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Improved equilibration rates to self-similarity for strong solutions of a thin-film and related evolution equations

Mario Bukal

TL;DR

The paper develops a time-dependent, second-moment-preserving rescaling for a family of fourth-order diffusion equations on $\mathbb{R}$, with a focus on the thin-film equation $\partial_t u = -(u u_{xxx})_x$. By embedding the dynamics into a nonlocal Fokker–Planck-type framework and employing relative Rényi entropies, entropy powers, and related inequalities, the authors derive a closed differential inequality for the relative entropy that yields sharp, early-time convergence toward self-similar profiles in $L^1$-norm. While the approach is formal for the general fourth-order family with $1<p\le 3/2$, it is rigorously justified for strong solutions of the thin-film case $p=3/2$, and is complemented by a parallel treatment of the DLSS equation ($p=1$). The results provide improved equilibration rates to self-similarity by leveraging Rényi-entropy methods, connecting high-order diffusion models to information-theoretic tools and yielding sharper convergence than classical entropy-based rates, with implications for accurate long-time behavior in thin-film dynamics.

Abstract

This paper investigates the asymptotic behavior of strong solutions to a family of nonlinear fourth-order evolution equations on the real line, with particular focus on the thin-film equation $\partial_tu = -(uu_{xxx})_x$. The method builds on the framework introduced by Carrillo and Toscani (Nonlinearity 27 (2014), 3159) for second-order nonlinear diffusion equations - by introducing a time-dependent rescaling that preserves the second moment, we establish sharp convergence rates toward the steady state in terms of the relative Rényi entropy. Compared to rates derived from the dissipation of the classical relative entropy, this approach yields improved estimates at early and intermediate times, and consequently a sharper convergence in the $L^1$-norm. The method is developed at a formal level for the family of fourth-order equations, including the well-known Derrida-Lebowitz-Speer-Spohn (DLSS) equation, but can be rigorously justified for strong solutions of the thin-film equation.

Improved equilibration rates to self-similarity for strong solutions of a thin-film and related evolution equations

TL;DR

The paper develops a time-dependent, second-moment-preserving rescaling for a family of fourth-order diffusion equations on , with a focus on the thin-film equation . By embedding the dynamics into a nonlocal Fokker–Planck-type framework and employing relative Rényi entropies, entropy powers, and related inequalities, the authors derive a closed differential inequality for the relative entropy that yields sharp, early-time convergence toward self-similar profiles in -norm. While the approach is formal for the general fourth-order family with , it is rigorously justified for strong solutions of the thin-film case , and is complemented by a parallel treatment of the DLSS equation (). The results provide improved equilibration rates to self-similarity by leveraging Rényi-entropy methods, connecting high-order diffusion models to information-theoretic tools and yielding sharper convergence than classical entropy-based rates, with implications for accurate long-time behavior in thin-film dynamics.

Abstract

This paper investigates the asymptotic behavior of strong solutions to a family of nonlinear fourth-order evolution equations on the real line, with particular focus on the thin-film equation . The method builds on the framework introduced by Carrillo and Toscani (Nonlinearity 27 (2014), 3159) for second-order nonlinear diffusion equations - by introducing a time-dependent rescaling that preserves the second moment, we establish sharp convergence rates toward the steady state in terms of the relative Rényi entropy. Compared to rates derived from the dissipation of the classical relative entropy, this approach yields improved estimates at early and intermediate times, and consequently a sharper convergence in the -norm. The method is developed at a formal level for the family of fourth-order equations, including the well-known Derrida-Lebowitz-Speer-Spohn (DLSS) equation, but can be rigorously justified for strong solutions of the thin-film equation.

Paper Structure

This paper contains 11 sections, 7 theorems, 117 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Nonlocal equations, steady states and Rényi entropies
  3. Change of variables
  4. Steady state
  5. Rényi entropies
  6. Improved equilibration rates to self-similarity
  7. The family of fourth-order equations for $1< p \leq 3/2$
  8. The thin-film equation
  9. Strong solutions: definition, basic properties and long-time behavior
  10. Justification of Theorem \ref{['3.tm']} for $p=3/2$
  11. The DLSS equation

Key Result

Proposition 2.1

Let $u(x,t)$ be a solution to equation 2.eq:lnp, then function $v(y,\tau)$ defined by 1.def:v satisfies a nonlocal fourth-order equation

Figures (1)

  • Figure 1: Comparison of exponential and super-exponential decay of re-la-tive entropies.

Theorems & Definitions (17)

  • Proposition 2.1
  • proof
  • Remark 2.1
  • Remark 2.2
  • Lemma 2.2: Lemma 4.1 in CaTo14
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • Remark 2.3
  • Lemma 2.5
  • ...and 7 more