On the probabilistic well-posedness of the two-dimensional periodic nonlinear Schrödinger equation with the quadratic nonlinearity |u|2

Abstract

We study the two-dimensional periodic nonlinear Schrödinger equation (NLS) with the quadratic nonlinearity |u|2. In particular, we study the quadratic NLS with random initial data distributed according to a fractional derivative (of order α≥0) of the Gaussian free field. After removing the singularity at the zeroth frequency, we prove that the quadratic NLS is almost surely locally well-posed for α<12 and is probabilistically ill-posed for α≥34 in a suitable sense. The probabilistic ill-posedness result shows that in the case of rough random initial data and a quadratic nonlinearity, the standard probabilistic well-posedness theory for NLS breaks down before reaching the critical value α=1 predicted by the scaling analysis due to Deng, Nahmod, and Yue (2019), and thus this paper is a continuation of the work by Oh and Okamoto (2021) on stochastic nonlinear wave and heat equations by building an analogue for NLS.