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

Abstract

We prove that if $u\left(t\right)$ is a log-log blow-up solution, of the type studied by Merle and Raphaël, to the $L^2$ critical focusing NLS equation $i{\partial }_{t}u+\Delta u+|u{|}^{4∕d}u=0$ with initial data $u_0\in H^1\left({ℝ}^d\right)$ in the cases $d=1,2$, then $u\left(t\right)$ remains bounded in $H^1$ away from the blow-up point. This is obtained without assuming that the initial data $u_0$ has any regularity beyond $H^1\left({ℝ}^d\right)$. As an application of the $d=1$ result, we construct an open subset of initial data in the radial energy space $H_{rad}^1\left({ℝ}^3\right)$ with corresponding solutions that blow up on a sphere at positive radius for the 3D quintic (${Ḣ}^1$-critical) focusing NLS equation $i{\partial }_{t}u+\Delta u+|u{|}^{4}u=0$. This improves the results of Raphaël and Szeftel [2009], where an open subset in $H_{rad}^3\left({ℝ}^3\right)$ 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.