A Global Existence Theorem for a Fourth-Order Crystal Surface Model with Gradient Dependent Mobility

TL;DR

This paper proves the global existence of weak solutions for a complex fourth-order nonlinear PDE modeling crystal surface growth, overcoming challenges posed by gradient-dependent mobility matrices.

Contribution

It introduces novel techniques to establish global existence results for a nonlinear PDE with gradient-dependent mobility, extending previous results where mobility was constant.

Findings

Established global existence of weak solutions

Overcame difficulties due to gradient-dependent mobility

Extended mathematical understanding of crystal surface models

Abstract

In this article we study the existence of solutions to a fourth-order nonlinear PDE related to crystal surface growth. The key difficulty in the equations comes from the mobility matrix, which depends on the gradient of the solution. When the mobility matrix is the identity matrix there are now many existence results, however when it is allowed to depend on the solution we lose crucial estimates in the time direction. In this work we are able to prove the global existence of weak solutions despite this lack of estimates in the time direction.