Weighted estimates for generalized steady Stokes systems in nonsmooth domains

TL;DR

This paper establishes weighted $L^q$ estimates for the gradient and pressure of solutions to generalized steady Stokes systems with discontinuous coefficients in nonsmooth domains, extending classical results to more complex settings.

Contribution

It provides the first global weighted $L^q$ estimates for the Stokes system with discontinuous coefficients in nonsmooth domains under minimal regularity assumptions.

Findings

Proved global weighted $L^q$ estimates for the gradient and pressure.

Extended Calderón-Zygmund estimates to nonsmooth domains with discontinuous coefficients.

Demonstrated applicability to Muckenhoupt weights and Reifenberg flat domains.

Abstract

We consider a generalized steady Stokes system with discontinuous coefficients in a nonsmooth domain when the inhomogeneous term belongs to a weighted $L^q$ space for $2<q<\infty$. We prove the global weighted $L^q$-estimates for the gradient of the weak solution and an associated pressure under the assumptions that the coefficients have small BMO (bounded mean oscillation) semi-norms and the domain is sufficiently flat in the Reifenberg sense. On the other hand, a given weight is assumed to belong to a Muckenhoupt class. Our result generalizes the global $W^{1.q}$ estimate of Calder\'{o}n-Zygmund with respect to the Lebesgue measure for the Stokes system in a Lipschitz domain.