Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Manuel V. Gnann, Slim Ibrahim

TL;DR

This work proves well-posedness and stability for perturbations of a non-explicit self-similar profile of a thin-film equation with gravity. It uses a mass-Lagrangian reformulation and a weighted $L^2$ gradient-flow framework around the self-similar state, establishing a coercive Hessian and maximal-regularity estimates for the linearized problem, then controlling the nonlinear terms to obtain global existence and decay. The main results show convergence toward the self-similar profile in weighted Sobolev norms, with explicit asymptotics in the original variables: the height decays like $t^{-1/5}$ and the contact-line velocity decays like $t^{-4/5}$, up to perturbative corrections. Importantly, the approach avoids requiring an explicit algebraic representation of the self-similar profile, offering a flexible toolbox for analyzing stability of special solutions in related degenerate-diffusion problems.

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.

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

TL;DR

This work proves well-posedness and stability for perturbations of a non-explicit self-similar profile of a thin-film equation with gravity. It uses a mass-Lagrangian reformulation and a weighted gradient-flow framework around the self-similar state, establishing a coercive Hessian and maximal-regularity estimates for the linearized problem, then controlling the nonlinear terms to obtain global existence and decay. The main results show convergence toward the self-similar profile in weighted Sobolev norms, with explicit asymptotics in the original variables: the height decays like and the contact-line velocity decays like , up to perturbative corrections. Importantly, the approach avoids requiring an explicit algebraic representation of the self-similar profile, offering a flexible toolbox for analyzing stability of special solutions in related degenerate-diffusion problems.

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 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 , 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.
Paper Structure (22 sections, 13 theorems, 159 equations)

This paper contains 22 sections, 13 theorems, 159 equations.

Table of Contents

  1. Introduction and heuristic motivation
  2. Thin-film equation with gravity and self-similar solutions
  3. Heuristics: the standard gradient-flow formulation
  4. Mass-Lagrangian formulation
  5. Heuristics: gradient-flow formulation in mass-Lagrangian coordinates
  6. Heuristics: lower bound on the Hessian
  7. A formal argument for stability
  8. Formal estimates on the Hessian: refined linear analysis and functional-analytic setting
  9. Main results and discussion
  10. Main results
  11. Asymptotics in the original variables
  12. Discussion
  13. Outline
  14. Notation and conventions
  15. The linear degenerate-parabolic equation
  16. ...and 7 more sections

Key Result

Theorem 2.1

For $V^{(0)} \in \mathcal{H}^6$ such that $|V|_0 \ll_\ell 1$, there exists exactly one solution to the nonlinear Cauchy problem nonlinear-cauchy. Additionally, this solution satisfies where $s_{2,\ell} = s_{3,\ell} = 0$ and $s_{k,\ell} \gg_{k,\ell} 1$ for $k \ge 4$. Furthermore, we have the à-priori estimate for any $k \in \mathbb{N}$ with $k \ge 2$.

Theorems & Definitions (26)

  • Theorem 2.1: Well-posedness
  • Theorem 2.2: Stability
  • Lemma 3.1
  • proof : Proof of Lemma \ref{['lem-der-l']}
  • Lemma 3.2
  • proof
  • Lemma 3.3: see gnann2015well
  • proof
  • Lemma 3.4: Interpolation inequality
  • proof
  • ...and 16 more