H\"older continuity of weak solutions to the thin-film equation in $d=2$

TL;DR

This paper proves that weak solutions to the two-dimensional thin-film equation are H"older continuous, addressing a longstanding open problem in the regularity theory of this degenerate fourth-order PDE.

Contribution

It establishes H"older continuity of weak solutions in 2D, overcoming the challenge posed by the equation's degeneracy and fourth-order structure.

Findings

Weak solutions are H"older continuous in 2D.

Addresses the boundedness problem for weak solutions.

Introduces a novel proof technique based on hole-filling.

Abstract

The thin-film equation $\partial_t u = -\nabla \cdot (u^n \nabla \Delta u)$ 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 H\"older 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.