On blow up for the energy super critical defocusing non linear Schr\"odinger equations

TL;DR

This paper demonstrates finite-time blow-up for certain energy supercritical defocusing nonlinear Schr"odinger equations in high dimensions, via a novel front mechanism rather than soliton concentration or self-similarity.

Contribution

It introduces a new blow-up mechanism for defocusing NLS equations, involving front formation and hydrodynamical compression, expanding understanding of singularity formation.

Findings

Existence of smooth, localized initial data leading to blow-up.

Blow-up occurs through a front mechanism, not soliton concentration.

Construction of blow-up solutions using self-similar profiles from Euler equations.

Abstract

We consider the energy supercritical defocusing nonlinear Schr\"odinger equation $i\partial_tu+\Delta u-u|u|^{p-1}=0$ in dimension $d\ge 5$. In a suitable range of energy supercritical parameters $(d,p)$, we prove the existence of $\mathcal C^\infty$ 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 $\mathcal C^\infty$ 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}.