On the probabilistic well-posedness of the two-dimensional periodic nonlinear Schr\"odinger equation with the quadratic nonlinearity $|u|^2$

TL;DR

This paper investigates the probabilistic well-posedness of the 2D periodic quadratic nonlinear Schrödinger equation with rough random initial data, revealing breakdown of standard theory before the critical regularity predicted by scaling.

Contribution

It establishes almost sure local well-posedness for initial data with fractional derivatives less than 1/2 and shows ill-posedness for data with derivatives at or above 3/4, extending probabilistic analysis to quadratic NLS.

Findings

Almost sure local well-posedness for lpha < 1/2

Probabilistic ill-posedness for lpha  3/4

Breakdown of probabilistic well-posedness theory before the critical lpha = 1

Abstract

We study the two-dimensional periodic nonlinear Schr\"odinger 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 $\alpha \geq 0$) 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 $\alpha < \frac{1}{2}$ and is probabilistically ill-posed for $\alpha \geq \frac{3}{4}$ in a suitable sense. These results show 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 $\alpha = 1$ 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.