Math 281: Hyperbolic PDE (Spring 2011)
This course will cover
linear and nonlinear wave motion, dispersion,
stationary phase,
foundations of continuum mechanics,
characteristics,
systems of hyperbolic conservation laws.
entropies and admissible solutions,
L1 theory for scalar conservation laws,
Riemann problem, shock waves, rarefaction waves, and contact discontinuties,
gas dynamics, compensated compactness,
second order nonlinear wave equations.
Prerequisites
Mathematics 232 or equivalent
Schedule
TuTh, 8:309:45am, Room 205 Physics Building
Instructor
JianGuo Liu,
Office: Room 285 Physics Building, 6602546, jliu "at" math.duke.edu
Office hours
Tuesday and Thursday 10:00amnoon, room 285 Physics,
or by email request
for an appointment for other times.
Textbook

Systems of Conservation Laws 1: Hyperbolicity, Entropies, Shock Waves
,
by Denis Serre,
Cambridge University Press, 1999
Syllabus

Continuum mechanics and balance laws, derivation of gas dynamics eqautions, weak solutions, shock front and RankineHugoniot jump condition,
nonlinear wave pattens, shock wave, rarefaction wave, contact discontinuous, method of charasteristics, symmetrizable system and convex entropy

Scalar consevation laws, Lax formula, ColeHopf transformation and vanishing viscosity limit,
admissible solutions and entropy condition, Krushkov theorem, CrandallTartar Lemma and contractive semigroup,
Global existence of BV solution, uniqueness

System of hyperbolic conservation laws, hyperbolicity, Riemann problem, travel waves and stability of nonlinear waves,
compensated compactness and divcurl lemma, Global solution of isentripic ga dynamics equation, Rienamm invariants,

Dispersive waves, dispersion relations, and the asymptotic calculation of solutions using the method of steepest descent.
KdV eqaution, CamassaHolm equation, integrable system, bihamiltonian structure, Laxpair, soliton solution,

Second order nonlinear wave equations, Strichartz estimates
Reference books

Front Tracking for Hyperbolic Conservation Laws
by
Helge Holden and Nils H. Risebro
,
Springer, 2002

Hyperbolic Partial Differential Equations
by
Peter D Lax
,
American Mathematical Society, 2006

Hyperbolic Conservation Laws in Continuum Physics
by
Constantine M. Dafermos
,
Springer; 3rd ed. edition, 2009

Partial Differential Equations: Second Edition
by
,
American Mathematical Society, 2010

Linear and Nonlinear Waves
by
G.B. Whitham,
,
WileyInterscience, 1999