# Hyperbolic Conservation Law

Lecture: Mo. 13:45-15:15 in C012

Tutorial: Mo. 15:30-17:00 in C012

Description: Hyperbolic conservation law has many applications (traffic flows, fluiddynamics, biology...), 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.

Language: English

Prerequisites: Analysis I, II, basic knowledge on differential equations will be helpful.

References:

[1] Constantine M. Dafermos; Hyperbolic Conservation Laws in Continuum Phyisics, Springer-Verlag 2010, Third Edition.

[2] Yuxi Zheng; System of Conservation Laws, Birkhäuser 2001.

# Schedule

**09/02/15 **

**Lecture 1: Introductions**

- Syllabus including text book, grade, assignment, tutorial class, office hour and key points.
- System, applications and examples.
- Formulation of the general Balance law and Euler equations.

Assignment 1: (the paper version with formula will be delivered on 16 Feb)

Problem 1. For 2D steady isentropic Euler equations, with irrotational flow condition and polytropic gas condition, derive Bernoulli's principle.

Problem 2. For isentropic incompressible flows, prove that curl u=0 implies NSE and Euler equations are in the same form for dimension d=2, 3.

3. Prepare oral discussion on tutorial class(Feb 23, Monday), the topic is about the models in the form of Balance law or conservation law. Please find out one specific model and discuss the meaning of such system and its application. 5-10min for each student.

**16/02/15 **

**Lecture 2: Basic concepts, research emphasis and a toy model**

- Definition of hyperbolicity.
- Focus points: well-posedness, asymptotic behavior and numerical methods.
- Burgers' equation: 1) Formulation of singularity 2) Definition of weak solution 3) Rakine-Hugoniot jump condition 4) Invalidity of nonlinear Transformation

**23/02/15 **

**Lecture 3: Existence and entropy conditions**

- Existence Theorem (by vanishing viscosity approach)
- Loss of uniqueness (shock wave, rarefaction wave)
- Physical concerns and entropy conditions: Oleinik entropy condition, Lax geometric entropy condition, Liu's entropy condition and the equivalency.

Assignment 2:

Problem 1. With Cauchy initial condition u_0(x) = 1/(x^2 + 1), try to determine the life span of a smooth solution to Burger's equation.

Problem 2. Rewrite 2D steady isentropic Euler system into the general form of Balance law as the following:

AU_x + BU_y =0,

where U=(\rho, u, v) and A, B are 3x3 matrices. If possible, try to solve det(\lambda A - B) = 0 for \lambda.

**2/03/15 **

**Lecture 4: Uniqueness and L^1 Contraction principle**

- Uniqueness of weak entropy solution (Potential Method)
- L^1 contraction principle
- Smoothing mollifier

Prepare oral presentation for the next class: Maximal principle and Comparison principle for Harmonic functions.

**9/03/15 **

**Lecture 5: Riemann problem, wave interaction and introduction of general conservation law**

- L^1 contraction principle
- Riemann problem and typical waves
- Interaction of waves

Assignment 3:

Problem 1. Solve Burger's equation with initial data u_0(x)= 10, when x<5; 5, when 5<x<10; 2, when x>10. And draw the solution curve in x-t plane.

Problem 2. Solve Burger's equation with initial data u_0(x)= 12, when x<5; 3, when 5<x<10; 6, when x>10. And draw the solution curve in x-t plane.

Prepare oral presentation: Let initial date be u_0(x)= 10, when x<5; 5, when 5<x<10; y, when x>10. Discuss the asymptotic behavior of a solution with respect to each y.

**16/03/15 **

**Lecture 6: General scalar conservation law I**

- Definition of entropy weak solution
- Definition of entropy-entropy flux and motivation
- Admissible conditions
- Kruzkov's theory

Maximum principle and comparison principle for elliptic/parabolic equations will be discussed in the tutorial class.

**23/03/15 **

**Lecture 7: General scalar conservation law II**

- Existence of entropy weak solution
- Kruzkov's stability estimate(part 1)

Problems of assignment 3 will be discussed in the tutorial.

**13/04/15 **

**Lecture 8: General scalar conservation law III**

- Kruzkov's stability estimate(part 2)
- Introduction of weak convergence method
- Preliminaries for compensated compactness framework

**20/04/15 **

**Lecture 9: Compensated compactness framework I**

- Preliminaries
- Div-Curl Lemma and discussion on the hypotheses
- Young measure

Assignment 4 will be given as part of the proof for the main theorem.

**04/05/15**

**Lecture 10: Compensated compactness framework II**

- Main steps of the framework.
- Application to scalar conservation law

Assignment 5 will be given.

**11/05/15**

**Lecture 11: Examples, applications and research I**

- Mixed or change type PDEs
- Blast wave equations
- Airfoil problem
- Nozzle flow problems
- Others

**18/05/15**

**Lecture 12: Examples, applications and research II(END)**

- Shock reflection
- Transonic nozzle flows with shock wave
- Smooth transonic flows in nozzles