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.


  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


Published In

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


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


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

MSC codes

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



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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