Further properties of steady-state solutions to the Navier-Stokes problem past a three-dimensional obstacle

We show that the weak formulation of the steady-state Navier-Stokes problem in the exterior of a three-dimensional compact set (closure of a bounded domain), corresponding to a nonzero velocity at infinity and subjected to a given body force, is equivalent to a nonlinear equation in appropriate Banach spaces. We thus show that the relevant nonlinear operator enjoys a number of fundamental properties that allow us to derive many significant results for the original problem. In particular, we prove that the manifold constituted by the pairs $(u,λ)$⁠, with $λ$ the nondimensional speed at infinity (Reynolds number) and $u$ weak solution corresponding to $λ$ and to a given body force $f$⁠, is, for “generic” $f$⁠, a $C∞$ one-dimensional manifold, and that, for almost any $λ$⁠, the number of solutions is finite. We also show that, for any given $f$ in the appropriate function space and any given $λ>0$⁠, the corresponding solutions can be “controlled” by their specification only in a suitable neighborhood, $I$⁠, of the boundary. The “size” of $I$ depends only on $λ$ and $f$⁠. Furthermore, we analyze the steady bifurcation properties of branches of these solutions and prove that, in some important cases, the sufficient conditions for bifurcation formally coincide with the analogous ones for flow in a bounded domain. Finally, the stability of these solutions is analyzed. The paper ends with a section on relevant open questions.