American Institute of Mathematical Sciences

Advanced
March  2008, 1(1): 1-14. doi: 10.3934/dcdss.2008.1.1

On sharp interface limits of Allen--Cahn/Cahn--Hilliard variational inequalities

John W. Barrett 1, , Harald Garcke 2, and  Robert Nürnberg 1,

1. 

Department of Mathematics, Imperial College London, South Kensington Campus, London, SW7 2AZ, United Kingdom, United Kingdom
2. 

NWF I - Mathematik, Universität Regensburg, 93040 Regensburg, Germany

Received  September 2006 Revised  February 2007 Published  December 2007

Using formally matched asymptotic expansions we identify the sharp interface asymptotic limit of an Allen-Cahn/Cahn-Hilliard system using a novel approach which enables us to handle the case of variational inequalities.
Keywords: matched asymptotic expansions, Degenerate Cahn--Hilliard/Allen--Cahn variational inequality, degenerate parabolic problem..
Mathematics Subject Classification: Primary: 35K65, 49J40, 35C20; Secondary:35Q6.
Citation: John W. Barrett, Harald Garcke, Robert Nürnberg. On sharp interface limits of Allen--Cahn/Cahn--Hilliard variational inequalities. Discrete & Continuous Dynamical Systems - S, 2008, 1 (1) : 1-14. doi: 10.3934/dcdss.2008.1.1
[1]

Alain Miranville, Ramon Quintanilla, Wafa Saoud. Asymptotic behavior of a Cahn-Hilliard/Allen-Cahn system with temperature. Communications on Pure & Applied Analysis, 2020, 19 (4) : 2257-2288. doi: 10.3934/cpaa.2020099
[2]

Amy Novick-Cohen, Andrey Shishkov. Upper bounds for coarsening for the degenerate Cahn-Hilliard equation. Discrete & Continuous Dynamical Systems, 2009, 25 (1) : 251-272. doi: 10.3934/dcds.2009.25.251
[3]

Mohammad Hassan Farshbaf-Shaker, Takeshi Fukao, Noriaki Yamazaki. Singular limit of Allen--Cahn equation with constraint and its Lagrange multiplier. Conference Publications, 2015, 2015 (special) : 418-427. doi: 10.3934/proc.2015.0418
[4]

Matthieu Alfaro, Hiroshi Matano. On the validity of formal asymptotic expansions in Allen-Cahn equation and FitzHugh-Nagumo system with generic initial data. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1639-1649. doi: 10.3934/dcdsb.2012.17.1639
[5]

Alp Eden, Varga K. Kalantarov. 3D convective Cahn--Hilliard equation. Communications on Pure & Applied Analysis, 2007, 6 (4) : 1075-1086. doi: 10.3934/cpaa.2007.6.1075
[6]

Jaemin Shin, Yongho Choi, Junseok Kim. An unconditionally stable numerical method for the viscous Cahn--Hilliard equation. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1737-1747. doi: 10.3934/dcdsb.2014.19.1737
[7]

Quan Wang, Dongming Yan. On the stability and transition of the Cahn-Hilliard/Allen-Cahn system. Discrete & Continuous Dynamical Systems - B, 2020, 25 (7) : 2607-2620. doi: 10.3934/dcdsb.2020024
[8]

Georgia Karali, Yuko Nagase. On the existence of solution for a Cahn-Hilliard/Allen-Cahn equation. Discrete & Continuous Dynamical Systems - S, 2014, 7 (1) : 127-137. doi: 10.3934/dcdss.2014.7.127
[9]

Christopher P. Grant. Grain sizes in the discrete Allen-Cahn and Cahn-Hilliard equations. Discrete & Continuous Dynamical Systems, 2001, 7 (1) : 127-146. doi: 10.3934/dcds.2001.7.127
[10]

Shixing Li, Dongming Yan. On the steady state bifurcation of the Cahn-Hilliard/Allen-Cahn system. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3077-3088. doi: 10.3934/dcdsb.2018301
[11]

Jie Shen, Xiaofeng Yang. Numerical approximations of Allen-Cahn and Cahn-Hilliard equations. Discrete & Continuous Dynamical Systems, 2010, 28 (4) : 1669-1691. doi: 10.3934/dcds.2010.28.1669
[12]

Alain Miranville, Wafa Saoud, Raafat Talhouk. On the Cahn-Hilliard/Allen-Cahn equations with singular potentials. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3633-3651. doi: 10.3934/dcdsb.2018308
[13]

Harald Garcke, Kei Fong Lam. Analysis of a Cahn--Hilliard system with non-zero Dirichlet conditions modeling tumor growth with chemotaxis. Discrete & Continuous Dynamical Systems, 2017, 37 (8) : 4277-4308. doi: 10.3934/dcds.2017183
[14]

Changchun Liu, Hui Tang. Existence of periodic solution for a Cahn-Hilliard/Allen-Cahn equation in two space dimensions. Evolution Equations & Control Theory, 2017, 6 (2) : 219-237. doi: 10.3934/eect.2017012
[15]

Irena Pawłow. Thermodynamically consistent Cahn-Hilliard and Allen-Cahn models in elastic solids. Discrete & Continuous Dynamical Systems, 2006, 15 (4) : 1169-1191. doi: 10.3934/dcds.2006.15.1169
[16]

Ahmad Makki, Alain Miranville. Existence of solutions for anisotropic Cahn-Hilliard and Allen-Cahn systems in higher space dimensions. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 759-775. doi: 10.3934/dcdss.2016027
[17]

Cristina Pocci. On singular limit of a nonlinear $p$-order equation related to Cahn-Hilliard and Allen-Cahn evolutions. Evolution Equations & Control Theory, 2013, 2 (3) : 517-530. doi: 10.3934/eect.2013.2.517
[18]

Juan Wen, Yaling He, Yinnian He, Kun Wang. Stabilized finite element methods based on multiscale enrichment for Allen-Cahn and Cahn-Hilliard equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021074
[19]

Pierluigi Colli, Gianni Gilardi, Paolo Podio-Guidugli, Jürgen Sprekels. An asymptotic analysis for a nonstandard Cahn-Hilliard system with viscosity. Discrete & Continuous Dynamical Systems - S, 2013, 6 (2) : 353-368. doi: 10.3934/dcdss.2013.6.353
[20]

Luyi Ma, Hong-Tao Niu, Zhi-Cheng Wang. Global asymptotic stability of traveling waves to the Allen-Cahn equation with a fractional Laplacian. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2457-2472. doi: 10.3934/cpaa.2019111

2020 Impact Factor: 2.425

Tools

Metrics

  • PDF downloads (81)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences