Strong Solutions for Stochastic Partial Differential Equations of Gradient Type

TL;DR

This paper proves the unique existence and regularity of strong solutions for a class of stochastic partial differential equations with quasi-convex drift and multiplicative noise, introducing a novel weighted Galerkin approximation method.

Contribution

It introduces a new weighted Galerkin approximation method based on a quasi-convex function's distance, providing a unified framework for various stochastic PDEs.

Findings

Existence and uniqueness of strong solutions established.

Higher regularity of solutions demonstrated.

Applicable to multiple classes of stochastic PDEs.

Abstract

Unique existence of analytically strong solutions to stochastic partial differential equations (SPDE) with drift given by the subdifferential of a quasi-convex function and with general multiplicative noise is proven. The proof applies a genuinely new method of weighted Galerkin approximations based on the "distance" defined by the quasi-convex function. Spatial regularization of the initial condition analogous to the deterministic case is obtained. The results yield a unified framework which is applied to stochastic generalized porous media equations, stochastic generalized reaction diffusion equations and stochastic generalized degenerated p-Laplace equations. In particular, higher regularity for solutions of such SPDE is obtained.