Existence Theorems for a Fourth-Order Exponential PDE Related to Crystal Surface Growth

TL;DR

This paper proves the global existence and uniqueness of solutions for a fourth-order exponential PDE modeling crystal surface growth, highlighting advantages of hyperbolic sine nonlinearity over Arrhenius rates in high dimensions.

Contribution

It establishes the first global existence and uniqueness results for this specific PDE derived from microscopic models, emphasizing the benefits of hyperbolic sine nonlinearity.

Findings

Proves global existence of strong solutions

Demonstrates better control with hyperbolic sine nonlinearity

Highlights advantages over Arrhenius rate-based PDEs

Abstract

In this article we prove the global existence of a unique strong solution to the initial boundary-value problem for a fourth-order exponential PDE. The equation we study was originally proposed to study the evolution of crystal surfaces, and was derived by applying a nonstandard scaling regime to a microscopic Markov jump process with Metropolis rates. Our investigation here finds that compared to the PDE's which use Arhenious rates, (and also have a fourth order exponential nonlinearity) the hyperbolic sine nonlinearity in our equation can offer much better control over the exponent term even in high dimensions.