Descripción del proyecto
Un enfoque de representación lagrangiana para estudiar las ecuaciones diferenciales parciales no lineales
Las ecuaciones diferenciales parciales no lineales desempeñan un papel fundamental en las matemáticas y aparecen en varios modelos físicos y de ingeniería. Muchos de estos modelos presentan una falta de regularidad. El procesamiento de soluciones irregulares que pueden recopilar la dinámica singular de los procesos físicos conlleva grandes retos matemáticos, ya que la mayoría de las herramientas desarrolladas en entornos fluidos no son eficaces. El objetivo del proyecto Lagrangian, financiado por las Acciones Marie Skłodowska-Curie, es ampliar el enfoque de representación lagrangiana recientemente introducido para leyes de conservación no lineales al estudio de soluciones débiles multidimensionales y no entrópicas. El equipo del proyecto empleará técnicas de representación de Lagrange para abordar preguntas complejas relacionadas con el análisis de las leyes de conservación en la teoría del control, que también tienen aplicación en modelos mixtos de flujo de tráfico.
Objetivo
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.
Ámbito científico
Programa(s)
Régimen de financiación
MSCA-IF - Marie Skłodowska-Curie Individual Fellowships (IF)Coordinador
35122 Padova
Italia