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Asymptotically self-similar graph-like solutions to a multi-dimensional surface diffusion flow equation under contact angle and no-flux boundary conditions

Yoshikazu Giga, Sho Katayama

TL;DR

The paper studies Mullins' model of a surface diffusion flow in a multidimensional half-space with a contact-angle boundary condition and a no-flux boundary, focusing on graph-like initial surfaces. By reformulating the problem as a quasilinear parabolic equation and solving it via integral equations built from layer- and volume-potentials for an anisotropic biharmonic operator, the authors develop a framework in scaled Hölder spaces $Z_\alpha^{k+\mu}$ to handle long-time behavior and boundary effects. They establish global existence for small perturbations of a base slope $\gamma_0=\tan\theta$, construct self-similar solutions for homogeneous data, and prove convergence to self-similar profiles for asymptotically homogeneous initial data, without requiring a small-angle restriction. The results extend prior one-dimensional works to higher dimensions and provide a robust nonlinear-analytic approach using contraction mappings and Schauder-type estimates, yielding a family of self-similar solutions dictated by boundary data with potential impact on thermal grooving and related geometric diffusion phenomena.

Abstract

This paper studies Mullins' model of thermal grooving which consists of a surface diffusion flow equation with contact angle and no-flux boundary conditions. We consider this problem in a multi-dimensional half space and prove that if the slope of the initial data is close to that consistent with the contact angle, then there exists a unique global-in-time solution. In particular, we show the existence of a self-similar solution for a given behavior at the space infinity. We also show that our global solution converges to a self-similar solution as the time tends to infinity if the initial data is asymptotically homogeneous at the space infinity. No assumption on the size of the contact angle is imposed.

Asymptotically self-similar graph-like solutions to a multi-dimensional surface diffusion flow equation under contact angle and no-flux boundary conditions

TL;DR

The paper studies Mullins' model of a surface diffusion flow in a multidimensional half-space with a contact-angle boundary condition and a no-flux boundary, focusing on graph-like initial surfaces. By reformulating the problem as a quasilinear parabolic equation and solving it via integral equations built from layer- and volume-potentials for an anisotropic biharmonic operator, the authors develop a framework in scaled Hölder spaces to handle long-time behavior and boundary effects. They establish global existence for small perturbations of a base slope , construct self-similar solutions for homogeneous data, and prove convergence to self-similar profiles for asymptotically homogeneous initial data, without requiring a small-angle restriction. The results extend prior one-dimensional works to higher dimensions and provide a robust nonlinear-analytic approach using contraction mappings and Schauder-type estimates, yielding a family of self-similar solutions dictated by boundary data with potential impact on thermal grooving and related geometric diffusion phenomena.

Abstract

This paper studies Mullins' model of thermal grooving which consists of a surface diffusion flow equation with contact angle and no-flux boundary conditions. We consider this problem in a multi-dimensional half space and prove that if the slope of the initial data is close to that consistent with the contact angle, then there exists a unique global-in-time solution. In particular, we show the existence of a self-similar solution for a given behavior at the space infinity. We also show that our global solution converges to a self-similar solution as the time tends to infinity if the initial data is asymptotically homogeneous at the space infinity. No assumption on the size of the contact angle is imposed.
Paper Structure (8 sections, 21 theorems, 182 equations)

This paper contains 8 sections, 21 theorems, 182 equations.

Table of Contents

  1. Introduction
  2. Function spaces and main results
  3. Integral equations
  4. Linear estimates
  5. Estimates of nonlinear terms
  6. Proof of main results
  7. Remarks on the structure of equations
  8. Graphical forms of differential operators on $\Gamma$

Key Result

Theorem 2.1

Let $\mu\in(0,1)$ and $\gamma_0\in\mathbb{R}$. Then there exist constants $\varepsilon^*>0$ and $\delta^*>0$ such that if $\gamma=\tan\theta\in\mathbb{R}$ and $u_0\in \operatorname{Lip}(\overline{\mathbb{R}^N_+})$ satisfies then the problem EU1--EU3 with $T=\infty$ admits a unique solution $u$ with initial data $u_0$ in the set where $e_N=(0,\ldots,0,1)\in\mathbb{R}^N$. Moreover, $u$ belongs to

Theorems & Definitions (34)

  • Theorem 2.1
  • Corollary 2.2
  • Remark 2.3
  • proof : Proof of Corollary \ref{['CSS']} admitting Theorem \ref{['TM']}
  • Corollary 2.4
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 4.1
  • ...and 24 more