Speaker: Tuoc Phan, UTK

Title: Local Lebesgue mixed-norm estimates for non-stationary Stokes systems with measurable coefficients and applications

Abstract: We discuss about non-stationary Stokes systems with singular measurable coefficients. Local interior Lebesgue mixed norm regularity estimates are established and proved under a smallness condition on the mean oscillations of the coefficients. The results imply Caccioppoli type inequalities for the Stokes systems with variable coefficients and they are new even when the coefficients are constants. 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. In addition, some results on local boundary estimates are also presented and the optimality of the smallness condition on the mean oscillations of the coefficients will be addressed if time permitted. The talk is based on several separated papers which are joint work with Hongjie Dong (Brown University), Juraj Foldes (University of Virginia), and Doyoon Kim (Korea University).