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The Porous Medium Equation: Large Deviations and Gradient Flow with Degenerate and Unbounded Diffusion

Benjamin Gess, Daniel Heydecker

TL;DR

The paper derives the porous medium equation (PME) as the macroscopic limit of a rescaled zero-range process with jump rate g(k)=k^α (α>1) and, crucially, establishes a dynamical large-deviations principle around this PME limit in the presence of degenerate and unbounded diffusion. A novel pathwise regularity approach yields a superexponential replacement lemma, enabling a full LDP with upper and lower bounds and linking the dynamic cost to an entropy-dissipation framework. The main conceptual payoff is the identification of a gradient-flow structure for PME in a thermodynamic metric, arising from microscopic large deviations via an energy-dissipation inequality (EDI). The results extend the thermodynamic gradient-flow viewpoint to degenerate diffusion, showing that the PME can be interpreted as a gradient flow of the thermodynamic entropy even when global Riemannian geometry is unavailable, and they provide an H-theorem for PME in this generalized setting. This work thus connects microscopic stochastic dynamics to macroscopic nonlinear diffusion and furnishes a robust variational viewpoint for PME grounded in large-deviation theory.

Abstract

The problem of deriving a gradient flow structure for the porous medium equation which is {\em 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 $g(k)=k^α, α>1$ is considered, and its hydrodynamic limit and dynamical large deviations are shown in the presence of both degenerate and unbounded diffusion. The key superexponential 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.

The Porous Medium Equation: Large Deviations and Gradient Flow with Degenerate and Unbounded Diffusion

TL;DR

The paper derives the porous medium equation (PME) as the macroscopic limit of a rescaled zero-range process with jump rate g(k)=k^α (α>1) and, crucially, establishes a dynamical large-deviations principle around this PME limit in the presence of degenerate and unbounded diffusion. A novel pathwise regularity approach yields a superexponential replacement lemma, enabling a full LDP with upper and lower bounds and linking the dynamic cost to an entropy-dissipation framework. The main conceptual payoff is the identification of a gradient-flow structure for PME in a thermodynamic metric, arising from microscopic large deviations via an energy-dissipation inequality (EDI). The results extend the thermodynamic gradient-flow viewpoint to degenerate diffusion, showing that the PME can be interpreted as a gradient flow of the thermodynamic entropy even when global Riemannian geometry is unavailable, and they provide an H-theorem for PME in this generalized setting. This work thus connects microscopic stochastic dynamics to macroscopic nonlinear diffusion and furnishes a robust variational viewpoint for PME grounded in large-deviation theory.

Abstract

The problem of deriving a gradient flow structure for the porous medium equation which is {\em 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 superexponential 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.
Paper Structure (76 sections, 24 theorems, 359 equations, 1 figure)

This paper contains 76 sections, 24 theorems, 359 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Hydrodynamic Limit.
  3. Large Deviations.
  4. A regularity-based approach to the replacement lemma
  5. From Large Deviations to Gradient Flows
  6. Plan of the Paper
  7. Discussion & Literature Review
  8. 1. PME from Particle Systems
  9. 2. Relation to the Zero-Range Process The limits of the zero-range process for general jump rates, assuming bounded diffusivity, as well as fluctuation theorems about these limits, have been widely studied. The hydrodynamic limit, see kipnis1998scaling, is a nonlinear parabolic equation $\partial_t u=\Delta \Phi(u)$, with globally bounded and locally nondegenerate diffusivity $\sup_\rho \Phi'(\rho)<\infty, \inf_{\rho\le M}\Phi'(\rho)>0$. Let us cite the works by Menegaki menegaki2021quantitative and Menegaki and Mouhot menegaki2022consistence for a more recent hydrodynamic limit with an explicit rate of convergence. Equilibrium large deviations for a variant of the zero-range process have been studied by Bernardin, Gonçalves, Jiménez-Oviedo and Scotta in bernardin2022non. In infinite volume, which we do not consider in the present work, the large deviations and hydrodynamic limit have been studied by Landim and Yau landim1995large and Landim and Mourragui landim1997hydrodynamic. Quastel, Rezakhanlou and Varadhan quastel1999large found the analagous rate function for the simple symmetric exclusion process. As already mentioned below Theorem \ref{['thrm: LDP']}, an important difference to previous works is that we obtain a full large deviation principle, with matching upper and lower bounds and the infimum of (\ref{['eq: LB statement']}) running over the whole open set $\mathcal{U}$, rather than being restricted to $\mathcal{U} \cap \mathcal{Q}$ for a class of 'good' paths $\mathcal{Q}$. This is achieved through two steps, namely the exclusion of paths outside ${\mathcal{R}}$ by Theorem \ref{['thrm: WtS']}i), and the characterisation of $\mathcal{I}_\rho|_{\mathcal{R}}$ as the lower semicontinuous envelope of a restriction $\mathcal{I}_\rho|_\mathcal{X}$(Proposition \ref{['prop: lsc envelope']}), which we recall from fehrman2019large. This property is known for relatively few of the models whose large deviations have been studied. In the case kipnis1989hydrodynamics, the rate function is globally convex and such approximations can be obtained by convolution, and in works on reaction-diffusion systems jona-lasino1993largebodineau2012landim2018largefarfan2019, the rate is a pertubation of a convex functional by a lower-order term. In quastel1999largebertini2009non, the required property is proven for exclusion-type processes, using a priori $L^\infty_{t,x}$ bounds and the boundedness and nondegeneracy of the diffusion.
  10. 3. Macroscopic Fluctuation Theory and Large Deviations of Conservative SPDE
  11. 4. The scaling hypothesis (\ref{['eq: scaling hypothesis']}).
  12. 5. PME as a gradient flow
  13. Preliminaries
  14. Formal Definitions.
  15. 1. Lebesgue Spaces on the Discrete Lattice and Torus; Particle State Space.
  16. ...and 61 more sections

Key Result

Theorem 1

Assume (eq: scaling hypothesis). Fix $u_0\in L^1_{\ge 0}({\mathbb{T}^d})$ with finite entropy $\mathcal{H}(u_0)<\infty$, and let $u_\bullet=(u_t)_{t\ge 0}$ be the unique weak solution to the PME (eq: PME) starting at $u_0$. Let $\mathbb{P}$ be a probability measure, let $\eta^N_\bullet$ be the ZRP w and, for all $\epsilon>0$, Then in probability in the topology of the Skorokhod space $\mathbb{D}

Figures (1)

  • Figure 1: All possible double-limits. The horizontal arrows represent $N\to \infty$ with the hydrodynamic rescaling of space and time, and the vertical arrows represent $\chi\to 0$ with the value-time rescaling. The two curved arrows are the two double limits, depending on whether $\chi_N\to 0$ sufficiently quickly (e) or sufficiently slowly (f) with $N$.

Theorems & Definitions (50)

  • Theorem 1: Hydrodynamic Limit
  • Theorem 2: Large Deviations Principle
  • Theorem 3: Large Deviations Weak-to-Strong Principle
  • Theorem 4
  • Lemma 3.1
  • Definition 3.2
  • Proposition 3.3
  • Proposition 3.4
  • Lemma 3.5
  • Remark 3.6
  • ...and 40 more