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.
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BibTeX
@misc{ruffenach2026numerical,
title = {{Numerical study of probabilistic well-posedness of one dimensional fractional nonlinear wave equations}},
author = {Wandrille Ruffenach and Nikolay Tzvetkov},
howpublished = {HAL},
year = {2026},
month = sep,
eprint = {2604.05938},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05583261v2},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Numerical study of probabilistic well-posedness of one dimensional fractional nonlinear wave equations
AU - Wandrille Ruffenach
AU - Nikolay Tzvetkov
PY - 2026
DA - 2026/09/04
PB - HAL
UR - https://hal.science/hal-05583261v2
AB - 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.
ER -