Weak solution of a continuum model for vicinal surface in the attachment-detachment-limited regime

TL;DR

This paper introduces a weak solution framework for a continuum model of vicinal surface evolution in the attachment-detachment-limited regime, proving existence, positivity, and long-term convergence to a constant state.

Contribution

It formulates a notion of weak solutions for the model and establishes their global existence, positivity, and asymptotic behavior, advancing understanding of surface dynamics in this regime.

Findings

Existence of a global weak solution proven.

Weak solutions are positive almost everywhere.

Solutions converge to a constant as time approaches infinity.

Abstract

We study in this work a continuum model derived from 1D attachment-detachment-limited (ADL) type step flow on vicinal surface, $ u_t=-u^2(u^3)_{hhhh}$, where $u$, considered as a function of step height $h$, is the step slope of the surface. We formulate a notion of weak solution to this continuum model and prove the existence of a global weak solution, which is positive almost everywhere. We also study the long time behavior of weak solution and prove it converges to a constant solution as time goes to infinity. The space-time H\"older continuity of the weak solution is also discussed as a byproduct.