Math 256B: (Graduate) Partial differential equations II

Course description

This course will be a continuation of Andras's 256A.

We will start with some Calderon-Zygmund theory, working towards the L^p boundedness of pseudo-differential operators. (I will assume familiarity with the L^2 case, which was what 256A ended with.)

The remainder of the course will be about parabolic PDEs: We start with the linear theory, weak solutions, maximum principles, semigroup methods. (The pre-requisites for this part are essentially just the first part of 256A).

Depending on interest, the last part of the course will vary:

Option 1: Hölder regularity and Harnack inequalities via probabilistic methods. (I will only chose this option if those attending at this point have the required probability background).

Option 2: Non-linear PDE's: Variational methods, reaction diffusion, Navier-Stokes equations. (This part has no pre-requisites except for the first half of the course)

Mailing list for the class: math256b-win0708-guests

Lecture schedule (and notes)

• Lectures 1--3: L^p boundedness of Pseudo differential operators. (Big Stein, Chapter 1 and 6. Small Stein Chapter 2)
• Lectures 4--6: Parabolic PDE's: Weak solutions, existence and regularity. (Evans chapter 7)
• Lectures 7--9: Maximum principles, Harnack inequalities. (Evans Chapter 7, and Niermberg's paper).
• Lectures 10--13: Navier-Stokes equations (Constantin, Foias).
• Lectures 14--17 (Part 1 Part 2) Reaction Diffusion Equations (Unpublished notes of Lenya Ryzhik).
• Lecture 18 (Part 1 Part 2) ABP maximum principle, and sliding method. (Unpublished notes of Lenya Ryzhik)
• Lecture 19 Littlewood Paley theory, and L^p estimates for parabolic PDEs. (Stein, Krylov's paper, Ladyzhenskaya)

Problems

1. Singular integrals
2. Parabolic PDE's

References

1. Constantin and Foias Navier Stokes Equations
2. Evans Partial Differential Equations
3. Krylov A generalization of the Littlewood-Paley inequality and some other results related to stochastic partial differential equations. (Ulam Quaterly, '94).
4. Ladyzhenskaya Linear and quasilinear equations of parabolic type
5. Niremberg A Strong Maximum Principle for Parabolic Equations (CPAM '53).
6. Stein Singular Integrals and Differentiability Properties of Functions
7. Stein Harmonic Analysis