DISCONTINUOUS TRAVELING WAVE ENTROPY SOLUTIONS OF A MODIFIED ALLEN-CAHN MODEL

doi:10.1016/S0252-9602(17)30104-2

数学物理学报(英文版) ›› 2017, Vol. 37 ›› Issue (6): 1740-1760.doi: 10.1016/S0252-9602(17)30104-2

DISCONTINUOUS TRAVELING WAVE ENTROPY SOLUTIONS OF A MODIFIED ALLEN-CAHN MODEL

徐田媛¹, 金春花¹, 季善明²

1. School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China;
2. School of Mathematics, South China University of Technology, Guangzhou 510641, China

收稿日期:2016-06-28 修回日期:2017-06-16 出版日期:2017-12-25 发布日期:2017-12-25
通讯作者: Shanming JI E-mail:jishanming7@126.com
作者简介:Tianyuan XU,tyxu93@126.com;Chunhua JIN,jinchhua@126.com
基金资助:
This work was supported by the Program for New Century Excellent Talents in University of the Ministry of Education (NCET-13-0804), NSFC (11471127, 11371153), Guangdong Natural Science Funds for Distinguished Young Scholar (2015A030306029), the Excellent Young Teachers Program of Guangdong Province, and Special support program of Guangdong Province. The research of Xu was supported by the Innovation Project of Graduate School of South China Normal University (2016lkxm70), the High-Level University Developments Innovation Project of Graduate School of South China Normal University (2016YN05) and the Top-Notch Graduate Foundation of South China Normal University (2016028). The research of Ji was supported by the Fundamental Research Funds for the Central Universities (2017BQ109) and China Postdoctoral Science Foundation (2017M610517).

DISCONTINUOUS TRAVELING WAVE ENTROPY SOLUTIONS OF A MODIFIED ALLEN-CAHN MODEL

Tianyuan XU¹, Chunhua JIN¹, Shanming JI²

1. School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China;
2. School of Mathematics, South China University of Technology, Guangzhou 510641, China

Received:2016-06-28 Revised:2017-06-16 Online:2017-12-25 Published:2017-12-25
Contact: Shanming JI E-mail:jishanming7@126.com
Supported by:
This work was supported by the Program for New Century Excellent Talents in University of the Ministry of Education (NCET-13-0804), NSFC (11471127, 11371153), Guangdong Natural Science Funds for Distinguished Young Scholar (2015A030306029), the Excellent Young Teachers Program of Guangdong Province, and Special support program of Guangdong Province. The research of Xu was supported by the Innovation Project of Graduate School of South China Normal University (2016lkxm70), the High-Level University Developments Innovation Project of Graduate School of South China Normal University (2016YN05) and the Top-Notch Graduate Foundation of South China Normal University (2016028). The research of Ji was supported by the Fundamental Research Funds for the Central Universities (2017BQ109) and China Postdoctoral Science Foundation (2017M610517).

摘要/Abstract

摘要：

This paper deals with the existence and the asymptotic behavior of discontinuous traveling wave entropy solutions for a modified Allen-Cahn model, in which, the usual Ficken-based model for phase transition is replaced with a more physical model with nonlinear diffusive flux. The discontinuous traveling waves correspond to the discontinuous phase transition phenomena.

关键词: discontinuous traveling wave, phase transition, entropy condition

Abstract:

Key words: discontinuous traveling wave, phase transition, entropy condition

徐田媛, 金春花, 季善明. DISCONTINUOUS TRAVELING WAVE ENTROPY SOLUTIONS OF A MODIFIED ALLEN-CAHN MODEL[J]. 数学物理学报(英文版), 2017, 37(6): 1740-1760.

Tianyuan XU, Chunhua JIN, Shanming JI. DISCONTINUOUS TRAVELING WAVE ENTROPY SOLUTIONS OF A MODIFIED ALLEN-CAHN MODEL[J]. Acta mathematica scientia,Series B, 2017, 37(6): 1740-1760.

参考文献

[1] Allen S M, Cahn J W. A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metall, 1979, 27:1085-1095
[2] Barbera E, Currò C, Valenti G. On discontinuous travelling wave solutions for a class of hyperbolic reactiondiffusion models. Physica D, 2015, 308:116-126
[3] Bertsch M, Passo R D. Hyperbolic phenomena in a strongly degenerate parabolic equation. Arch Rat Mech Anal, 1992, 117:349-387
[4] Cahn J W, Hilliard J E. Free energy of a nonuniform system, I:interfacial free energy. J Chem Phys, 1958, 28(2):258-267
[5] Chmaj A, Ren X. Multiple layered solutions of the nonlocal bistable equation. Physica D, 2000, 147:135-154
[6] Dickman R. Discontinuous phase transition in a dimer lattice gas. J Chem Phys, 2012, 136:4678-4686
[7] Galenko P K, Sanches F I, Elder K R. Traveling wave profiles for a crystalline front invading liquid states:analytical and numerical solutions. Physica D, 2015, 308:1-10
[8] Garrione M, Sanchez L. Monotone traveling waves for reaction-diffusion equations involving the curvature operator. Bound Value Probl, 2015, 2015:1-31
[9] Holden H, Karlsen K H, Mitrovic D, Panov E Y. Strong compactness of approximate solutions to degenerate elliptic-hyperbolic equations with discontinuous flux function. Acta Math Sci, 2009, 29B(6):1573-1612
[10] Jordan P M. Growth and decay of shock and acceleration waves in a traffic flow model with relaxation. Physica D, 2005, 207:220-229
[11] Kruzhkov S N. First order quasilinear equations in several independent variables. Mat Sb, 1970, 81:228-255(English transl Math USSR-Sb, 1970, 10:217-243)
[12] Landman K A, Pettet G J, Newgreen D F. Chemotactic cellular migration:smooth and discontinuous travelling wave solutions. SIAM J Appl Math, 2003, 63(5):1666-1681
[13] Li M, Geng H, Long F et al. Discontinuous structural phase transition of Sn-Bi melts. J Mol Liq, 2015, 204:27-32
[14] Liu H X, Pan T. Pointwise convergence rate of vanishing viscosity approximations for scalar conservation laws with boundary. Acta Math Sci, 2009, 29B(1):111-128
[15] Mai D T. Traveling waves of an elliptic-hyperbolic model of phase transitions via varying viscositycapillarity. J Differ Equ, 2011, 251:439-456
[16] Pang P, Wang Y H, Yin J X. Free-boundary problem for a singular diffusion equation. J Math Anal Appl, 2002, 265:414-429
[17] Passo R D. Uniqueness of the entropy solution of a strongly degenerate parabolic equation. Commun Partial Differ Equ, 1993, 18:265-279
[18] Rosenau P. Free-energy functionals at the high-gradient limit. Phys Rev A, 1990, 41:2227-2230
[19] Tacan F, Bekir A. Travelling wave solutions of the Cahn-Allen equation by using first integral method. Appl Math Comput, 2009, 207:279-282
[20] Wang C, Yin J, Wang Z. Similar entropy solutions of a singular diffusion equation. Comput Math Appl, 2005, 49:1059-1068
[21] Wang C, Yin J, Lu B. Anti-shifting phenomenon of a convective nonlinear diffusion equation. Discrete Contin Dyn Syst Ser B, 2010, 3:1211-1236
[22] Wu Z Q, Yin J X. Some properties of functions in BV_x and their applications to the uniqueness of solutions for degenerate quasilinear parabolic equations. Northeastern Math J, 1989, 5:395-422
[23] Yin J X, Jin C H. Travelling wavefronts for a non-divergent degenerate and singular parabolic equation with changing sign sources. Proc R Soc Edinb Sect A-Math, 2009, 139:1179-1207
[24] Yin J X, Li H L, Pang P Y H. BV solutions of a singular diffusion equation. Math Nachr, 2003, 253:92-106

DISCONTINUOUS TRAVELING WAVE ENTROPY SOLUTIONS OF A MODIFIED ALLEN-CAHN MODEL

DISCONTINUOUS TRAVELING WAVE ENTROPY SOLUTIONS OF A MODIFIED ALLEN-CAHN MODEL

PDF (PC)

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 3

Metrics

本文评价

推荐阅读 10

[1]	Debora AMADORI, Paolo BAITI, Andrea CORLI, Edda DAL SANTO. GLOBAL EXISTENCE OF SOLUTIONS FOR A MULTI-PHASE FLOW: A BUBBLE IN A LIQUID TUBE AND RELATED CASES[J]. 数学物理学报(英文版), 2015, 35(4): 832-854.
[2]	Yilun SHANG. CONSENSUS FORMATION OF TWO-LEVEL OPINION DYNAMICS[J]. 数学物理学报(英文版), 2014, 34(4): 1029-1040.
[3]	刘小民, 王振. THE RIEMANN PROBLEM FOR THE NONLINEAR DEGENERATE WAVE EQUATIONS[J]. 数学物理学报(英文版), 2011, 31(6): 2313-2322.