Partial differential equations, Probability theory
On the two-dimensional singular stochastic viscous nonlinear wave equations
Comptes Rendus. Mathématique, Volume 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 ξ. In particular, we consider SvNLW with the singular additive forcing D 1 2 ξ 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.

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DOI: 10.5802/crmath.377
Classification: 35L71, 60H15
Liu, Ruoyuan 1; Oh, Tadahiro 1

1 School of Mathematics, The University of Edinburgh, and The Maxwell Institute for the Mathematical Sciences, James Clerk Maxwell Building, The King’s Buildings, Peter Guthrie Tait Road, Edinburgh, EH9 3FD, United Kingdom
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Liu, Ruoyuan; Oh, Tadahiro. On the two-dimensional singular stochastic viscous nonlinear wave equations. Comptes Rendus. Mathématique, Volume 360 (2022) no. G11, pp. 1227-1248. doi : 10.5802/crmath.377. http://www.numdam.org/articles/10.5802/crmath.377/

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