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Hyperbolic heat/mass transport and stochastic modelling - Three simple problems

Massimiliano Giona Luigi Pucci

*Corresponding author: Massimiliano Giona massimiliano.giona@uniroma1.it

MinE2019,2,224doi:10.3934/mine.2019.2.224

Starting from the correspondence between the Cattaneo hyperbolic heat equation and the stochastic formulation based on Poisson-Kac processes, that holds solely for one-dimensional spatial models, this article analyzes three paradigmatic problems in the hyperbolic theory of heat and mass transport. The problems considered involve unbounded, semi-bounded and bounded domains, and are aimed at : (i) highlighting analogies and differences between the two approaches (Cattaneo vs Poisson-Kac), (ii) addressing the role of a bounded propagation velocity in order to regularize the properties of the solutions of heat/mass transport problems. A typical example of the latter phenomenology is expressed by boundary-layer regulatization of interfacial fluxes. The case of transport in bounded domains permits to pinpoint unambiguously the need of a stochastic interpretation of the transport equation in order to unveil the occurrence of physical inconsistencies that may occur in the linear Cattaneo hyperbolic model in some range of parameter values.

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