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Consistency for the surface diffusion flat flow in three dimensions

Marco Cicalese, Nicola Fusco, Vesa Julin, Andrea Kubin

TL;DR

This work establishes the short-time consistency of a constrained minimizing movements scheme for the surface diffusion flow in three dimensions. By proving a higher-order geometric inequality and deriving uniform $H^4$-regularity for the discrete flat flow, the authors show that the scheme converges to the unique smooth solution of $V_t = \Delta_{\partial E_t} H_{E_t}$ for $C^5$-regular initial sets, with aδ-constraint that becomes irrelevant at small times. An Iteration Lemma provides a robust dissipation-control mechanism, enabling a bootstrap to high regularity and enabling convergence in $C^{2,\alpha}$ while preserving volume. The results offer a first rigorous bridge between the minimizing movements framework and classical surface-diffusion dynamics, and they extend to the flat torus, with implications for weak solutions and weak-strong uniqueness in this geometric flow context.

Abstract

We investigate the flat flow solution for the surface diffusion equation via the discrete minimizing movements scheme proposed by Cahn and Taylor. We prove that in dimension three the scheme converges to the unique smooth solution of the equation, provided that the initial set is sufficiently regular.

Consistency for the surface diffusion flat flow in three dimensions

TL;DR

This work establishes the short-time consistency of a constrained minimizing movements scheme for the surface diffusion flow in three dimensions. By proving a higher-order geometric inequality and deriving uniform -regularity for the discrete flat flow, the authors show that the scheme converges to the unique smooth solution of for -regular initial sets, with aδ-constraint that becomes irrelevant at small times. An Iteration Lemma provides a robust dissipation-control mechanism, enabling a bootstrap to high regularity and enabling convergence in while preserving volume. The results offer a first rigorous bridge between the minimizing movements framework and classical surface-diffusion dynamics, and they extend to the flat torus, with implications for weak solutions and weak-strong uniqueness in this geometric flow context.

Abstract

We investigate the flat flow solution for the surface diffusion equation via the discrete minimizing movements scheme proposed by Cahn and Taylor. We prove that in dimension three the scheme converges to the unique smooth solution of the equation, provided that the initial set is sufficiently regular.

Paper Structure

This paper contains 14 sections, 16 theorems, 269 equations.

Table of Contents

  1. Introduction
  2. Overview of the proof of Theorem \ref{['thm:thm2']}
  3. Preliminaries
  4. Notation
  5. Preliminary Results
  6. Useful formulas
  7. Almost minimizers of the perimeter
  8. Geometric estimate
  9. Regularity estimates for the flow
  10. Definition of the flat flow solution
  11. The first regularity estimates
  12. Proof of the main theorems
  13. Iteration Lemma
  14. Proof of Theorem \ref{['thm:thm2']} and Theorem \ref{['thm:thm1']}

Key Result

Theorem 1.1

Assume $E_0 \subset \mathbb{R}^3$ is a bounded and $C^5$-regular set. There exist $\delta_0, \sigma_0$ and $C_0$ with the following property: for every $\delta < \delta_0$ there exists $h_0$ such that the sets $E^{h,\delta}(t)$ are uniformly bounded in $H^4$, i.e., for all $t \in [0,\sigma_0]$ and $h \leq h_0$. Moreover, the functions $u^{h,\delta}$ converge in $L^\infty(0,\sigma_0; C^{2,\alpha}

Theorems & Definitions (32)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Definition 2.6
  • ...and 22 more