Construction of $L^2$ log-log blowup solutions for the mass critical nonlinear Schrödinger equation

Abstract

In this article, we study the log-log blowup dynamics for the mass critical nonlinear Schrödinger 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 $H^s({\mathbb{R}}^2)$ for any $s>0$, and which blowup according to the log-log dynamics.