On the Stationary Oseen Equations with Nonvanishing Divergence in Exterior Domains of IR 3

Abstract

Consider the Navier-Stokes equations describing the flow of an incompressible and viscous fluid past one or several obstacles in IR 3 such that the velocity converges to a vector u ∞, at infinity. In the most interesting case where u ∞≠0 we know by experiments that the flow gets turbulent in a region behind the obstacle, in the so-called wake. Linearizing the Navier-Stokes equations in the velocity field with respect to the constant vector u ∞ = k(1, 0, 0), k > 0, we get the Oseen equations[10] $$\left\{ {\begin{array}{*{20}{c}} \hfill { - \nu \Delta u + k{{\partial }_{1}}u + \nabla p = f in \Omega } \\ \hfill {div u = g in \Omega } \\ \hfill {u = 0 on \partial \Omega } \\ \end{array} } \right.$$ (1) where u(x) → 0 at infinity. Here Ω ⊂ ℝ3 denotes an exterior domain with boundary ∂Ω of class C 2, and ∂1 = ∂/∂x 1. We include a prescribed nonvanishing divergence g, since a priori estimates of the generalized Oseen equations (1) are a crucial step in the study of the stationary Navier-Stokes equations of a compressible, viscous fluid in an exterior domain, see e.g. Matsumura, Nishida [9], when u∞ = 0, and Padula [11], when u∞≠0. Contrary to the exterior Stokes problem there are only few papers on the exterior Oseen equations (with g = 0). Faxen [6] and Bemelmans [2] used the theory of hydrodynamical potentials to solve the homogeneous Oseen equations, i.e. when f = 0 and g = 0, while Finn [7] exploited the skew-symmetry of the operator ∂1 and investigated Green’s function. If g = 0 and Ω = ℝ n , let u = S f denote the solution of (1). Babenko [1] was the first to observe that ∂1S is a bounded linear operator on L q (ℝ3)3 for each q ∈ (1, ∞). This consequence of the Lizorkin-Marcinkiewicz multiplier theorem is also used by Galdi [8] proving that for each f ∈ L q (Ω) n , Ω ⊂ ℝ n an exterior domain, there is a unique solution (u, ∇ p) of (1) such that D2u and ∂1u are in L q (Ω).