THE NAVIER-STOKES EQUATIONS WITH THE KINEMATIC AND VORTICITY BOUNDARY CONDITIONS ON NON-FLAT BOUNDARIES

doi:10.1016/S0252-9602(09)60078-3

Abstract

Abstract:

We study the initial-boundary value problem of the Navier-Stokes equations for incompressible fluids in a general domain in Rⁿwith 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 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 Rⁿ(n ≥ 3) with nonhomogeneous vorticity boundary condition converge in L² to the corresponding Euler equations satisfying the kinematic condition.

Key words: Navier-Stokes equations, incompressible, vorticity boundary condition, kine-matic boundary condition, absolute boundary conditio, non-flat boundary, general domain, Stokes operator, Neumann problem, Poisson equation, vor-ticity, strong solutions, inviscid limit, slip boundary condition

CLC Number:

35Q30

Gui-Qiang Chen, Dan Osborne, Zhongmin Qian. THE NAVIER-STOKES EQUATIONS WITH THE KINEMATIC AND VORTICITY BOUNDARY CONDITIONS ON NON-FLAT BOUNDARIES[J].Acta mathematica scientia,Series B, 2009, 29(4): 919-948.

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