4/26/2010

Justin Holmer
Brown University

Improved regularity for $H1$ log-log blow-up solutions to the $L2$ critical NLS

We prove that if $u(t)$ is a log-log blow-up solution, of the type studied by Merle-Raphael (2001--2005) to the 1-d $L2$ critical focusing NLS equation $i\partial_t u +\Delta u + |u|^4 u=0$ with initial data $u_0\in H1$ then $u(t)$ remains bounded in $H1$ away from the blow-up point. This is obtained without assuming that the initial data $u_0$ has any regularity above $H1$. As an application, we construct an open subset of initial data in $H1_{\rad}(\mathbb{R}3)$ (radial $H1(\mathbb{R}3)$ functions) with corresponding solutions that blow-up on a sphere at positive radius for the 3d quintic ($\dot H1$-critical) focusing NLS equation $i\partial_tu + \Delta u +|u|^4u=0$. This is joint work with Svetlana Roudenko.