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

Li, Dong; Zhang, Xiaoyi

doi:10.2140/apde.2010.3.175

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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

DOI: 10.2140/apde.2010.3.175

Abstract

We consider the $L_{x}^{2}$ solution $u$ to mass-critical NLS $i u_{t} + Δ u = \pm | u |^{4 ∕ d} u$ . We prove that in dimensions $d \geq 4$ , if the solution is spherically symmetric and is almost periodic modulo scaling, then it must lie in $H_{x}^{1 + ε}$ 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 $L_{x}^{2}$ initial data with ground state mass. We prove that if a radial $L_{x}^{2}$ 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 $Δ Q - Q + Q^{1 + 4 ∕ d} = 0$ . This is the first rigidity type result in scale invariant space $L_{x}^{2}$ .

Keywords

Schrödinger equation, mass-critical

Mathematical Subject Classification 2000

Primary: 35Q55

Milestones

Received: 10 August 2009

Revised: 18 November 2009

Accepted: 17 December 2009

Published: 8 June 2010

Authors

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 Beijing China	Department of Mathematics University of Iowa 14 MacLean Hall Iowa City, IA 52240 United States

Vol. 3, No. 2, 2010