For the quadratic stochastic nonlinear wave and heat equations on the two-dimensional torus, standard Da Prato-Debussche solution theory fails at noise roughness alpha = 1/2 (wave) and alpha = 1 (heat), before the scaling-critical values 3/4 and 2.
Global dynamics for the two-dimensional stochastic nonlinear wave equations
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abstract
We study global-in-time dynamics of the stochastic nonlinear wave equations (SNLW) with an additive space-time white noise forcing, posed on the two-dimensional torus. Our goal in this paper is two-fold. (i) By introducing a hybrid argument, combining the $I$-method in the stochastic setting with a Gronwall-type argument, we first prove global well-posedness of the (renormalized) cubic SNLW in the defocusing case. Our argument yields a double exponential growth bound on the Sobolev norm of a solution. (ii) We then study the stochastic damped nonlinear wave equations (SdNLW) in the defocusing case. In particular, by applying Bourgain's invariant measure argument, we prove almost sure global well-posedness of the (renormalized) defocusing SdNLW with respect to the Gibbs measure and invariance of the Gibbs measure.
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Comparing the stochastic nonlinear wave and heat equations: a case study
For the quadratic stochastic nonlinear wave and heat equations on the two-dimensional torus, standard Da Prato-Debussche solution theory fails at noise roughness alpha = 1/2 (wave) and alpha = 1 (heat), before the scaling-critical values 3/4 and 2.