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

TL;DR

This paper studies nonstationary Stokes equations with slip boundary conditions on curved domains, establishing new regularity and boundary estimates even with constant viscosity, advancing understanding of fluid flow near complex boundaries.

Contribution

It provides novel local regularity and boundary Hessian estimates for nonstationary Stokes equations with slip boundary conditions, including cases with shape operator slip tensor.

Findings

Established a priori local regularity estimates near curved boundaries.

Derived boundary Hessian estimates when slip tensor is the shape operator.

Results are new even for constant viscosity coefficients.

Abstract

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.