| Date | Topics |
| W 1/8 | Test functions and distributions. |
| F 1/10 | Operations on distributions. Support and order. |
| W 1/15 | Local structure of distributions. The Fourier transform. |
| F 1/17 | Tempered distributions. Paley-Wiener theorem. |
| W 1/22 | Compactly supported distributions. Convolution of distribution-function pairs. |
| F 1/24 | Convolution of distributions. |
| W 1/29 | Fundamental solutions. Laurent expansion and hypoellipticity. |
| F 1/31 | Schwartz theorem on hypoellipticity. |
| W 2/5 | Malgrange-Ehrenpreis theorem. |
| F 2/7 | Hypoelliptic polynomials. |
| W 2/12 | Hörmander's theorem. |
| F 2/14 | Ellipticity. Petrowsky's theorem. |
| W 2/19 | Microlocal regularity. |
| F 2/21 | Propagation of singularities. |
| W 2/26 | Gårding hyperbolicity. |
| F 2/28 | Fundamental solutions supported in a cone. |
| 3/2–3/6 | Study break |
| W 3/11 | Petrowsky well posedness. |
| F 3/13 | Class cancelled |
| W 3/18 | Class cancelled |
| F 3/20 | Class cancelled |
| W 3/25 | Class cancelled |
| F 3/27 | Class cancelled |
| W 4/1 | Strong well posedness. |
| F 4/3 | Duhamel's principle. Semilinear evolution equations. |
| W 4/8 | The Navier-Stokes equations in 2D. |
| F 4/10 | Good Friday |
| T 4/14 | Weak solutions of the Navier-Stokes equations. |
Instructor: Dr. Gantumur Tsogtgerel
Prerequisite: MATH 580 (PDE1), MATH 355 (Honours Analysis 4) or equivalent
Note: If you plan to take this course without taking MATH 580, please consult with the instructor.
Topics: The main focus of the course is going to be on nonlinear problems. Sobolev spaces, the Fourier transform, and functional analytic methods will be heavily used.
Calendar description: Systems of conservation laws and Riemann invariants. Cauchy-Kowalevskaya theorem, powers series solutions. Distributions and transforms. Weak solutions; introduction to Sobolev spaces with applications. Elliptic equations, Fredholm theory and spectra of elliptic operators. Second order parabolic and hyperbolic equations. Further advanced topics may be included.
Homework: Assigned and graded roughly every other week.
Weakly seminars: We will organize weekly seminars on standard results from analysis and geometry, and other stuff related to the course.
Course project: The course project consists of the student reading a paper or monograph on an advanced topic, typing up notes, and giving a lecture.
Grading: Homework assignments 50% + Course project 50%.