Vol. 5, No. 3, 2012

Download this article
Download this article For screen
For printing
Recent Issues

Volume 17
Issue 5, 1501–1870
Issue 4, 1127–1500
Issue 3, 757–1126
Issue 2, 379–756
Issue 1, 1–377

Volume 16, 10 issues

Volume 15, 8 issues

Volume 14, 8 issues

Volume 13, 8 issues

Volume 12, 8 issues

Volume 11, 8 issues

Volume 10, 8 issues

Volume 9, 8 issues

Volume 8, 8 issues

Volume 7, 8 issues

Volume 6, 8 issues

Volume 5, 5 issues

Volume 4, 5 issues

Volume 3, 4 issues

Volume 2, 3 issues

Volume 1, 3 issues

The Journal
About the journal
Ethics and policies
Peer-review process
Submission guidelines
Submission form
Editorial board
Editors' interests
ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
Author index
To appear
Other MSP journals
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