1,424 research outputs found
Global wellposedness and scattering for 3D Schr\"odinger equations with harmonic potential and radial data
In this paper,we show that spherical bounded energy solution of the
defocusing 3D energy critical Schr\"odinger equation with harmonic potential,
, exits globally and
scatters to free solution in the space . We
preclude the concentration of energy in finite time by combining the energy
decay estimates.Comment: 39 page
Dynamics for the energy critical nonlinear Schr\"odinger equation in high dimensions
In \cite{duck-merle}, T. Duyckaerts and F. Merle studied the variational
structure near the ground state solution of the energy critical NLS and
classified the solutions with the threshold energy in dimensions
under the radial assumption. In this paper, we extend the results to
all dimensions . The main issue in high dimensions is the non-Lipschitz
continuity of the nonlinearity which we get around by making full use of the
decay property of .Comment: 30 Pages. To appear JF
Regularity of almost periodic modulo scaling solutions for mass-critical NLS and applications
In this paper, we consider the solution to mass critical NLS
. We prove that in dimensions , if
the solution is spherically symmetric and is \emph{almost periodic modulo
scaling}, then it must lie in H_x^{1+\eps} for some \eps>0. Moreover, the
kinetic energy of the solution is localized uniformly in time. One important
application of the theorem is a simplified proof of the scattering conjecture
for mass critical NLS without reducing to three enemies(see the work of
Killip-Tao-Visan, and Killip-Visan-Zhang). As another important application, we
establish a Liouville type result for initial data with ground state
mass. We prove that if a radial solution to focusing mass critical
problem has the ground state mass and does not scatter in both time directions,
then it must be global and coincide with the solitary wave up to symmetries.
Here the ground state is the unique, positive, radial solution to elliptic
equation . This is the first rigidity type result
in scale invariant space .Comment: 22 page
Global well-posedness and scattering for a class of nonlinear Schrodinger equations below the energy space
We prove global well-posedness and scattering for the nonlinear Schr\"odinger
equation with power-type nonlinearity \begin{equation*} \begin{cases} i u_t
+\Delta u = |u|^p u, \quad \frac{4}{n}<p<\frac{4}{n-2}, u(0,x) = u_0(x)\in
H^s(\R^n), \quad n\geq 3, \end{cases} \end{equation*} below the energy space,
i.e., for . In \cite{ckstt:low7}, J. Colliander, M. Keel, G. Staffilani,
H. Takaoka, and T. Tao established polynomial growth of the -norm of the
solution, and hence global well-posedness for initial data in , provided
is sufficiently close to 1. However, their bounds are insufficient to yield
scattering. In this paper, we use the \emph{a priori} interaction Morawetz
inequality to show that scattering holds in whenever is larger
than some value
On a nonlocal aggregation model with nonlinear diffusion
We consider a nonlocal aggregation equation with nonlinear diffusion which
arises from the study of biological aggregation dynamics. As a degenerate
parabolic problem, we prove the well-posedness, continuation criteria and
smoothness of local solutions. For compactly supported nonnegative smooth
initial data we prove that the gradient of the solution develops
-norm blowup in finite time.Comment: Submitted Jun 200
Dynamics for the energy critical nonlinear wave equation in high dimensions
In the work by T. Duyckaerts and F. Merle, they studied the variational
structure near the ground state solution of the energy critical wave
equation and classified the solutions with the threshold energy in
dimensions . In this paper, we extend the results to all dimensions
. The main issue in high dimensions is the non-Lipschitz continuity of
the nonlinearity which we get around by making full use of the decay property
of .Comment: 24 pages, to appear in Transactions AM
The mass-critical nonlinear Schr\"odinger equation with radial data in dimensions three and higher
We establish global well-posedness and scattering for solutions to the
mass-critical nonlinear Schr\"odinger equation for large spherically symmetric L^2_x(R^d) initial data in dimensions . In the focusing case we require that the mass is strictly less than that of
the ground state. As a consequence, we obtain that in the focusing case, any
spherically symmetric blowup solution must concentrate at least the mass of the
ground state at the blowup time
Minimal-mass blowup solutions of the mass-critical NLS
We consider the minimal mass required for solutions to the
mass-critical nonlinear Schr\"odinger (NLS) equation to blow up. If is finite, we show that there exists a
minimal-mass solution blowing up (in the sense of an infinite spacetime norm)
in both time directions, whose orbit in is compact after
quotienting out by the symmetries of the equation. A similar result is obtained
for spherically symmetric solutions. Similar results were previously obtained
by Keraani, \cite{keraani}, in dimensions 1, 2 and Begout and Vargas,
\cite{begout}, in dimensions for the mass-critical NLS and by Kenig
and Merle, \cite{merlekenig}, in the energy-critical case. In a subsequent
paper we shall use this compactness result to establish global existence and
scattering in for the defocusing NLS in three and higher
dimensions with spherically symmetric data.Comment: Contains updated references and related remark
Energy-critical NLS with quadratic potentials
We consider the defocusing -critical nonlinear Schr\"odinger
equation in all dimensions () with a quadratic potential . We show global well-posedness for radial initial data obeying
. In view of the potential , this is the
natural energy space. In the repulsive case, we also prove scattering.
We follow the approach pioneered by Bourgain and Tao in the case of no
potential; indeed, we include a proof of their results that incorporates a
couple of simplifications discovered while treating the problem with quadratic
potential.Comment: Incorporates corrections to Lemma 6.
Global well-posedness and scattering for defocusing energy-critical NLS in the exterior of balls with radial data
We consider the defocusing energy-critical NLS in the exterior of the unit
ball in three dimensions. For the initial value problem with Dirichlet boundary
condition we prove global well-posedness and scattering with large radial
initial data in the Sobolev space . We also point out that the same
strategy can be used to treat the energy-supercritical NLS in the exterior of
balls with Dirichlet boundary condition and radial initial data.Comment: 19 page
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