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Gradient flow for a class of diffusion equations with Dirichlet boundary data

Matthias Erbar, Giulia Meglioli

TL;DR

This work establishes a gradient-flow framework for nonlinear diffusion equations with constant Dirichlet boundary data on a bounded domain by employing the modified Wasserstein distance $Wb_p$, which accommodates mass exchange with the boundary. It develops a dynamic Benamou–Brenier-type formulation for $Wb_p$, proves a complete AC-curve characterization in this geometry via continuity equations, and shows that nonlinear diffusion equations with Dirichlet data are exactly the curves of maximal slope of a suitable internal energy $\mathcal{F}$ in $(\mathcal{M}_2(\Omega), Wb_2)$, with Dirichlet boundary conditions encoded through traces of $G(\rho)$ on $\partial\Omega$. A chain-rule argument yields a De Giorgi-type energy-dissipation identity, linking the slope to the dissipation functional $\overline{\mathcal{I}}$, and the resulting gradient-flow description explains the boundary mass exchange as a consequence of the interaction between the energy landscape and the transport geometry. The paper also proves a dynamic characterization of $Wb_p$ via the continuity equation and demonstrates convergence of the minimizing movement (JKO) scheme to weak solutions, thereby providing a robust variational construction for boundary-driven diffusion models. These results extend gradient-flow theory to boundary-driven diffusion and offer new tools for analyzing PDEs with Dirichlet reservoirs in measure spaces.

Abstract

In this paper we provide a variational characterisation for a class of non-linear evolution equations with constant non-negative Dirichlet boundary conditions on a bounded domain as gradient flows in the space of non-negative measures. The relevant geometry is given by the modified Wasserstein distance introduced by Figalli and Gigli that allows for a change of mass by letting the boundary act as a reservoir. We give a dynamic formulation of this distance as an action minimisation problem for curves of non-negative measures satisfying a continuity equation in the spirit of Benamou-Brenier. Then we characterise solutions to non-linear diffusion equations with Dirichlet boundary conditions as metric gradient flows of internal energy functionals in the sense of curves of maximal slope.

Gradient flow for a class of diffusion equations with Dirichlet boundary data

TL;DR

This work establishes a gradient-flow framework for nonlinear diffusion equations with constant Dirichlet boundary data on a bounded domain by employing the modified Wasserstein distance , which accommodates mass exchange with the boundary. It develops a dynamic Benamou–Brenier-type formulation for , proves a complete AC-curve characterization in this geometry via continuity equations, and shows that nonlinear diffusion equations with Dirichlet data are exactly the curves of maximal slope of a suitable internal energy in , with Dirichlet boundary conditions encoded through traces of on . A chain-rule argument yields a De Giorgi-type energy-dissipation identity, linking the slope to the dissipation functional , and the resulting gradient-flow description explains the boundary mass exchange as a consequence of the interaction between the energy landscape and the transport geometry. The paper also proves a dynamic characterization of via the continuity equation and demonstrates convergence of the minimizing movement (JKO) scheme to weak solutions, thereby providing a robust variational construction for boundary-driven diffusion models. These results extend gradient-flow theory to boundary-driven diffusion and offer new tools for analyzing PDEs with Dirichlet reservoirs in measure spaces.

Abstract

In this paper we provide a variational characterisation for a class of non-linear evolution equations with constant non-negative Dirichlet boundary conditions on a bounded domain as gradient flows in the space of non-negative measures. The relevant geometry is given by the modified Wasserstein distance introduced by Figalli and Gigli that allows for a change of mass by letting the boundary act as a reservoir. We give a dynamic formulation of this distance as an action minimisation problem for curves of non-negative measures satisfying a continuity equation in the spirit of Benamou-Brenier. Then we characterise solutions to non-linear diffusion equations with Dirichlet boundary conditions as metric gradient flows of internal energy functionals in the sense of curves of maximal slope.
Paper Structure (14 sections, 20 theorems, 187 equations)

This paper contains 14 sections, 20 theorems, 187 equations.

Table of Contents

  1. Introduction
  2. The space $(\mathcal{M}_p(\Omega),\,Wb_p)$
  3. The set of admissible and optimal plans
  4. Properties of $Wb_p$
  5. Characterisation of absolutly continuous curves
  6. Proof of Theorem \ref{['teo1']}
  7. Proof of Proposition \ref{['prop23']}
  8. Non-linear diffusion equations with Dirichlet bc as gradient flows
  9. Internal energy and dissipation
  10. Variational characterisation
  11. Gradient flow characterisation
  12. Convergence of the minimising movement scheme
  13. A boundary estimate
  14. Existence of weak solutions

Key Result

Theorem 1.1

For any absolutely continuous curve $(\mu_t)_{t\in[0,T]}$ in $(\mathcal{M}_2(\Omega),Wb_2)$ such that ${\mathcal{F}}(\mu_0)$ is finite, we have Moreover, we have $\mathcal{L}_T(\mu_t)=0$ if and only if $\mu_t=\rho_t{\sf Leb}|_\Omega$ such $t\mapsto G(\rho_t)-G(\lambda)$ belongs to $L^2([0,T];W^{1,2}_0(\Omega))$ and $(\rho_t)$ is a weak solution to $\partial_t\rho= \Delta f(\rho)$.

Theorems & Definitions (53)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Lemma 2.2
  • Proposition 2.3
  • Proposition 2.4
  • proof
  • Definition 3.1: Continuity equation
  • Remark 3.2
  • Proposition 3.3
  • ...and 43 more