The porous medium equation: Large deviations and gradient flow with degenerate and unbounded diffusion

The problem of deriving a gradient flow structure for the porous medium equation which is thermodynamic, in that it arises from the large deviations of some microscopic particle system is studied. To this end, a rescaled zero‐range process with jump rate g(k)=kα,α>1$g(k)=k^\alpha, \alpha >1$ i...

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Bibliographic Details
Published inCommunications on pure and applied mathematics Vol. 78; no. 9; pp. 1609 - 1655
Main Authors Gess, Benjamin, Heydecker, Daniel
Format Journal Article
LanguageEnglish
Published New York John Wiley and Sons, Limited 01.09.2025
Subjects
Deviation
Gradient flow
Porous media
Thermodynamics
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Summary:The problem of deriving a gradient flow structure for the porous medium equation which is thermodynamic, in that it arises from the large deviations of some microscopic particle system is studied. To this end, a rescaled zero‐range process with jump rate g(k)=kα,α>1$g(k)=k^\alpha, \alpha >1$ is considered, and its hydrodynamic limit and dynamical large deviations are shown in the presence of both degenerate and unbounded diffusion. The key super‐exponential estimate is obtained using pathwise discretised regularity estimates in the spirit of the Aubin–Lions–Simons lemma. This allows to exhibit the porous medium equation as the gradient flow of the entropy in a thermodynamic metric via the energy‐dissipation inequality.
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content type line 14
ISSN:0010-3640
1097-0312
DOI:10.1002/cpa.22251