- Lecturer: Gui-Qiang Chen

General Prerequisites:

A good background in Multivariate Calculus and Lebesgue Integration is expected (for instance as covered in the Oxford Prelims and Part A Integration). It would be useful to know some basic Functional Analysis and Distribution Theory; however, this is not strictly necessary as the presentation will be self-contained.

Course Term: Hilary

Course Lecture Information: 16 lectures.

Course Weight: 1

Course Level: M

Assessment Type: Written Examination

Course Overview:

We introduce analytical and geometric approaches to hyperbolic equations, by discussing model problems from transport equations, wave equations, and conservation laws. These approaches have been applied and extended extensively in recent research and lie at the heart of the theory of hyperbolic PDEs.

Course Synopsis:

1. Transport equations and nonlinear first order equations: Method of characteristics, formation of singularities

2. Introduction to nonlinear hyperbolic conservation laws: Discontinuous solutions, Rankine-Hugoniot relation, Lax entropy condition, shock waves, rarefaction waves, Riemann problem, entropy solutions, Lax-Oleinik formula, uniqueness.

3. Linear wave equations: The solution of Cauchy problem, energy estimates, finite speed of propagation, domain of determination, light cone and null frames, hyperbolic rotation and Lorentz vector fields, Sobolev inequalities, Klainerman inequality.

4. Nonlinear wave equations: local well-posedness, weak solutions

If time permits, we might also discuss parabolic approximation (viscosity method), compactness methods, Littlewood-Paley theory, and harmonic analysis techniques for hyperbolic equations/systems (off syllabus - not required for exam)

2. Introduction to nonlinear hyperbolic conservation laws: Discontinuous solutions, Rankine-Hugoniot relation, Lax entropy condition, shock waves, rarefaction waves, Riemann problem, entropy solutions, Lax-Oleinik formula, uniqueness.

3. Linear wave equations: The solution of Cauchy problem, energy estimates, finite speed of propagation, domain of determination, light cone and null frames, hyperbolic rotation and Lorentz vector fields, Sobolev inequalities, Klainerman inequality.

4. Nonlinear wave equations: local well-posedness, weak solutions

If time permits, we might also discuss parabolic approximation (viscosity method), compactness methods, Littlewood-Paley theory, and harmonic analysis techniques for hyperbolic equations/systems (off syllabus - not required for exam)