Math 667

  Topics in

Differential Equations

Winter 2005, MWF 9:00-9:50,  CAB 563

University of Alberta
Mathematical and Statistical Sciences
Dr. Thomas Hillen
492-3395, 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 "In-class" 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 Navier-Stokes equations and reaction-diffusion equations. We will cover:

  • Infinite dimensional dynamical systems and partial differential equations (PDE's), reaction-diffusion equations, Navier-Stokes equations
  • some functional analysis
  • weak solutions, Sobolev spaces
  • existence theory for some PDE's
  • global attractors, general properties
  • global attractors for reaction-diffusion equations
  • global attractors for the Navier-Stoles equations
  • finite dimensional attractors, fractal- and Hausdorff-dimensions
  • squeezing property and inertial manifolds
  • application to reaction-diffusion equations
  • application to Navier-Stokes 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:   

     Homework 70%, in class presentation 30%

    Contact:

     Dr. Thomas Hillen, 492-3395, 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.

 

 

 

 

  

  

Course Notes 

   

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

 
  2. Some Functional Analysis
  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
  3. Reaction-Diffusion Equations
  1. Modelling
  2. Basic Assumptions
  3. Weak Solutions (Galerkin Approximation)
  4. Strong Solutions
  4. The Navier Stokes Equation
  1. Preassure 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 
  5. Global Attractors
  1. Dissipation, Limit Stes and Attractors
  2. Structure of the Attractor 
  3. Shadowing
  4. Continuous Dependence on Parameters

 
  6. Global Attractor for Reaction-Diffusion Equations in 1-D
  1. Absorbing Sets and the Attractor
  2. Injectivity
  3. A Lyapunov Function
  4. The Chaffee-Infante Equation
  7. Global Attractor for Navier-Stokes Equations in 2-D
  1. Global Attractor
  2. Injectivity
  8. Finite Dimensional Attractors

 

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