Dynamics of partial differential equations

Description

This book contains two review articles on the dynamics of partial differential equations that deal with closely related topics but can be read independently. Wayne reviews recent results on the global dynamics of the two-dimensional Navier-Stokes equations. This system exhibits stable vortex solutions: the topic of Wayne's contribution is how solutions that start from arbitrary initial conditions evolve towards stable vortices. Weinstein considers the dynamics of localized states in nonlinear Schrödinger/Gross-Pitaevskii equations that describe many optical and quantum systems. In this contribution, Weinstein reviews recent results on the bifurcation of solitary waves, their linear and nonlinear stability properties, and the role of radiation damping in the synamics of energy flow for the energy-conserving PDE. The articles, written independently, are combined into one volume to showcase the tools of dynamical systems theory at work in explaining qualitative phenomena associated with two classes of partial differential equations with very different physical origins and mathematical properties.  Read more... Abstract: This book contains two review articles on the dynamics of partial differential equations that deal with closely related topics but can be read independently.Wayne reviews recent results on the global dynamics of the two-dimensional Navier-Stokes equations.  Read more... Content: Review 1. Dynamical systems and the two-dimensional Navier-Stokes equations / C. Eugene Wayne -- Review 2. Localized states and dynamics in the nonlinear Schrödinger/Gross-Pitaevskii equation / Michael I. Weinstein.