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

Holmer, Justin; Roudenko, Svetlana

doi:10.2140/apde.2012.5.475

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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

DOI: 10.2140/apde.2012.5.475

Abstract

We prove that if $u (t)$ is a log-log blow-up solution, of the type studied by Merle and Raphaël, to the $L^{2}$ critical focusing NLS equation $i \partial_{t} u + Δ u + | u |^{4 ∕ d} u = 0$ with initial data $u_{0} \in H^{1} (ℝ^{d})$ in the cases $d = 1, 2$ , then $u (t)$ 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 $H^{1} (ℝ^{d})$ . As an application of the $d = 1$ result, we construct an open subset of initial data in the radial energy space $H_{rad}^{1} (ℝ^{3})$ with corresponding solutions that blow up on a sphere at positive radius for the 3D quintic ( $Ḣ^{1}$ -critical) focusing NLS equation $i \partial_{t} u + Δ u + | u |^{4} u = 0$ . This improves the results of Raphaël and Szeftel [2009], where an open subset in $H_{rad}^{3} (ℝ^{3})$ 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

Mathematical Subject Classification 2000

Primary: 35Q55

Milestones

Received: 23 July 2010

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

Vol. 5, No. 3, 2012