Spatial $C^1$, $C^2$, and Schauder estimates for nonstationary Stokes equations with Dini mean oscillation coefficients

We establish the spatial differentiability of weak solutions to nonstationary Stokes equations in divergence form with variable viscosity coefficients having $L_2$-Dini mean oscillations. As a corollary, we derive local spatial Schauder estimates for such equations if the viscosity coefficient belongs to $C_x^{\alpha }$. Similar results also hold for strong solutions to nonstationary Stokes equations in nondivergence form.