| Abstract: |
| We analyze in this lecture macroscopic models for heterogeneous media (mixtures, suspensions,
crowd) in dense regimes. These regimes exhibit interesting behaviors such as transition phases
(congestion, jamming, glass transition, etc.).Two different approaches are generally considered in the literature to model congestion phenomena at the macroscopic level.The first one, usually called {\it hard approach}, consists in coupling compressible dynamics in the free domain $\{\rho < \rho^*\}$, with incompressible dynamics in the congested domain $\{\rho = \rho^*\}$.The second one which, by opposition, is called {\it soft approach}, prevents the apparition of congested phases by introducing in the compressible dynamics repulsive forces which become singular as $\rho$ approaches $\rho^*$.
An intuitive link can be made between the two approaches: if the scope of action of the repulsive forces tends to $0$, one expects that the soft congestion model degenerates towards a hard congestion model.The goal of this lecture is to investigate rigorously this link, more precisely we aim at understanding how the choice of the constitutive laws (pressure and/or viscosities as functions of the density) in the soft model impacts the behavior of the limit hard system in congested regions. |
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