2000 Mt. Baldy Conference on Applied Mathematics: Abstracts
Andrea Bertozzi (Duke University): Undercompressive Shocks in Thin Film Flow
Nonlinear hyperbolic conservation laws have solutions with propagating `shocks' or discontinuities. Classically admissible shocks in scalar hyperbolic conservation laws satisfy a well known `entropy condition', in which characteristics enter the shock from both sides. Undercompressive shocks, in which characteristics pass through the shock, arise in e.g. combined dispersive/diffusive limits of scalar laws with non-convex flux functions. We show that fourth order diffusion alone produces undercompressive fronts, yielding such unusual behavior as double shock structures from simple jump initial data. Thermal/gravity driven thin film flow is described by such equations and the signature of undercompressive fronts have been observed in recent experiments. Mathematically, the undercompressive front is an accumulation point for a countable family of compressive waves having the same speed. The family appears through a cascade of bifurcations parameterized by the shock speed.
Michael Cross (CalTech): Theoretical Challenges of Spatiotemporal Chaos
Spatially extended nonequilibrium systems may show deterministic disordered dynamics which can be characterized as chaos with an extensive fractal dimension (i.e. dimension scaling with the system size). We call this dynamics spatiotemporal chaos. Natural examples, such as the earth's atmosphere, are widespread. In this talk I discuss some of the interesting issues in defining, characterizing, and understanding spatiotemporal chaos, and the theoretical challenges in addressing these issues, drawing on ideas from the study of low dimensional chaos and the statistical mechanics of equilibrium systems. Experimental examples and comparisons between theory and experiment will also be discussed.
L. Mahadevan (MIT): Folding, Wrinkling and Crumpling
Bending a thin elastic sheet is easier than stretching it, a fact that arises more from geometry than physics. Keeping this simple idea from structural mechanics in mind, I will give various examples involving the large deformations of thin films that are relevant to phenomena over a range of scales, from micro-patterning to geology. A feature that often arises in the ensuing deformation patterns is strain localization in the vicinity of a point. Therefore, I will discuss the shape, response and stability of these singularities, and their influence on the evolution of the patterns. Finally, I will explain and exploit an analogy that allows us to understand the folding, wrinkling and crumpling of thin fluid sheets.
Michael Ward (University of British Columbia): The Stability and Dynamics of Spikes for a Reaction-Diffusion System
Many classes of singularly perturbed reaction-diffusion systems possess localized solutions where the gradient of the solution is very large only in the vicinity of certain points in the domain. An example of a problem where such spikes occur is the Geirer-Meinhardt (GM) activator-inhibitor system modeling biological morphogenesis. In the limit of a small activator diffusivity, this system has been used to model many situations including spot-type patterns on sea-shells and head formation in the Hydra. Most of the previous work on this system over the past twenty years has been based either on full numerical simulations or on a linearized Turing-type stability analysis around spatially homogeneous steady-state solutions. However, this type of linearized analysis is not appropriate for determining the stability of spike-type patterns. In this talk we will survey some recent results on the existence and stability of symmetric and asymmetric equilibrium spike patterns for the GM model. The inhibitor diffusivity is found to be a critical parameter. A key result that is obtained is that there exists a sequence of critical values Dn of the inhibitor diffusivity D for which an n-spike symmetric equilibrium solution is stable if D < Dn and unstable if D > Dn. An explicit formula for Dn is given. The dynamics of spike patterns is also characterized in a one-dimensional domain and partial results are obtained in a multi-dimensional context. The mathematical tools used include asymptotic analysis, spectral analysis of nonlocal eigenvalue operators, dynamical systems, and numerical and matrix analysis.
Some of this work is joint with David Iron (graduate student at UBC), and Prof. Juncheng Wei (Chinese University of Hong Kong).