# Class schedule

*Note*: Click on the date to get

**Ibrahim's class notes**. He has also written some supplementray notes:

Wed 1/11 | Idea of distributions. Topological vector spaces. |

Fri 1/13 | Topological vector spaces. Hausdorff property. |

Wed 1/18 | Locally convex spaces. Seminorms. Fréchet spaces. |

Fri 1/20 | LF spaces. |

Wed 1/25 | Distributions. Radon measures. |

Fri 1/27 | Subspaces of distributions. Basic operations on distributions. |

Wed 2/1 | Sheaf structure of distributions. |

Fri 2/3 | Local structure of distributions. |

Wed 2/8 | Convolution. |

Fri 2/10 | Constant coefficient operators. Fundamental solutions. Hypoellipticity. |

Wed 2/15 | Schwartz theorem. Laurent expansion. Analytic hypoellipticity. |

Fri 2/17 | Fourier transform. Liouville's theorem. |

2/20–2/24 | Study break |

Mon 2/27 | Hörmander's characterization of hypoelliptic polynomials. |

Fri 3/2 | Problems in half-space. Cauchy problem. Petrowsky well-posedness. |

Wed 3/7 | Boundary value problems. Lopatinsky-Shapiro condition. |

Fri 3/9 | Strongly hyperbolic and p-parabolic systems. |

Wed 3/14 | Strong hyperbolicity. Inhomogeneous Cauchy problem. |

Fri 3/16 | Well-posedness of a general class of Cauchy problems. Parabolicity. |

Wed 3/21 | Semilinear evolution equations. |

Fri 3/23 | Multiplication in Sobolev spaces. Derivative nonlinearities. |

Wed 3/28 | Elliptic boundary value problems. Gårding inequality. |

Fri 3/30 | Gårding inequality proof. Dirichlet problem. Lax-Milgram lemma. |

Wed 4/4 | Friedrichs inequality. Rellich-Kondrashov compactness. |

Fri 4/6 | Good Friday |

Wed 4/11 | L2-regularity theory. |

Fri 4/13 | Spectral theory. Semigroups. |

# Assignments

# Final project

Date | ||

4/16 | Morgane Henry | Wave maps |

4/16 | Ibrahim Al Balushi | Nonlinear diffusion |

4/16 | Olga Yakovlenko | Introduction to the Ricci flow |

4/30 | Yang Guo | Einstein equations |

4/30 | Sebastien Picard | Introduction to the Yang-Mills equations |

4/30 | Spencer Frei | DeGiorgi-Nash-Moser regularity theory |

4/30 | Mario Palasciano | A nonlocal aggregation model |

4/30 | Olivier Mercier | Mean curvature flow |

4/30 | Joshua Lackman | Pseudodifferential operators |

4/30 | Andrew MacDougall | Some topics in semiclassical analysis |

The final project consists of the student studying an advanced topic, typing up expository notes, and presenting it in class. Here are some ideas for the project:

# Weekly seminars

PDE questions from previous qualifying exams for download.Date | |||

1/16 | Baire's theorem and consequences | Rudin Ch2, Tao | Gantumur |

1/23 | Hahn-Banach theorem | Rudin §3.1-3.7, Tao | Spencer |

1/30, 2/6 | Closed range theorem | Rudin §4.1-4.15 | Ibrahim |

2/13, 2/29 | Fredholm operators | Rudin §4.16-4.25, McLean 2.14-2.17, Tao | Mario |

3/5 | Banach-Alaoglu theorem | Rudin §3.8-3.18, Tao | Andrew |

3/12, 3/19 | Spectral theorem | Rudin Ch13, Jaksic | Sébastien |

3/26 | Hille-Yosida theorem | Rudin §13.34-13.37 | Yang |

4/2 | Navier-Stokes equations | Tao, Clay, Lei-Lin | Gantumur |

# References

*Functional analysis*. McGraw-Hill.

*Topological vector spaces, distributions and kernels*. Dover 1995.

*Lectures on linear partial differential equations*. AMS 2011.

*Generalized functions III*. Academic Press 1967.

*Basic linear partial differential equations*. Dover 2006.

*Initial-boundary value problems and the Navier-Stokes equations*. SIAM 2004.

*Introduction to partial differential equations*. Princeton 1995.

*Partial differential equations*. Cambridge 1987.

# Topics to be covered

*Distributions and constant coefficient operators*: Introduction to distributions, fundamental solutions, parametrices, hypoellipticity and ellipticity, introduction to Fourier transform, hyperbolicity and parabolicity;

*Linear elliptic operators*: Dirichlet principle, Sobolev spaces, Poincaré inequality, variable coefficients, stationary Stokes problem and linear elasticity, Harnack estimates, regularity theory, Sobolev embedding, Hodge-type decompositions, introduction to spectral and Fredholm theories;

*Nonlinear elliptic equations*: Lagrange multipliers, semilinear problems with subcritical, critical, and supercritical exponents, direct method of calculus of variations, De Giorgi-Nash-Moser regularity theory;

*Evolution equations*: Heat, wave, Schrödinger, and Stokes propagators, Navier-Stokes, Euler and magnetohydrodynamics equations, turbulence models, nonlinear heat and wave examples, weak and strong solutions, regularity theory, viscosity solutions, linear scattering theory (if time permits)

# Instructor

Dr. Gantumur Tsogtgerel*Office*: Burnside Hall 1123. Phone: (514) 398-2510.

*Office hours*: Just drop by or make an appointment

# Online resources

PDE Lecture notes by Bruce Driver (UCSD)Xinwei Yu's page (Check the Intermediate PDE Math 527 pages)

John Hunter's teaching page at UC Davis (218B is PDE)

Textbook by Ralph Showalter on Hilbert space methods

Lecture notes by Georg Prokert on elliptic equations