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 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.

We find minimal and maximal length of intersections of lines at a fixed distance to the origin with the cross-polytope. We also find maximal volume noncentral sections of the cross-polypote by hyperplanes which are at a fixed large distance to the origin and minimal volume sections by symmetric slabs of a large fixed width. This parallels recent results about noncentral sections of the cube due to Moody, Stone, Zach and Zvavitch.

We study well-posedness of viscous nonlinear wave equations (vNLW) on the two-dimensional torus with a stochastic forcing or randomized initial data. In particular, we prove pathwise global well-posedness of the stochastic defocusing vNLW with an additive stochastic forcing D α ξ, where α < 1 2 and ξ denotes the space-time white noise. We also study the deterministic vNLW with randomized initial data and prove its almost sure global well-posedness in the defocusing case. Contents 1. Introduction 1.1. Viscous nonlinear wave equations 1.2. SvNLW with an additive stochastic forcing 1.3. Deterministic vNLW with randomized initial data 2. Preliminary lemmas 2.1. Sobolev spaces and Besov spaces 2.2. On the stochastic term 2.3. Linear estimates 2.4. Probabilistic estimates 3. Local well-posedness of SvNLW 4. Almost sure local well-posedness of vNLW with randomized initial data 5. Global well-posedness of SvNLW 5.1. Case 1 < p ≤ 3 5.2. Case 3 < p ≤ 5 5.3. Case p > 5 6. Almost sure global well-posedness of vNLW with randomized initial data Appendix A. On local well-posedness of subcritical vNLW A.1. The inhomogeneous Strichartz estimates A.2. Local well-posedness of subcritical vNLW References

No abstract