Math 525

  Ordinary Differential Equations II

University of Alberta
Mathematical and Statistical Sciences
Dr. Thomas Hillen
492-3395, thillen@ualberta.ca

Cab 575

Syllabus: LINK

Summary:

In this course we will study asymptotics of ordinary differential equations and boundary value problems. The Poincare-Bendixson theory has been covered in Math 524. We cover the theory of dynamical systems and differential equations in Banach spaces. The concepts of stability and bifurcations can be generalized from ODEs to PDEs. We will  systematically derive a theory of finite dimensional  compact global attractors, and we will investigate two examples in detail: the Navier-Stokes equations and reaction-diffusion equations.

Texts:

• J.C. Robinson.  Infinite-Dimensional Dynamical Systems. Cambridge University Press, 2001.

• R. Temam.  Infinite-Dimensional 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. North-Holland, 1992

  Grading:   

    6 Homework Assignments of equal  weight = 100%
  

  Contact:

   Dr. Thomas Hillen, 492-3395, thillen@ualberta.ca
  office hours: after calss 10:00-10:45, or by appointment,   CAB 575.
  
  

Lecture Notes 

1. Assignment 1 (Jan 16, 2015)
   
2. Assignment 2 (Jan 30, 2015)
   
3. Assignment 3 (Wednesday, Feb 25, 2015)
   
4. Assignment 4 (Mar 13, 2015)
   
5. Assignment 5 (Mar 27, 2015)
   
6. Assignment 6 (Apr 10, 2015)
   

1. ODE's
2. Discrete Dynamical Systems
3. Connection of Discrete and Continuous
   
4. Abel's formula and the Wronskian 

We skip the following sections (1.5) -(1.7) since they were covered in Math 524:

5. Floquet Theory
6. Periodic Attractors
7. The Lorenz Equations

1. Banach Spaces
2. Mollifiers
3. Some useful integral estimates
4. Hilbert Spaces
5. Linear Operators 
6. Dual Spaces and Weak Convergence
7. Sobolev Spaces
8. Banach-Space Valued Functions

1. Modelling
2. Basic Assumptions
3. Weak Solutions (Galerkin Approximation)
4. Strong Solutions

1. Pressure and Fluid Velocity
2. The Stokes Operator
3. Weak Formulation of the N-S eq.
4. Weak Solutions
5. Uniqueness in 2-D
6. Strong Solutions 

1. Dissipation, Limit Sets and Attractors
2. Structure of the Attractor 
3. Shadowing
4. Continuous Dependence on Parameters

1. Absorbing Sets and the Attractor
2. Injectivity
3. A Lyapunov Function
4. The Chaffee-Infante Equation

1. Global Attractor
2. Injectivity

1. Fractal and Hausdorff Dimesnion
2. Evolution of n-Dimensional Volumes
3. Reaction-Diffusion Equations
4. Navier-Stokes equations