Spring School on Conservation Laws and Applications at LmB

18-20 mars 2025 Besançon (France)

FR EN

Une intervention sur la base de données est programmée jeudi 24 avril entre 11h00 et 12h00.
Elle peut occasionner des perturbations sur Sciencesconf.

Accueil

Main purpose

This thematic school focuses on a class of nonlinear partial differential equations, namely conservation laws. They are considered to be fundamental laws of nature, with broad application in physics, as well as in other fields such as chemistry, biology, geology, and engineering. It will be primarily aimed at M2 students, PhD students, as well as post-doctoral researchers, but also at more experienced researchers.

Video conference

Please click the link below:

https://cnrs.zoom.us/j/94011333652?pwd=EY61Dx09kMEQkLFgVezmObYTlVV1u7.1

Ps: You might need to install Zoom Workplace. If you use linux (e.g. ubuntu) and encounter sound problem, you might eventually install the snap version (see e.g. https://doc.ubuntu-fr.org/zoom).

Speakers

We will host four speakers:

Giuseppe Maria Coclite (Polytechnic University of Bari),
Andrea Corli (University of Ferrara),
Cyril Imbert (ENS-PSL),
Massimiliano Daniele Rosini (UMCS in Poland and University of Chieti-Pescara in Italy).

Topics and abstracts

They will each give lectures, approximately 3 hours long, on the following topics:

Compensated Compactness for Scalar Conservation Laws (Giuseppe Maria Coclite)

Abstract. Compensated Compactness is a useful measure theory tool based on the compression effects of nonlinear equations. It works only in the nonlinear case and applies in several convergence problems.

The lectures will be organized as follows:

1. Motivation
2. Compensated compactness
3. Young measures
4. Murat Lemma
5. Div-Curl Lemma
6. Proof of Compensated Compactness
7. Global Existence of Bounded Solutions
8. Long Time Behavior of Periodic Solutions
9. Diffusive-Dispersive Limits
Traveling waves for parabolic equations with degenerate diffusivities (Andrea Corli)

Abstract. These lectures are intended to provide an introduction to traveling waves for parabolic equations. After a short recap of the classical theory, the focus is on degenerate diffusivities, both in the case they simply vanish but remain otherwise positive, and in the case they become negative. Also the occurrence of discontinuous traveling waves is discussed. Some examples from biology and physics are provided to provide insight to the topic.
Discontinuous Hamilton-Jacobi equations (Cyril Imbert)

Abstract. The literature about Hamilton-Jacobi equations with discontinuous coefficients grew up since the beginning of the years 2010. In these lectures, I will focus on a few results of this large body of articles. We will first assume that the Hamiltonian is convex in the gradient variable and we will quickly review the literature about flux-limited solutions. We will then turn to non-convex Hamiltonians and describe Guerand's relaxation operator. A third part will be dedicated to the twin blow up method for proving strong uniqueness of solutions.

References: biblio.pdf
Introduction to Microscopic and Macroscopic Modeling of Vehicular Traffic (Massimiliano Daniele Rosini)

Abstract. We introduce fundamental concepts and key properties of vehicular traffic. We begin with a brief overview of state-of-the-art modeling approaches across different scales of description, with a particular focus on macroscopic models. We highlight the limitations of classical conservation law theory in capturing certain phenomena, such as traffic flow through bottlenecks. To address these challenges, we present the framework of non-classical conservation laws.

In the second part, we apply these models to realistic traffic scenarios, demonstrating their ability to capture complex and counterintuitive phenomena. Specifically, we illustrate how they can reproduce effects such as Braess’ paradox, the "faster is slower" effect, and capacity drop, providing deeper insights into real-world traffic dynamics.