Descrizione del progetto
Un approccio di rappresentazione lagrangiana allo studio delle equazioni differenziali alle derivate parziali non lineari
Le equazioni differenziali alle derivate parziali non lineari svolgono un ruolo importante in matematica e si presentano in diversi modelli fisici e ingegneristici, molti dei quali mostrano una mancanza di regolarità. La gestione di soluzioni irregolari in grado di catturare le dinamiche peculiari dei processi fisici pone grandi sfide matematiche: la maggior parte degli strumenti sviluppati in ambienti regolari sono inefficaci. Finanziato dal programma di azioni Marie Skłodowska-Curie, il progetto Lagrangian si propone di estendere l’approccio di rappresentazione lagrangiana recentemente introdotto per le leggi di conservazione non lineari allo studio di soluzioni deboli multidimensionali e non entropiche. Il progetto sfrutterà inoltre le tecniche di rappresentazione lagrangiana per affrontare questioni impegnative relativamente all’analisi delle leggi di conservazione nella teoria del controllo, applicate anche nei modelli misti del flusso di traffico.
Obiettivo
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.
Campo scientifico
Programma(i)
Argomento(i)
Meccanismo di finanziamento
MSCA-IF - Marie Skłodowska-Curie Individual Fellowships (IF)Coordinatore
35122 Padova
Italia