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Global exponential stability of stationary profiles in a thin film equation with second-order diffusion

Christian Parsch

TL;DR

The paper addresses the global exponential stability of stationary profiles for a 1D thin-film equation with confinement and a small second-order diffusion term. It casts the dynamics as a Wasserstein gradient flow of the energy ${\mathcal E}_\varepsilon(\rho)$ on ${\mathcal P}_2(\mathbb{R})$ and proves existence of weak solutions for any $\varepsilon>0$, as well as the existence of a unique global minimizer ${\bar{\rho}}^\varepsilon$ of ${\mathcal E}_\varepsilon$. For small $\varepsilon$, the authors construct a uniformly convex Lyapunov functional ${\mathcal L}_\varepsilon$ via a carefully extended potential $W_\varepsilon$, and prove a set of functional inequalities that control the remainder in a splitting of the equation. This leads to exponential decay of ${\mathcal L}_\varepsilon(\rho(t))$ and, through a Csiszár–Kullback-type bound, to exponential convergence of the solution to ${\bar{\rho}}^\varepsilon$ in $L^1(\mathbb{R})$, with rate $2\lambda-C\varepsilon$. The results quantify the long-time behavior on the unbounded domain and highlight how small lower-order perturbations perturb the gradient-flow structure to preserve stability.

Abstract

We study existence and long-time behavior of weak solutions to a thin-film equation with a confinement potential and a second-order degenerate diffusion term. It is known that in absence of second order effects, solutions for general initial data converge at an exponential rate in time to the unique stationary profile. Our main result is that if the strength of the additional forces is sufficiently small, this global exponential equilibration behavior persists, at a slightly smaller rate. Our proof uses the formulation of the equation as a Wasserstein gradient flow, and an auxiliary lower-order Lyapunov functional.

Global exponential stability of stationary profiles in a thin film equation with second-order diffusion

TL;DR

The paper addresses the global exponential stability of stationary profiles for a 1D thin-film equation with confinement and a small second-order diffusion term. It casts the dynamics as a Wasserstein gradient flow of the energy on and proves existence of weak solutions for any , as well as the existence of a unique global minimizer of . For small , the authors construct a uniformly convex Lyapunov functional via a carefully extended potential , and prove a set of functional inequalities that control the remainder in a splitting of the equation. This leads to exponential decay of and, through a Csiszár–Kullback-type bound, to exponential convergence of the solution to in , with rate . The results quantify the long-time behavior on the unbounded domain and highlight how small lower-order perturbations perturb the gradient-flow structure to preserve stability.

Abstract

We study existence and long-time behavior of weak solutions to a thin-film equation with a confinement potential and a second-order degenerate diffusion term. It is known that in absence of second order effects, solutions for general initial data converge at an exponential rate in time to the unique stationary profile. Our main result is that if the strength of the additional forces is sufficiently small, this global exponential equilibration behavior persists, at a slightly smaller rate. Our proof uses the formulation of the equation as a Wasserstein gradient flow, and an auxiliary lower-order Lyapunov functional.
Paper Structure (19 sections, 35 theorems, 165 equations)

This paper contains 19 sections, 35 theorems, 165 equations.

Table of Contents

  1. Introduction
  2. Splitting the equation
  3. Assumptions and main results
  4. Outline
  5. Notations
  6. Construction of weak solution as Wasserstein gradient flow
  7. Basic properties of the energy functional
  8. Analysis of the minimizing movement scheme
  9. Interpolation and convergence to weak solution
  10. Global minimizer of the energy
  11. Existence
  12. Higher derivatives of the square-root
  13. Exponential convergence of solution
  14. Construction of the Lyapunov functional
  15. Splitting of the equation
  16. ...and 4 more sections

Key Result

Theorem 1.3

For all $\varepsilon > 0$ and initial data $\rho_0 \in {\mathcal{P}_2}({\mathbb R}) \cap H^1({\mathbb R})$, there exists a weakly continuous curve $\rho: [0, +\infty[ \,\to {\mathcal{P}_2}({\mathbb R})$ with $\rho(0) = \rho_0$ that satisfies equation eq:thinfilm in the following weak sense: We have

Theorems & Definitions (68)

  • Remark 1.2
  • Theorem 1.3: Existence of solution
  • Theorem 1.4: Global minimizer
  • Theorem 1.5: Exponential convergence to equilibrium
  • Proposition 2.1
  • proof
  • Lemma 2.2
  • proof
  • Proposition 2.3: Existence
  • proof
  • ...and 58 more