Dynamical Systems Approaches to Partial Differential Equations

Trustees Of Boston University, Boston

Investigators

Abstract

NSF Award Abstract - DMS-0103915 Mathematical Sciences: Dynamical Systems Approaches to Partial Differential Equations Abstract DMS-0103915 Wayne This project explores the long-time behavior of partial differential equations using tools from the theory of dynamical systems. The equations to be studied include both dissipative and Hamiltonian systems. As an example of the former, Professor Wayne will study the long-time behavior of the Navier-Stokes equations in the neighborhood of vortex solutions. He will construct finite dimensional invariant manifolds in the phase space of these equations and use them to study both the long-time asymptotics of solutions of the equations and the existence of vortex solutions for which no explicit formulas exist. In related work he will derive and rigorously justify approximate equations for the motion of waves on the surface of a fluid, studying in particular the interaction of colliding waves. In addition, by using ideas first developed to study finite dimensional, nearly integrable, Hamiltonian systems, he will investigate the validity of commonly used beam and plate models for motion of elastic materials in thin domains. Finally, in collaboration with a mathematical biologist at Brown University, Professor Wayne will study the existence and stability of pulses in models of neural tissue. The systems under study arise in many applications, including materials science, fluid mechanics, and biology. Even though it is impossible to solve the equations that govern their motion explicitly, applications require at least a qualitative understanding of the behavior of their solutions. For instance, the equations that describe vibrations of elastic materials are so complicated that their solution remains very time consuming, even with modern computational tools. Consequently, engineers have derived many approximate models for the behavior of such systems, particularly in situations where one dimension of the system is much smaller than others, as is the case for beams and plates. Very little is known rigorously, however, about how well these models actually mimic the true behavior of the beam or plate. This research aims both to provide accurate estimates of the errors that occur in using such models and to develop an algorithm that permits one to systematically improve the models. In a similar vein, Professor Wayne will also investigate models for waves on the ocean. Recently it has been realized that the wake of high-speed ferries can produce solitary waves of sufficient magnitude to cause significant damage at the shore. The mechanism by which these waves are created in the wake is not yet understood. Understanding the relationship between the solitary waves of the model problem and the solutions of the actual water wave problem may shed light on the creation and propagation of this type of wake.