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

We prove that if u(t) is a log-log blow-up solution, of the type studied by Merle and Raphaël, to the L2 critical focusing NLS equation itu + Δu + |u|4du = 0 with initial data u0 H1(d) 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(d). As an application of the d = 1 result, we construct an open subset of initial data in the radial energy space Hrad1(3) with corresponding solutions that blow up on a sphere at positive radius for the 3D quintic (1-critical) focusing NLS equation itu + Δu + |u|4u = 0. This improves the results of Raphaël and Szeftel [2009], where an open subset in Hrad3(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.

blow-up, nonlinear Schrödinger equation
Mathematical Subject Classification 2000
Primary: 35Q55
Received: 23 July 2010
Revised: 10 January 2011
Accepted: 21 February 2011
Published: 15 October 2012
Justin Holmer
Brown University
Box 1917
151 Thayer St
Providence, RI 02912
United States
Svetlana Roudenko
George Washington University
2115 G Street NW
George Washington University
Washington, DC 20052
United States