Convergence of a Degenerate Microscopic Dynamics to the Porous Medium Equation

Autor: Oriane Blondel, Clément Cancès, Makiko Sasada, Marielle Simon
Přispěvatelé: Probabilités, statistique, physique mathématique (PSPM), Institut Camille Jordan [Villeurbanne] (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS), Reliable numerical approximations of dissipative systems (RAPSODI ), Laboratoire Paul Painlevé - UMR 8524 (LPP), Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Inria Lille - Nord Europe, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Graduate School of Mathematical Sciences[Tokyo], The University of Tokyo (UTokyo), Méthodes quantitatives pour les modèles aléatoires de la physique (MEPHYSTO-POST), Inria Lille - Nord Europe, We thank Patrícia Gonçalves, Claudio Landim, Cristina Toninelli and Kenkichi Tsunoda for helpful discussions. O.B. and M.S. are grateful to the University of Tokyo for its hospitality. O.B. acknowledges support from ANR-15-CE40-0020-03 grant LSD. C.C. and M.S. thank Labex CEMPI (ANR-11-LABX-0007-01), and C.C. is grateful to ANR-13-JS01-0007-01 (project GEOPOR) for their support. O.B. and M.S. thank INSMI (CNRS) for its support through the PEPS project 'Dérivation et Étude Mathématique de l’Équation des Milieux Poreux' (2016). M.S. was supported by JSPS Grant-in-Aid for Young Scientists (B) No. JP25800068. The research leading to the present results benefited from the financial support of the seventh Framework Program of the European Union (7ePC/2007-2013), grant agreement no266638. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovative programme (grant agreement no715734)., ANR-15-CE40-0020,LSD,Modèles stochastiques en grande dimension pour la physique statistique hors équilibre(2015), ANR-11-LABX-0007,CEMPI,Centre Européen pour les Mathématiques, la Physique et leurs Interactions(2011), ANR-13-JS01-0007,GEOPOR,Approche géométrique pour les écoulements en milieux poreux: théorie et numérique(2013), ANR-10-LABX-0070,MILYON,Community of mathematics and fundamental computer science in Lyon(2010), ANR-16-CE93-0003,MALIN,Marches aléatoires en interaction(2016), ANR-14-CE25-0011,EDNHS,Diffusion de l'énergie dans des systèmes hamiltoniens bruitésés(2014), European Project: 266638,EC:FP7:SiS,FP7-SCIENCE-IN-SOCIETY-2010-1,INTEGER(2011), European Project: 715734,H2020,HyLEF(2016), Institut Camille Jordan (ICJ), Université de Lyon-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS), Laboratoire Paul Painlevé (LPP)
Jazyk: angličtina
Rok vydání: 2019
Předmět:
Zdroj: Annales de l'Institut Fourier
In press
inPress
HAL
ISSN: 0373-0956
1777-5310
Popis: 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.
Comment: Uncomplete version!
Databáze: OpenAIRE
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