A $W^n_2$-Theory of Stochastic Parabolic Partial Differential Systems on $C^1$-domains

TL;DR

This paper develops a $W^n_2$-theory for stochastic parabolic PDE systems on $C^1$-domains, addressing existence, uniqueness, and regularity in weighted Sobolev spaces even with boundary blow-up and oscillating coefficients.

Contribution

It introduces a new $W^n_2$-theory for stochastic parabolic systems on $C^1$-domains, accommodating boundary blow-up and coefficient oscillations.

Findings

Established existence and uniqueness in weighted Sobolev spaces.

Allowed derivatives to blow up near the boundary.

Handled coefficients with significant oscillations or blow-up.

Abstract

In this article we present a $W^n_2$-theory of stochastic parabolic partial differential systems. In particular, we focus on non-divergent type. The space domains we consider are $\bR^d$, $\bR^d_+$ and eventually general bounded $C^1$-domains $\mathcal{O}$. By the nature of stochastic parabolic equations we need weighted Sobolev spaces to prove the existence and the uniqueness. In our choice of spaces we allow the derivatives of the solution to blow up near the boundary and moreover the coefficients of the systems are allowed to oscillate to a great extent or blow up near the boundary.