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

• Published in: Acta mathematica scientia Vol. 29; no. 4; pp. 919 - 948
• Main Authors: Chen, Gui-Qiang, Osborne, Dan, Qian, Zhongmin
• Format: Journal Article
• Language: English
• Published: Elsevier Ltd 01.07.2009; School of Mathematical Sciences, Fudan University, Shanghai 200433, China; Department of Mathematics, Northwestern University, Evanston, IL 60208-2730, USA%Mathematical Institute, University of Oxford, 24-29 St Giles, Oxford OX1 3LB, UK
• Subjects: 35A35; 35B35; 35P05; 35Q30; 36M60; 58C35; 76D03; 76D05; 76D07; absolute boundary condition; Apriori; general domain; incompressible; inviscid limit; kinematic boundary condition; Navier-Stokes equations; Neumann problem; Neumann问题; non-flat boundary; Poisson equation; Poisson方程; slip boundary condition; Stokes operator; strong solutions; vorticity; vorticity boundary condition; 初边值问题; 定常Stokes方程; 涡量边界条件; 运动学; 非齐次边界条件; kinematic boundary condition; absolute boundary condition; Neumann problem; general domain; vorticity; 35B35; 35A35; 58C35; 76D03; Poisson equation; slip boundary condition; 76D05; 35Q30; incompressible; 76D07; 36M60; inviscid limit; 35P05; vorticity boundary condition; strong solutions; Stokes operator; Navier-Stokes equations; non-flat boundary; kine-matic boundary condition; invisci; vor-ticity
• ISSN: 0252-9602; 1572-9087
• DOI: 10.1016/S0252-9602(09)60078-3

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 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 ≥ 3） with nonhomogeneous vorticity boundary condition converge in L^2 to the corresponding Euler equations satisfying the kinematic condition.