Mixed-norm $L_p$-estimates for non-stationary Stokes systems with singular VMO coefficients and applications

TL;DR

This paper establishes mixed-norm Sobolev estimates for non-stationary Stokes systems with VMO coefficients, leading to new regularity criteria for Navier-Stokes solutions and extending classical results to variable coefficient settings.

Contribution

It provides the first mixed-norm estimates for Stokes systems with VMO coefficients and derives a novel epsilon-regularity criterion for Navier-Stokes equations.

Findings

Mixed-norm Sobolev estimates for Stokes systems with VMO coefficients.

Caccioppoli-type estimates for variable coefficient Stokes systems.

New epsilon-regularity criterion for Leray-Hopf solutions of Navier-Stokes.

Abstract

We prove the mixed-norm Sobolev estimates for solutions to both divergence and non-divergence form time-dependent Stokes systems with unbounded measurable coefficients having small mean oscillations with respect to the spatial variable in small cylinders. As a special case, our results imply Caccioppoli's type estimates for the Stokes systems with variable coefficients. A new $\epsilon$-regularity criterion for Leray-Hopf weak solutions of Navier-Stokes equations is also obtained as a consequence of our regularity results, which in turn implies some borderline cases of the well-known Serrin's regularity criterion.