Blow-up Solutions on a Sphere for the 3d Quintic Nls in the Energy Space

We prove that if u(t) is a log-log blow-up solution, of the type studied by Merle-Raphaël [14], to the L critical focusing NLS equation i∂tu+∆u+|u|u = 0 with initial data u0 ∈ H(R) in the cases d = 1, 2, then u(t) remains bounded in H away from the blow-up point. This is obtained without assuming that the initial data u0 has any regularity beyond H(R). As an application of the d = 1 result, we construct an open subset of initial data in the radial energy space H rad(R) with corresponding solutions that blow-up on a sphere at positive radius for the 3d quintic (Ḣ-critical) focusing NLS equation i∂tu + ∆u + |u|4u = 0. This improves Raphaël-Szeftel [17], where an open subset in H rad(R) 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.