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A functional inequalities approach for the field-road diffusion model with (symmetric) nonlinear exchanges

Matthieu Alfaro, Claire Chainais-Hillairet, Flore Nabet

TL;DR

The paper addresses the long-time behavior of a field-road diffusion model with symmetric nonlinear exchanges by introducing a Tsallis-type relative entropy and proving its exponential decay to the mass-determined steady state. The authors develop a direct functional-inequalities framework, combining an adapted Poincaré-Wirtinger inequality on an enlarged domain and a generalized Beckner inequality to bound the dissipation from below by the entropy itself. This yields explicit exponential convergence and, as a by-product, decay of the $L^{\alpha+1}$ norms towards the steady state. Numerical experiments based on a TPFA finite-volume scheme corroborate the theory and suggest robustness of the decay rate against variations in the exchange exponent and initial data, while highlighting potential extensions to nonsymmetric exchanges and structure-preserving discretizations.

Abstract

In this note, we consider the so-called field-road diffusion model in a bounded domain, consisting of two parabolic PDEs posed on sets of different dimensions and coupled through (symmetric) nonlinear exchange terms. We propose a new and rather direct functional inequalities approach to prove the exponential decay of a relative entropy, and thus the convergence of the solution towards the stationary state selected by the total mass of the initial datum.

A functional inequalities approach for the field-road diffusion model with (symmetric) nonlinear exchanges

TL;DR

The paper addresses the long-time behavior of a field-road diffusion model with symmetric nonlinear exchanges by introducing a Tsallis-type relative entropy and proving its exponential decay to the mass-determined steady state. The authors develop a direct functional-inequalities framework, combining an adapted Poincaré-Wirtinger inequality on an enlarged domain and a generalized Beckner inequality to bound the dissipation from below by the entropy itself. This yields explicit exponential convergence and, as a by-product, decay of the norms towards the steady state. Numerical experiments based on a TPFA finite-volume scheme corroborate the theory and suggest robustness of the decay rate against variations in the exchange exponent and initial data, while highlighting potential extensions to nonsymmetric exchanges and structure-preserving discretizations.

Abstract

In this note, we consider the so-called field-road diffusion model in a bounded domain, consisting of two parabolic PDEs posed on sets of different dimensions and coupled through (symmetric) nonlinear exchange terms. We propose a new and rather direct functional inequalities approach to prove the exponential decay of a relative entropy, and thus the convergence of the solution towards the stationary state selected by the total mass of the initial datum.
Paper Structure (4 sections, 3 theorems, 22 equations, 2 figures, 1 table)

This paper contains 4 sections, 3 theorems, 22 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Setting of the result
  3. A functional inequalities approach
  4. Numerical experiments

Key Result

Theorem 2.1

Assume $\alpha=\beta>1$. Let $v_0\in L^\infty(\Omega)$ and $u_0\in L^\infty(\omega)$ be both nonnegative and not simultaneously trivial. Let $(v=v(t,x,y), u=u(t,x))$ be the solution to syst starting from $(v_0=v_0(x,y), u_0=u_0(x))$, and $(v_\infty,u_\infty)$ the associated steady-state defined by e for some positive $\lambda=\lambda(N,\Omega,\mu,\nu,d,D,v_0,u_0,\alpha)$. As a by-product, for som

Figures (2)

  • Figure 1: Exponential decay of the relative entropy in the symmetric case, $\alpha=\beta$, for Test case 1 (left) and Test case 2 (right).
  • Figure 2: Comparison of the decay of the relative entropy in the symmetric and nonsymmetric cases, for Test case 1 (left) and Test case 2 (right).

Theorems & Definitions (3)

  • Theorem 2.1: Exponential decay of entropy
  • Lemma 3.1: Adapted Poincaré-Wirtinger inequality
  • Lemma 3.2: Generalized Beckner inequality II