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

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 type inequalities 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. (C) 2020 Elsevier Inc. All rights reserved.

Automated summary

The article proves mixed-norm Sobolev estimates for time-dependent Stokes systems with unbounded measurable coefficients having small mean oscillations in small cylinders, which also implies Caccioppoli type inequalities for Stokes systems with variable coefficients and a new ε-regularity criterion for Leray-Hopf weak solutions of Navier-Stokes equations. This new criterion further implies some borderline cases of the well-known Serrin's regularity criterion.