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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.
Entropy inequalities, correlation functions, couplings between
stochastic processes are powerful techniques which have been
extensively used to give arigorous foundation to the theory of
complex, many component systems and to its many applications in a
variety of fields as physics, biology, population dynamics,
economics, ... The purpose of the book is to make theseand other
mathematical methods accessible to readers with a limited
background in probability and physics by examining in detail a few
models where the techniques emerge clearly, while extra
difficulties arekept to a minimum. Lanford's method and its
extension to the hierarchy of equations for the truncated
correlation functions, the v-functions, are presented and applied
to prove the validity of macroscopic equations forstochastic
particle systems which are perturbations of the independent and of
the symmetric simple exclusion processes. Entropy inequalities are
discussed in the frame of the Guo-Papanicolaou-Varadhan technique
and of theKipnis-Olla-Varadhan super exponential estimates, with
reference to zero-range models. Discrete velocity Boltzmann
equations, reaction diffusion equations and non linear parabolic
equations are considered, as limits of particles models. Phase
separation phenomena are discussed in the context of
Glauber+Kawasaki evolutions and reaction diffusion equations.
Although the emphasis is onthe mathematical aspects, the physical
motivations are explained through theanalysis of the single models,
without attempting, however to survey the entire subject of
hydrodynamical limits.
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