Teaching / Course archive
Math 581: Winter 2020
PDE 2
Course Record
Overview
PDE II course on distributions, Fourier transform, fundamental solutions, hypoellipticity, well-posedness, semilinear evolution equations, and the Navier-Stokes equations.
Schedule and Topics
| Date | Topic |
|---|---|
| W 1/8 | Test functions and distributions. |
| W 1/15 | Local structure of distributions and the Fourier transform. |
| W 1/29 | Fundamental solutions, Laurent expansion, and hypoellipticity. |
| W 2/5 | Malgrange-Ehrenpreis theorem. |
| W 2/12 | Hormander's theorem. |
| W 2/19 | Microlocal regularity. |
| W 3/11 | Petrowsky well-posedness. |
| W 4/1 | Strong well-posedness. |
| F 4/3 | Duhamel's principle and semilinear evolution equations. |
| T 4/14 | Weak solutions of the Navier-Stokes equations. |
Topic List
- Cauchy-Kovalevskaya theorem, Holmgren's theorem, and characteristic surfaces
- Distributions, convolution, Fourier transform, and fundamental solutions
- Constant coefficient operators, hypoellipticity, and the Malgrange-Ehrenpreis theorem
- Ellipticity, parabolicity, hyperbolicity, and well-posedness
- Sobolev spaces, semilinear evolution equations, and applications to Navier-Stokes type problems
Books and Related Pages
Books
- Gerald B. Folland, Introduction to partial differential equations. Princeton, 1995.
- Gregory I. Eskin, Lectures on linear partial differential equations. AMS, 2011.
- Joseph Wloka, Partial differential equations. Cambridge, 1987.
- Francois Treves, Basic linear partial differential equations. Dover, 2006.
Assignments and Projects
Assignments and Exams
- Assignments 1-5.
- Course project.