Course projects

  • Olivier Hénot - Ricci flow
  • Zhenhe Zhang - Burgers' equation
  • Peter Yuen - Schrödinger equation
  • Dana Berman - Method of moving planes
  • Gabriel Rioux - Monge-Ampere equations
  • Damien Tageddine - Nash-De Giorgi-Moser regularity theory
  • Ian Weih-Wadman - Eigenvalue problems for the p-Laplacian
  • Thomas Tendron - Kolmogorov-Petrowsky-Piskunov equations
  • Matthieu Cadiot - Periodic solutions of the Navier-Stokes equations
  • David Knapik - Cloaking in acoustic and electromagnetic scattering
  • Edward Chernysh - Global compactness results for p-Laplace equations
  • Alexandros Kontogiannis - Shape optimization using the Navier-Stokes equations
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    Assignments

  • Assignment 1 [tex] due Thursday January 31
  • Assignment 2 [tex] due Thursday February 14
  • Assignment 3 [tex] due Tuesday March 12
  • Assignment 4 [tex] due Thursday March 28
  • Assignment 5 [tex] due Thursday April 11
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    Reading material

  • Elements of distributions (Updated on Feb 9)
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    Class schedule

  • TR 11:05–12:25, Burnside Hall 1214

    Date Topics
    T 1/8 Green's identities for the heat operator. The Euclidean heat kernel.
    R 1/10 Green's representation formula. The Cauchy problem. Heat spheres.
    T 1/15 Mean value property for caloric functions. Maximum principles.
    R 1/17 Tychonov's uniqueness theorem. Functional calculus.
    T 1/22 Spectral resolution of the heat equation. Smoothing. Decay. Backward ill-posedness.
    R 1/24 Backward uniqueness. Caloric measure. Gradient estimates.
    T 1/29 Li-Yau Harnack inequality.
    R 1/31 Distributions.
    T 2/5 Support. Local structure of distributions.
    R 2/7 Fourier transform. Tempered distributions.
    T 2/12 Cauchy problem for wave equations.
    R 2/14 Fundamental solutions. Energy.
    T 2/19 Finite speed of propagation. Existence results.
    R 2/21 Morawetz estimate on local energy decay.
    T 2/26 The Cauchy-Kovalevskaya theorem.
    R 2/28 Characteristics. Holmgren's uniqueness theorem.
    3/4–3/8 Study break
    T 3/12 Well posedness. Fundamental solutions. Point supported distributions. Liouville's theorem.
    R 3/14 Hypoellipticity. Laurent expansion. Convolutions.
    T 3/19 Convolution between distributions. Schwartz's theorem.
    R 3/21 Malgrange-Ehrenpreis theorem. Hörmander's theorem on hypoellipticity.
    T 3/26 Petrowsky's theorem on ellipticity.
    R 3/28 Gårding hyperbolicity. Petrowsky well-posedness.
    T 4/2 Strong well-posedness.
    R 4/4 Strong hyperbolicity and Petrowsky parabolicity.
    T 4/9 Duhamel's principle. Semilinear problems.
    R 4/11 Sobolev algebras. Ladyzhenskaya inequality. Navier-Stokes equations.

    Student seminar

  • F 10:35–11:55, Burnside Hall 1234

    Date Topics Speaker
    F 2/22 Moser's Harnack inequality
    Damien
    F 3/1 Moser's iteration
    Damien
    F 3/15 Critical p-Laplace equations
    Edward
    F 3/22 No seminar ~
    F 3/29 Monge-Ampère equation
    Gabriel
    F 3/29 KPP equations
    Thomas
    F 4/5 Schrödinger equation
    Peter

    Online resources

  • Previous incarnations: 2012, 2013, 2014
  • Lecture notes by Bruce Driver (UCSD)
  • Teaching page of John Hunter (UC Davis)

    Reference books

  • Lawrence Craig Evans, Partial differential equations. AMS 1998.
  • Fritz John, Partial differential equations. Springer 1982.

    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 evolution equations and nonlinear problems. Sobolev spaces, the Fourier transform, and functional analytic methods will be heavily used. The planned topics are

  • Heat and wave equations
  • Tempered distributions, convolution, Fourier transform
  • Fourier analytic treatment of Sobolev spaces
  • Problems in half-space, shades of hyperbolicity, parabolicity, and ellipticity
  • Semilinear evolution equations
  • The Navier-Stokes equations and related turbulence models
  • Overview of elliptic theory, regularity
  • Semilinear elliptic equations, monotonicity methods, variational problems (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.

    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%.