On the two-dimensional singular stochastic viscous nonlinear wave equations

Abstract

We study the stochastic viscous nonlinear wave equations (SvNLW) on $\mathbb{T}^2$, forced by a fractional derivative of the space-time white noise $\xi $. In particular, we consider SvNLW with the singular additive forcing $D^\frac{1}{2}\xi $ such that solutions are expected to be merely distributions. By introducing an appropriate renormalization, we prove local well-posedness of SvNLW. By establishing an energy bound via a Yudovich-type argument, we also prove pathwise global well-posedness of the defocusing cubic SvNLW. Lastly, in the defocusing case, we prove almost sure global well-posedness of SvNLW with respect to certain Gaussian random initial data.