Approximation of the Global Attractors of Evolution Equations

Indiana University, Bloomington IN

Investigators

Abstract

The investigator and his colleague atudy the computation of global attractors and invariant manifolds for dissipative partial differential equations. Global attractors are detected by interpolation of analytic (in complex time) rather than direct simulations. The performance of this approach is connected to the growth rate of solutions backward in time. Computations are carried out to determine if a set of solutions with a particular rate is in fact a manifold. Adaptations of a succesful contraction mapping algorthm for inertial manifolds are made for center manifolds and foliations. Specific applications are made to physical models including the Camassa-Holm, Kuramoto-Sivashinsky and Navier-Stokes equations, including issues from turbulence theory. This project concerns a variety of mathematical equations describing such physical phenomena as combustion, fluid flow, and turbulence. Part of the effort is to develop a method to reliably determine whether a particular computed solution represents permanent or merely temporary behavior. Another concerns the visualization of geometric objects corresponding to distinguished sets of solutions, which determine the ultimate behavior of the physical system. Though the work is carried out for several specific, physically significant equations, it is general enough to extend, in a natural way, to many others of a similar form, including some modeling the weather.