Nonlinear Multidimensional Systems of Hyperbolic Partial Differential Equations

University Of California-Santa Barbara, Santa Barbara CA

Investigators

Abstract

The proposed research will focus on models involving multidimensional systems of nonlinear partial differential equations, arising in elastodynamics, incompressible fluid flow, and electromagnetics. The emphasis of this work will be on existence and long time behavior of solutions using analytical methods which exploit the fundamental structure of the models, based on the null condition. In the area of nonlinear elastodynamics, the connection between the structure of the strain-energy relations and global well-posedness for the initial value problem will be studied. The research aims at a better understanding of the relationship between compressible and incompressible materials, the formation of shock waves, and the behavior of elastic bodies in astrophysical applications. The recent methods developed for elastodynamics will be applied to the Born-Infeld system from nonlinear electromagnetics and recently revived in string theory. The confinement of vortical motion in planar incompressible ideal fluids will be studied. According to the classical principle of determinism, the the evolutionary laws of a physical system together with an initial configuration should determine the state of the system at all future times. Mathematically speaking, the the laws of physics are encoded in a system of partial differential equations solutions of which should exist and depend uniquely and continuously on the initial configuration. Creating the mathematical framework for a given system and verifying its well-posedness (in the sense above) is thus a problem of fundamental importance. Much of the research in this proposal is devoted to such basic questions for models describing the dynamics of elastic materials. Material properties are expressed through the specification of the relation between strain and energy, using phenomenological arguments. However, phenomenology alone does not fully characterize the structure of the model. The desire to describe dynamics for all future times leads to the imposition of further structural assumptions which are deeply connected with the mathematical analysis of the problem. In this sense, mathematical analysis offers important guidance in the selection of the model. With a mathematically tractable theory in hand, it is possible to consider qualitative questions, such as how the behavior of the system varies under changes in material properties. In particular, the stability of the system as the material becomes incompressible will be studied. This includes rubber-like materials, for example. The techniques will also be adapted to allow for the inclusion of relativistic effects, necessary for astrophysical applications. The Born-Infeld model from nonlinear electrodynamics (and recently revived in the theory of strings) will also be investigated, since it shares similar structural properties.