# Mathematics Taught Course Centre

## Hyperbolic Partial Differential Equations

#### October - December 2011, Monday 10:00-12:00Prof. G-Q Chen.

The mathematical theory of hyperbolic partial differential equations is a large subject, which has close connections with the other branches of mathematics including analysis, mechanics, mathematical physics, and differential geometry/topology.  Besides its mathematical importance, it has a wide range of applications in engineering, mechanics, physics, biology, and economics.

This course will be focused on the nonlinear theory of hyperbolic PDE as an introduction.  No previous knowledge of hyperbolic PDE will be assumed.   However, some familiarity with basic PDE theory, analysis, and algebra is desirable.

Lecture 8

##### Topics:

1. Introduction: Conservation laws, Euler equations; connections (Einstein equations, calculus of variations, differential geometry, among others); hyperbolic systems, prototypes; basic features and phenomena; ...
2. One-Dimensional Theory: Riemann problem, Cauchy problem; elementary waves: shock waves, rarefaction waves, and contact discontinuities; Lax entropy conditions; Glimm scheme, front-tracking approximations, BV solutions; compensated compactness, entropy analysis, L-p solutions; vanishing viscosity methods; uniqueness and continuous dependence; ....
3. Multidimensional Theory: Basic features/phenomena (re-visit); important models; steady problems; self-similar problems; discontinuities and free boundary problems; stability of shock waves, rarefaction waves, vortex sheets; divergence-measure fields; ...
4. Nonlinear Wave Equations: Existence of solutions; semilinear wave equations; critical power nonlinearity; nonexistence of solutions.
5. Further Connections and Applications
##### References:

1. Richard Courant and David Hilbert, "Methods of Mathematical Physics", Vol. II. Reprint of the 1962 original. John Wiley & Sons, Inc.: New York, 1989.
2. Constantine Dafermos, "Hyperbolic Conservation Laws in Continuum Physics", Third edition. Springer-Verlag: Berlin, 2010.
3. Lawrence C. Evans, "Partial Differential Equations", Second edition. AMS: Providence, RI, 2010.
4. Lars Hormander, "Lectures on Nonlinear Hyperbolic Differential Equations", Springer-Verlag: Berlin-Heidelberg, 1997
5. Peter D. Lax, "Hyperbolic Differential Equations", AMS: Providence, 2000
6. Alberto Bressan, Gui-Qiang Chen, Marta Lewicka, and Dehua Wang, ``Nonlinear Conservation Laws and Applications", IMA Volume 153, Springer: New York, 2011.
7. Denis Serre, "Systems of Conservation Laws", 1, 2, Cambridge University Press: Cambridge, 1999, 2000.
8. Christopher D. Sogge, Lectures on Nonlinear Wave Equations,Second edition. International Press, Boston, MA, 2008.