On the two-dimensional singular stochastic viscous nonlinear wave equations

Liu, Ruoyuan; Oh, Tadahiro

Comptes Rendus. Mathématique, Tome 360 (2022) no. G11, pp. 1227-1248.

We study the stochastic viscous nonlinear wave equations (SvNLW) on ${𝕋}^2$, 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.

Classification : 35L71, 60H15

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