American Institute of Mathematical Sciences

Advanced
May  2006, 15(2): 505-528. doi: 10.3934/dcds.2006.15.505

A descent method for the free energy of multicomponent systems

Herbert Gajewski 1, and  Jens A. Griepentrog 1,

1. 

Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstrasse 39, D-10117 Berlin, Germany, Germany

Received  March 2005 Revised  December 2005 Published  March 2006

Equilibrium distributions of multicomponent systems minimize the free energy functional under the constraint of mass conservation of the components. However, since the free energy is not convex in general, usually one tries to characterize and to construct equilibrium distributions as steady states of an adequate evolution equation, for example, the nonlocal Cahn-Hilliard equation for binary alloys. In this work a direct descent method for nonconvex functionals is established and applied to phase separation problems in multicomponent systems and image segmentation.
Keywords: asymptotic behaviour, Lyapunov function, phase separation, image segmentation., Nonconvex functionals, Cahn-Hilliard equation.
Mathematics Subject Classification: 90C26, 82B26, 94A0.
Citation: Herbert Gajewski, Jens A. Griepentrog. A descent method for the free energy of multicomponent systems. Discrete & Continuous Dynamical Systems, 2006, 15 (2) : 505-528. doi: 10.3934/dcds.2006.15.505
[1]

Mauro Fabrizio, Claudio Giorgi, Angelo Morro. Phase transition and separation in compressible Cahn-Hilliard fluids. Discrete & Continuous Dynamical Systems - B, 2014, 19 (1) : 73-88. doi: 10.3934/dcdsb.2014.19.73
[2]

Jan Prüss, Vicente Vergara, Rico Zacher. Well-posedness and long-time behaviour for the non-isothermal Cahn-Hilliard equation with memory. Discrete & Continuous Dynamical Systems, 2010, 26 (2) : 625-647. doi: 10.3934/dcds.2010.26.625
[3]

Andrea Signori. Optimal treatment for a phase field system of Cahn-Hilliard type modeling tumor growth by asymptotic scheme. Mathematical Control & Related Fields, 2020, 10 (2) : 305-331. doi: 10.3934/mcrf.2019040
[4]

Pierluigi Colli, Gianni Gilardi, Elisabetta Rocca, Jürgen Sprekels. Asymptotic analyses and error estimates for a Cahn-Hilliard type phase field system modelling tumor growth. Discrete & Continuous Dynamical Systems - S, 2017, 10 (1) : 37-54. doi: 10.3934/dcdss.2017002
[5]

Desheng Li, Xuewei Ju. On dynamical behavior of viscous Cahn-Hilliard equation. Discrete & Continuous Dynamical Systems, 2012, 32 (6) : 2207-2221. doi: 10.3934/dcds.2012.32.2207
[6]

Laurence Cherfils, Alain Miranville, Sergey Zelik. On a generalized Cahn-Hilliard equation with biological applications. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 2013-2026. doi: 10.3934/dcdsb.2014.19.2013
[7]

Álvaro Hernández, Michał Kowalczyk. Rotationally symmetric solutions to the Cahn-Hilliard equation. Discrete & Continuous Dynamical Systems, 2017, 37 (2) : 801-827. doi: 10.3934/dcds.2017033
[8]

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
[9]

Ciprian G. Gal, Hao Wu. Asymptotic behavior of a Cahn-Hilliard equation with Wentzell boundary conditions and mass conservation. Discrete & Continuous Dynamical Systems, 2008, 22 (4) : 1041-1063. doi: 10.3934/dcds.2008.22.1041
[10]

Pierluigi Colli, Gianni Gilardi, Danielle Hilhorst. On a Cahn-Hilliard type phase field system related to tumor growth. Discrete & Continuous Dynamical Systems, 2015, 35 (6) : 2423-2442. doi: 10.3934/dcds.2015.35.2423
[11]

Tian Ma, Shouhong Wang. Cahn-Hilliard equations and phase transition dynamics for binary systems. Discrete & Continuous Dynamical Systems - B, 2009, 11 (3) : 741-784. doi: 10.3934/dcdsb.2009.11.741
[12]

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
[13]

Sergey P. Degtyarev. On Fourier multipliers in function spaces with partial Hölder condition and their application to the linearized Cahn-Hilliard equation with dynamic boundary conditions. Evolution Equations & Control Theory, 2015, 4 (4) : 391-429. doi: 10.3934/eect.2015.4.391
[14]

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
[15]

Tomáš Roubíček. Cahn-Hilliard equation with capillarity in actual deforming configurations. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 41-55. doi: 10.3934/dcdss.2020303
[16]

Dimitra Antonopoulou, Georgia Karali, Georgios T. Kossioris. Asymptotics for a generalized Cahn-Hilliard equation with forcing terms. Discrete & Continuous Dynamical Systems, 2011, 30 (4) : 1037-1054. doi: 10.3934/dcds.2011.30.1037
[17]

Annalisa Iuorio, Stefano Melchionna. Long-time behavior of a nonlocal Cahn-Hilliard equation with reaction. Discrete & Continuous Dynamical Systems, 2018, 38 (8) : 3765-3788. doi: 10.3934/dcds.2018163
[18]

Dimitra Antonopoulou, Georgia Karali. Existence of solution for a generalized stochastic Cahn-Hilliard equation on convex domains. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 31-55. doi: 10.3934/dcdsb.2011.16.31
[19]

Alain Miranville, Sergey Zelik. The Cahn-Hilliard equation with singular potentials and dynamic boundary conditions. Discrete & Continuous Dynamical Systems, 2010, 28 (1) : 275-310. doi: 10.3934/dcds.2010.28.275
[20]

S. Maier-Paape, Ulrich Miller. Connecting continua and curves of equilibria of the Cahn-Hilliard equation on the square. Discrete & Continuous Dynamical Systems, 2006, 15 (4) : 1137-1153. doi: 10.3934/dcds.2006.15.1137

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (66)
  • HTML views (0)
  • Cited by (20)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences