Gradient Estimates for Stokes and Navier--Stokes Systems with Piecewise DMO Coefficients
Jongkeun Choi, Hongjie Dong, Longjuan Xu · SIAM Journal on Mathematical Analysis · 2022
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.