On higher regularity of Stokes systems with piecewise H\"{o}lder continuous coefficients

TL;DR

This paper establishes higher regularity results for solutions to stationary Stokes systems with piecewise Hölder continuous coefficients, showing that derivatives and pressure are also piecewise Hölder continuous under certain smoothness assumptions.

Contribution

It proves new higher regularity results for Stokes systems with piecewise Hölder coefficients, even in the 2D constant coefficient case.

Findings

Solutions have piecewise $C^{s,eta}$ regularity for derivatives and pressure.

Regularity depends on the smoothness of coefficients and interfaces.

Results apply to systems with multiple subdomains and complex interfaces.

Abstract

In this paper, we consider higher regularity of a weak solution $({\bf u},p)$ to stationary Stokes systems with variable coefficients. Under the assumptions that coefficients and data are piecewise $C^{s,\delta}$ in a bounded domain consisting of a finite number of subdomains with interfacial boundaries in $C^{s+1,\mu}$, where $s$ is a positive integer, $\delta\in (0,1)$, and $\mu\in (0,1]$, we show that $D{\bf u}$ and $p$ are piecewise $C^{s,\delta_{\mu}}$, where $\delta_{\mu}=\min\big\{\frac{1}{2},\mu,\delta\big\}$. Our result is new even in the 2D case with piecewise constant coefficients.