9404570 Levermore This project will support research in the collective large-scale behavior of dynamical systems. There are four projects that will be undertaken as components of this research program. The "Zero Dispersion Limits of Nonlinear Wave Equations" and "Macroscopic Lattice Dynamics" projects both involve the study of the limiting behavior of dynamical systems with small amounts of dispersion. The goal is to construct macroscopic models that consist of modulation equations for families of exact solutions of these systems, and to identify those regions of phase space in which such models are valid. Both analytical and numerical tools will be employed, and both integrable and non-integrable systems will be considered. The "NLS Limit of the CGL Equation" project studies the complex Ginzburg-Landau (CGL) equation as a model for "turbulent" dynamics in nonlinear partial differential equations in which the "inviscid" limit is the nonlinear Schroedinger (NLS) equation. This relationship is in direct analogy to that of the driven Navier-Stokes equations to the Euler equations in incompressible fluid turbulence. The structures of the NLS equation (which is even integrable in one space dimension for the cubic case) will be used to provide coordinates for the CGL phase space. The "Moment Closure Hierarchies for Kinetic Equations", and "Numerical Schemes for Hyperbolic Systems" projects study a systematic nonperturbative derivation of a hierarchy of closed systems of moment equations corresponding to any classical kinetic theory and the numerical schemes to implement them. Unlike earlier moment closures, every member of the hierarchy is a hyperbolic system with relaxation terms and entropy, and hence, is formally well-posed. Every member also formally recovers the Euler limit, while members beyond the second recover a Navier-Stokes correction consistent with the true Navier-Stokes system. The related "Kinetic Theory" project is more theoretical in natur e. It will try to rigorously establish the linearized Euler limit of the Boltzmann equation and the existence of the Boltzmann shock profile. Both of these are long standing open problems that may be amenable to the techniques developed in earlier work with Bardos and Golse. The "Global Ocean Circulation Modeling" project studies the long-time effects of topography and hydrostatic imbalance on shallow inviscid incompressible flows, with a free boundary, over a varying bottom. An asymptotic expansion of the three-D Euler equations leads to a system of shallow water equations that captures weak hydrostatic imbalances and dispersion due to topography, yet inherits several structural properties of the two-D Euler equations. For example, it conserves an energy norm, possesses a circulation theorem, and has a Hamiltonian form. This structure will facilitate both the analytical study of this system and the development of sound numerical schemes. The unifying theme that runs through the projects supported is to understand how collective large-scale behavior of dynamical systems arise. From a physical perspective this effort could be viewed as a study of non-equilibrium statistical mechanics. But it is more than that, for it strives to discover and study new nonlinear phenomena that could impact the way models are developed for many important physical processes. The projects are divided into four groups. The first group impacts the theories of super-fluidity and super-conductivity. The second contributes to our understanding of turbulent fluids. The third will find application in a wide range of problems such as hypersonic flight and nano-scale semiconductors. The fourth is a first step in the effort to develop global climate models that are both accurate and economical.
