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 language | English |
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Pages (from-to) | 771-830 |
Number of pages | 60 |
Journal | Annales de l'Institut Fourier |
Volume | 72 |
Issue number | 2 |
DOIs | |
Publication status | Published - 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