EPSRC Centre for Doctoral Training In Partial Differential Equations: Analysis and Applications

Core Course: Analysis of Partial Partial Differential Equations-3

Trinity Term 2015, 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 II – 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 Nonservation 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.

*Optional

Prerequisites: Introduction to PDE foundation module and Analsyis of PDEs, parts 1 and 2

Lecture Notes:

Lecture 0

Lecture 1

Lecture 2

Lecture 3

Lecture 4

Homework Problem Sets:

Problem Set-1

Problem Set-2

Announcement

Topics & References: See Lecture 0

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