A non-local approach to the generalized Stokes operator with bounded measurable coefficients

TL;DR

This paper develops a non-local framework for analyzing the Stokes operator with bounded measurable coefficients, establishing key properties like resolvent bounds and regularity estimates in $L^p$ spaces.

Contribution

It introduces a non-local approach to the Stokes operator with measurable coefficients, proving resolvent bounds, maximal regularity, and gradient estimates.

Findings

Established optimal resolvent bounds for the Stokes operator.

Proved maximal $L^q$-regularity for the operator.

Derived regularity estimates for solutions' gradients.

Abstract

We establish functional analytic properties of the Stokes operator with bounded measurable coefficients on $L^p_{\sigma} (\mathbb{R}^d)$, $d \geq 2$, for $\lvert 1 / p - 1 / 2 \rvert < 1 / d$. These include optimal resolvent bounds and the property of maximal $L^q$-regularity. We further give regularity estimates on the gradient of the solution to the Stokes resolvent problem with bounded measurable coefficients. As a key to these results we establish the validity of a non-local Caccioppoli inequality to solutions of the Stokes resolvent problem.