10/9/2006

Pierre Raphael
Princeton University

Blow up of the critical norm for some L^2 super critical nonlinear Schrodinger equations

Let the focusing (NLS) $iu_t+\Delta u+u|u|^{2}=0$ in dimension $N\geq 1$ with initial data in the energy space $H1$. For $N=1$, the problem is $L2$ subcritical and all $H1$ solutions are global. On the contrary, in dimensions $N=2,3$, there exist finite time blow up solutions but the understanding of the blow up dynamics is still very poor. In dimension $N=2$, the equation is $L2$ critical and thus the critical -that is scaling invariant- $L2$ norm is bounded because it is conserved. In dimension $N=3$, the problem is $H^{1/2}$ critical and numerics suggest that this norm should blow up if the solution blows up in finite time. We prove this result for radially symmetric initial data together with a lower bound $|u(t)|_{H^{1/2}}\geq |log(T-t)|^{\alpha}$. This is joint work with Frank Merle.