Probabilistic local Cauchy theory 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 wellposedness 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 (unrenormalized) 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
Pages (from-to)771-830
Number of pages60
JournalAnnales de l'Institut Fourier
Volume72
Issue number2
DOIs
Publication statusPublished - 7 Jul 2022

Keywords

  • Gaussian measure
  • local well-posedness
  • nonlinear wave equation
  • renormalization
  • triviality

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Geometry and Topology

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