math 238 Fall 2001

Room 216 Physics, Mon Wed 2:15-3:30 (NOTE NEW TIME!!!)

Instructor: Andrea L. Bertozzi, Professor of Mathematics and Physics

138A Physics Building, 660-2818

This course is a special advance graduate course for students interested in nonlinear PDE. If you have not taken Math 232, graduate PDE, you do not belong in this class. Please note that we will not be following the syllabus described in the course catalogue. Below I list some topics that I am interested in discussing. Students will have the opportunity to suggest topics.

I. Introduction to nonlinear PDE and complexity in continuum models. Finite time singularities, self-similarity, pattern formation and instabilitity, weak or viscosity solutions.

II. Basic higher order problems: Kuramoto-Sivashinsky, Cahn Hilliard, and related equations for phase transitions. Stability of states, bifurcation theory, boundedness of solutions, pattern formation, turbulence and coarsening instabilities, attracting manifolds.

III. Higher order degenerate diffusion equations: lubrication theory and other models including low curvature image simplifiers for image processing. Weak solutions for lubrication equations. Entropies and energy methods. Existence and regularity, nonuniqueness results, connection to the physics of contact line motion. Numerical schemes and positivity preserving methods.

The course will include lectures by Bertozzi as well as student presentations of papers from the literature.

Grades in this course will be based on class participation, including presentations given by students, and on a final project which includes a written report of approximately ten pages.

Reference List for This Course.

Lecture schedule for the course.