Hyperbolic Partial Differential Equations
October - December 2011, Monday 10:00-12:00 Prof. G-Q Chen.
No lecture on 17 October as Prof Chen is away
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.
Topics:
-
Introduction:
Conservation laws,
Euler equations; connections (Einstein equations, calculus of
variations,
differential geometry, among others); hyperbolic systems, prototypes;
basic
features and phenomena; ...
-
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;
....
-
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; ...
- Nonlinear
Wave Equations:
Existence of solutions; semilinear wave equations; critical power
nonlinearity;
nonexistence of solutions.
- Further
Connections and
Applications
References:
- Richard Courant and David Hilbert, "Methods of Mathematical
Physics", Vol. II. Reprint of the 1962 original. John Wiley & Sons,
Inc.: New York, 1989.
- Constantine Dafermos, "Hyperbolic Conservation Laws in Continuum
Physics", Third edition. Springer-Verlag: Berlin, 2010.
- Lawrence C. Evans, "Partial Differential Equations", Second
edition. AMS: Providence, RI, 2010.
- Lars Hormander, "Lectures on Nonlinear Hyperbolic Differential
Equations", Springer-Verlag: Berlin-Heidelberg, 1997
- Peter D. Lax, "Hyperbolic Differential Equations", AMS:
Providence, 2000
- Alberto
Bressan, Gui-Qiang Chen, Marta
Lewicka, and Dehua Wang, ``Nonlinear
Conservation Laws and Applications",
IMA Volume 153,
Springer: New York, 2011.
- Denis Serre, "Systems of Conservation Laws", 1, 2, Cambridge
University Press: Cambridge, 1999, 2000.
- Christopher D. Sogge, Lectures on Nonlinear Wave Equations,Second
edition.
International Press, Boston, MA, 2008.
