Existence of self-similar solutions for the surface diffusion flow with nonlinear boundary conditions in the half space

TL;DR

This paper proves the existence and uniqueness of self-similar solutions for a surface diffusion flow model with nonlinear boundary conditions, specifically addressing Mullins' problem related to thermal grooving, under small contact angles.

Contribution

It establishes the first rigorous proof of self-similar solutions for Mullins' problem with nonlinear boundary conditions in the half space.

Findings

Existence of self-similar solutions under small contact angles.

Uniqueness of these solutions.

Application to Mullins' problem in thermal grooving.

Abstract

We study the Mullins' problem that was proposed by Mullins in 1957 and is one of the models of the thermal grooving by surface diffusion. Mathematically, this is the problem of evolving curves in the half space that is governed by the surface diffusion flow with the contact angle condition and the no-flux condition on the boundary. The no-flux condition is represented as the equation that the first order derivative of the curvature with respect to the arc-length parameter is equal to zero, so that it is the nonlinear boundary condition. For this original Mullins' problem, we show the existence and the uniqueness of the self-similar solution. The self-similar solution is obtained as the mild solution under the smallness assumption on the contact angle.