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
We prove that if is a log-log blow-up solution, of the type studied by Merle and Raphaël, to the critical focusing NLS equation with initial data in the cases , then remains bounded in away from the blow-up point. This is obtained without assuming that the initial data has any regularity beyond . As an application of the result, we construct an open subset of initial data in the radial energy space with corresponding solutions that blow up on a sphere at positive radius for the 3D quintic (-critical) focusing NLS equation . This improves the results of Raphaël and Szeftel , where an open subset in 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.
Anal. PDE, Volume 5, Number 3 (2012), 475-512.
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
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 35Q55: NLS-like equations (nonlinear Schrödinger) [See also 37K10]
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