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

TL;DR

This paper develops a new rescaling method to improve the convergence rates of strong solutions to the thin-film and related fourth-order evolution equations, providing sharper early and intermediate time estimates.

Contribution

It introduces a time-dependent rescaling preserving the second moment, leading to improved convergence rates toward steady states for nonlinear fourth-order equations.

Findings

Sharp convergence rates in relative Rényi entropy

Enhanced early and intermediate time estimates

Rigorous justification for strong solutions of the thin-film equation

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 R\'enyi 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.