Nonlinear and stochastic phenomena. The grand challenge for partial differential equations

The major problems for partial differential equations are either nonlinear or stochastic or both. 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. Stochastic solutions of partial differential equations may result from stochastic data. 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.