Well-posedness and stability of the self-similar profile for a thin-film equation with gravity
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arXiv
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| Format: | Preprint |
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2026
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| author | Gnann, Manuel V. Ibrahim, Slim |
| author_facet | Gnann, Manuel V. Ibrahim, Slim |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_16549 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Well-posedness and stability of the self-similar profile for a thin-film equation with gravity Gnann, Manuel V. Ibrahim, Slim Analysis of PDEs Mathematical Physics 35C06, 35K65, 35K35, 76D08 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. |
| title | Well-posedness and stability of the self-similar profile for a thin-film equation with gravity |
| topic | Analysis of PDEs Mathematical Physics 35C06, 35K65, 35K35, 76D08 |
| url | https://arxiv.org/abs/2602.16549 |