Well-posedness and stability of the self-similar profile for a thin-film equation with gravity

TL;DR

This paper analyzes the stability of self-similar solutions to a thin-film equation with gravity, establishing convergence rates without relying on explicit profile formulas, and introduces a flexible analytical framework.

Contribution

It develops a novel stability analysis method for self-similar solutions of the thin-film equation that does not depend on explicit profile representations.

Findings

Proves convergence of perturbations to the self-similar profile at rate t^{-1/5}

Establishes coercivity of the linearized operator in a weighted inner product

Provides a systematic approach applicable to broader classes of equations

Abstract

We consider the thin-film equation with linear mobility and a stabilizing second-order porous-medium type term modeling gravity. The model admits self-similar solutions, and our goal is to analyze their stability. We reformulate the problem in mass-Lagrangian coordinates and exploit the underlying gradient-flow structure of the equation with respect to a weighted $L^2$ inner product, where the weight is given by the self-similar source-type profile. This framework allows us to establish a coercivity result for the Hessian (the linearization around the self-similar solution) in a suitably weighted inner product. As a consequence, we prove the convergence of perturbations toward the self-similar profile at an algebraic rate of order $t^{-\frac 1 5}$, in arbitrary scales of weighted Sobolev norms. The analysis relies on maximal-regularity estimates for the linearized evolution, combined with appropriate estimates for the nonlinear terms. Notably, beyond perturbative regimes and in contrast to previous results for the thin-film equation (convergence to the Smyth-Hill profile) or the porous-medium equation (convergence to the Barenblatt-Pattle solution), our analysis does not rely on an explicit (algebraic) representation of the self-similar profile. Instead, it is based solely on a systematic use of the ordinary differential equation satisfied by the self-similar solution, together with a careful analysis of its boundary asymptotics. As a result, we expect that the approach developed here can serve as a flexible toolbox for the study of more general classes of equations and for the stability analysis of special solutions in future work.