Blow-up solutions on a sphere for the 3D quintic NLS in the energy space

Abstract

We prove that if u(t) is a log-log blow-up solution, of the type studied by Merle and Raphael, to the L2 critical focusing NLS equation i∂tu+Δu+|u|4∕du=0 with initial data u0∈H1(ℝd) in the cases d=1,2, then u(t) remains bounded in H1 away from the blow-up point. This is obtained without assuming that the initial data u0 has any regularity beyond H1(ℝd). As an application of the d=1 result, we construct an open subset of initial data in the radial energy space Hrad1(ℝ3) with corresponding solutions that blow up on a sphere at positive radius for the 3D quintic (Ḣ1-critical) focusing NLS equation i∂tu+Δu+|u|4u=0. This improves the results of Raphael and Szeftel [2009], where an open subset in Hrad3(ℝ3) is obtained. The method of proof can be summarized as follows: On the whole space, high frequencies above the blow-up scale are controlled by the bilinear Strichartz estimates. On the other hand, outside the blow-up core, low frequencies are controlled by finite speed of propagation.