Analysis & PDE
- Anal. PDE
- Volume 3, Number 2 (2010), 175-195.
Regularity of almost periodic modulo scaling solutions for mass-critical NLS and applications
We consider the solution to mass-critical NLS . We prove that in dimensions , if the solution is spherically symmetric and is almost periodic modulo scaling, then it must lie in for some . 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 initial data with ground state mass. We prove that if a radial 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 . This is the first rigidity type result in scale invariant space .
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
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 35Q55: NLS-like equations (nonlinear Schrödinger) [See also 37K10]
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. https://projecteuclid.org/euclid.apde/1513731055