# Math 285J : Topics in Nonlinear PDEs with
application to Fluid Dynamics and Image Processing

Winter 2004

Room 5217 Mathematical Sciences

11am Monday/Wednesday/Friday

Instructor: Prof. Andrea Bertozzi, 7619D Mathematical Sciences

phone: 310-825-4340

email: bertozzi@math.ucla.edu

Office Hours: 1-2pm Thursdays

This course will showcase methods from applied analysis
of nonlinear evolution equations by considering some
problems of recent and current interest.
Students should have basic graduate courses in Partial Differential
Equations and Analysis. We will also consider a few ideas
from numerical analysis although no prior knowledge of numerics is required.

Format: Lectures Monday and Wednesday, Friday is group discussion
and student presentations.

Here is an approximate syllabus for the course. The plan
may change slightly depending on the interest of the students.

** Friday 1/10 ** Introduction to the course, Overview
** Monday 1/12 Week 1 ** Sobolev Spaces, mollifiers, estimates, introduction to nonlinear PDEs
* Examples of nonlinear evolution equations. What are the issues?
L^2 Sobolev spaces and the physical interpretation of the Fourier transform.
Mollifiers as solutions of the heat equation and as smoothing operators.
Brief discussion of psuedodifferential operators. ODEs on Banach spaces
and the generalized Picard theorem. *
** Monday 1/19 Week 2 ** Euler and Navier-Stokes equations, basic estimates
* Kinetic energy. Vorticity transport and estimates. Conserved and dissipated quantities. Eulerian vs. Lagrangian variables. Particle trajectories.
Integrodifferential equations. Some special exact solutions. *
** Monday 1/26 Week 3 ** Euler and Navier-Stokes energy estimates and local existence of smooth solutions
* A constructive local existence proof. We use mollifiers to
construct an approximating problem involving bounded operators on Sobolev
spaces. We derive estimates that result in a global existence
theory for the regularized problem. We derive estimates independent
of the regularization in order to find a limiting solution that solves
the Euler or Navier-Stokes equations. Differences between the viscous
and inviscid case will be discussed. *
** Monday 2/2 Week 4 ** Euler and Navier-Stokes vorticity and global existence of smooth solutions.
* We discuss the famous millennium problem and results to date.
Energy methods are used to prove the well-known Beale-Kato-Majda
theorem that connects growth of vorticity to global existence of a smooth solution. Our proof is valid for both Euler and Navier-Stokes.
*
** Monday 2/9 Week 5 ** Euler equations in Lagrangian form, Singular Integral Operators and Holder spaces
* Particle trajectory map and Lagrangian variables. Integrodifferential
equation for particle paths as an ODE on a Banach space. This theory
uses some ideas from Singular Integrals (Harmonic Analysis).
*
** Monday 2/16 Week 6 ** Nonlinear diffusion equations, introduction
* The Navier-Stokes equations involve linear diffusion with a quadratic nonlinearity in the inertial term. We explore some problems in which
the diffusion itself is nonlinear. The porous media equation is a classical
second order example. Some techniques will be discussed such as the maximum
principle, similarity solutions and energy estimates.
These problems are excellent case studies to showcase how rigorous analysis
can be combined with asymptotics, numerics, and other tools from applied mathematics to obtain intuition for the behavior of evolution equations.
*
** Monday 2/23 Week 7 ** Lubrication equations, energy and entropy estimates
* We discuss a class of equations that arise in the study of thin
films and surface tension. These are fourth order nonlinear PDEs that
are fairly new to the world of applied analysis. In general they do not
possess a maximum principle and the nonlinearity plays a very interesting
and different role from that in the analogous second order porous media equation. Some connections to the physics will be discussed alongside the analysis.
*
** Monday 3/1 Week 8 ** Finite difference and finite element methods for lubrication equations
* We build on the analytical results to develop finite difference and finite element methods for nonlinear diffusion equations. We show how entropy
estimates can lead to the design of special schemes. *
** Monday 3/8 Week 9 ** Image processing PDEs, basic energy estimates
* Many PDEs in image processing are derived from a variational formulation.
We discuss these problems in a different and larger context, as evolution
equations with dissipation. Again we can use energy methods to
prove results about existence of solutions and to draw conclusions about
dynamics of the equations. *
** Monday 3/15 Week 10 ** Image processing PDEs, traveling wave solutions, nonlinear estimates
* We will discuss some very recent research results involving
higher order PDEs in imaging and the role of nonlinearity.
This discussion will showcase some recent results of UCLA applied math faculty
and students (Osher, Vese, Greer, Bertozzi, Chan, etc.)
*
References
Majda and Bertozzi, Vorticity and Incompressible Flow, Cambridge University
Press, 2002, Chapters 3 and 4

Michael Taylor, PDEs Volume III, Springer Verlag

Some papers from the recent literature on image processing
and on lubrication equations