# In Partial Differential Equations: Analysis and Applications

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

#### Trinity Term   2016,Prof. Gui-Qiang G. Chen and Professor Qian Wang

Duration:     16 hours

#### 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 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 3

Lecture 4

Homework Problem Sets: