January - February 2013

Conservation Laws & Applications

R.M. Colombo - M. Garavello - G. Guerra

A PhD course in Mathematical Analysis, with emphasis on the modeling.

Aim of the course is to bring students to the boundary between what is known and what is not known in the theory of conservation laws and their applications.

This web page will be regularly updated with news about the course: timetable, contents of the lectures delivered, useful links, ...

Presentation Schedule Preliminary Program References Examination


First lecture: Wednesday, January 16th, 10:00 a.m., Building U5, Room 3014 (the same of the presentation).

Possible next lectures:
Wednesday, 23.01, 10:00-13:00
Wednesday, 30.01, 10:00-13:00
Friday, 01.02, 10:00-12:00, 13:15-15:15
Thursday, 07.02, 10:00-13:00, Garavello
Thursday, 14.02, 09:30-11:30, 13:30-15:30, Guerra
Wednesday, 20.02, 9:30-12:30, Guerra, Room U5-3014
Wednesday, 20.02, 13:30-15:30, Garavello, Room U5-2107
Thursday, 21.02, 13:30-15:30, Guerra, Room U5-3014
Monday 25.02, 9:30-12:30, Garavello, Room U5-3014
Monday 04.03, 14:00-16:00, Garavello, Room U5-3014

Preliminary Program

(The real program will consist of a selection of the topics below).

Introduction to Conservation Laws: examples, motivations, state of the art.

1D equation: the rise of discontinuities. Weak formulation, lack of uniqueness.

1D systems: the linear case. The Riemann Problem, how to choose the good solutions.

1D systems: wavefront tracking: existence theorems.

1D systems: problems with boundary and/or with source

1D systems: junctions, networks and nonlocal flows.

MultiD equations: Kruzkov approach.

MultiD equations: stability and nonlocal flows.

Mixed problems: conservation laws coupled with other equations.

Throughout, analytical results are motivated / illustrated by means of applications to: fluid dynamics, vehicular traffic, crowd dynamics, supply chains, irrigation channels, blood circulation, granular matter dynamics, phase transitions, ...


Here is an introductory paper in English.

An introductory paper (in Italian):
Alberto Bressan
Leggi di Conservazione
Boll. Unione Mat. Ital. Sez. A Mat. Soc. Cult. (8) 6 (2003), no. 3, 415-439.

A reference book about conservation laws:
Constantine M. Dafermos
Hyperbolic conservation laws in continuum physics. Third edition.
Grundlehren der Mathematischen Wissenschaften 325. Springer-Verlag, Berlin, 2010.

A reference book about 1D systems:
Alberto Bressan
Hyperbolic systems of conservation laws. The one-dimensional Cauchy problem.
Oxford Lecture Series in Mathematics and its Applications, 20. Oxford University Press, Oxford, 2000.

An overview of 1D systems:
Rinaldo M. Colombo
Wave Front Tracking in Systems of Conservation Laws

Course delivered at the school Mathematical Theory in Fluid Mechanics
Applications of Mathematics, 49, 6, 501-537, 2004


The examination wll meet the standards of the PhD courses requiring it. Presumably, when an examination will be required, the candidates will be asked to deliver a seminar.