The boundary value problems for the scalar Oseen equation

Abstract

The scalar Oseen equation represents a linearized form of the Navier Stokes equations, well‐known in hydrodynamics. In the present paper we develop an explicit potential theory for this equation and solve the interior and exterior Oseen Dirichlet and Oseen Neumann boundary value problems via a boundary integral equation method in spaces of continuous functions on a C2‐boundary, extending the classical approach for the isotropic selfadjoint Laplace operator to the anisotropic non‐selfadjoint scalar Oseen operator. It turns out that the solution to all boundary value problems can be presented by boundary potentials with source densities constructed as uniquely determined solutions of boundary integral equations.

The exterior Dirichlet and the interior Neumann boundary value problems for the scalar Oseen equation

“…Analogously to the classical potential theory we can prove the existence of the direct values of Eϕ, Dϕ, H * ϕ on the boundary. Moreover, the Oseen potentials have the same continuity behaviour and jump relations as the classical Laplace potentials (see [3][4][5]). …”

“…With help of the jump relations of Dϕ and H * ϕ (see [3], [4]) we can reduce (ED) and (IN) to the following boundary integral equations:…”

“…The unique solution u ∈ C 2 (Ω) ∩ C(Ω) with ∂ n u ∈ C(Ω) to (IN) is given by the single layer potential E * ϕ, where ϕ ∈ C(∂Ω) is the uniquely determined solution to the boundary integral equation (3).…”

The Robin problem for the scalar Oseen equation

“…have been studied by the method of integral equations [8,9]. Here, the authors study the Dirichlet problem, that is, they prescribe the boundary condition u D g on @ , and the Oseen Neumann problem, prescribing the boundary condition @u @n n 1 u D g on @ .…”

“…where h denotes a positive function, and the Robin problem corresponding to the Oseen Neumann condition studied in [8,9], that is, we prescribe the boundary condition @u @n…”