University of Oxford                

EPSRC - Engineering and Physical Sciences Research Council


EPSRC Centre for Doctoral Training

In Partial Differential Equations: Analysis and Applications


Core Course:  Analysis of Partial Partial Differential Equations-3

Trinity Term   2017,   Prof. Gui-Qiang G. Chen and Professor Wei Xiang


Duration:     16 hours


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.



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 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.





Prerequisites:  Introduction to PDE foundation module and Analysis of PDEs, Parts 1 and 2


Lecture Notes:

Lecture 0
Lecture 1
Lecture 2
Lecture 3


Lecture 4-a


Lecture 4-b


Homework Problem Sets:

Problem Set-1
Problem Set-2





Topics & References:  See  Lecture 0