Approximation of the Global Attractors of Evolution Equations

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9706903 M. Jolly Abstract This project concerns the long time behavior of certain dissipative physical systems. One major component seeks to locate global attractors by interpolatory means rather than by direct numerical solution of initial value problems. It would extend our previous work which used a Taylor expansion in complexified time at a single point in phase space. The new approach will use several points, typically those on solutions provided either by independent computations or from experimental data. The main mathematical tool in this effort will be Nevalinna-Pick interpolation. The systems on which this will be tested include the 2-D Navier-Stokes (NS), Kuramoto-Sivashinsky (KS) Lorenz equations as well as simple geostrophic models. Another major component is to compute invariant manifolds to arbitrary accuracy. We will use the visualization of 2-D (un)stable manifolds to help understand the geometric mechanisms behind certain global bifurcations in 3-D phase space. This involves computing a major portion of global manifolds. Other applications require only that the manifold be computed along particular trajectories. In one case a center manifold will be treated in this way to compute bounded solutions to an elliptic partial differential equation (PDE) in an infinite cylinder. In another, the dimension of phase space for the KS equation will be effectively reduced to three dimensions by restricting the flow to an inertial manifold of that dimension. Such a reduction will allow us to study global bifurcations as described above. The algorithms developed to compute these manifolds will also be applied to the sets (conjectured to be manifolds) of a prescribed exponential growth rate backward in time for the NS equation. In fact we will construct such sets as stable "manifolds" for an inverted form of the NS equation in which infinity and the origin of phase space are swapped. These sets play a role in the interpolatory approach to locating global attractors, and thus bring our research full circle. The main purpose of this work is to develop reliable methods to determine whether certain dynamic behavior in physical systems is permanent, or merely temporary. The ultimate application will be to climatology. Since the earth's weather system has been evolving for millions of years, one would expect that unless sudden external events take place, the patterns we are living through now will more or less continue for a reasonable period of time. This is not about accurate long time forecasting, rather it is about confirming basic assumptions regarding the mathematical models used in making those predictions. The scientific community makes a tremendous effort in deriving appropriate mathematical equations, and discretizing them so they can be solved on a computer, all to produce a function of time, which should describe some aspect of the weather. We all know how often this computed function of time deviates from the actual weather after a relatively short time period. The major source of this error is not clear. Is it in the model itself? Is it from the numerical approximation in the computer solution? Or is it that both the model and the approximation are valid, but the actual solution is very sensitive to small changes in the initial data, and we simply need to tighten the tolerance of error in that data and in the algorithm used at each time step. Our work is directed at distinguishing between the first two cases and the third. Indeed we seek to validate the model-algorithm pair which produces the forecast, as producing a pattern which is of a permanent nature, even if it is not the particular pattern we are experiencing after several days time. The failure of such a test will indicate that either the model and/or the method of solution are faulty. This approach can be applied to other physical problems. Indeed the initial testing of the methods will be done on systems less invol ved than that of the weather, but which are nevertheless of current scientific interest. In particular we consider fundamental models of combustion, fluid flow, and turbulence.