Approximation of the Global Attractors of Evolution Equations

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The investigator and his colleague atudy the computation ofglobal attractors and invariant manifolds for dissipative partialdifferential equations. Global attractors are detected byinterpolation of analytic (in complex time) rather than directsimulations. The performance of this approach is connected tothe growth rate of solutions backward in time. Computations arecarried out to determine if a set of solutions with a particularrate is in fact a manifold. Adaptations of a succesfulcontraction mapping algorthm for inertial manifolds are made forcenter manifolds and foliations. Specific applications are madeto physical models including the Camassa-Holm,Kuramoto-Sivashinsky and Navier-Stokes equations, includingissues from turbulence theory. This project concerns a variety of mathematical equationsdescribing such physical phenomena as combustion, fluid flow, andturbulence. Part of the effort is to develop a method toreliably determine whether a particular computed solutionrepresents permanent or merely temporary behavior. Anotherconcerns the visualization of geometric objects corresponding todistinguished sets of solutions, which determine the ultimatebehavior of the physical system. Though the work is carried outfor several specific, physically significant equations, it isgeneral enough to extend, in a natural way, to many others of asimilar form, including some modeling the weather.