In this project, the PI will apply spectral and variational methods, as well as the techniques of Fourier analysis, to study some outstanding open problems in the theory of Hamiltonian and dispersive partial differential equations. The first part of the project focuses on the long-time behavior of the solutions of the Kuramoto-Sivashinsky and Burgers-Sivashinsky equations. Questions to be addressed concern the global well-posedness, the existence of attracting sets, and the regularity of the solutions. The PI will continue her work on developing a systematic approach to the questions of existence and regularity of attractors for dissipative partial differential equations in the case of unbounded domains. Of particular interest are the two-dimensional Navier-Stokes equation and the "alpha models" from fluid dynamics. An innovative approach, using modulation equations and spectral decomposition techniques will be used to study the existence and stability of solitary waves, standing waves, and similar particular solutions in the vicinity of the central manifold for a class of parabolic problems. In solving these problems the PI will use a variety of techniques including evolution semigroups, spectral and frequency analysis. The goal here is to investigate the Green-Naghdi system, the coupled-mode system from nonlinear optics as well as to establish nonlinear stability and complete invariant manifolds description for a class of abstract Hamiltonian partial differential equations.
This proposal deals with a variety of problems concerning solutions of a large class of partial differential equations of mathematical physics. Such equations are viewed as dynamical systems on an infinite-dimensional space. Of particular interest are the Kuramoto-Sivashinsky and Burgers-Sivashinsky equations, the two-dimensional Navier-Stokes equation, and the ?alpha models? of fluid dynamics. The PI will use a variety of techniques, including spectral analysis, variational methods, the techniques of Fourier analysis, and evolution semigroups. The Hamiltonian partial differential equations addressed in this project arise in numerous physical situations - some have their origins in the study of quantum mechanical systems and nonlinear optics, others in fluid dynamics and combustion. Understanding the local behavior of solutions nearby a special solution is an issue of paramount importance for the applications, since only waves that are stable can be expected to be physically realizable. On the other hand, identifying instability or the source of it is of great practical importance. Any progress on these questions will not only be important mathematically, but will find immediate applications in physical sciences. Another goal of the PI is to involve some undergraduate students in this fascinating area of research. The PI also plans a topics class on nonlinear dynamical systems jointly with other faculty of the Mathematics and Physics Departments.
