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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 PDE's. 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
analysed. 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 PDE's and stochastic processes can find in this
volume an extension of this correspondence to modern mathematical
physics.
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