Convergence of a Degenerate Microscopic Dynamics to the Porous Medium Equation

Abstract : We derive the porous medium equation from an interacting particle system which belongs to the family of exclusion processes, with nearest neighbor exchanges. The particles follow a degenerate dynamics, in the sense that the jump rates can vanish for certain configurations, and there exist blocked configurations that cannot evolve. In [7] it was proved that the macroscopic density profile in the hydrodynamic limit is governed by the porous medium equation (PME), for initial densities uniformly bounded away from 0 and 1. In this paper we consider the more general case where the density can take those extreme values. In this context, the PME solutions display a richer behavior, like moving interfaces, finite speed of propagation and breaking of regularity. As a consequence, the standard techniques that are commonly used to prove this hydrodynamic limits cannot be straightforwardly applied to our case. We present here a way to generalize the relative entropy method, by involving approximations of solutions to the hydrodynamic equation, instead of exact solutions.
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https://hal.archives-ouvertes.fr/hal-01710628
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Submitted on : Thursday, April 26, 2018 - 10:50:57 AM
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Oriane Blondel, Clément Cancès, Makiko Sasada, Marielle Simon. Convergence of a Degenerate Microscopic Dynamics to the Porous Medium Equation. 2018. ⟨hal-01710628v2⟩

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