Spatial $C^1$, $C^2$, and Schauder estimates for nonstationary Stokes equations with Dini mean oscillation coefficients

TL;DR

This paper proves spatial differentiability and Schauder estimates for solutions to nonstationary Stokes equations with variable viscosity coefficients exhibiting Dini mean oscillations, advancing understanding of regularity under minimal smoothness conditions.

Contribution

It establishes new regularity results for nonstationary Stokes equations with Dini mean oscillation coefficients, including Schauder estimates under minimal smoothness assumptions.

Findings

Spatial differentiability of weak solutions proven.

Local Schauder estimates derived for coefficients in $C^\alpha_x$.

Results extend to nondivergence form equations.

Abstract

We establish the spatial differentiability of weak solutions to nonstationary Stokes equations in divergence form with variable viscosity coefficients having $L_2$-Dini mean oscillations. As a corollary, we derive local spatial Schauder estimates for such equations if the viscosity coefficient belongs to $C^\alpha_x$. Similar results also hold for strong solutions to nonstationary Stokes equations in nondivergence form.