Existence of self-similar solutions for the surface diffusion flow with nonlinear boundary conditions in the half space
Tomoro Asai, Yoshihito Kohsaka
TL;DR
The paper proves the existence and uniqueness of self-similar solutions for Mullins' surface-diffusion flow in a half-space under a contact-angle boundary condition and a nonlinear no-flux boundary condition. It combines a linear theory with an explicit Green-function framework and a fixed-point argument in weighted Hölder spaces to construct a mild self-similar solution $w_*(x,t)=t^{1/4}W(x/t^{1/4})$ in a setting with small boundary angle $\beta$ and small nonlinear perturbation from the linear problem. Central to the analysis are detailed Green-function estimates for the fourth-order diffusion operator on the half-line and precise bounds for the nonlinear term $\Xi(w_x,w_{xx},w_{xxx})$, enabling a contraction mapping on a carefully chosen function space. The results extend prior linearized analyses to a nonlinear boundary condition, providing a rigorous framework for nontrivial self-similar profiles and establishing the existence of corresponding weak solutions. These contributions advance understanding of nonlinear boundary effects in geometric surface-diffusion flows and offer a rigorous method for constructing self-similar solutions in unbounded domains.
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 and the gradient difference between the solutions to the nonlinear and the linear problems.