American Institute of Mathematical Sciences

Advanced
November  2006, 15(4): 1137-1153. doi: 10.3934/dcds.2006.15.1137

Connecting continua and curves of equilibria of the Cahn-Hilliard equation on the square

S. Maier-Paape 1, and  Ulrich Miller 2,

1. 

Institut für Mathematik, RWTH Aachen, D-52062 Aachen
2. 

Institut für Mathematik, Universität Augsburg, 86135 Augsburg, Germany

Received  December 2004 Revised  January 2006 Published  May 2006

We state an alternative for paths of equilibria of the Cahn-Hilliard equation on the square, bifurcating from the trivial solution at eigenfunctions of the form $w_{ij}=\cos(\pi ix)\cos(\pi j y)$, for $i,j \in \N$. We show that the paths either only connect the bifurcation point $m_{ij}$ with $-m_{ij}$ and are separated from all other paths with even more symmetry, or they contain a loop of nontrivial solutions connecting the bifurcation point $m_{ij}$ with itself. In any case the continua emerging at $m_{ij}$ and $-m_{ij}$ are equal. For fixed mass $m_0=0$ we furthermore prove that the continua bifurcating from the trivial solution at eigenfunctions of the form $w_{i0}+w_{0i}$ or $w_{ij}$, for $i,j \in \N$ are smooth curves parameterized over the interaction length related parameter $\lambda$.
Keywords: Cahn-Hilliard equation., Bifurcation theory.
Mathematics Subject Classification: Primary: 35B32,35Q55; Secondary: 35B4.
Citation: S. Maier-Paape, Ulrich Miller. Connecting continua and curves of equilibria of the Cahn-Hilliard equation on the square. Discrete & Continuous Dynamical Systems - A, 2006, 15 (4) : 1137-1153. doi: 10.3934/dcds.2006.15.1137
[1]

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
[2]

Desheng Li, Xuewei Ju. On dynamical behavior of viscous Cahn-Hilliard equation. Discrete & Continuous Dynamical Systems - A, 2012, 32 (6) : 2207-2221. doi: 10.3934/dcds.2012.32.2207
[3]

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
[4]

Álvaro Hernández, Michał Kowalczyk. Rotationally symmetric solutions to the Cahn-Hilliard equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (2) : 801-827. doi: 10.3934/dcds.2017033
[5]

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
[6]

Annalisa Iuorio, Stefano Melchionna. Long-time behavior of a nonlocal Cahn-Hilliard equation with reaction. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 3765-3788. doi: 10.3934/dcds.2018163
[7]

Dimitra Antonopoulou, Georgia Karali, Georgios T. Kossioris. Asymptotics for a generalized Cahn-Hilliard equation with forcing terms. Discrete & Continuous Dynamical Systems - A, 2011, 30 (4) : 1037-1054. doi: 10.3934/dcds.2011.30.1037
[8]

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
[9]

Alain Miranville, Sergey Zelik. The Cahn-Hilliard equation with singular potentials and dynamic boundary conditions. Discrete & Continuous Dynamical Systems - A, 2010, 28 (1) : 275-310. doi: 10.3934/dcds.2010.28.275
[10]

Laurence Cherfils, Madalina Petcu, Morgan Pierre. A numerical analysis of the Cahn-Hilliard equation with dynamic boundary conditions. Discrete & Continuous Dynamical Systems - A, 2010, 27 (4) : 1511-1533. doi: 10.3934/dcds.2010.27.1511
[11]

Amy Novick-Cohen, Andrey Shishkov. Upper bounds for coarsening for the degenerate Cahn-Hilliard equation. Discrete & Continuous Dynamical Systems - A, 2009, 25 (1) : 251-272. doi: 10.3934/dcds.2009.25.251
[12]

Gianni Gilardi, A. Miranville, Giulio Schimperna. On the Cahn-Hilliard equation with irregular potentials and dynamic boundary conditions. Communications on Pure & Applied Analysis, 2009, 8 (3) : 881-912. doi: 10.3934/cpaa.2009.8.881
[13]

Tomáš Roubíček. Cahn-Hilliard equation with capillarity in actual deforming configurations. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020303
[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]

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
[16]

Ciprian G. Gal, Maurizio Grasselli. Longtime behavior of nonlocal Cahn-Hilliard equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (1) : 145-179. doi: 10.3934/dcds.2014.34.145
[17]

Alain Miranville. Existence of solutions for Cahn-Hilliard type equations. Conference Publications, 2003, 2003 (Special) : 630-637. doi: 10.3934/proc.2003.2003.630
[18]

Sergey Zelik, Jon Pennant. Global well-posedness in uniformly local spaces for the Cahn-Hilliard equation in $\mathbb{R}^3$. Communications on Pure & Applied Analysis, 2013, 12 (1) : 461-480. doi: 10.3934/cpaa.2013.12.461
[19]

Aibo Liu, Changchun Liu. Cauchy problem for a sixth order Cahn-Hilliard type equation with inertial term. Evolution Equations & Control Theory, 2015, 4 (3) : 315-324. doi: 10.3934/eect.2015.4.315
[20]

Irena Pawłow, Wojciech M. Zajączkowski. A sixth order Cahn-Hilliard type equation arising in oil-water-surfactant mixtures. Communications on Pure & Applied Analysis, 2011, 10 (6) : 1823-1847. doi: 10.3934/cpaa.2011.10.1823

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (33)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences