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

2022/05/16 by Liu, Ruoyuan · 1 citation
#35Q55 #35R60 #Analysis of PDEs (math.AP) #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2205.07797

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< (1)/(2) and is probabilistically ill-posed for α≥ (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 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.

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• Probabilistic well-posedness of dispersive PDEs beyond variance blowup I: Benjamin-Bona-Mahony equation

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