Minimal mass blow-up solutions to rough nonlinear Schroedinger equations

TL;DR

This paper investigates the threshold for global well-posedness and blow-up in rough nonlinear Schrödinger equations, establishing the existence of minimal mass blow-up solutions and identifying the ground state mass as a critical threshold.

Contribution

It introduces a framework for analyzing rough stochastic NLS equations and proves the existence of minimal mass blow-up solutions, extending classical results to stochastic settings.

Findings

Global well-posedness holds below the ground state mass.

Existence of minimal mass blow-up solutions in 1D and 2D.

Ground state mass is the exact threshold for blow-up.

Abstract

We study the focusing mass-critical rough nonlinear Schroedinger equations, where the stochastic integration is taken in the sense of controlled rough path. We obtain the global well-posedness if the mass of initial data is below that of the ground state. Moreover, the existence of minimal mass blow-up solutions is also obtained in both dimensions one and two. In particular, these yield that the mass of ground state is exactly the threshold of global well-posedness and blow-up of solutions in the stochastic focusing mass-critical case. Similar results are also obtained for a class of nonlinear Schroedinger equations with lower order perturbations.