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On higher-order anisotropic perturbed Caginalp phase field systems
Faculté des Sciences et Techniques, Université Marien Ngouabi, B.P 69, Brazzaville, Congo |
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.
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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