Mathematical Modelling in Applied Analysis

Mathematical modelling in applied analysis refers to exploiting mathematical knowledge about (classes of basic) equations with the aim to describe and analyse phenomena that appear in the natural and technical sciences. As an example we will show that low dimensional models can be derived for phenomena that appear in (perturbations of) dynamical Poisson systems with symmetries. We show that the speciic Poisson structure makes it possible to deene a manifold of relative equilibria. This manifold, consisting of solutions of the unperturbed equation, is used as a model manifold into which evolutions of perturbed equations can be projected. It turns out that for small perturbations that can have a large eeect on the time-and space scales we are interested in, Fredholm solvability conditions determine the projection and hence the nal model. We illustrate the ideas to two spatially inhomogeneous systems from uid dynamics: distorting waves over uneven bottom and swirling ows in expanding pipes. More details about these and other problems can be found in 3, 6]. 1. General outline: perturbation theory on model manifold Consider quite generally an innnite dimensional evolution equation (a partial diierential equation in the applications) written like E 0 (u) @ t u ? K(u) = 0 where u is the state variable, and K the vector eld. We assume that for this unperturbed system a smooth manifold of exact solutions is known; with p the parameters characterising the solutions, we get a parameterised manifold of solutions that will be used as the model-manifold in the following: