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

Tadahiro Oh, Oana Pocovnicu, Nikolay Tzvetkov

Research output: Contribution to journalArticlepeer-review

Abstract

We study the three-dimensional cubic nonlinear wave equation (NLW) with random initial data below L^2(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.
Original languageEnglish
Number of pages43
JournalAnnales de l'Institut Fourier
Publication statusAccepted/In press - 28 Oct 2020

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