Optimal regularity in time and space for stochastic porous medium equations

Abstract

In this thesis, we study the regularity of the solutions to a particular class of degenerate quasilinear stochastic partial differential equations. These equations, known as stochastic porous medium equations, model nonlinear diffusion dynamics perturbed by noise appearing in a variety of problems in population dynamics and gas and water propagation.

For this class of equations, we obtain two optimal regularity results: the first for the spatial regularity and the second for the time regularity. The optimality for the first result is argued by comparing it with the optimal spatial regularity obtained for the deterministic porous medium equation in Gess [35] and Gess, Sauer and Tadmor [37]. This is the first regularity result which gives more than one spatial derivative for any degenerate quasilinear stochastic partial differential equation. The time differentiability that we obtain is consistent with the presence of the temporal white noise. The optimality of the time integrability is shown using a scaling argument for the deterministic porous medium equation.

At the end of this thesis, we detail possible directions for future work using the techniques and tools developed in the proofs of these optimal regularity results.

Date of Award	23 Mar 2022
Original language	English
Awarding Institution	University of Bath
Supervisor	Hendrik Weber (Supervisor) & Manuel Del Pino (Supervisor)