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 ![](https://www.scirp.org/html/8-5300438\53b0791d-82a8-4b5f-873a-e9714c63cb15.jpg)
then, for any initial data
, there exist
such that there exists a unique solution
![](https://www.scirp.org/html/8-5300438\189458cc-cfb9-4cbd-9abe-907f81cf713d.jpg)
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, ![](https://www.scirp.org/html/8-5300438\8de3af0d-5d80-4c99-bac8-55530138050b.jpg)
This means that a solution u
and called critical solution.
Definition 1.1. Let
,
is a solution to (1.1) if for any compact
![](https://www.scirp.org/html/8-5300438\3e94af0d-19fa-463d-bed7-67975827e86b.jpg)
and for all ![](https://www.scirp.org/html/8-5300438\69be5a3c-30cd-42c5-8154-3c8d01e15364.jpg)
(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 ![](https://www.scirp.org/html/8-5300438\c77d4c1c-016f-4253-ae91-6f86f6d9a4d7.jpg)
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
![](https://www.scirp.org/html/8-5300438\cfba2434-d446-403b-9788-c23ca757a863.jpg)
Then for any u0 in the ball
![](https://www.scirp.org/html/8-5300438\ebde2db2-e5d0-4638-9585-573aeee5ab91.jpg)
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 ![](https://www.scirp.org/html/8-5300438\7c19a29e-31d2-49cf-a78d-fd8e051dafca.jpg)
We defined the Fourier transform on
by
![](https://www.scirp.org/html/8-5300438\002e2357-b513-45dd-a875-2ed19381d87d.jpg)
We use
to denote the Banach space for any space time slab
of function
with norm is
![](https://www.scirp.org/html/8-5300438\243a8dce-ca46-417f-9766-de40fe64c8cd.jpg)
With the usual amendments when q or r is equal to infinity. When
we cut short
as
.
Defined the fractional differentiation operators
,
by
![](https://www.scirp.org/html/8-5300438\b85733cc-3a37-4faf-8925-1d025e18f1d5.jpg)
![](https://www.scirp.org/html/8-5300438\b86fa98a-9a6b-4f73-9555-ee186786f37b.jpg)
where
, specially, we will use
to signify the spatial gradient
and define the Sobolev norms as
![](https://www.scirp.org/html/8-5300438\e679582f-8895-45bf-a103-6a3c96efa4ae.jpg)
![](https://www.scirp.org/html/8-5300438\47c4032c-596b-4c1b-b140-9563167b8ae5.jpg)
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,
![](https://www.scirp.org/html/8-5300438\28bd2c9b-4afa-493a-8cb8-ee19ca393006.jpg)
For all
and
, where ![](https://www.scirp.org/html/8-5300438\8a9abd6d-5509-4f7e-b03b-6b85fa6b96e6.jpg)
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 ![](https://www.scirp.org/html/8-5300438\e7fca5de-5ed2-4a50-b003-36f5c1c9e2dc.jpg)
Theorem 2.2. If
solves the initial value problem
![](https://www.scirp.org/html/8-5300438\6c2874b9-93ab-48ea-b506-876f8a081678.jpg)
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
![](https://www.scirp.org/html/8-5300438\90a38914-fb4b-4ddf-a572-9d28481beed3.jpg)
Then
(2.8)
So by (2.6), when ![](https://www.scirp.org/html/8-5300438\36f727fc-d63d-431e-93df-816100f9f98c.jpg)
(2.9)
Thus, the limit
(2.10)
Exists, and,
(2.11)
A conformable argument can be made for ![](https://www.scirp.org/html/8-5300438\88c82911-8dfc-4fa4-83cc-7959b9d23fa4.jpg)
indeed, if
, then
can be division into
subintervals K with ![](https://www.scirp.org/html/8-5300438\a479c349-69ca-4b62-b410-0e3e32fa01d7.jpg)
on each subinterval. Using the Duhamel formula on each interval individually, we obtain global well-posedness and scattering. ![](https://www.scirp.org/html/8-5300438\0ba975f2-aa5c-4e7b-b069-f3ef04ec1c09.jpg)
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
![](https://www.scirp.org/html/8-5300438\e4d45141-3526-4c48-9e72-2c13567c2df1.jpg)
And the mapping,
(2.13)
We want to prove that the δ small adequate,
is contraction. We use first Strichartz estimates, to compute that
![](https://www.scirp.org/html/8-5300438\9a264265-dc42-43c1-a824-fac1f38560d5.jpg)
where ![](https://www.scirp.org/html/8-5300438\9d3689f3-9997-460b-8fd9-59608d7fd289.jpg)
Then, distinctly,
![](https://www.scirp.org/html/8-5300438\c8073a14-ce4d-41ba-a8b3-199a10cb02b1.jpg)
![](https://www.scirp.org/html/8-5300438\86c03ae6-48ed-42da-9f4b-d526e86f68d0.jpg)
So that for
is small enough,
is settled under
. In addition,
![](https://www.scirp.org/html/8-5300438\5a06c19e-ef88-443b-b2bd-2c4ec69973e5.jpg)
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
![](https://www.scirp.org/html/8-5300438\bafc47c0-c873-42c3-b925-ce3a7391fe4c.jpg)
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
![](https://www.scirp.org/html/8-5300438\01ef0d74-01f2-4fab-bcc2-1b38ccac10e9.jpg)
Since v and
lies in the
we may calling Duhamel’s and conclude
![](https://www.scirp.org/html/8-5300438\6e5c93c2-32bb-4d0c-9849-5263335e1b95.jpg)
for all. By Minkowski’s inequality, and the unitarity of
, conclude that,
![](https://www.scirp.org/html/8-5300438\56a3973d-0aa4-454f-bbc3-97326c2fedf1.jpg)
Since u and v are in
, and the function
is locally Lipcshitz, we have the bound
![](https://www.scirp.org/html/8-5300438\b66b8af2-8174-4ea1-a924-d1ad374c0e77.jpg)
Apply Gronwall’s inequality to conclude that
for all
and hence
. ![](https://www.scirp.org/html/8-5300438\297b3f95-e629-4df5-80d4-85a93e642348.jpg)
3. Decay Estimates
Consider the defocusing nonlinear Schrödinger Equation (1.1), in
, where
and
, for
. We suppose that at ![](https://www.scirp.org/html/8-5300438\6697874e-e4e6-4adb-a695-5ecbab64284d.jpg)
(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
![](https://www.scirp.org/html/8-5300438\04e69861-2b31-4bdd-be04-d6efc687b682.jpg)
,
then
![](https://www.scirp.org/html/8-5300438\4d4c2ea3-db5a-4250-8e51-facff2783ebb.jpg)
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
![](https://www.scirp.org/html/8-5300438\1bb53ff2-5ec0-45c2-a59f-2474e8f1ad34.jpg)
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,
![](https://www.scirp.org/html/8-5300438\6033f377-d5a5-495d-8ccd-84560e46a045.jpg)
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
![](https://www.scirp.org/html/8-5300438\9a4ea02b-f7fc-4694-82ed-a4034da5c771.jpg)
for all
.
Also we note that if
and
then
(3.4)
To extract the controlling impacts as
we fix
and R such that,
![](https://www.scirp.org/html/8-5300438\efdabdda-52bd-4cfd-bb5a-4c0b730d0abb.jpg)
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]. ![](https://www.scirp.org/html/8-5300438\8b120a58-c0b5-43a0-b688-84d26b7b57d1.jpg)
Proof of Theorem 3.1: Suppose that p is critical. By Lemma 3.2 and pursuant to the time-dependent rescaling (3.2),
![](https://www.scirp.org/html/8-5300438\4052fb21-c8b3-4e37-ac6a-0e0f6d4331b1.jpg)
Thus
![](https://www.scirp.org/html/8-5300438\5a25a1e8-3936-47c1-81db-8e2f12f96390.jpg)
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. ![](https://www.scirp.org/html/8-5300438\8a32d3c9-9a35-4ca8-85bc-95285525f249.jpg)
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,
![](https://www.scirp.org/html/8-5300438\cc843325-ac19-4ecf-ba9b-b7c8fb4544c0.jpg)
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
![](https://www.scirp.org/html/8-5300438\3e3e210b-981d-4455-9d61-b6db9c08069a.jpg)
From
we see that v has finite energy:
![](https://www.scirp.org/html/8-5300438\f4c6859b-b3bd-406a-aa83-37599eabc8e6.jpg)
And, the pseudoconformal transformation saves mass and hence
![](https://www.scirp.org/html/8-5300438\c9d34029-8f54-4fdc-87d1-c86d49893e46.jpg)
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, ![](https://www.scirp.org/html/8-5300438\bed4dee3-52d4-4f04-8592-b6206b99932c.jpg)
We reverse the pseudoconformal transformation, which defines the original field u on the new slab ![](https://www.scirp.org/html/8-5300438\2d451355-b3cf-4ccf-b575-63445efc2a1b.jpg)
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:
![](https://www.scirp.org/html/8-5300438\43bc85c8-0035-4acd-8ace-c155fe3f7775.jpg)
Conversely, energy and mass are controlled by the
norm (the Gagliardo-Nirenberg inequality showed that):
![](https://www.scirp.org/html/8-5300438\dbe26ce3-0c3f-4e50-bab4-acecff228782.jpg)
![](https://www.scirp.org/html/8-5300438\554ec166-4255-4ad1-9c8b-3a8361a7ce2c.jpg)
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
![](https://www.scirp.org/html/8-5300438\9a1e8226-b4af-493a-a0c6-c3bbece4ba52.jpg)
Lemma 4.2. If
the following holds:
.
Proof: The proof depends on the noticing that;
.
With
![](https://www.scirp.org/html/8-5300438\86bebb61-67b7-42a2-b41b-fcbc4f43ab2f.jpg)
Thus
![](https://www.scirp.org/html/8-5300438\b6370777-5a40-435f-a487-26b3f25f95a2.jpg)
By standard Gagliardo-Nirenberg inequality,
![](https://www.scirp.org/html/8-5300438\4ebe06ca-e404-4cbf-900b-2155bd68cd23.jpg)
Lemma.4.3. Let
. For any spacetime slab
,
, and for any ![](https://www.scirp.org/html/8-5300438\ac017faf-47ac-4b4f-a4eb-01df16e2e030.jpg)
(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 ![](https://www.scirp.org/html/8-5300438\01f07fae-b07f-4add-8e5a-50c87776f0e3.jpg)
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 ![](https://www.scirp.org/html/8-5300438\44793b88-0d48-4b32-a33f-b1ada053fb07.jpg)
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):
![](https://www.scirp.org/html/8-5300438\cec14e9b-198c-4188-a2d0-7b1f281a6013.jpg)
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
![](https://www.scirp.org/html/8-5300438\b8580be2-7959-42b2-8474-2597f4f74c6b.jpg)
And ![](https://www.scirp.org/html/8-5300438\ccff7488-5cc5-439d-99d1-403efa3c30ee.jpg)
We show that by calculation
![](https://www.scirp.org/html/8-5300438\7fab6217-f581-48ee-9c52-505b2d09744d.jpg)
Thus, upon changing variables in the inner two integrals, we encounter
![](https://www.scirp.org/html/8-5300438\b87f1b09-9567-4e5f-b780-2a29eef80c82.jpg)
where
![](https://www.scirp.org/html/8-5300438\c45fe0f0-4d33-47c5-9a3f-1cb5c5a73dc8.jpg)
Apply the Cauchy-Schwarz on the u, v integration and change back to the original variables to obtain
![](https://www.scirp.org/html/8-5300438\2798af16-4e04-49f7-9725-7f7c108f83f9.jpg)
We recall that
and use Cauchy-Schwarz in the integration, taking into consideration the localization
, to get
![](https://www.scirp.org/html/8-5300438\ae8ca94c-ef44-49fb-9842-d29464ff78fb.jpg)
Choose
and
. with
to obtain
![](https://www.scirp.org/html/8-5300438\b31f80f2-aa32-4a30-9de1-341493bf4438.jpg)
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
![](https://www.scirp.org/html/8-5300438\14b27395-e438-4fb1-aa86-7a83a4d7cadb.jpg)
![](https://www.scirp.org/html/8-5300438\09575348-aade-4ce8-bb59-865e34b5351d.jpg)
We obtain
![](https://www.scirp.org/html/8-5300438\bb786dad-0fa9-485c-8a81-fbf6c9a314aa.jpg)
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,
![](https://www.scirp.org/html/8-5300438\4bbec2ef-d0b5-43ce-bb1c-897344d4c67f.jpg)
And in this case the lemma follows from the homogeneous estimate proved above. Finally, again by Minkowski’s inequality we have
![](https://www.scirp.org/html/8-5300438\15dd02a6-264b-46d2-aaea-f148bfdc8032.jpg)
And the proof follows by inserting in the integral the homogeneous estimate above. ![](https://www.scirp.org/html/8-5300438\9b9c3471-c277-409d-a3ae-e3369b95fcb4.jpg)
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 ![](https://www.scirp.org/html/8-5300438\d29a1d3b-5988-40b3-b05c-da22a4af545c.jpg)
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
![](https://www.scirp.org/html/8-5300438\0784611e-db1a-41fb-963c-5f506ec8f2ec.jpg)
And hesitancy concentration estimate
![](https://www.scirp.org/html/8-5300438\1546ada6-4e33-4a70-a885-cc2dc855d9fa.jpg)
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. ![](https://www.scirp.org/html/8-5300438\acb1c38b-45bd-48a6-b943-cc201a699dea.jpg)
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 ![](https://www.scirp.org/html/8-5300438\dcc51056-0f34-4929-9756-cdb4afa98102.jpg)