Numerical study of probabilistic well-posedness of one dimensional fractional nonlinear wave equations

Abstract

The three dimensional cubic defocusing nonlinear wave equation is known to be ill-posed for general low regularity initial data. Well-posedness can however be recovered globally in time on a probabilistic level, provided the Fourier coefficients of the random initial data follow a sub-Gaussian law, an assumption on which the available proofs rely heavily. In this article we perform numerical simulations of the one dimensional fractional cubic defocusing wave equation in a periodic setting, with low regularity random initial data drawn either from a Gaussian or from a heavy tailed Student law. Besides illustrating probabilistic well-posedness and norm inflation in both the energy subcritical and supercritical regimes, our simulations display no qualitative difference between the heavy tailed and the sub-Gaussian cases. This leads us to conjecture that probabilistic well-posedness holds under the sole assumption that the Fourier coefficients are centered with finite variance.