Abstract

We study stationary Stokes systems in divergence form with piecewise Dini mean oscillation (DMO) coefficients and data in a bounded domain containing a finite number of subdomains with $C^{1,{Dini}}$ 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. Our results can be applied to stationary Stokes systems and Navier--Stokes systems with the second-order term $\operatorname{div} (\tau \mathcal{S}u)$, where $\mathcal{S}u=\frac{1}{2}(Du+(Du)^{\top})$ is the strain tensor and $\tau$ is a positive piecewise DMO scalar function.

Keywords

  1. Stokes system
  2. piecewise Dini mean oscillation
  3. gradient estimate

MSC codes

  1. 76D07
  2. 35B65
  3. 35J47

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Information & Authors

Information

Published In

cover image SIAM Journal on Mathematical Analysis
SIAM Journal on Mathematical Analysis
Pages: 3609 - 3635
ISSN (online): 1095-7154

History

Submitted: 1 June 2021
Accepted: 25 February 2022
Published online: 16 June 2022

Keywords

  1. Stokes system
  2. piecewise Dini mean oscillation
  3. gradient estimate

MSC codes

  1. 76D07
  2. 35B65
  3. 35J47

Authors

Affiliations

Funding Information

National Research Foundation of Korea https://doi.org/10.13039/501100003725 : NRF-2019R1F1A1058826
Simons Foundation https://doi.org/10.13039/100000893 : 709545

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