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Convergence of a degenerate microscopic dynamics to the porous medium equation

Oriane Blondel,C. Cancès,M. Sasada,M. Simon

Published 2018 in arXiv: Probability

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 [Gon\c{c}alves-Landim-Toninelli '09] 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 \emph{relative entropy method}, by involving approximations of solutions to the hydrodynamic equation, instead of exact solutions.

PUBLICATION RECORD

  • Publication year

    2018

  • Venue

    arXiv: Probability

  • Publication date

    2018-02-16

  • Fields of study

    Mathematics, Physics

  • Identifiers
    arXiv 1802.05912
  • External record

    Open on Semantic Scholar

  • Source metadata

    Semantic Scholar

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