Core Course: Analysis of Partial Partial Differential Equations-3
Trinity
Term 2019, Prof. Gui-Qiang G. Chen
and Professor Qian Wang
Duration: 16 hours
Overview:
This is a CDT Course on the
theory of hyperbolic PDEs and related PDEs, as one of the PDE-CDT series of
core courses on the analysis of PDEs, which forms the backbone of the
first-year CDT training programme.
Synopsis:
Linear Theory: Spaces involving
time; Second-order
hyperbolic equations, hyperbolic
systems of first-order equations, examples; Weak solutions, well-posedness;
Galerkin method, Vanishing viscosity method, energy methods, Fourier transform
method.
Nonlinear Theory I – Multidimensional Scalar Conservation Laws: L¹ - well-posedness
theory, test function methods, vanishing viscosity method; *Other methods
(numerical methods, kinetic method, relaxation method, the layering method,
…); *Further results
(compactness, regularity, decay, trace, structure).
Nonlinear Theory II – One-Dimensional Systems of Conservation
Laws: Riemann problem,
Cauchy problem; Elementary waves: shock waves, rarefaction waves, contac discontinuities; Lax entropy conditions; Glimm
scheme, front-tracking,
BV solutions; *Compensated compactness, entropy analysis, Lᵖ
solutions, vanishing viscosity methods; *Uniqueness and continuous
dependence.
Nonlinear Theory III – Noninear Wave
Equations: Local existence and energy estimates, Galerkin method;
Global existence of semi-linear wave equations with small data (Quasilinear
case could be similarly treated); Lower regularity results for large data; *Littlewood-Paley theory and Strichartz
estimates.
*Nonlinear Theory IV - Multidimensional Systems of Conservation
Laws: Basic
features/phenomena (re-visit); Local existence and stability; formation of
singularities; Discontinuities and free boundary problems; Stability of shock
waves, rarefaction waves, vortex sheets, entropy waves.
*Optional
Prerequisites:
Introduction to PDE foundation module and Analysis of
PDEs, Parts 1 and 2
Lecture Notes:
Lecture 0
Lecture 1
Lecture 2
Lecture 3
Homework Problem Sets: