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We study the stationary Stokes system with Dini mean oscillation coefficients in a domain having $C^{1,rm{Dini}}$ boundary. We prove that if $(u, p)$ is a weak solution of the system with zero Dirichlet boundary condition, then $(Du,p)$ is continuous up to the boundary. We also prove a weak type-$(1,1)$ estimate for $(Du, p)$.
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We prove the existence and pointwise bounds of the Green functions for stationary Stokes systems with measurable coefficients in two dimensional domains. We also establish pointwise bounds of the derivatives of the Green functions under a regularity assumption on the $L_1$-mean oscillations of the coefficients.
We study stationary Stokes systems in divergence form with piecewise Dini mean oscillation coefficients and data in a bounded domain containing a finite number of subdomains with $C^{1,rm{Dini}}$ boundaries. We prove that if $(u, p)$ is a weak solution of the system, then $(Du, p)$ is bounded and piecewise continuous. The corresponding results for stationary Navier-Stokes systems are also established, from which the Lipschitz regularity of the stationary $H^1$-weak solution in dimensions $d=2,3,4$ is obtained.
We study the stationary Stokes system in divergence form. The coefficients are assumed to be merely measurable in one direction and have Dini mean oscillations in the other directions. We prove that if $(u,p)$ is a weak solution of the system, then $(Du,p)$ is bounded and its certain linear combinations are continuous. We also prove a weak type-$(1,1)$ estimate for $(Du,p)$ under a stronger assumption on the $L^1$-mean oscillation of the coefficients. The corresponding results up to the boundary on a half ball are also established. These results are new even for elliptic equations and systems.
This paper is build around the stationary anisotropic Stokes and Navier-Stokes systems with an $L^infty$-tensor coefficient satisfying an ellipticity condition in terms of symmetric matrices in ${mathbb R}^{ntimes n}$ with zero matrix traces. We analyze, in $L^2$-based Sobolev spaces, the non-homogeneous boundary value problems of Dirichlet-transmission type for the anisotropic Stokes and Navier-Stokes systems in a compressible framework in a bounded Lipschitz domain with a Lipschitz interface in ${mathbb R}^n$, $nge 2$ ($n=2,3$ for the nonlinear problems). The transversal interface intersects the boundary of the Lipschitz domain. First, we use a mixed variational approach to prove well-posedness results for the linear anisotropic Stokes system. Then we show the existence of a weak solution for the nonlinear anisotropic Navier-Stokes system by implementing the Leray-Schauder fixed point theorem and using various results and estimates from the linear case, as well as the Leray-Hopf and some other norm inequalities. Explicit conditions for uniqueness of solutions to the nonlinear problems are also provided.
We study boundary gradient estimates for second-order divergence type parabolic and elliptic systems in $C^{1,alpha}$ domains. The coefficients and data are assumed to be Holder in the time variable and all but one spatial variables. This type of systems arises from the problems of linearly elastic laminates and composite materials.
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