Construction of $L^2$ log-log blowup solutions for the mass critical nonlinear Schr\"odinger equation

TL;DR

This paper constructs solutions to the mass critical nonlinear Schrödinger equation that blow up in finite time following a log-log rate, using probabilistic methods to handle rough initial data at the $L^2$ level.

Contribution

It introduces a novel probabilistic approach to construct $L^2$ solutions exhibiting log-log blowup, even with rough initial data not in any Sobolev space.

Findings

Constructed $L^2$ blowup solutions with log-log dynamics.

Solutions are rough, not in any $H^s$, $s>0$.

Used probabilistic methods to handle rough initial data.

Abstract

In this article, we study the log-log blowup dynamics for the mass critical nonlinear Schr\"odinger equation on $\mathbb{R}^{2}$ under rough but structured random perturbations at $L^{2}(\mathbb{R}^2)$ regularity. In particular, by employing probabilistic methods, we provide a construction of a family of $L^{2}(\mathbb{R}^2)$ regularity solutions which do not lie in any $H^{s}(\mathbb{R}^2)$ for any $s>0$, and which blowup according to the log-log dynamics.