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A classification of solitons for the surface diffusion flow of entire graphs

Piotr Rybka, Glen Wheeler

TL;DR

The paper classifies solitons for the surface diffusion flow acting on entire graphs, proving that under general conditions every graphical soliton—steady states, forward/backward self-similar profiles, or travelling waves—must be linear, namely $\\phi(y)=Ay$. The authors develop a curvature-based framework with $k[\\phi]$ and $v[\\phi]$, derive a key angle bound, and employ convex functionals along self-similar profiles to enforce rigidity; for travelling waves, they reduce the profile to a linear form $\\phi(y)=\\frac{b}{a}y$. These results clarify that nontrivial graphical solitons do not exist in this setting, highlighting a sharp contrast with non-graphical solitons and emphasizing the need for unbounded initial data theory to capture richer dynamics.

Abstract

In this article we classify solitons (equilibria, self-similar solutions and travelling waves) for the surface diffusion flow of entire graphs of function over the real line.

A classification of solitons for the surface diffusion flow of entire graphs

TL;DR

The paper classifies solitons for the surface diffusion flow acting on entire graphs, proving that under general conditions every graphical soliton—steady states, forward/backward self-similar profiles, or travelling waves—must be linear, namely . The authors develop a curvature-based framework with and , derive a key angle bound, and employ convex functionals along self-similar profiles to enforce rigidity; for travelling waves, they reduce the profile to a linear form . These results clarify that nontrivial graphical solitons do not exist in this setting, highlighting a sharp contrast with non-graphical solitons and emphasizing the need for unbounded initial data theory to capture richer dynamics.

Abstract

In this article we classify solitons (equilibria, self-similar solutions and travelling waves) for the surface diffusion flow of entire graphs of function over the real line.
Paper Structure (4 sections, 8 theorems, 86 equations, 2 figures)

This paper contains 4 sections, 8 theorems, 86 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Steady states
  3. Self-similar solutions
  4. Travelling waves

Key Result

Theorem 1

Let $\phi:\mathbb{R}\rightarrow\mathbb{R}$ be a locally smooth function. Assume that either for the surface diffusion flow. Then $\phi(y) = Ay$ for some $A\in\mathbb{R}$.

Figures (2)

  • Figure 1: (a) A straight line with gradient equal to one. (b) A circle with curvature equal to two. (c) An Eulerian clothoid with curvature equal to arc-length.
  • Figure 2: The lemniscate of Bernoulli shrinking under surface diffusion flow. The figure overlays images of the flow at times $0, 1, 2, 3, 4, 5$, and $6$, by which time it has vanished to the origin. Its initial parametrisation is $t\mapsto \frac{\sqrt{6}}{1+\sin^{2}t}\left(\cos\ t,\ \frac{1}{2}\sin\ 2t\right) .$

Theorems & Definitions (26)

  • Theorem 1
  • Remark 2
  • Remark 3
  • Lemma 4
  • Definition 5
  • Definition 6
  • Lemma 7
  • proof
  • proof : Proof of Lemma \ref{['LMsteadystateclassification']}
  • Definition 8
  • ...and 16 more