Theory and computation of periodic solutions of autonomous partial differential equation boundary value problems, with application to the driven cavity problem

In ordinary differential equations, one distinguishes two cases: autonomous and nonautonomous. Roughly speaking, the theory of the latter is built upon the theory of the former. The same distinction should be applied to partial differential equations, where much less is known. Here I will focus on the question of the generation of periodic solutions for autonomous partial differential equation boundary value problems. Specifically, I consider the incompressible Navier-Stokes equations, and the important driven cavity problem. mplicity, attention is restricted to two bifurcation parameters, the Reynolds number and the Aspect ratio. Only Dirichlet velocity boundary conditions are considered. Both the known theory and known computational results for the driven cavity are surveyed. The importance of computationally adhering to the div u = 0 condition to accurately simulate unsteady flows which will be qualitatively correct for the incompressible Navier-Stokes equations is stressed. The dependence of sustained periodicity upon the existence of highly localized vortex shedding sequences somewhere along the boundary is pointed out. A new analysis of the pressure boundary condition, based upon a general regularity principle, is given. A conjectured Hopf bifurcation criticality curve is explained.