American Institute of Mathematical Sciences

Advanced
  2019, 26: 36-53. doi: 10.3934/era.2019.26.004

On higher-order anisotropic perturbed Caginalp phase field systems

Clesh Deseskel Elion Ekohela , and  Daniel Moukoko

Faculté des Sciences et Techniques, Université Marien Ngouabi, B.P 69, Brazzaville, Congo

* Corresponding author: Clesh Deseskel Elion Ekohela

Received  March 2019 Revised  June 2019 Published  July 2019

Our aim in this paper is to study the existence and uniqueness of solution for hyperbolic relaxations of higher-order anisotropic Caginalp phase field systems with homogeous Dirichlet boundary conditions with regular potentials.

Keywords: Caginalp system, hyperbolic relaxation, Dirichlet boundary condition, higher-order systems, regular potential, anisotropy.
Mathematics Subject Classification: Primary: 35B20, 35B25.
Citation: Clesh Deseskel Elion Ekohela, Daniel Moukoko. On higher-order anisotropic perturbed Caginalp phase field systems. Electronic Research Announcements, 2019, 26: 36-53. doi: 10.3934/era.2019.26.004
References:
[1]

S. Agmon, Lectures on Elliptic Boundary Value Problems, princeton, 1965.  Google Scholar

[2]

S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, I. Com. on Pure and Appl. Math., 12 (1959), 623–727. doi: 10.1002/cpa.3160120405.  Google Scholar

[3]

S. Aizicovici and E. Feireisl, Long-time stabilization of solutions to a phase-field model with memory, Journal of Evolution Equations, 1 (2001), 69–84. doi: 10.1007/PL00001365.  Google Scholar

[4]

S. Aizicovici, E. Feireisl and F. Issard-Roch, Long-time convergence of solutions to a phase-field system, Mathematical Methods in the Applied Sciences, 24 (2001), 277–287. doi: 10.1002/mma.215.  Google Scholar

[5]

D. Brochet, D. Hilhorst, A. Novick-Cohen, et al., Maximal attractor and inertial sets for a conserved phase field model, Advances in Differential Equations, 1 (1996), 547–578.  Google Scholar

[6]

G. Caginalp, An analysis of a phase field model of a free boundary, Archive for Rational Mechanics and Analysis, 92 (1986), 205–245. doi: 10.1007/BF00254827.  Google Scholar

[7]

G. Caginalp, Conserved-phase field system, Implications for Kinetic Undercooling. Physical, Review B, 38 (1988), 789. Google Scholar

[8]

G. Caginalp, The dynamics of a conserved phase field system: Stefan-like, hele-shaw, and cahn-hilliard models as asymptotic limits, IMA Journal of Applied Mathematics, 44 (1990), 77–94. doi: 10.1093/imamat/44.1.77.  Google Scholar

[9]

G. Caginalp and E. Esenturk, Anisotropic phase field equations of arbitrary order, Discrete and Continuous Dynamical Systems-S, 4 (2011), 311–350. doi: 10.3934/dcdss.2011.4.311.  Google Scholar

[10]

J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system, i. interfacial free energy, The Journal of Chemical Physics, 28 (1958), 258-267.  doi: 10.1002/9781118788295.ch4.  Google Scholar

[11]

P. J. Chen and M. E. Gurtin, On a theory of heat conduction involving two temperatures, Zeitschrift Für Angewandte Mathematik und Physik (ZAMP), 19 (1968), 614-627.  doi: 10.1007/BF01594969.  Google Scholar

[12]

X. Chen, G. Caginalp and E. Esenturk, Interface conditions for a phase field model with anisotropic and non-local interactions, Archive for Rational Mechanics and Analysis, 202 (2011), 349–372. doi: 10.1007/s00205-011-0429-8.  Google Scholar

[13]

L. Cherfils and A. Miranville, On the caginalp system with dynamic boundary conditions and singular potentials, Applications of Mathematics, 54 (2009), 89–115. doi: 10.1007/s10492-009-0008-6.  Google Scholar

[14]

G. Giacomin and J. L. Lebowitz, Phase segregation dynamics in particle systems with long range interactions. Ⅰ. Macroscopic limits, Journal of Statistical Physics, 87 (1997), 37–61. doi: 10.1007/BF02181479.  Google Scholar

[15]

G. Giacomin and J. L. Lebowitz, Phase segregation dynamics in particle systems with long range interactions Ⅱ: Interface motion, SIAM Journal on Applied Mathematics, 58 (1998), 1707–1729. doi: 10.1137/S0036139996313046.  Google Scholar

[16]

M. Grasselli and H. Wu, Well-posedness and long-time behavior for the modified phase-field crystal equation, Mathematical Models and Methods in Applied Sciences, 24 (2014), 2743–2783. doi: 10.1142/S0218202514500365.  Google Scholar

[17]

R. Kobayashi, Modeling and numerical simulations of dendritic crystal growth, Physica D: Nonlinear Phenomena, 63 (1993), 410-423.  doi: 10.1016/0167-2789(93)90120-P.  Google Scholar

[18]

C. Laurence, A. Miranville and S. Peng, Higher-order models in phase separation, Journal of Applied Analysis and Computation, 7 (2017), 39–56.  Google Scholar

[19]

A. Miranville, Some mathematical models in phase transition, Discrete and Continuous Dynamical Systems-S, 7 (2014), 271–306. doi: 10.3934/dcdss.2014.7.271.  Google Scholar

[20]

A. Miranville, Higher-order anisotropic caginalp phase-field systems, Mediterranean Journal of Mathematics, 13 (2016), 4519–4535. doi: 10.1007/s00009-016-0760-2.  Google Scholar

[21]

A. Miranville, On higher-order anisotropic conservative caginalp phase-field systems, Applied Mathematics and Optimization, 77 (2018), 297–314. doi: 10.1007/s00245-016-9375-z.  Google Scholar

[22]

A. Miranville and R. Quintanilla, A Caginalp phase field system based on type Ⅲ heat conduction with two temperatures, Quarterly of Applied Mathematics, 74 (2016), 375–398. doi: 10.1090/qam/1430.  Google Scholar

[23]

A. J. Ntsokongo, On higher-order anisotropic caginalp phase-field systems with polynomial nonlinear terms, J. Appl. Anal. Comput, 7 (2017), 992–1012.  Google Scholar

[24]

R. Quintanilla, A well-posed problem for the dual-phase-lag heat conduction, Journal of Thermal Stresses, 31 (2008), 260-269.   Google Scholar

[25]

R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences, 68. Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4684-0313-8.  Google Scholar

show all references

References:
[1]

S. Agmon, Lectures on Elliptic Boundary Value Problems, princeton, 1965.  Google Scholar

[2]

S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, I. Com. on Pure and Appl. Math., 12 (1959), 623–727. doi: 10.1002/cpa.3160120405.  Google Scholar

[3]

S. Aizicovici and E. Feireisl, Long-time stabilization of solutions to a phase-field model with memory, Journal of Evolution Equations, 1 (2001), 69–84. doi: 10.1007/PL00001365.  Google Scholar

[4]

S. Aizicovici, E. Feireisl and F. Issard-Roch, Long-time convergence of solutions to a phase-field system, Mathematical Methods in the Applied Sciences, 24 (2001), 277–287. doi: 10.1002/mma.215.  Google Scholar

[5]

D. Brochet, D. Hilhorst, A. Novick-Cohen, et al., Maximal attractor and inertial sets for a conserved phase field model, Advances in Differential Equations, 1 (1996), 547–578.  Google Scholar

[6]

G. Caginalp, An analysis of a phase field model of a free boundary, Archive for Rational Mechanics and Analysis, 92 (1986), 205–245. doi: 10.1007/BF00254827.  Google Scholar

[7]

G. Caginalp, Conserved-phase field system, Implications for Kinetic Undercooling. Physical, Review B, 38 (1988), 789. Google Scholar

[8]

G. Caginalp, The dynamics of a conserved phase field system: Stefan-like, hele-shaw, and cahn-hilliard models as asymptotic limits, IMA Journal of Applied Mathematics, 44 (1990), 77–94. doi: 10.1093/imamat/44.1.77.  Google Scholar

[9]

G. Caginalp and E. Esenturk, Anisotropic phase field equations of arbitrary order, Discrete and Continuous Dynamical Systems-S, 4 (2011), 311–350. doi: 10.3934/dcdss.2011.4.311.  Google Scholar

[10]

J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system, i. interfacial free energy, The Journal of Chemical Physics, 28 (1958), 258-267.  doi: 10.1002/9781118788295.ch4.  Google Scholar

[11]

P. J. Chen and M. E. Gurtin, On a theory of heat conduction involving two temperatures, Zeitschrift Für Angewandte Mathematik und Physik (ZAMP), 19 (1968), 614-627.  doi: 10.1007/BF01594969.  Google Scholar

[12]

X. Chen, G. Caginalp and E. Esenturk, Interface conditions for a phase field model with anisotropic and non-local interactions, Archive for Rational Mechanics and Analysis, 202 (2011), 349–372. doi: 10.1007/s00205-011-0429-8.  Google Scholar

[13]

L. Cherfils and A. Miranville, On the caginalp system with dynamic boundary conditions and singular potentials, Applications of Mathematics, 54 (2009), 89–115. doi: 10.1007/s10492-009-0008-6.  Google Scholar

[14]

G. Giacomin and J. L. Lebowitz, Phase segregation dynamics in particle systems with long range interactions. Ⅰ. Macroscopic limits, Journal of Statistical Physics, 87 (1997), 37–61. doi: 10.1007/BF02181479.  Google Scholar

[15]

G. Giacomin and J. L. Lebowitz, Phase segregation dynamics in particle systems with long range interactions Ⅱ: Interface motion, SIAM Journal on Applied Mathematics, 58 (1998), 1707–1729. doi: 10.1137/S0036139996313046.  Google Scholar

[16]

M. Grasselli and H. Wu, Well-posedness and long-time behavior for the modified phase-field crystal equation, Mathematical Models and Methods in Applied Sciences, 24 (2014), 2743–2783. doi: 10.1142/S0218202514500365.  Google Scholar

[17]

R. Kobayashi, Modeling and numerical simulations of dendritic crystal growth, Physica D: Nonlinear Phenomena, 63 (1993), 410-423.  doi: 10.1016/0167-2789(93)90120-P.  Google Scholar

[18]

C. Laurence, A. Miranville and S. Peng, Higher-order models in phase separation, Journal of Applied Analysis and Computation, 7 (2017), 39–56.  Google Scholar

[19]

A. Miranville, Some mathematical models in phase transition, Discrete and Continuous Dynamical Systems-S, 7 (2014), 271–306. doi: 10.3934/dcdss.2014.7.271.  Google Scholar

[20]

A. Miranville, Higher-order anisotropic caginalp phase-field systems, Mediterranean Journal of Mathematics, 13 (2016), 4519–4535. doi: 10.1007/s00009-016-0760-2.  Google Scholar

[21]

A. Miranville, On higher-order anisotropic conservative caginalp phase-field systems, Applied Mathematics and Optimization, 77 (2018), 297–314. doi: 10.1007/s00245-016-9375-z.  Google Scholar

[22]

A. Miranville and R. Quintanilla, A Caginalp phase field system based on type Ⅲ heat conduction with two temperatures, Quarterly of Applied Mathematics, 74 (2016), 375–398. doi: 10.1090/qam/1430.  Google Scholar

[23]

A. J. Ntsokongo, On higher-order anisotropic caginalp phase-field systems with polynomial nonlinear terms, J. Appl. Anal. Comput, 7 (2017), 992–1012.  Google Scholar

[24]

R. Quintanilla, A well-posed problem for the dual-phase-lag heat conduction, Journal of Thermal Stresses, 31 (2008), 260-269.   Google Scholar

[25]

R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences, 68. Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4684-0313-8.  Google Scholar

[1]

George J. Bautista, Ademir F. Pazoto. Decay of solutions for a dissipative higher-order Boussinesq system on a periodic domain. Communications on Pure & Applied Analysis, 2020, 19 (2) : 747-769. doi: 10.3934/cpaa.2020035
[2]

Robert Jankowski, Barbara Łupińska, Magdalena Nockowska-Rosiak, Ewa Schmeidel. Monotonic solutions of a higher-order neutral difference system. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 253-261. doi: 10.3934/dcdsb.2018017
[3]

Weisheng Niu, Yao Xu. Convergence rates in homogenization of higher-order parabolic systems. Discrete & Continuous Dynamical Systems, 2018, 38 (8) : 4203-4229. doi: 10.3934/dcds.2018183
[4]

Daomin Cao, Guolin Qin. Liouville type theorems for fractional and higher-order fractional systems. Discrete & Continuous Dynamical Systems, 2021, 41 (5) : 2269-2283. doi: 10.3934/dcds.2020361
[5]

Huijun He, Zhaoyang Yin. On the Cauchy problem for a generalized two-component shallow water wave system with fractional higher-order inertia operators. Discrete & Continuous Dynamical Systems, 2017, 37 (3) : 1509-1537. doi: 10.3934/dcds.2017062
[6]

Ciprian G. Gal, M. Grasselli. On the asymptotic behavior of the Caginalp system with dynamic boundary conditions. Communications on Pure & Applied Analysis, 2009, 8 (2) : 689-710. doi: 10.3934/cpaa.2009.8.689
[7]

Eduardo Martínez. Higher-order variational calculus on Lie algebroids. Journal of Geometric Mechanics, 2015, 7 (1) : 81-108. doi: 10.3934/jgm.2015.7.81
[8]

Kazuyuki Yagasaki. Higher-order Melnikov method and chaos for two-degree-of-freedom Hamiltonian systems with saddle-centers. Discrete & Continuous Dynamical Systems, 2011, 29 (1) : 387-402. doi: 10.3934/dcds.2011.29.387
[9]

Robert Baier, Thuy T. T. Le. Construction of the minimum time function for linear systems via higher-order set-valued methods. Mathematical Control & Related Fields, 2019, 9 (2) : 223-255. doi: 10.3934/mcrf.2019012
[10]

Ke Yang, Wencheng Zou, Zhengrong Xiang, Ronghao Wang. Fully distributed consensus for higher-order nonlinear multi-agent systems with unmatched disturbances. Discrete & Continuous Dynamical Systems - S, 2021, 14 (4) : 1535-1551. doi: 10.3934/dcdss.2020396
[11]

Angelo Favini, Yakov Yakubov. Regular boundary value problems for ordinary differential-operator equations of higher order in UMD Banach spaces. Discrete & Continuous Dynamical Systems - S, 2011, 4 (3) : 595-614. doi: 10.3934/dcdss.2011.4.595
[12]

Laurence Cherfils, Stefania Gatti, Alain Miranville. Long time behavior of the Caginalp system with singular potentials and dynamic boundary conditions. Communications on Pure & Applied Analysis, 2012, 11 (6) : 2261-2290. doi: 10.3934/cpaa.2012.11.2261
[13]

Franck Davhys Reval Langa, Morgan Pierre. A doubly splitting scheme for the Caginalp system with singular potentials and dynamic boundary conditions. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 653-676. doi: 10.3934/dcdss.2020353
[14]

Simão P. S. Santos, Natália Martins, Delfim F. M. Torres. Noether's theorem for higher-order variational problems of Herglotz type. Conference Publications, 2015, 2015 (special) : 990-999. doi: 10.3934/proc.2015.990
[15]

Xinmin Yang, Xiaoqi Yang, Kok Lay Teo. Higher-order symmetric duality in multiobjective programming with invexity. Journal of Industrial & Management Optimization, 2008, 4 (2) : 385-391. doi: 10.3934/jimo.2008.4.385
[16]

Pedro D. Prieto-Martínez, Narciso Román-Roy. Higher-order mechanics: Variational principles and other topics. Journal of Geometric Mechanics, 2013, 5 (4) : 493-510. doi: 10.3934/jgm.2013.5.493
[17]

Feliz Minhós, Hugo Carrasco. Solvability of higher-order BVPs in the half-line with unbounded nonlinearities. Conference Publications, 2015, 2015 (special) : 841-850. doi: 10.3934/proc.2015.0841
[18]

Anthony Bloch, Leonardo Colombo, Fernando Jiménez. The variational discretization of the constrained higher-order Lagrange-Poincaré equations. Discrete & Continuous Dynamical Systems, 2019, 39 (1) : 309-344. doi: 10.3934/dcds.2019013
[19]

Feng Wang, Fengquan Li, Zhijun Qiao. On the Cauchy problem for a higher-order μ-Camassa-Holm equation. Discrete & Continuous Dynamical Systems, 2018, 38 (8) : 4163-4187. doi: 10.3934/dcds.2018181
[20]

Qilin Wang, S. J. Li. Higher-order sensitivity analysis in nonconvex vector optimization. Journal of Industrial & Management Optimization, 2010, 6 (2) : 381-392. doi: 10.3934/jimo.2010.6.381

2019 Impact Factor: 0.5

Tools

Article outline

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences