All lectures will be from 10:00-12:00 on Mondays
This course will focus on the local and global analysis of transport and wave equations, both linear and nonlinear. After introducing some key concepts from ODE and PDE theory (the method of characteristics, Picard iteration, Sobolev spaces…), we will study linear free transport equations in phase space and linear wave equations. In the second half of the course, we will specialise to the analysis of global existence theory for two basic equations from kinetic theory (Vlasov-Poisson and Vlasov-Maxwell) where transport equations play a central role, and then to analysis of existence and uniqueness of nonlinear wave equations.
Useful resources:
- Lecture notes of Prof. Clément Mouhot on kinetic theory (Cambridge)
- Lecture notes of Prof. François Golse on kinetic theory (École polytechnique)
- Lecture notes of Prof. Jean Dolbeault on kinetic theory (Paris Dauphine)
- Lecture notes of Prof. Sigmund Selberg on PDEs (NTNU)
- Prof. Terence Tao's book on dispersive equations (UCLA)
Course Plan
12/10: ODEs and Connections to Evolution Equations (AS)
19/10: PDEs and Kinetic Theory (JBA)
26/10: Vlasov-Poisson: Local Existence (JBA)
2/11: No lecture (technical difficulties)
9/11: No lecture (technical difficulties)
16/11: Vlasov-Poisson: Global Existence (JBA)
23/11: Linear wave equations (AS)
30/11: Vlasov-Maxwell: Global Conditional Existence (JBA)
7/12: Nonlinear wave equations: classial existence and uniqueness (AS)
14/12: Nonlinear wave equations: the vector-field method, global and long-time existence (AS)