Global existence results for pattern forming processes in infinite cylindrical domains — Applications to 3D Navier–Stokes problems

Abstract

We consider translational invariant systems on unbounded cylindrical domains in R 3 which are described by the Navier–Stokes equations. The examples which we have in mind are the Taylor–Couette problem and Bénard's problem. For certain parameter ranges these systems exhibit pattern of almost spatial periodic nature. Although classical energy methods fail on unbounded domains the so called Ginzburg–Landau formalism allows us to show the global existence of strong solutions to all initial conditions in a neighborhood U of the weakly unstable ground state. For all times the bifurcating solutions of the original system can be shadowed by pseudo-orbits of the associated formally derived Ginzburg–Landau equation. This allows us to control the size of the solutions in the original system in terms of the bifurcation parameter for t→∞.