Analysis & PDE

  • Anal. PDE
  • Volume 3, Number 2 (2010), 175-195.

Regularity of almost periodic modulo scaling solutions for mass-critical NLS and applications

Dong Li and Xiaoyi Zhang

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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.

Article information

Anal. PDE, Volume 3, Number 2 (2010), 175-195.

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

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Schrödinger equation mass-critical


Li, Dong; Zhang, Xiaoyi. Regularity of almost periodic modulo scaling solutions for mass-critical NLS and applications. Anal. PDE 3 (2010), no. 2, 175--195. doi:10.2140/apde.2010.3.175.

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