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Register a new userFundamental solutions for stationary Stokes systems with measurable coefficients
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We establish the existence and the pointwise bound of the fundamental solution for the stationary Stokes system with measurable coefficients in the whole space $mathbb{R}^d$, $d ge 3$, under the assumption that weak solutions of the system are locally Holder continuous. We also discuss the existence and the pointwise bound of the Green function for the Stokes system with measurable coefficients on $Omega$, where $Omega$ is an unbounded domain such that the divergence equation is solvable. Such a domain includes, for example, half space and an exterior domain.
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We study the stationary Stokes system with variable coefficients in the whole space, a half space, and on bounded Lipschitz domains. In the whole and half spaces, we obtain a priori $dot W^1_q$-estimates for any $qin [2,infty)$ when the coefficients are merely measurable functions in one fixed direction. For the system on bounded Lipschitz domains with a small Lipschitz constant, we obtain a $W^1_q$-estimate and prove the solvability for any $qin (1,infty)$ when the coefficients are merely measurable functions in one direction and have locally small mean oscillations in the orthogonal directions in each small ball, where the direction is allowed to depend on the ball.
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 of the spatial second-order derivatives of solutions hold, assuming the smallness of the mean oscillations of the coefficients with respect to the spatial variables in small cylinders. In the un-mixed norm case with $s=q=2$, the result is still new and provides local boundary Caccioppoli-type estimates, which are important in applications. The main challenges in the work arise from the lack of regularity of the pressure and time derivatives of the solutions and from interaction of the boundary with the nonlocal structure of the system. To overcome these difficulties, our approach relies heavily on several newly developed regularity estimates for parabolic equations with coefficients that are only measurable in the time variable and in one of the spatial variables.
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 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 small mean oscillations in the other directions. Using this result, we establish the unique solvability in Muckenhoupt type weighted Sobolev spaces for the system with partially BMO coefficients on a Reifenberg flat domain. We also present weighted a priori $L_q$-estimates for the system when the domain is the whole Euclidean space or a half space.
Auscher, McIntosh and Tchamitchian studied the heat kernels of second order elliptic operators in divergence form with complex bounded measurable coefficients on $mathbb{R}^n$. In particular, in the case when $n=2$ they obtained Gaussian upper bound estimates for the heat kernel without imposing further assumption on the coefficients. We study the fundamental solutions of the systems of second order parabolic equations in the divergence form with bounded, measurable, time-independent coefficients, and extend their results to the systems of parabolic equations.
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