Hölder continuity of weak solutions to the thin-film equation in $d=2$
Federico Cornalba, Julian Fischer, Erika Maringová Kokavcová
TL;DR
This work establishes local Hölder continuity for energy-dissipating weak solutions of the 2D thin-film equation $\partial_t u = -\nabla\cdot(u^n\nabla\Delta u)$ in the physically relevant range $2-\sqrt{4/5}<n<3$, a regime where previous regularity results were unavailable. The authors develop a hole-filling strategy adapted to the degenerate fourth-order mobility $u^n$, centered on a tilt-excess functional whose evolution is controlled separately on good and bad times; the analysis combines Bernis–Grün inequalities, Morrey embeddings, and weighted energy dissipation. The core result is a space-time Hölder estimate $u\in C^{\sigma}_{loc}(\mathbb{R}^2\times(0,T))$ for some $\sigma>0$, with global Hölder regularity up to $t=0$ under additional data on $\nabla u_0$. This advances the theory beyond integral estimates and provides a rigorous regularity framework for a broad class of weak solutions in two dimensions, leveraging a novel adaptation of the hole-filling technique to a degenerate, higher-order PDE.
Abstract
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.