REGULARITY OF ALMOST PERIODIC MODULO SCALING SOLUTIONS FOR MASS-CRITICAL NLS AND APPLICATIONS

Abstract

We consider the (Formula Presented) solution u to mass-critical NLS iut C1u D (Formula Presented). We prove that in dimensions d ≥ 4, if the solution is spherically symmetric and is almost periodic modulo scaling, then it must lie in (Formula Presented) for some ∈ > 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 L2x initial data with ground state mass. We prove that if a radial L2x solution to focusing masscritical 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 ∆Q-Q+Q1+4/d = 0. This is the first rigidity type result in scale invariant space L2x