A $W^1_2$-theory of Stochastic Partial Differential Systems of Divergence type on $C^1$ domains

TL;DR

This paper develops a $W^1_2$-theory for stochastic divergence-type PDE systems on $C^1$ domains, establishing existence and uniqueness despite coefficients and derivatives potentially blowing up near the boundary.

Contribution

It introduces a Sobolev space framework with weights to handle solutions with boundary blow-up and allows measurable, boundary-blowing coefficients.

Findings

Established existence and uniqueness of solutions.

Extended theory to coefficients that blow up near the boundary.

Provided a weighted Sobolev space approach for boundary behavior.

Abstract

In this paper we study the stochastic partial differential systems of divergence type with $C^1$ space domains in $\bR^d$. Existence and uniqueness results are obtained in terms of Sobolev spaces with weights so that we allow the derivatives of the solution to blow up near the boundary. The coefficients of the systems are only measurable and are allowed to blow up near the boundary.