Interior and boundary mixed norm derivative estimates for nonstationary Stokes equations

TL;DR

This paper establishes weighted mixed norm Sobolev estimates for nonstationary Stokes equations with variable viscosity, providing interior and boundary derivative estimates under minimal regularity assumptions.

Contribution

It introduces new weighted mixed norm Sobolev estimates for nonstationary Stokes equations with measurable time coefficients and small mean oscillation in space, including boundary estimates.

Findings

Weighted mixed norm Sobolev estimates in the whole space.

Interior mixed norm derivative estimates for solutions.

Boundary mixed norm Hessian estimates under Lions boundary conditions.

Abstract

We obtain weighted mixed norm Sobolev estimates in the whole space for nonstationary Stokes equations in divergence and nondivergence form with variable viscosity coefficients that are merely measurable in time variable and have small mean oscillation in spatial variables in small cylinders. As an application, we prove interior mixed norm derivative estimates for solutions to both equations. We also discuss boundary mixed norm Hessian estimates for solutions to equations in nondivergence form under the Lions boundary conditions.