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

Hongjie Dong; Hyunwoo Kwon

doi:10.3934/dcds.2025092

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2026, Volume 46: 80-113. Doi: 10.3934/dcds.2025092

This volume Previous Article Boundary estimates for a fully nonlinear Yamabe problem on Riemannian manifolds Next Article Infinitely many normalized solutions of $ L^2 $-supercritical NLS equations on noncompact metric graphs with localized nonlinearities

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

Hongjie Dong and
Hyunwoo Kwon^,

Division of Applied Mathematics, Brown University, 182 George Street, Providence, RI 02912, USA

^*Corresponding author: Hyunwoo Kwon
^*Corresponding author: Hyunwoo Kwon

Received: March 2025

Revised: May 2025

Early access: June 19, 2025

Published online: June 19, 2025

Both authors are partially supported by the NSF under agreement DMS-2350129.

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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.

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Mathematics Subject Classification: 76D03, 76D07, 35K51, 35B45.

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