Math 525 

Differential Equations II

Texts:

1. J.C. Robinson.  Infinite-Dimensional Dynamical Systems. Cambridge University Press, 2001.
2. L. Perko.  Differential Equations and Dynamical Systems. Springer, 3rd ed., 2001.
3. R. Temam.  Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Springer, 1988.
4. M.W. Hirsh, S. Smale.  Differential Equations, Dynamical Systems, and Linear Algebra. Academic Press, 1974.
5. C. Sparrow.  The Lorenz Equations: Bifurcation, Chaos, and Strange Attractors. Springer, 1982.
   

Syllabus:

•  Poincare' map and stability of periodic orbits
•  Floquet theory
•  The Lorenz Equations
•  Infinite dimensional Dynamical Systems, PDE's, reaction-diffusion equations, Navier-Stokes equations
•  Some Functional Analysis
•  Weak solutions, Sobolev spaces
•  Existence theory for some PDE's
•  Global Attractors
•  Lyapunov exponents and Lyapunov multipliers 
•  Fractal- and Hausdorff-dimensions, finite dimensional attractors
•  Squeezing property and inertial manifolds
•  Application to reaction-diffusion equations
•  Application to Navier-Stokes equations

Grading:   

 Contact:

Seminar:

Assignments: 1, 2, 3, 4, 5          

Contents and Lecture 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