Communications in Mathematical Sciences

Volume 22 (2024)

Number 2

Weak solutions for a modified degenerate Cahn–Hilliard model for surface diffusion

Pages: 487 – 517

DOI: https://dx.doi.org/10.4310/CMS.2024.v22.n2.a8

Authors

Xiaohua Niu (School of Mathematics and Statistics, Xiamen University of Technology, Xiamen, Fujian, China)

Yang Xiang (Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong; and HKUST Shenzhen-Hong Kong Collaborative Innovation Research Institute, Futian, Shenzhen, China)

Xiaodong Yan (Department of Mathematics, University of Connecticut, Storrs, Ct., U.S.A.)

Abstract

In this paper, we study the weak solutions of a modified degenerate Cahn–Hilliard type model for surface diffusion. With degenerate phase-dependent diffusion mobility and additional stabilizing function, this model is able to give the correct sharp interface limit. We introduce a notion of weak solutions for the nonlinear model. The existence of such solutions is obtained by approximations of the proposed model with non-degenerate mobilities. We also employ this method to prove the existence of weak solutions to a related model where the chemical potential contains a nonlocal term originating from self-climb of dislocations in crystalline materials.

Keywords

phase field model, degenerate Cahn–Hilliard equation, surface diffusion, weak solutions

2010 Mathematics Subject Classification

35A01, 35G20, 35K25, 74N20, 82C26

Full Text (PDF format)

X.H. Niu’s research is supported by National Natural Science Foundation of China under the grant number 11801214 and the Natural Science Foundation of Fujian Province of China under the grant number 2021J011193.

Y. Xiang’s research is supported by the Hong Kong Research Grants Council General Research Fund 16307319, and the Project of Hetao Shenzhen-HKUST Innovation Cooperation Zone HZQB-KCZYB-2020083.

X. Yan’s research is supported by Simons Collaboration Grant #947054, a Research Excellence Grant, and CLAS Dean’s Summer Research Grant from University of Connecticut.

Received 7 April 2023

Received revised 7 July 2023

Accepted 13 July 2023

Published 1 February 2024