二维情况下薄膜方程弱解的Hölder连续性
Hölder continuity of weak solutions to the thin-film equation in $d=2$
AI总结：
本文研究了二维情况下薄膜方程弱解的Hölder连续性，通过孔填充技术克服了四阶偏微分方程的退化抛物性挑战。
摘要：
薄膜方程∂_t u = -∇·(u^n ∇Δu)描述了粘性薄液膜在平坦固体表面展开时高度u=u(x,t)≥0的演变。我们证明了在物理上最相关的情况下，二维空间维度d=2中薄膜方程弱解的Hölder连续性。尽管早在十多年前就建立了弱解的广泛存在理论，但即使在d=2中弱解的有界性仍然是薄膜方程理论中的主要未解问题。由于薄膜方程的四阶结构，De Giorgi-Nash-Moser理论不适用。我们的证明基于孔填充技术，挑战来自于四阶偏微分方程的退化抛物性。
英文摘要：
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.
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