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

H. Dong; T. Phan

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arXiv · 1805.04143

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

H. Dong·T. Phan

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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.

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H. Dong, T. Phan. 2018-05-10. Mixed-norm $L_p$-estimates for non-stationary Stokes systems with singular VMO coefficients and applications. https://arxiv.org/abs/1805.04143

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