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

Abstract

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 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.