Acta mathematica scientia, Series B >
THE NAVIER-STOKES EQUATIONS WITH THE KINEMATIC AND VORTICITY BOUNDARY CONDITIONS ON NON-FLAT BOUNDARIES
Received date: 2008-12-31
Online published: 2009-07-20
Supported by
Gui-Qiang Chen’s research was supported in part by the National Science Foundation under Grants DMS-0807551, DMS-0720925, and DMS-0505473, and the Natural Science Foundation of China (10728101). Zhongmin Qian’s research was supported in part by EPSRC grant EP/F029578/1
We study the initial-boundary value problem of the Navier-Stokes equations for incompressible fluids in a general domain in Rn 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 Rn(n ≥ 3) with nonhomogeneous vorticity boundary condition converge in L2 to the corresponding Euler equations satisfying the kinematic condition.
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 . DOI: 10.1016/S0252-9602(09)60078-3
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