Gradient estimates for Stokes and Navier-Stokes systems with piecewise DMO coefficients

TL;DR

This paper establishes gradient estimates and regularity results for stationary Stokes and Navier-Stokes systems with piecewise Dini mean oscillation coefficients in bounded domains, demonstrating boundedness and piecewise continuity of solutions.

Contribution

It provides new gradient bounds and regularity results for Stokes and Navier-Stokes systems with piecewise Dini coefficients, including Lipschitz regularity in low dimensions.

Findings

Solutions have bounded and piecewise continuous gradients.

Lipschitz regularity of stationary solutions in dimensions 2, 3, and 4.

Extension of regularity results to systems with piecewise Dini mean oscillation coefficients.

Abstract

We study stationary Stokes systems in divergence form with piecewise Dini mean oscillation coefficients and data in a bounded domain containing a finite number of subdomains with $C^{1,\rm{Dini}}$ boundaries. We prove that if $(u, p)$ is a weak solution of the system, then $(Du, p)$ is bounded and piecewise continuous. The corresponding results for stationary Navier-Stokes systems are also established, from which the Lipschitz regularity of the stationary $H^1$-weak solution in dimensions $d=2,3,4$ is obtained.