On $W^{1,p}$-regularity estimate for a class of generalized Stokes systems and its applications to the Navier-Stokes equations

TL;DR

This paper proves weighted regularity estimates for generalized Stokes systems with discontinuous coefficients, including singular skew-symmetric parts, and applies these results to establish criteria for the global regularity of Navier-Stokes solutions.

Contribution

It introduces new weighted $W^{1,p}$-regularity estimates for Stokes systems with discontinuous and singular coefficients, advancing understanding of Navier-Stokes regularity.

Findings

Weighted Calderón-Zygmund estimates established

Criteria for Navier-Stokes regularity derived

Handles singular skew-symmetric coefficients

Abstract

This paper establishes global weighted Calder\'on-Zygmund type regularity estimates for weak solutions of a class of generalized Stokes systems in divergence form. The focus of the paper is on the case that the coefficients in the divergence-form Stokes operator consist of symmetric and skew-symmetric parts, which are both discontinuous. Moreover, the skew-symmetric part is not required to be bounded and therefore it could be singular. Sufficient conditions on the coefficients are provided to ensure the global weighted $W^{1,p}$-regularity estimates for weak solutions of the systems. As a direct application, we show that our $W^{1,p}$-regularity results give some criteria in critical spaces for the global regularity of weak Leray-Hopf solutions of the Navier-Stokes system of equation