On time regularity of stochastic evolution equations with monotone coefficients

TL;DR

This paper establishes fractional Sobolev type time regularity results up to order 1/2 for solutions of stochastic evolution equations with monotone coefficients, aiding numerical analysis and complementing existing regularity results.

Contribution

It provides new time regularity estimates for stochastic PDEs with monotone coefficients, especially in the borderline case, enhancing understanding of solution regularity.

Findings

Achieves fractional Sobolev regularity of order up to 1/2 for certain functionals of solutions.

Extends regularity results to stochastic convolution solutions in the linear case.

Supports optimal convergence rates for time discretization schemes.

Abstract

We report on a time regularity result for stochastic evolutionary PDEs with monotone coefficients. If the diffusion coefficient is bounded in time without additional space regularity we obtain a fractional Sobolev type time regularity of order up to $\tfrac{1}{2}$ for a certain functional $ G( u )$ of the solution. Namely, $ G( u )=\nabla u $ in the case of the heat equation and $G( u )=|\nabla u |^{\frac{p-2}{2}}\nabla u $ for the $p$-Laplacian. The motivation is twofold. On the one hand, it turns out that this is the natural time regularity result that allows to establish the optimal rates of convergence for numerical schemes based on a time discretization. On the other hand, in the linear case, i.e. where the solution is given by a stochastic convolution, our result complements the known stochastic maximal space-time regularity results for the borderline case not covered by other methods.