The Hopf bifurcation for nonlinear semigroups

Abstract

Several authors, have shown by perturbation techniques that the Hopf theorem on the development of periodic stable solutions is valid for the Navier-Stokes equations; in particular, solutions near the stable periodic ones remain defined and smooth for all t ≥ 0 . The principal difficulty is that the Hopf theorem deals with flows of smooth vector fields on finite-dimensional spaces, whereas the Navier-Stokes equations define a flow (or evolution operator) for a nonlinear partial differential operator (actually it is a nonlocal operator). The aim of this note is to outline a method for overcoming this difficulty which is entirely different in appearance from the perturbation approach. The method depends on invariant manifold theory plus certain smoothness properties of the flow which actually hold for the Navier-Stokes flow.