-
Previous Article
Finite difference scheme for 2D parabolic problem modelling electrostatic Micro-Electromechanical Systems
- ERA-MS Home
- This Volume
-
Next Article
Fractal Weyl bounds and Hecke triangle groups
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. |
[1] |
George J. Bautista, Ademir F. Pazoto. Decay of solutions for a dissipative higher-order Boussinesq system on a periodic domain. Communications on Pure and 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 and 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 and 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 and Continuous Dynamical Systems, 2021, 41 (5) : 2269-2283. doi: 10.3934/dcds.2020361 |
[5] |
Ciprian G. Gal, M. Grasselli. On the asymptotic behavior of the Caginalp system with dynamic boundary conditions. Communications on Pure and Applied Analysis, 2009, 8 (2) : 689-710. doi: 10.3934/cpaa.2009.8.689 |
[6] |
Huijun He, Zhaoyang Yin. On the Cauchy problem for a generalized two-component shallow water wave system with fractional higher-order inertia operators. Discrete and Continuous Dynamical Systems, 2017, 37 (3) : 1509-1537. doi: 10.3934/dcds.2017062 |
[7] |
Kui Li, Zhitao Zhang. Liouville-type theorem for higher-order Hardy-Hénon system. Communications on Pure and Applied Analysis, 2021, 20 (11) : 3851-3869. doi: 10.3934/cpaa.2021134 |
[8] |
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 |
[9] |
Kazuyuki Yagasaki. Higher-order Melnikov method and chaos for two-degree-of-freedom Hamiltonian systems with saddle-centers. Discrete and Continuous Dynamical Systems, 2011, 29 (1) : 387-402. doi: 10.3934/dcds.2011.29.387 |
[10] |
Robert Baier, Thuy T. T. Le. Construction of the minimum time function for linear systems via higher-order set-valued methods. Mathematical Control and Related Fields, 2019, 9 (2) : 223-255. doi: 10.3934/mcrf.2019012 |
[11] |
Ke Yang, Wencheng Zou, Zhengrong Xiang, Ronghao Wang. Fully distributed consensus for higher-order nonlinear multi-agent systems with unmatched disturbances. Discrete and Continuous Dynamical Systems - S, 2021, 14 (4) : 1535-1551. doi: 10.3934/dcdss.2020396 |
[12] |
Angelo Favini, Yakov Yakubov. Regular boundary value problems for ordinary differential-operator equations of higher order in UMD Banach spaces. Discrete and Continuous Dynamical Systems - S, 2011, 4 (3) : 595-614. doi: 10.3934/dcdss.2011.4.595 |
[13] |
Laurence Cherfils, Stefania Gatti, Alain Miranville. Long time behavior of the Caginalp system with singular potentials and dynamic boundary conditions. Communications on Pure and Applied Analysis, 2012, 11 (6) : 2261-2290. doi: 10.3934/cpaa.2012.11.2261 |
[14] |
Franck Davhys Reval Langa, Morgan Pierre. A doubly splitting scheme for the Caginalp system with singular potentials and dynamic boundary conditions. Discrete and Continuous Dynamical Systems - S, 2021, 14 (2) : 653-676. doi: 10.3934/dcdss.2020353 |
[15] |
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 |
[16] |
Xinmin Yang, Xiaoqi Yang, Kok Lay Teo. Higher-order symmetric duality in multiobjective programming with invexity. Journal of Industrial and Management Optimization, 2008, 4 (2) : 385-391. doi: 10.3934/jimo.2008.4.385 |
[17] |
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 |
[18] |
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 |
[19] |
Anthony Bloch, Leonardo Colombo, Fernando Jiménez. The variational discretization of the constrained higher-order Lagrange-Poincaré equations. Discrete and Continuous Dynamical Systems, 2019, 39 (1) : 309-344. doi: 10.3934/dcds.2019013 |
[20] |
Feng Wang, Fengquan Li, Zhijun Qiao. On the Cauchy problem for a higher-order μ-Camassa-Holm equation. Discrete and Continuous Dynamical Systems, 2018, 38 (8) : 4163-4187. doi: 10.3934/dcds.2018181 |
2020 Impact Factor: 0.929
Tools
Article outline
[Back to Top]