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