Analysis & PDE

  • Anal. PDE
  • Volume 5, Number 3 (2012), 475-512.

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

Justin Holmer and Svetlana Roudenko

Full-text: Open access

Abstract

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 u0H1(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.

Article information

Source
Anal. PDE, Volume 5, Number 3 (2012), 475-512.

Dates
Received: 23 July 2010
Revised: 10 January 2011
Accepted: 21 February 2011
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.apde/1513731228

Digital Object Identifier
doi:10.2140/apde.2012.5.475

Mathematical Reviews number (MathSciNet)
MR2994505

Zentralblatt MATH identifier
1329.35280

Subjects
Primary: 35Q55: NLS-like equations (nonlinear Schrödinger) [See also 37K10]

Keywords
blow-up nonlinear Schrödinger equation

Citation

Holmer, Justin; Roudenko, Svetlana. Blow-up solutions on a sphere for the 3D quintic NLS in the energy space. Anal. PDE 5 (2012), no. 3, 475--512. doi:10.2140/apde.2012.5.475. https://projecteuclid.org/euclid.apde/1513731228


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