General Prerequisites:
A4 Integration is highly recommended and a basic knowledge of metric spaces is assumed. Ideas from B4.1 and B4.2 (Functional Analysis I and II) are lightly used but not required since the course is self-contained. A brief study of L^p spaces and Sobolev embedding is sufficient pre-requisite understanding for the course.
Course Term: Hilary
Course Weight: 1
Course Level: M
Assessment Type: Written Examination
Course Overview:
We introduce geometric and analysis approaches to hyperbolic equations, by discussing model problems from wave equations and conservation laws. These approaches have been applied and extended extensively in recent research, and lie in the heart of theory of hyperbolic PDEs.
Lecturer(s):
Dr Jan Sbierski
Course Synopsis:
- Sobolev space and Sobolev inequalities
- Nonlinear first order equations: Eikonal equations and method of characteristics
- Introduction to conservation laws in one space dimension (shocks, simple waves, rarefaction waves, Riemann problem)
- Theory of linear wave equation : The solution of Cauchy problem, energy estimates, finite speed of propagation, domain of determination, ligntcone and null frames, hyperbolic
rotation and Lorentz vector elds, Klainerman inequality. - Weak solution of wave equation, and local well-posedness
- Littlewood-Paley theory and harmonic analysis technique for wave equation (off syllabus - not required for exam)