Boundary Lebesgue mixed-norm estimates for non-stationary Stokes systems with VMO coefficients

TL;DR

This paper establishes new boundary mixed-norm estimates for non-stationary Stokes systems with VMO coefficients, addressing challenges from irregular coefficients and boundary interactions, and extending regularity results.

Contribution

It provides the first local boundary mixed-norm estimates for second derivatives of solutions under Lions-type boundary conditions with VMO coefficients.

Findings

Boundary mixed-norm $L_{s,q}$-estimates for second derivatives

Local boundary Caccioppoli-type estimates for $s=q=2$

New regularity estimates for parabolic equations with measurable coefficients

Abstract

We consider Stokes systems with measurable coefficients and Lions-type boundary conditions. We show that, in contrast to the Dirichlet boundary conditions, local boundary mixed-norm $L_{s,q}$-estimates hold for the spatial second-order derivatives of solutions, assuming the smallness of the mean oscillations of the coefficients with respect to the spatial variables in small cylinders. In the un-mixed norm case with $s=q=2$, the result is still new and provides local boundary Caccioppoli-type estimates. The main challenges in the work arise from the lack of regularity of the pressure and time derivatives of the solutions and from interaction of the boundary with the nonlocal structure of the system. To overcome these difficulties, our approach relies heavily on several newly developed regularity estimates for both divergence and non-divergence form parabolic equations with coefficients that are only measurable in the time variable and in one of the spatial variables.