GRADIENT ESTIMATES FOR STOKES AND NAVIER-STOKES SYSTEMS WITH PIECEWISE DMO COEFFICIENTS

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 sub domains with C1,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 H1-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 div(\tau Su), where Su = 21 (Du + (Du)T) is the strain tensor and \tau is a positive piecewise DMO scalar function.