, to get This s es to get the claimed homogeneous estimate.Now we discuss the inhomogeneous estimate (4.1).For simplicity we set,   The first term was treated in the first part of the proof.The second and the third are similar and so we consider I 2 only.By the Minkowski inequality, We consider the Cauchy problem for the nonlinear Schrödinger equation in dimension d = 2.
5) with respect to t gives and decay of the energy.
and any time interval, K the Cauchy problem (1.1) has a 1 x H well posed solution


and d, and as me that m 0 is finite.Then there exists a maximal-lifespan solution su  , d, m 0 ) such that we have the concentration esti- And in this case the lemma follows from the homogeneous estimate proved above.Finally, again by Minkowski's inequality we haveREFCauchy ProbERENCES[1] T. Cazenave and F. Weissler, "The lem for And the proof follows by inserting in the integral the homogeneous estimate above.