Some remarks on the nonlinear Schr\"odinger equation with fractional dissipation

TL;DR

This paper investigates the effects of fractional dissipation on the $L^2$-critical nonlinear Schrödinger equation, establishing conditions for global existence or finite-time blowup with log-log speed based on dissipation order.

Contribution

It provides new results on the global behavior and blowup dynamics of the nonlinear Schrödinger equation with fractional dissipation, depending on the dissipation order.

Findings

Global existence for certain dissipation orders

Finite time blowup with log-log speed for others

Characterization based on fractional dissipation order

Abstract

We consider the Cauchy problem for the $L^2$-critical nonlinear Schr\"odinger equation with a fractional dissipation. According to the order of the fractional dissipation, we prove the global existence or the existence of finite time blowup dynamics with the log-log blow-up speed for $\|\nabla u(t)\|_{L^2}$.