Instructor: Yannick Sire

Email: sire**AT**math.jhu.edu

Office Hours: Krieger 208, By appointment- Main Textbooks: Partial Differential Equations by L.C. Evans supplemented by Functional Analysis, Sobolev Spaces and PDEs by H. Brezis
Syllabus : I will start by giving a review on (space-time) Sobolev spaces and the necessary functional analysis for the course (from Brezis' book and Evans' book). Then, I will cover Evans' chapter on linear evolution equations. The rest of the course will be devoted to several topics dealing with evolutionary nonlinear equations:

1- Nonlinear parabolic theory. Fixed point Theorems. Topological degree.

2- Strichartz estimates for Schrodinger and wave propagators. Elements of Littlewood-Paley theory

3- Infinite dimensional systems: elements of symplectic geometry, invariant manifolds for evolutionary ODEs and PDEs, Lyapunov-Schmidt reduction for PDEs, normal forms.

4- KAM theorem. Small divisors.Homeworks are due every Monday. I will assign all the HW from Evans' book. Firstly, on chapter 7 covered in class and then I will ask you to read yourself the following chapters of Evans' dealing with nonlinear equations and few problems related to them. At the beginning of each monday lecture, I will answer questions related to these chapters not covered in class. I will send the HWs by email.