##

## Core Course: Analysis of
Partial Partial Differential Equations-3

**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, contact 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 e**nergy** estimates, Galerkin method;
G**lobal** 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 4*

**Homework
Problem Sets:**

** **

**Announcement**

**Topics & References:** **See
****Lecture 0**