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2010 Regularity of almost periodic modulo scaling solutions for mass-critical NLS and applications
Dong Li, Xiaoyi Zhang
Anal. PDE 3(2): 175-195 (2010). DOI: 10.2140/apde.2010.3.175

Abstract

We consider the Lx2 solution u to mass-critical NLS iut+Δu=±|u|4du. We prove that in dimensions d4, if the solution is spherically symmetric and is almost periodic modulo scaling, then it must lie in Hx1+ε for some ε>0. Moreover, the kinetic energy of the solution is localized uniformly in time. One important application of the theorem is a simplified proof of the scattering conjecture for mass-critical NLS without reducing to three enemies. As another important application, we establish a Liouville type result for Lx2 initial data with ground state mass. We prove that if a radial Lx2 solution to focusing mass-critical problem has the ground state mass and does not scatter in both time directions, then it must be global and coincide with the solitary wave up to symmetries. Here the ground state is the unique, positive, radial solution to elliptic equation ΔQQ+Q1+4d=0. This is the first rigidity type result in scale invariant space Lx2.

Citation

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Dong Li. Xiaoyi Zhang. "Regularity of almost periodic modulo scaling solutions for mass-critical NLS and applications." Anal. PDE 3 (2) 175 - 195, 2010. https://doi.org/10.2140/apde.2010.3.175

Information

Received: 10 August 2009; Revised: 18 November 2009; Accepted: 17 December 2009; Published: 2010
First available in Project Euclid: 20 December 2017

zbMATH: 1225.35220
MathSciNet: MR2657453
Digital Object Identifier: 10.2140/apde.2010.3.175

Subjects:
Primary: 35Q55

Rights: Copyright © 2010 Mathematical Sciences Publishers

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