Course outline

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. The planned topics are

  • Tempered distributions, convolution, Fourier transform
  • Fourier analytic treatment of Sobolev spaces
  • Problems in half-space, shades of hyperbolicity, parabolicity, and ellipticity
  • Overview of elliptic theory, regularity
  • Semilinear elliptic equations, monotonicity methods
  • Variational problems, compactness methods
  • Semilinear evolution equations, Duhamel's principle
  • The Navier-Stokes equations and related turbulence models
  • Semilinear elliptic problems with critical exponents (if time permits)

    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.

    Books: There is no required textbook. The following are recommended.

  • Gerald Budge Folland, Introduction to partial differential equations. Princeton 1995.
  • Joseph Theodor Wloka, Partial differential equations. Cambridge 1987.

    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%.

    Online resources

  • Lecture notes by Bruce Driver (UCSD)
  • Teaching page of John Hunter (UC Davis)

    Class schedule

  • TR 10:05–11:25, Burnside Hall 1205

    Date Topics
    T 1/8 ...
    R 1/10 ...
    ... ...
    3/4–3/8 Study break
    ... ...
    R 4/11 ...

    Student seminar

  • TBD

    Date Topics Speaker
    ... ...
    ...