Lyapunov functionals for boundary-driven nonlinear drift-diffusions

TL;DR

This paper develops a broad class of Lyapunov functionals for boundary-driven nonlinear drift-diffusion equations, linking them to stochastic systems and establishing exponential convergence rates under Dirichlet boundary conditions.

Contribution

It introduces new Lyapunov functionals for nonlinear drift-diffusions with boundary conditions, generalizing large deviation functionals and applying them to prove relaxation rates.

Findings

Established linear inequalities between entropy-like functionals and entropy production.

Proved exponential relaxation rates related to the first Dirichlet eigenvalue.

Derived Lyapunov functions for nonlinear diffusion systems and non-reversible Markov processes.

Abstract

We exhibit a large class of Lyapunov functionals for nonlinear drift-diffusion equations with non-homogeneous Dirichlet boundary conditions. These are generalizations of large deviation functionals for underlying stochastic many-particle systems, the zero range process and the Ginzburg-Landau dynamics, which we describe briefly. As an application, we prove linear inequalities between such an entropy-like functional and its entropy production functional for the boundary-driven porous medium equation in a bounded domain with positive Dirichlet conditions: this implies exponential rates of relaxation related to the first Dirichlet eigenvalue of the domain. We also derive Lyapunov functions for systems of nonlinear diffusion equations, and for nonlinear Markov processes with non-reversible stationary measures.