On the rigidity of minimal mass solutions to the focusing mass-critical NLS for rough initial data

Li, Dong ; Zhang, Xiaoyi

Electronic journal of differential equations, Tome 2009 (2009)

Résumé

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.

Détail

Classification : 35Q55
Keywords: mass-critical, nonlinear Schrödinger equation