Periodic Reporting for period 1 - Lagrangian (A Lagrangian approach: from conservation laws to line-energy Ginzburg-Landau models)
Reporting period: 2022-09-01 to 2024-08-31
The project Lagrangian has been devoted to the development and exploitation of ‘Lagrangian tools’ in the study of conservation laws.
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.
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.
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.