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.