#### Vol. 5, No. 3, 2012

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

### Justin Holmer and Svetlana Roudenko

Vol. 5 (2012), No. 3, 475–512
##### 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.

##### Keywords
blow-up, nonlinear Schrödinger equation
Primary: 35Q55
##### Milestones
Revised: 10 January 2011
Accepted: 21 February 2011
Published: 15 October 2012
##### Authors
 Justin Holmer Mathematics Brown University Box 1917 151 Thayer St Providence, RI 02912 United States Svetlana Roudenko Mathematics George Washington University 2115 G Street NW George Washington University Washington, DC 20052 United States