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

TL;DR

We address the global existence of weak solutions to a fourth-order nonlinear parabolic equation modeling crystal surface evolution with gradient-dependent mobility. The problem is formulated with $E(z)=\frac{1}{p}|z|^p+\beta_0|z|$ leading to the evolution $\partial_t u + \nabla\cdot\left(M(\nabla u)\nabla\operatorname{div}[\nabla_z E(\nabla u) ]\right)=0$, under homogeneous Neumann boundary conditions. The authors develop a strategy based on time discretization, regularization by a parameter $\tau$, a priori energy estimates, and monotone operator theory, together with the Leray–Schauder fixed-point theorem for the approximate problems, and they pass to the limit using Aubin–Lions type compactness to obtain a global weak solution of the original problem. This framework addresses the challenge posed by the gradient-dependent mobility and provides a rigorous existence theory for a class of fourth-order, nonlinear parabolic crystal surface models with potential implications for related gradient-flow PDEs.

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.

Theorems & Definitions (20)

• Definition 1.1: Definition of a weak solution
• Theorem 1.2: Main Theorem
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5: Lions-Aubin
• Definition 3.1
• Theorem 3.2
• proof