A Lagrangian approach: from conservation laws to line-energy Ginzburg-Landau models

Informations projet

Lagrangian

N° de convention de subvention: 101025032

DOI

10.3030/101025032

Projet clôturé

Date de signature de la CE 8 Juillet 2021

Date de début 1 Septembre 2022

Date de fin 2 Mars 2025

Financé au titre de

• EXCELLENT SCIENCE - Marie Skłodowska-Curie Actions

Coût total

€ 171 473,28

Contribution de l’UE

€ 171 473,28

Coordonné par

UNIVERSITA DEGLI STUDI DI PADOVA
Italy

Periodic Reporting for period 1 - Lagrangian (A Lagrangian approach: from conservation laws to line-energy Ginzburg-Landau models)

Période du rapport: 2022-09-01 au 2024-08-31

Résumé du contexte et des objectifs généraux du projet

The project Lagrangian has been devoted to the development and exploitation of ‘Lagrangian tools’ in the study of conservation laws.

Travail effectué depuis le début du projet jusqu’à la fin de la période considérée dans le rapport et principaux résultats atteints jusqu’à présent

Three main lines of research have been developed. The first one belongs to the classical theory of entropy solutions of conservation laws we obtained fine regularity results for continuous solutions to balance laws which are relevant in geometric measure theory on sub-Riemannian structures. We moreover proved decay estimates for entropy solutions of scalar conservation laws and proved a uniqueness results for 2x2 systems for a class of data including functions with unbounded variation, following an approach proposed by A. Bressan.

A second direction has been devoted to the study of a conjecture by Aviles and Giga: this work has been done in collaboration with X. Lamy and we obtained new results on the structure of solutions to the eikonal equation and a quantitative stability result of the vortex.

The third direction was about applications of conservation laws to traffic models: we investigated with different groups three ways of producing approximate solutions, one through nonlocal equations and is specific for traffic models. We obtained moreover a general stability result for scalar equation reviewing the classical theory, this allowed to treat a particle model for traffic. Eventually we studied a general class of 2x2 systems and in particular solutions obtained by vanishing viscosity: this result is general and it is not limited to the cases suitable for traffic.

Progrès au-delà de l’état des connaissances et impact potentiel prévu (y compris l’impact socio-économique et les conséquences sociétales plus larges du projet jusqu’à présent)

In conclusion we believe that the Lagrangian techniques introduced and developed in the framework of this project gave new insights in the structure of solutions of conservation laws and related variational problems.

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