Hölder continuity of weak solutions to the thin-film equation in $d=2$
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866914228159381504 |
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| author | Cornalba, Federico Fischer, Julian Kokavcová, Erika Maringová |
| author_facet | Cornalba, Federico Fischer, Julian Kokavcová, Erika Maringová |
| contents | The thin-film equation $\partial_t u = -\nabla \cdot (u^n \nabla Δu)$ describes the evolution of the height $u=u(x,t)\geq 0$ of a viscous thin liquid film spreading on a flat solid surface. We prove Hölder continuity of energy-dissipating weak solutions to the thin-film equation in the physically most relevant case of two spatial dimensions $d=2$. While an extensive existence theory of weak solutions to the thin-film equation was established more than two decades ago, even boundedness of weak solutions in $d=2$ has remained a major unsolved problem in the theory of the thin-film equation. Due the fourth-order structure of the thin-film equation, De Giorgi-Nash-Moser theory is not applicable. Our proof is based on the hole-filling technique, the challenge being posed by the degenerate parabolicity of the fourth-order PDE. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_24809 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hölder continuity of weak solutions to the thin-film equation in $d=2$ Cornalba, Federico Fischer, Julian Kokavcová, Erika Maringová Analysis of PDEs The thin-film equation $\partial_t u = -\nabla \cdot (u^n \nabla Δu)$ describes the evolution of the height $u=u(x,t)\geq 0$ of a viscous thin liquid film spreading on a flat solid surface. We prove Hölder continuity of energy-dissipating weak solutions to the thin-film equation in the physically most relevant case of two spatial dimensions $d=2$. While an extensive existence theory of weak solutions to the thin-film equation was established more than two decades ago, even boundedness of weak solutions in $d=2$ has remained a major unsolved problem in the theory of the thin-film equation. Due the fourth-order structure of the thin-film equation, De Giorgi-Nash-Moser theory is not applicable. Our proof is based on the hole-filling technique, the challenge being posed by the degenerate parabolicity of the fourth-order PDE. |
| title | Hölder continuity of weak solutions to the thin-film equation in $d=2$ |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2512.24809 |