Nonlinear and Stochastic Phenomena: The Grand Challenge for Partial Differential Equations

Abstract

The major problems for partial differential equations are either nonlinear or stochastic or both. Considerable progress has occurred recently in these areas, much of which has resulted from a striking interplay among theory, computation, and applications. The nonlinear structures emphasized here are the nonlinear hyperbolic waves, which occur in the solutions of nonlinear hyperbolic conservation laws and associated dissipative equations. These equations are those most commonly used to model physical and chemical processes in continuum systems. In one space dimension these nonlinear waves form traveling waves, for which the methods of dynamical systems and bifurcation theory are very useful. Established points of view toward mathematical theories have been challenged by recent work (e.g., uniqueness of solutions, entropy conditions, and elliptic regions in hyperbolic equations). A type of geometry is defined in the state space of the conservation law by the nonlinear waves. This geometry is generically singular in the case of nonlinear resonance. A regularizing space, called the Fundamental Wave Manifold, has been constructed which removes nonessential singularities in this geometry; the remaining, essential, sin- gularities serve as organizing centers for the bifurcation of the nonlinear hyperbolic waves. Stochastic solutions of partial differential equations may result from stochastic data, for example, in the form of random initial conditions or equation coefficients. This phenomenon can be called forced, driven, or external chaos. Stochastic solutions can also result from internal considerations, i.e., from instabilities inherent in the equation itself, if the equation is nonlinear. The instabilities which give rise to chaotic solutions are generally unstable on a very large range of length scales, or in some approximation, on all length scales. Fractal geometries provide one model for the typically intermittent, or blotchy, chaos which can occur, and the renormalization group is an analytic tool which in some cases has given a quantitatively correct theory of chaotic solutions of partial differential equations.