# American Institute of Mathematical Sciences

July  2011, 16(1): 173-188. doi: 10.3934/dcdsb.2011.16.173

## Dynamic phase transition for binary systems in cylindrical geometry

 1 Department of Mathematics and Taidar Institute of Mathematical Science, National Taiwan University, Taipei, 10617, Taiwan 2 Department of Mathematics and Taidar Institute of Mthematical Science, National Taiwan University, Taipei, 10617, Taiwan

Received  February 2010 Revised  August 2010 Published  April 2011

In this article, we present a dynamic phase transition and stability analysis for the Cahn-Hilliard equations in cylindrical geometry. Two types of phase transitions (the continuous type and the jump type) are determined explicitly in terms of relevant physical and geometric parameters.
Citation: I-Liang Chern, Chun-Hsiung Hsia. Dynamic phase transition for binary systems in cylindrical geometry. Discrete and Continuous Dynamical Systems - B, 2011, 16 (1) : 173-188. doi: 10.3934/dcdsb.2011.16.173
##### References:
 [1] Stephen J. Blundell and Katherine M. Blundell, "Concepts in Thermal Physics," Oxford University Press, 2008. [2] C. M. Elliott and H. Garcke, Diffusional phase transitions in multicomponent systems with a concentration dependent mobility matrix, Physica D, 109 (1997), 242-256. doi: 10.1016/S0167-2789(97)00066-3. [3] J. E. Hilliard, Spinodal decomposition, in Phase Transformations, American Society for Metal, Cleveland, (1970), 497-560. [4] T. Ma and S. Wang, "Bifurcation Theory and Applications," vol. 53 of World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises, [5] T. Ma and S. Wang, Dynamic phase transition theory in PVT systems, Indiana Univ. Math. J., 57 (2008), 2861-2889. doi: 10.1512/iumj.2008.57.3630. [6] T. Ma and S. Wang, Cahn-Hilliard equations and phase transition dynamics for binary systems, Discrete Contin. Dyn. Syst. Ser. B, 11 (2009), 741-784. doi: 10.3934/dcdsb.2009.11.741. [7] T. Ma and S. Wang, Phase separation of binary systems, Phys. Rev. A., 388 (2009), 4811-4817. [8] J. Moser, A rapidly convergent iteration method and nonlinear partial differential equation. I, Ann. Sc. Norm. Super. Pisa, 20 (1966), 265-315. [9] L. E. Reichl, "A Modern Course in Statistical Physics," (Second ed.) A Wiley-Interscinece Publication. New York: John Wiley & Sons Inc. 1998.

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##### References:
 [1] Stephen J. Blundell and Katherine M. Blundell, "Concepts in Thermal Physics," Oxford University Press, 2008. [2] C. M. Elliott and H. Garcke, Diffusional phase transitions in multicomponent systems with a concentration dependent mobility matrix, Physica D, 109 (1997), 242-256. doi: 10.1016/S0167-2789(97)00066-3. [3] J. E. Hilliard, Spinodal decomposition, in Phase Transformations, American Society for Metal, Cleveland, (1970), 497-560. [4] T. Ma and S. Wang, "Bifurcation Theory and Applications," vol. 53 of World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises, [5] T. Ma and S. Wang, Dynamic phase transition theory in PVT systems, Indiana Univ. Math. J., 57 (2008), 2861-2889. doi: 10.1512/iumj.2008.57.3630. [6] T. Ma and S. Wang, Cahn-Hilliard equations and phase transition dynamics for binary systems, Discrete Contin. Dyn. Syst. Ser. B, 11 (2009), 741-784. doi: 10.3934/dcdsb.2009.11.741. [7] T. Ma and S. Wang, Phase separation of binary systems, Phys. Rev. A., 388 (2009), 4811-4817. [8] J. Moser, A rapidly convergent iteration method and nonlinear partial differential equation. I, Ann. Sc. Norm. Super. Pisa, 20 (1966), 265-315. [9] L. E. Reichl, "A Modern Course in Statistical Physics," (Second ed.) A Wiley-Interscinece Publication. New York: John Wiley & Sons Inc. 1998.
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