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A solution to the focusing 3d NLS that blows up on a contracting sphere

Authors: Justin Holmer, Galina Perelman and Svetlana Roudenko
Journal: Trans. Amer. Math. Soc. 367 (2015), 3847-3872
MSC (2010): Primary 35Q55
Published electronically: February 20, 2015
MathSciNet review: 3324912
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Abstract: We rigorously construct radial $ H^1$ solutions to the 3d cubic focusing NLS equation $ i\partial _t \psi + \Delta \psi + 2 \vert\psi \vert^2\psi =0$ that blow-up along a contracting sphere. With blow-up time set to $ t=0$, the solutions concentrate on a sphere at radius $ \sim t^{1/3}$ but focus towards this sphere at the faster rate $ \sim t^{2/3}$. Such dynamics were originally proposed heuristically by Degtyarev-Zakharov-Rudakov in 1975 and independently later by Holmer-Roudenko in 2007, where it was demonstrated to be consistent with all conservation laws of this equation. In the latter paper, it was proposed as a solution that would yield divergence of the $ L_x^3$ norm within the ``wide'' radius $ \sim \Vert\nabla u(t)\Vert _{L_x^2}^{-1/2}$ but not within the ``tight'' radius $ \sim \Vert\nabla u(t)\Vert _{L_x^2}^{-2}$, the second being the rate of contraction of self-similar blow-up solutions observed numerically and described in detail by Sulem-Sulem in 1999.

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Additional Information

Justin Holmer
Affiliation: Department of Mathematics, Brown University, Box 1917, 151 Thayer Street, Providence, Rhode Island 02912

Galina Perelman
Affiliation: Laboratoire d’Analyse et de Mathématiques Appliquées, Université Paris-Est Créteil, Créteil Cedex, France

Svetlana Roudenko
Affiliation: Department of Mathematics, Munroe Hall, The George Washington University, 2115 G Street NW, Washington, DC 20052

Received by editor(s): October 4, 2012
Received by editor(s) in revised form: December 14, 2012
Published electronically: February 20, 2015
Article copyright: © Copyright 2015 American Mathematical Society

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