Electronic Journal of Differential Equations,
Vol. 2009(2009), No. 78, pp. 1-19.
Title: On the rigidity of minimal mass solutions to the focusing
mass-critical NLS for rough initial data
Authors: Dong Li (Institute for Advanced Study, Princeton, NJ, USA)
Xiaoyi Zhang (Academy of Math. and System Sciences, Beijing, China)
Abstract:
For the focusing mass-critical nonlinear Schrodinger equation
$iu_t+\Delta u=-|u|^{4/d}u$, an important problem is to
establish Liouville type results for solutions with ground state mass.
Here the ground state is the positive solution to elliptic equation
$\Delta Q-Q+Q^{1+\frac 4d}=0$.
Previous results in this direction were established in
[12, 16, 17, 29] and they all require
$u_0\in H_x^1(\mathbb{R}^d)$. In this paper, we consider the rigidity results
for rough initial data $u_0 \in H_x^s(\mathbb{R}^d)$ for any $s>0$.
We show that in dimensions $d\ge 4$ and under the radial assumption,
the only solution that does not scatter in both time directions
(including the finite time blowup case) must be global and coincide
with the solitary wave $e^{it}Q$ up to symmetries of the equation.
The proof relies on a non-uniform local iteration scheme, the refined
estimate involving the $P^{\pm}$ operator and a new smoothing estimate
for spherically symmetric solutions.
Submitted April 15, 2009. Published June 16, 2009.
Math Subject Classifications: 35Q55.
Key Words: Mass-critical; nonlinear Schrodinger equation.