## Analysis & PDE

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

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

#### Abstract

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

#### Article information

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

Dates
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
https://projecteuclid.org/euclid.apde/1513731055

Digital Object Identifier
doi:10.2140/apde.2010.3.175

Mathematical Reviews number (MathSciNet)
MR2657453

Zentralblatt MATH identifier
1225.35220

#### Citation

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

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