$L_q$-estimates for stationary Stokes system with coefficients measurable in one direction

TL;DR

This paper establishes $L_q$-estimates for the stationary Stokes system with coefficients measurable in one direction, extending regularity results to variable coefficient cases in various domains.

Contribution

It provides new a priori estimates and solvability results for the Stokes system with coefficients measurable in one direction, in whole space, half space, and bounded Lipschitz domains.

Findings

A priori $ abla u$ estimates in whole and half spaces for $q o finite$

Solvability of the Stokes system on Lipschitz domains with small oscillation coefficients

Extension of regularity theory to coefficients measurable in one direction

Abstract

We study the stationary Stokes system with variable coefficients in the whole space, a half space, and on bounded Lipschitz domains. In the whole and half spaces, we obtain a priori $\dot W^1_q$-estimates for any $q\in [2,\infty)$ when the coefficients are merely measurable functions in one fixed direction. For the system on bounded Lipschitz domains with a small Lipschitz constant, we obtain a $W^1_q$-estimate and prove the solvability for any $q\in (1,\infty)$ when the coefficients are merely measurable functions in one direction and have locally small mean oscillations in the orthogonal directions in each small ball, where the direction is allowed to depend on the ball.