Hyperbolic Conservation Law
Course Description:Hyperbolic conservation law has many applications (traffic flows, fluid dynamics, biology...), and there are also many beautiful theories in this subject. In this course, we will discuss the typical problems in this field. As the introduction part, the Burgers’ equation will be studied as a toy model, the detailed and important issues will be given, for example, like singularity formulation, Rankine-Hugoniot condition, entropy condition, L^1 contraction, Riemann problem, wave interaction and large time behavior. Based on such issues and the methods, traces and ideas therein, we will discuss the general conservation law systems in the second part of this course, Kruzkov’s theory, compensated compactness framework and Glimm Scheme will be introduced. The third part is for the application on the topic of nozzle flows, both irrotational and rotational flows will be studied.
Prerequisites: Analysis I, II, basic knowledge on differential equations will be helpful.
 Constantine M. Dafermos; Hyperbolic Conservation Laws in Continuum Phyisics, Springer-Verlag 2010, Third Edition.
 ZHENG Yuxi; System of Conservation Laws, Birkhäuser 2001.
Lecture 1: Introductions
1. Syllabus including text book, grade, assignment, tutorial class, office hour and key points.
2. System, applications and examples.
3. Formulation of the general Balance law and Euler equations.
Lecture 2: Basic concepts, research emphasis and a toy model
1. Definition of hyperbolicity.
2. Focus points: well-posedness, asymptotic behavior and numerical methods.
3. Burgers' equation: 1) Formulation of singularity 2) Definition of weak solution 3) Rakine-Hugoniot jump condition 4) Invalidity of nonlinear Transformation
Lecture 3: Existence and entropy conditions
1. Existence Theorem (by vanishing viscosity approach)
2. Loss of uniqueness (shock wave, rarefaction wave)
3. Physical concerns and entropy conditions: Oleinik entropy condition, Lax geometric entropy condition, Liu's entropy condition and the equivalency.
Lecture 4: Uniqueness and L^1 Contraction principle
1. Uniqueness of weak entropy solution (Potential Method)
2. L^1 contraction principle
3. Smoothing mollifier
Lecture 5: Riemann problem, wave interaction and introduction of general conservation law
1. L^1 contraction principle
2. Riemann problem and typical waves
3. Interaction of waves
Lecture 6: General scalar conservation law I
1. Definition of entropy weak solution
2. Definition of entropy-entropy flux and motivation
3. Admissible conditions
4. Kruzkov's theory
Lecture 7: General scalar conservation law II
1. Existence of entropy weak solution
2. Kruzkov's stability estimate(part 1)
Lecture 8: General scalar conservation law III
1. Kruzkov's stability estimate(part 2)
2. Introduction of weak convergence method
3. Preliminaries for compensated compactness framework
Lecture 9: Compensated compactness framework I
2. Div-Curl Lemma and discussion on the hypotheses
3. Young measure
Lecture 10: Compensated compactness framework II
1. Main steps of the framework.
2. Application to scalar conservation law
Lecture 11: Examples, applications and research I
1. Mixed or change type PDEs
2. Blast wave equations
3. Airfoil problem
4. Nozzle flow problems
Lecture 12: Examples, applications and research II(END)
1. Shock reflection
2. Transonic nozzle flows with shock wave
3. Smooth transonic flows in nozzles