The Navier-Stokes Equations with the Kinematic and Vorticity Boundary Conditions on Non-Flat Boundaries

TL;DR

This paper analyzes the Navier-Stokes equations with kinematic and vorticity boundary conditions on non-flat boundaries, establishing well-posedness and convergence to Euler solutions as viscosity tends to zero.

Contribution

It introduces a novel approach to handle nonhomogeneous boundary conditions for Navier-Stokes equations on non-flat domains, proving well-posedness and zero-viscosity limits.

Findings

Established well-posedness of the initial-boundary value problem.

Developed a velocity mapping with a priori estimates.

Proved convergence of Navier-Stokes solutions to Euler solutions as viscosity approaches zero.

Abstract

We study the initial-boundary value problem of the Navier-Stokes equations for incompressible fluids in a general domain in $\R^n$ with compact and smooth boundary, subject to the kinematic and vorticity boundary conditions on the non-flat boundary. We observe that, under the nonhomogeneous boundary conditions, the pressure $p$ can be still recovered by solving the Neumann problem for the Poisson equation. Then we establish the well-posedness of the unsteady Stokes equations and employ the solution to reduce our initial-boundary value problem into an initial-boundary value problem with absolute boundary conditions. Based on this, we first establish the well-posedness for an appropriate local linearized problem with the absolute boundary conditions and the initial condition (without the incompressibility condition), which establishes a velocity mapping. Then we develop \emph{apriori} estimates for the velocity mapping, especially involving the Sobolev norm for the time-derivative of the mapping to deal with the complicated boundary conditions, which leads to the existence of the fixed point of the mapping and the existence of solutions to our initial-boundary value problem. Finally, we establish that, when the viscosity coefficient tends zero, the strong solutions of the initial-boundary value problem in $\R^n (n\ge 3)$ with nonhomogeneous vorticity boundary condition converges in $L^2$ to the corresponding Euler equations satisfying the kinematic condition.