Probabilistic local well-posedness of the cubic nonlinear wave equation in negative Sobolev spaces

Tadahiro Oh; Oana Pocovnicu; Nikolay Tzvetkov

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arXiv · 1904.06792

Probabilistic local well-posedness of the cubic nonlinear wave equation in negative Sobolev spaces

Tadahiro Oh·Oana Pocovnicu·Nikolay Tzvetkov

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Abstract

We study the three-dimensional cubic nonlinear wave equation (NLW) with random initial data below $L^2(\mathbb{T}^3)$. By considering the second order expansion in terms of the random linear solution, we prove almost sure local well-posedness of the renormalized NLW in negative Sobolev spaces. We also prove a new instability result for the defocusing cubic NLW without renormalization in negative Sobolev spaces, which is in the spirit of the so-called triviality in the study of stochastic partial differential equations. More precisely, by studying (un-renormalized) NLW with given smooth deterministic initial data plus a certain truncated random initial data, we show that, as the truncation is removed, the solutions converge to $0$ in the distributional sense for any deterministic initial data.

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Tadahiro Oh, Oana Pocovnicu, Nikolay Tzvetkov. 2019-04-15. Probabilistic local well-posedness of the cubic nonlinear wave equation in negative Sobolev spaces. https://arxiv.org/abs/1904.06792

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