Nonstationary Stokes equations on a domain with curved boundary under slip boundary conditions

We consider nonstationary Stokes equations in nondivergence form with variable viscosity coefficients and generalized Navier slip boundary conditions with slip tensor $ \mathcal{A} $ in a domain $ \Omega $ in $ \mathbb{R}^d $. First, under the assumption that slip matrix $ \mathcal{A} $ is sufficiently smooth, we establish a priori local regularity estimates for solutions near a curved portion of the domain boundary. Second, when $ \mathcal{A} $ is the shape operator, we derive local boundary estimates for the Hessians of the solutions, where the right-hand side does not involve the pressure. Notably, our results are new even if the viscosity coefficients are constant.