Decay Estimates for Steady Solutions of the Navier--Stokes Equations in Two Dimensions in the Presence of a Wall

Christoph Boeckle, Peter Wittwer
2012 SIAM Journal on Mathematical Analysis  
Let ω be the vorticity of a stationary solution of the two-dimensional Navier-Stokes equations with a drift term parallel to the boundary in the half-plane Ω + = {(x, y) ∈ R 2 | y > 1}, with zero Dirichlet boundary conditions at y = 1 and at infinity, and with a small force term of compact support. Then |xyω(x, y)| is uniformly bounded in Ω + . The proof is given in a specially adapted functional framework, and the result is a key ingredient for obtaining information on the asymptotic behavior
more » ... f the velocity at infinity. 3347 is then based on a new linear fixed point problem involving the solution obtained in [14] and the derivative of the vorticity with respect to k. Since the original equation is elliptic, the dynamical system under consideration contains stable and unstable modes and no spectral gap, so that standard versions of the center manifold theorem are not sufficient to prove existence of solutions. Functional techniques that allow one to deal with such a situation go back to [6] and were adapted to the case of the Navier-Stokes equations in [16] and in [17] , [18] . For a general review, see [11] . The linearized version of the current problem was studied in [13] . A related problem in three dimensions was discussed in [9] . The results of the present paper are the basis for the work described in [2], where we extract several orders of an asymptotic expansion of the vorticity and the velocity field at infinity. The asymptotic velocity field obtained this way is divergence-free and may be used to define artificial boundary conditions of Dirichlet type when the system of equations is restricted to a finite subdomain to be solved numerically. The use of asymptotic terms as artificial boundary conditions was pioneered in [3], [4] for the related problem of an exterior flow in the whole space in two dimensions, and in [10] for the case in three dimensions. Let x = (x, y), and let Ω + = {(x, y) ∈ R 2 | y > 1}. The model under consideration is given by the Navier-Stokes equations with a drift term parallel to the boundary,
doi:10.1137/110852565 fatcat:mitfp4itzbfozcoqrexmdnjrti