A regularity theory for quasi-linear Stochastic Partial Differential Equations in weighted Sobolev spaces

TL;DR

This paper develops a regularity theory for second-order quasi-linear SPDEs on $C^1$ domains, establishing existence, uniqueness, and estimates for solutions and their gradients in weighted Sobolev spaces.

Contribution

It introduces a novel regularity framework for quasi-linear SPDEs in weighted Sobolev spaces, including existence, uniqueness, and a priori estimates.

Findings

Proved existence and uniqueness of solutions in weighted Sobolev spaces.

Established $L_p$ and H"older estimates for solutions and their gradients.

Extended regularity theory to quasi-linear SPDEs on $C^1$ domains.

Abstract

We study the second-order quasi-linear stochastic partial differential equations (SPDEs) defined on $C^1$ domains. The coefficients are random functions depending on $t,x$ and the unknown solutions. We prove the uniqueness and existence of solutions in appropriate Sobolev spaces, and in addition, we obtain $L_p$ and H\"older estimates of both the solution and its gradient.