Vol. 3, No. 2, 2010

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

Dong Li and Xiaoyi Zhang

Vol. 3 (2010), No. 2, 175–195

We consider the Lx2 solution u to mass-critical NLS iut + Δu = ±|u|4du. We prove that in dimensions d 4, 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 ΔQ Q + Q1+4d = 0. This is the first rigidity type result in scale invariant space Lx2.

Schrödinger equation, mass-critical
Mathematical Subject Classification 2000
Primary: 35Q55
Received: 10 August 2009
Revised: 18 November 2009
Accepted: 17 December 2009
Published: 8 June 2010
Dong Li
Department of Mathematics
University of Iowa
14 MacLean Hall
Iowa City, IA 52240
United States
Xiaoyi Zhang
Academy of Mathematics and System Sciences
Department of Mathematics
University of Iowa
14 MacLean Hall
Iowa City, IA 52240
United States