On the vorticity of the Oseen problem in a half plane

We derive the equation for the vorticity of the incompressible Oseen problem in a half plane with homogeneous (no slip) boundary conditions. The resulting equation is a scalar Oseen equation with certain Dirichlet boundary conditions which are determined by the incompressibility condition and the boundary conditions of the original problem. We prove existence and uniqueness of solutions for this equation in function spaces that provide detailed information on the asymptotic behavior of the solution. We show that, in contrast to the Oseen problem in the whole space where the vorticity decays exponentially fast outside the wake region, the vorticity only decays algebraically in the present case. This algebraic decay is however faster than what one would expect for a generic problem, since the dominant volume and boundary contributions cancel each other as a consequence of the incompressibility and the no slip boundary conditions of the original problem. Contents 1 Introduction and main results 2 2 The vorticity boundary condition 4 3 Solution for data with compact support 6 3.1 Construction of a solution ! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3.2 Expected asymptotic behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 4 Formulation of results in function spaces 9 4.1 The space _ W 1;p (R+) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 4.2 Exact formulation of main theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 4.3 Uniqueness of solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 5 Technical lemmas 13 5.1 Volume terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 5.2 Boundary terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 6 Proof of Theorem 12 17 6.1 Proof of Lemma 23. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 6.2 Proof of Lemma 24. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 7 Proof of Theorem 13 20 7.1 Proof of Lemma 25 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 7.2 Proof of Lemma 26 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 7.3 Proof of Lemma 27 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Supported in part by the Fonds National Suisse de la recherche Scienti…que.