Synopsis for B5.1a: Dynamics and Energy Minimization

Level: H-level Method of Assessment: Written examination.
Weight: Half-unit (OSS Number to be confirmed)

Recommended Prerequisites:

Elements from the following Part A courses: Differential Equations (Picard's theorem and phase plane arguments), Integration (convergence theorems and spaces). Topology (metric spaces). However this material will be reviewed in the course.

Overview

The aim of this course is to discuss the mathematics needed to describe the approach to equilibrium in dissipative dynamical systems. It will describe the basic ideas of dynamical systems, Lyapunov functions and stability, and also provide an introduction to the theory of local minimizers in the one-dimensional calculus of variations. The ideas will be applied to the dynamical system generated by a semilinear PDE.

As well as addressing aspects of an important scientific question, the course is an introduction to some of the rigorous techniques that are central to the modern study of nonlinear PDE.

Synopsis

Part I, Dynamical systems (7 lectures). Introduction to the problem of the approach to equilibrium and its thermodynamic origins. ODEs in ; local and global existence, continuous dependence on initial data, Lyapunov functions. Dynamical systems in and metric spaces. limit sets, invariance, La Salle invariance principle, Lyapunov stability.

Part II, Local minimizers in the 1D calculus of variations (5 lectures) Introduction to Sobolev spaces in 1D. Simplifed theory of global, weak and strong local minimizers in the calculus of variations.

Part III, Applications to PDE. (4 lectures) Discussion of one-dimensional semilinear parabolic PDE, the approach to equi- librium, and stability.

There is no single book that covers the course, which is a new compilation of material, and the lecturer aims to provide comprehensive notes. The follow- ing books contain useful material (but go well beyond the course in different directions):

For Part I

1. Nonlinear differential equations and dynamical systems, Ferdinand Verhulst, 2nd Edition, Springer, 1996.
2. Nonlinear Ordinary Differential Equations,An Introduction to Dynamical Systems, 4th Edition, D.W. Jordan and P. Smith (Oxford University Press, 2007).
3. Dynamics and Bifurcations, J.K. Hale and H. Kocak (Springer, 1991).

For Part II,

1. One-dimensional Variational Problems, G. Buttazzo, M. Giaquinta, S. Hilde- brandt, Oxford Lecture Series in Mathematics, Vol. 15 (Oxford University Press, 1998)
2. Introduction to the Calculus of Variations, U. Brechtken-Manderscheid (Chap- man and Hall, 1991).
3. Introduction to the Calculus of Variations, H. Sagan, (Dover, 1992).

For Part III,

1. Infnite-Dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, James C. Robinson, (Cambridge Texts in Applied Mathematics, Cambridge University Press, 2001).