Korean J. Math.  Vol 22, No 4 (2014)  pp.621-632
DOI: https://doi.org/10.11568/kjm.2014.22.4.621

Dynamical bifurcation of the one-dimensional convective Cahn-Hilliard equation

Yuncherl Choi

Abstract


In this paper, we study the dynamical behavior of the one-dimensional convective Cahn-Hilliard equation(CCHE) on a periodic cell $[-\pi,\pi]$.
We prove that as the control parameter passes through the critical number,the CCHE bifurcates from the trivial solution to an attractor.
We describe the bifurcated attractor in detail which gives the final patterns of solutions near the trivial solution.


Keywords


convective Cahn-Hilliard equation, dynamic bifurcation, center manifold reduction

Subject classification

37G35, 35B32

Sponsor(s)

This work was supported by the Research Grant of Kwangwoon University in 2014.

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References


Y. Choi, Dynamical bifurcation of a one dimensional modified Swift-Hohenberg equation, submitted. (Google Scholar)

Y. Choi and J. Han, Dynamical bifurcation of the damped Kuramoto-Sivashinsky equation, J. Math. Anal. Appl. 421 (2015), 383–398. (Google Scholar)

A. Edena and V. K. Kalantarovb, The convective Cahn–Hilliard equation, Appl. Math. Lett. 20 (2007), 455–461. (Google Scholar)

H. Gao and Q. Xiao, Bifurcation analysis of the 1D and 2D generalized Swift-Hohenberg equation, Intern. J. Bifur. Chaos, 20 (2010), 619–643. (Google Scholar)

J. Han and C.-H. Hsia, Dynamical bifurcation of the two dimensional Swift- Hohenberg equation with odd periodic condition, Dis. Cont. Dyn. Sys. B 17 (2012), 2431–2449. (Google Scholar)

J. Han and M. Yari, Dynamic bifurcation of the periodic Swift-Hohenberg equation, Bull. Korean Math. Soc. 49 (2012), 923–937. (Google Scholar)

T. Ma and S. Wang, Bifurcation Theory and Applications, World Scientific, 2005. (Google Scholar)

T. Ma and S. Wang, Cahn-Hilliard equations and phase transition dynamics for (Google Scholar)

binary systems, Disc. Cont. Dyn. Sys. A 11 (2009), 741-784. (Google Scholar)

L. A. Peletier and V. Rottscha ̈fer, Pattern selection of solutions of the Swift-Hohenberg equations, Phys. D 194 (2004), 95–126. (Google Scholar)

L. Peletier and J. Williams, Some canonical bifurcations in the Swift-Hohenberg (Google Scholar)

equation, SIAM J. Appl. Dyn. Sys. 6 (2007), 208–235. (Google Scholar)

A. Podolnya, M.A. Zaksb, B.Y. Rubinsteinc, A.A. Golovin, and A.A. Nepom-nyashchya, Dynamics of domain walls governed by the convective Cahn–Hilliard equation, Phys. D 201 (2005), 291–305. (Google Scholar)

M. Yari, Attractor bifurcation and final patterns of the Swift-Hohenberg and generalized Swift-Hohenberg equations, Dis. Cont. Dyn. Sys. B 7 (2007), 441– 456. (Google Scholar)

M. A. Zaks, A. Podolny, A. A. Nepomnyashchya, and A.A. Golovin, Periodic stationary patterns governed by a convective Cahn-Hilliard equation, SIAM J. Appl. Math. 66 (2006), 700–720. (Google Scholar)


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