Free boundary problems in PDEs and particle systems

TL;DR

This paper develops a comprehensive theory for free boundary problems in PDEs and particle systems, introducing a new notion of relaxed solutions and establishing their properties and connections to stochastic models across various scientific fields.

Contribution

It introduces a new framework for free boundary problems, including a relaxed solution concept and its hydrodynamic limit, extending the PDE-stochastic process correspondence.

Findings

Proved global existence and uniqueness of the relaxed solution.

Established the hydrodynamic limit linking particle systems to PDE models.

Extended the PDE-stochastic process correspondence to new models.

Abstract

In this volume a theory for models of transport in the presence of a free boundary is developed. Macroscopic laws of transport are described by PDEs. When the system is open, there are several mechanisms to couple the system with the external forces. Here a class of systems where the interaction with the exterior takes place in correspondence of a free boundary is considered. Both continuous and discrete models sharing the same structure are analyzed. In Part I a free boundary problem related to the Stefan Problem is worked out in all details. For this model a new notion of relaxed solution is proposed for which global existence and uniqueness is proven. It is also shown that this is the hydrodynamic limit of the empirical mass density of the associated particle system. In Part II several other models are discussed. The expectation is that the results proved for the basic model extend to these other cases. All the models discussed in this volume have an interest in problems arising in several research fields such as heat conduction, queuing theory, propagation of fire, interface dynamics, population dynamics, evolution of biological systems with selection mechanisms.In general researchers interested in the relations between PDEs and stochastic processes can find in this volume an extension of this correspondence to modern mathematical physics.