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

Project description

A Lagrangian representation approach to studying non-linear partial differential equations

Non-linear partial differential equations play an important role in mathematics and arise in several physical and engineering models. Many of these models exhibit a lack of regularity. Handling irregular solutions that can capture the peculiar dynamics of physical processes poses great mathematical challenges: most of the tools developed in smooth settings are ineffective. Funded by the Marie Skłodowska-Curie Actions programme, the Lagrangian project aims to extend the recently introduced Lagrangian representation approach for non-linear conservation laws to the study of multi-dimensional and non-entropic weak solutions. The project will also leverage Lagrangian representation techniques to address challenging questions in the analysis of conservation laws in control theory, which also have application in mixed models of traffic flow.

Objective

The core of this project is the Lagrangian Representation (LR) and the interplay of this novel Geometric Measure Theory (GMT) tool with the study of 1st-order, nonlinear Partial Differential Equations (PDEs). Several nonlinear PDEs arise in important models from physics, engineering, biology and chemistry. The lack of regularity is an intrinsic feature of these models and reflects actual properties of the underlying real-world systems, as for example shock waves in fluid dynamics or traffic flow. Handling irregular solutions capable to capture the peculiar features of these systems poses great mathematical challenges since most of the tools developed in the smooth setting (specifically the method of characteristics) cannot be employed in this context. In the first line of research of the project I propose a new and innovative extension and exploitation, for the multidimensional case and for non entropic weak solutions, of the recently introduced LR for nonlinear conservation laws. Such a (characteristic-like) representation has proved to be a powerful technique to analyze the geometric structure and the regularity of solutions to nonlinear PDEs. In the second line of research, I will employ the LR to investigate fine properties of the 2d eikonal equation in the context of a surprisingly related celebrated conjecture in the calculus of variations by Aviles-Giga. In the last line of research, I will exploit the LR techniques to address challenging questions in the analysis of nonlinear conservation laws from the point of view of control theory, concerning controllability issues and necessary conditions for optimality, which have also application in recent mixed models of traffic flow (involving for example E-scooters in addition to cars). The Marie Skłodowska-Curie fellowship and the consequent close collaboration with Prof. Ancona and the top research group in PDEs and GMT of University of Padova are a great and unique opportunity of fulfillment of this project.

Coordinator

UNIVERSITA DEGLI STUDI DI PADOVA
Net EU contribution
€ 171 473,28
Address
VIA 8 FEBBRAIO 2
35122 Padova
Italy

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Region
Nord-Est Veneto Padova
Activity type
Higher or Secondary Education Establishments
Links
Total cost
€ 171 473,28