[PAST EVENT] Mathematics Colloquium and EXTREEMS-QED Lecture: Michael Jolly (Indiana University)
Abstract: For certain evolutionary equations the long time behavior consists of bounded trajectories in a set called the global attractor. Examples of such equations range from the Lorenz system of three ordinary differential equations to the partial differential equations that model fluid flow. A determining form for such an equation is an entirely different ODE in a certain function space of trajectories where the solutions on the global attractor of the original evolutionary equation are readily recognized. It is an ODE in the true sense of defining a vector field which is (globally) Lipschitz. We will see how the construction of one type of determining form is related to a certain approach to data assimilation, i.e. the injection of a coarse-grain time series into the model in order to recover the matching full solution. We will then find that the dynamic behavior of this determining form is quite simple. All terms will be defined during the talk to make it accessible to a general audience, including undergraduates.