Weighted estimates for generalized steady Stokes systems in nonsmooth 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.