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

Informacje na temat projektu

Lagrangian

Identyfikator umowy o grant: 101025032

DOI

10.3030/101025032

Projekt został zamknięty

Data podpisania przez KE 8 Lipca 2021

Data rozpoczęcia 1 Września 2022

Data zakończenia 2 Marca 2025

Finansowanie w ramach

• EXCELLENT SCIENCE - Marie Skłodowska-Curie Actions

Koszt całkowity

€ 171 473,28

Wkład UE

€ 171 473,28

Koordynowany przez

UNIVERSITA DEGLI STUDI DI PADOVA
Italy

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

Okres sprawozdawczy: 2022-09-01 do 2024-08-31

Podsumowanie kontekstu i ogólnych celów projektu

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

Prace wykonane od początku projektu do końca okresu sprawozdawczego oraz najważniejsze dotychczasowe rezultaty

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.

Innowacyjność oraz oczekiwany potencjalny wpływ (w tym dotychczasowe znaczenie społeczno-gospodarcze i szersze implikacje społeczne projektu)

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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