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

TL;DR

This paper numerically investigates the probabilistic well-posedness and norm inflation phenomena in one-dimensional fractional nonlinear wave equations across different energy regimes.

Contribution

It provides the first numerical exploration of probabilistic well-posedness and norm inflation in fractional wave equations in one dimension.

Findings

Probabilistic well-posedness observed numerically in sub-critical regimes.

Norm inflation phenomena detected in super-critical regimes.

Numerical results align with theoretical predictions about energy regimes.

Abstract

The three dimensional cubic defocusing nonlinear wave equation is known to be ill-posed for general low regularity initial data. However, well-posedness can be recovered globally in time on a probabilistic level when considering random Gaussian initial data approximated by truncation of Fourier modes. These fine behaviors of nonlinear wave equations have not yet been observed numerically . In this article we perform numerical simulations of the one dimensional fractional cubic defocusing wave equation in a periodic setting. This allows us to explore energy subcritical and supercritial regimes. Our numerical results suggest that both norm inflation and probabilistic well-posedness can be observed numerically in energy sub-critical and super-critical regimes.