Large time behavior of exponential surface diffusion flows on ℝ

2024/11/26 by Yoshikazu Giga, Giga, Yoshikazu, Michael Gößwein +2 · 1 citation
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Mathematical Dynamics and Fractals #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2411.17175

openalex publication_date 2024/11/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider a surface diffusion flow of the form V=∂_s²f(-κ) with a strictly increasing smooth function f typically, f(r)=e^r, for a curve with arc-length parameter s, where κ denotes the curvature and V denotes the normal velocity. The conventional surface diffusion flow corresponds to the case when f(r)=r. We consider this equation for the graph of a function defined on the whole real line ℝ. We prove that there exists a unique global-in-time classical solution provided that the first and the second derivatives are bounded and small. We further prove that the solution behaves like a solution to a self-similar solution to the equation V=-f'(0)κ. Our result justifies the explanation for grooving modeled by Mullins (1957) directly obtained by Gibbs--Thomson law without linearization of f near κ=0.

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• Remarks on graph-like forward self-similar solutions to the surface diffusion flow equations

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