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.
Categories: math.AP