Dynamics of Partial Differential Equations

Volume 16 (2019)

Number 1

Gradient estimates for Stokes systems in domains

Pages: 1 – 24

DOI: http://dx.doi.org/10.4310/DPDE.2019.v16.n1.a1

Authors

Jongkeun Choi (Division of Applied Mathematics, Brown University, Providence, Rhode Island, U.S.A.)

Hongjie Dong (Division of Applied Mathematics, Brown University, Providence, Rhode Island, U.S.A.)

Abstract

We study the stationary Stokes system with Dini mean oscillation coefficients in a domain having $C^{1,\mathrm{Dini}}$ 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 $(Du, p)$.

Keywords

Stokes system, Dini mean oscillation condition, $C^1$-estimate, weak type-$(1,1)$ estimate

2010 Mathematics Subject Classification

35B65, 35J57, 76D07

Full Text (PDF format)

J. Choi was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2017R1A6A3A03005168). H. Dong was partially supported by the NSF under agreement DMS-1600593.

Received 3 May 2018

Published 5 December 2018