Energy Critical NLS in two space dimensions

TL;DR

This paper studies the energy-critical nonlinear Schrödinger equation in two dimensions, identifying regimes of well-posedness and establishing global solutions in subcritical and critical cases.

Contribution

It classifies the energy regimes for the equation and proves global well-posedness in subcritical and critical regimes, highlighting failure in the supercritical case.

Findings

Global well-posedness in subcritical regime

Global well-posedness in critical regime

Ill-posedness in supercritical regime

Abstract

We investigate the initial value problem for a defocusing nonlinear Schr\"odingerequation with exponential nonlinearity. We identify subcritical, critical and supercritical regimes in the energy space. We establish global well-posedness in the subcritical and critical regimes. Well-posedness fails to hold in the supercritical case.