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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 Caccioppolis 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 Serrins regularity criterion.
We consider Stokes systems in non-divergence form with measurable coefficients and Lions-type boundary conditions. We show that for the Lions conditions, in contrast to the Dirichlet boundary conditions, local boundary mixed-norm $L_{s,q}$-estimates
We study a class of elliptic and parabolic equations in non-divergence form with singular coefficients in an upper half space with the homogeneous Dirichlet boundary condition. Intrinsic weighted Sobolev spaces are found in which the existence and un
In this paper, we study both elliptic and parabolic equations in non-divergence form with singular degenerate coefficients. Weighted and mixed-norm $L_p$-estimates and solvability are established under some suitable partially weighted BMO regularity
This paper is a comprehensive study of $L_p$ estimates for time fractional wave equations of order $alpha in (1,2)$ in the whole space, a half space, or a cylindrical domain. We obtain weighted mixed-norm estimates and solvability of the equations in
We prove the unique solvability of solutions in Sobolev spaces to the stationary Stokes system on a bounded Reifenberg flat domain when the coefficients are partially BMO functions, i.e., locally they are merely measurable in one direction and have s