On blow up for the energy super critical defocusing non linear Schrödinger equations

Frank Merle; Pierre Raphael; Igor Rodnianski; Jeremie Szeftel

Paper

On blow up for the energy super critical defocusing non linear Schrödinger equations

Abstract

We consider the energy supercritical defocusing nonlinear Schrödinger equation

in dimension

. In a suitable range of energy supercritical parameters

, we prove the existence of

well localized spherically symmetric initial data such that the corresponding unique strong solution blows up in finite time. Unlike other known blow up mechanisms, the singularity formation does not occur by concentration of a soliton or through a self similar solution, which are unknown in the defocusing case, but via a front mechanism. Blow up is achieved by compression in the associated hydrodynamical flow which in turn produces a highly oscillatory singularity. The front blow up profile is chosen among the countable family of

spherically symmetric self similar solutions to the compressible Euler equation whose existence and properties in a suitable range of parameters are established in the companion paper \cite{MRRSprofile}.