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

On higher-order anisotropic perturbed Caginalp phase field systems

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.

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.

[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.

[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.

[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.

[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.

[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.

[7]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[18]

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

[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.

[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.

[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.

[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.

[23]

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

[24]

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

[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.

show all references

References:
[1]

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

[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.

[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.

[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.

[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.

[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.

[7]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[18]

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

[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.

[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.

[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.

[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.

[23]

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

[24]

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

[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.

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