
Math 667
Topics in
Differential Equations
Winter
2005, MWF 9:009:50, CAB 563

University of
Alberta
Mathematical and Statistical Sciences
Dr. Thomas Hillen
4923395,
thillen@ualberta.ca
Cab 575

Outline


Assignments:
1,
due Jan 31, 05. in
class 9am.
2,
due Feb 14, 05. in
class 9am.
3,
due Mar 07, 05. in
class 9am.
4, due Mar 21, 05. in
class 9am.
5, due April 06, 05. in
class 9am.
List of "Inclass" presentations

Syllabus: In this course we study infinite dimensional dynamical systems. We
will systematically derive a theory of finite dimensional
compact global attractors, and we will investigate two examples in
detail: the NavierStokes equations and reactiondiffusion equations.
We will cover:
 Infinite dimensional dynamical systems and partial
differential equations (PDE's), reactiondiffusion equations,
NavierStokes equations
 some functional analysis
 weak solutions, Sobolev spaces
 existence theory for some PDE's
 global attractors, general properties
 global attractors for reactiondiffusion equations
 global attractors for the NavierStoles equations
 finite dimensional attractors, fractal and
Hausdorffdimensions
 squeezing property and inertial manifolds
 application to reactiondiffusion equations
 application to NavierStokes equations
Texts:

J.C. Robinson.
InfiniteDimensional Dynamical Systems. Cambridge University Press,
2001.
 R. Temam.
InfiniteDimensional Dynamical Systems
in Mechanics and Physics. Springer, 1988
 O.A. Ladyzhenskaya, Attractors for Semigroups and
Evolution Equations, Cambridge 1991.
 M.W. Hirsh, S. Smale. Differential
Equations, Dynamical Systems, and Linear Algebra. Academic Press, 1974.
 L. Perko. Differential Equations and Dynamical
Systems. Springer, 3rd ed., 2001
 A.V. Babin, M.I. Vishik, Attractors of Evolution
Equations. NorthHolland, 1992
Grading:
Homework 70%, in class
presentation 30%
Contact:
Dr. Thomas Hillen, 4923395,
thillen@ualberta.ca
office hours: TBA, CAB 575.
Policies:
Policy about
course outlines can be found in Section 23.4(2) of the University
Calendar.
Academic
honesty:
The University of
Alberta is committed to the highest standards of academic integrity and
honesty. Students are expected to be familiar with these standards
regarding academic honesty and to uphold the policies of the University
in this respect. Students are particularly urged to familiarize
themselves with the provisions of the Code of Student Behavior
(online at www.ualberta.ca/secretariat/appeals.htm)
and avoid any behavior which could potentially result in suspicions of
cheating, plagiarism, misinterpretation of facts and/or participation
in
an offence. Academic dishonesty is a serious offence and can result in
suspension or expulsion from the University.


1. Introduction 
 ODE's
 Discrete Dynamical Systems
 Connection
of Discrete and Continuous
 Abel's formula and the Wronskian
 Floquet Theory
 Periodic Attractors
 The Lorenz Equations

2. Some Functional Analysis 
 Banach Spaces
 Mollifiers
 Some useful integral estimates
 Hilbert Spaces
 Linear
Operators
 Dual Spaces and Weak Convergence
 Sobolev Spaces
 BanachSpace
Valued Functions

3. ReactionDiffusion Equations 
 Modelling
 Basic Assumptions
 Weak Solutions (Galerkin Approximation)
 Strong Solutions

4. The Navier Stokes Equation 
 Preassure and Fluid Velocity
 The Stokes Operator
 Weak
Formulation of the NS eq.
 Weak Solutions
 Uniqueness in 2D
 Strong Solutions

5. Global Attractors 
 Dissipation, Limit Stes and Attractors
 Structure of the Attractor
 Shadowing
 Continuous Dependence on Parameters

6. Global Attractor for ReactionDiffusion Equations in 1D 
 Absorbing Sets and the Attractor
 Injectivity
 A Lyapunov Function
 The ChaffeeInfante Equation

7. Global Attractor for NavierStokes Equations in 2D 
 Global Attractor
 Injectivity

8. Finite Dimensional Attractors

 Fractal and Hausdorff Dimesnion
 Evolution of nDimensional Volumes
 ReactionDiffusion Equations
 NavierStokes equations

