## 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

#### 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 $i∂tu+Δu+|u|4∕du=0$ with initial data $u0∈H1(ℝd)$ 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 $H1(ℝd)$. As an application of the $d=1$ result, we construct an open subset of initial data in the radial energy space $Hrad1(ℝ3)$ with corresponding solutions that blow up on a sphere at positive radius for the 3D quintic ($Ḣ1$-critical) focusing NLS equation $i∂tu+Δu+|u|4u=0$. This improves the results of Raphaël and Szeftel [2009], where an open subset in $Hrad3(ℝ3)$ 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
Revised: 10 January 2011
Accepted: 21 February 2011
First available in Project Euclid: 20 December 2017

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

#### 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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