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Hyperbolic heat/mass transport and stochastic modelling – three simple problems. (English) Zbl 1435.80005

Summary: 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 regularization 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.

MSC:

80A19 Diffusive and convective heat and mass transfer, heat flow
80A10 Classical and relativistic thermodynamics
35L02 First-order hyperbolic equations
35L10 Second-order hyperbolic equations
35Q79 PDEs in connection with classical thermodynamics and heat transfer
82C70 Transport processes in time-dependent statistical mechanics
82C35 Irreversible thermodynamics, including Onsager-Machlup theory
35Q82 PDEs in connection with statistical mechanics
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
35R60 PDEs with randomness, stochastic partial differential equations
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