Interior $L_{p}$ regularity for Stokes systems

TL;DR

This paper introduces an iteration method to establish interior $L_{p}$ regularity for Stokes systems, enhancing the integrability of solutions' derivatives through a step-by-step process that leverages maximal function techniques.

Contribution

It develops a novel iterative approach for $L_{p}$ regularity of Stokes systems, accommodating coefficients with Hölder continuity in space.

Findings

Achieves $L_{p}$ regularity for Stokes systems in divergence and non-divergence forms.

Uses maximal function method to handle solutions and derivatives simultaneously.

Improves integrability of derivatives iteratively until $L_{p}$ regularity is obtained.

Abstract

A new iteration method is represented to study the interior $L_{p}$ regularity for Stokes systems both in divergence form and in non-divergence form. By the iteration, we improve the integrability of derivatives of solutions for Stokes systems step by step; after infinitely many steps, $L_{p}$ regularity is achieved; and in each step, the maximal function method is used where solutions and their derivatives are involved simultaneously in each scale. The H\"{o}lder continuity of the coefficients in spatial variables is assumed to compensate the different scalings between the solutions and their derivatives.