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

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.

43 pages. Expanded the introduction. To appear in Ann. Inst. Fourier (Grenoble)

References in corpus (4)

• Ill-posedness for nonlinear Schrodinger and wave equations
• Solving the 4NLS with white noise initial data
• Random tensors, propagation of randomness, and nonlinear dispersive equations
• Comparing the stochastic nonlinear wave and heat equations: a case study

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• Non relativistic and ultra relativistic limits in 2d stochastic nonlinear damped Klein-Gordon equation
• Comparing the stochastic nonlinear wave and heat equations: a case study
• On the parabolic and hyperbolic Liouville equations
• On the two-dimensional hyperbolic stochastic sine-Gordon equation