On the two-dimensional singular stochastic viscous nonlinear wave equations

TL;DR

This paper investigates the local and global well-posedness of two-dimensional stochastic viscous nonlinear wave equations driven by singular fractional noise, introducing renormalization techniques and energy bounds.

Contribution

It establishes local and global well-posedness results for the singular stochastic viscous nonlinear wave equations on a2^2, including renormalization methods and almost sure global well-posedness for random initial data.

Findings

Proved local well-posedness of SvNLW with singular fractional noise.

Established global well-posedness for the defocusing cubic SvNLW.

Demonstrated almost sure global well-posedness with Gaussian initial data.

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 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.