Dynamical Systems Methods for Fluid Mechanics and Hamiltonian Mechanics

Trustees Of Boston University, Boston

Investigators

Abstract

The investigator studies the behavior of partial differential equations and other infinite dimensional dynamical systems, and adapts and develops ideas originating in the study of finite-dimensional dynamics to make qualitative and quantitative predictions about the behavior of solutions of such systems. He focuses primarily on questions arising in physical applications. Among the questions he studies are the metastable behavior of fluid systems, whereby long-lived structures like vortices appear in the flow on a relatively short time scale, and then determine the subsequent evolution of the fluid for very long times. He also studies the influence of localized structures like vortices on the behavior of rotating, stratified fluid layers like the atmosphere. The differential equations arise in a variety of different physical contexts and are characterized by the fact that while the equations themselves are well known, they are too complicated to solve explicitly except in very special or physically unrealistic cases. Nevertheless, applications require at least a qualitative understanding of the behavior of their solutions; this research project aims to develop such an understanding. The project incorporates the training of graduate students and post-doctoral fellows into all facets of this research. Among the specific systems that the investigator studies are: (i) nearly inviscid fluids and weakly damped Hamiltonian systems like the Fermi-Pasta-Ulam system using normal forms and the theory of hypercoercivity; (ii) invariant manifolds and their implications for the behavior of: (a) unstably stratified, rotating fluid systems, (b) the compressible Navier-Stokes equations and (c) as a means of understanding the continuum approximation of kinetic systems, and (iii) small-divisors in fluid mechanics and other infinite dimensional systems. One feature that makes it difficult to apply dynamical systems ideas to partial differential equations on unbounded domains is the interplay of effects due to continuous spectrum with those coming from discrete spectrum. The planned work on stratified fluids and approximation theorems for kinetic systems further develops the theory of invariant manifolds to treat these problems. The study of small divisor problems in the vortex sheet equations extends KAM-like methods to a new class of infinite dimensional systems. The study of weakly viscous fluids and weakly damped oscillator systems both leads to a deeper intrinsic understanding of the origin of intermediate time scales in these systems and illuminates the mathematical relationships between these two apparently quite different physical systems. Graduate students and post-doctoral fellows are included in the work of the project. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.