Date | Topics |
W 9/3 | Introduction. Convergence of Fourier series in L^{2}. |
F 9/5 | Weak and strong derivatives. Sobolev spaces. Differentiation of vector valued functions. |
W 9/10 | L^{2}-theory of the heat equation. Existence and uniqueness of strong solutions. |
F 9/12 | Regularity of the strong solutions. Instantaneous smoothing and long term decay. |
W 9/17 | Extension to arbitrary dimensions. |
F 9/19 | Riemann integration of vector valued functions. |
W 9/24 | Duhamel's principle in L^{2}. Uniqueness. |
F 9/26 | Duhamel's principle in L^{2}. Mild solutions. |
W 10/1 | Duhamel's principle in L^{2}. Strong solutions. |
F 10/3 | Semilinear parabolic equations. Local existence and uniqueness. |
W 10/8 | Semilinear parabolic equations. Maximal solutions and blow-up criteria. |
F 10/10 | Regularity of mild solutions. Comparison principles. Finite time blow-up. |
W 10/15 | Global existence. Viscous Burgers' equations: Basic estimate. |
F 10/17 | Viscous Burgers' equations: Higher order estimates. |
W 10/22 | Multiplication in Sobolev spaces. Leray projector. |
F 10/24 | Local theory of the Navier-Stokes equations. |
W 10/29 | Global solutions in two dimensions. Ladyzhenskaya inequality. |
F 10/31 | Global solutions in three dimensions. Scaling properties. |
W 11/5 | Leray's regularization. |
F 11/7 | Negative order Sobolev spaces. |
W 11/12 | Distributions. |
F 11/14 | Pettis theorem. Bochner integral. |
W 11/19 | Bochner-Lebesgue spaces. |
F 11/21 | Weak dual topology. Banach-Alaoglu theorem. |
W 11/26 | Weak convergence. Strong derivatives. Bochner-Sobolev spaces. |
F 11/28 | Weak solutions to the Navier-Stokes equations. |
W 12/3 | Rellich's lemma for Bochner-Sobolev spaces (cancelled). |
Instructor: Dr. Gantumur Tsogtgerel
Topics: The course will be sufficiently self-contained to be accessible to undergraduates with some background in real analysis, and can serve as an introduction to the basic tools used in the analysis of nonlinear parabolic systems, or even as an introduction to rigorous treatments of PDEs. At the same time, the students will have a taste of the difficulties involved in the Navier-Stokes regularity problem. Our starting point will be the heat equation, Fourier series, and Sobolev spaces. Then we will cover the classical Leray-Hopf existence theory, and the partial regularity theory of Scheffer-Caffarelli-Kohn-Nirenberg. If time permits, further specialized topics will be included.
Grading: Homework 50% and the final project 50%.