| Peer-Reviewed

On the Caginalp for a Conserved Phase-Field with Two Temperatures

Received: 15 December 2021     Accepted: 7 March 2022     Published: 24 October 2022
Views:       Downloads:
Abstract

The general theme of this article is the theorical study of phase field systems, more precisely that of Caginalp. This work is motivated by their immense applications in many physical fields, industriels… The Caginalp problem gives the authors a formulation based on the fact that the phases separated by an unknown regular interface, which evolves in a regular way. The authors’ aim in this paper is to study on Caginalp for a conserved Phase-field with two temperatures. The authors have worked on the existence and uniqueness of the Caginalp phase field in a conservative version. Moreover, the authors have also used Dirichlet type boundary conditions with a regular potential; existence and uniqueness are analyzed by means of absorbing bounded sets. The authors build the solution of the conservative problem on the estimates which lead authors to treat the problem well to arrive at the result. These equations are known as the conserved phase-field based on type II heat conduction and two temperatures. The authors consider a regular potential, more precisely a polynomial with edge conditions of Dirichlet type. More precisely, the authors prove the existence and uniqueness of solutions.

Published in American Journal of Applied Mathematics (Volume 10, Issue 5)
DOI 10.11648/j.ajam.20221005.12
Page(s) 205-211
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2022. Published by Science Publishing Group

Keywords

A Conserved Phase-Field, Two Temperatures, Dirichlet Boundary Conditions, Regular Potential

References
[1] Y. B Altundas and G. Caginalp, Velocity selection in 3D dentries: phase field computations and microgravity experments, nonlinear Anal., 62 (2005), 467-481.
[2] Bos, F., &Ruijs, A. (2021). Quantifying the Non-Use Value of Biodiversity in Cost–Benefit Analysis: The Dutch Biodiversity Points. Journal Of Benefit-Cost Analysis, 12 (2), 287-312. doi: 10.1017/bca.2020.27.
[3] A. C. Aristotelous, O. A. Karakashian, S. M. Wise, Adaptive, second-order in time, primitivevariable discontinuous Galerkin schemes for a CahnHilliard equation with a mass source, IMA J. Numer. Anal. 35 (2015), 11671198. MR3407258; url.
[4] N. Batangouna and M. Pierre. Convergence of exponential attractors for a time splitting approximation of the caginalp phase-field system. Comm. Pure Appl. Anal, 17 (1): 1–19, 2018.
[5] F. Bai, C. M. Elliott, A. Gardiner, A. Spence and A. M. Stuart, The viscous Cahn-Hilliard equation. I. Computations, Nonlinearity, 8 (1995), 131–160.
[6] P. W. Bates and S. M. Zheng, Inertial manifolds and inertial sets for the phase-field equations, J. Dynam. Differential Equations, 4 (1992), 375–398.
[7] B. Doumbe, Etude de modeles de champ de phase de type Caginalp, PhD thesis, Universit´e de Poiters, 2013.
[8] H. Brezis, Op´erateursmaximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973.
[9] D. Brochet, D. Hilhorst and X. Chen, Finite-dimensional exponential attractor for the phase field model, Appl. Anal., 49 (1993), 197–212.
[10] A. Miranville, R. Quintanilla, A Caginalp phase-field system based on type III heat conduction with two temperatures, Quart. Appl. Math. 74 (2016), 375-398.
[11] A. Miranville, Some mathematical models in phase transition, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), 271–306.
[12] Ntsokongo, A. J. and Batangouna, N. (2016) Existence and Uniqueness of Solutions for a Conserved Phase-Field Type Model. AIMS Mathematics, 1, 144-155.
[13] R. Temam, Infinite-dimensional dynamical systems in mechanics and physics, Second edition, Applied Mathematical Sciences, vol. 68, Springer-Verlag, New York, 1997.
[14] A. Miranville, R. Quintanilla, Some generalizations of the Caginalp phase-field system, Appl. Anal. 88 (2009), 877-894.
[15] G. Caginalp, An analysis of a phase-field model of a freeboundary, Arch. Rational Mech. Anal. 92 (1986), 205-245.
[16] R. Quintanilla, A well-posed problem for the three-dual-phase-lag heat conduction, J. Thermal Stresses 32 (2009), 1270-1278. Series S 7 (2014), 271-306.
[17] P. J. Chen, M. E. Gurtin, and W. O. Williams, A note on non-simple heat conduction, J. Appl. Phys. (ZAMP) 19 (1968), 969-970. with two temperatures, Quart. Appl. Math. 74 (2016), 375-398.
[18] L. Cherfils, A. Miranville, On the Caginalp system with dynamic boundary conditions and singular potentials, Appl. Math., 54 (2009), 89115.
[19] R. Chill, E. Fasangova, J. Pruss, Convergence to steady states of solutions of the Cahn-Hilliard equation with dynamic boundary conditions, Math. Nachr., 279 (2006), 14481462.
[20] C. I. Christov, P. M. Jordan, Heat conduction paradox involving second-sound propagation in moving media, Phys. Rev. Lett., 94 (2005), 154301.
[21] J. N. Flavin, R. J. Knops, L. E. Payne, Decay estimates for the constrained elastic cylinder of variable cross-section, Quart. Appl. Math., 47 (1989), 325350.
[22] C. Giorgi, M. Grasselli, V. Pata, Uniform attractors for a phase-field model with memory and quadratic nonlinearity, Indiana U. Math. J., 48 (1999), 13951445.
[23] M. Grasseli, A. Miranville, V. Pata, et al. Well-posedness and long time behavior of a parabolichyperbolic phase-field system with singular potentials, Math. Nachr., 280 (2007), 14751509.
Cite This Article
  • APA Style

    Narcisse Batangouna, Cyr Séraphin Ngamouyih Moussata, Urbain Cyriaque Mavoungou. (2022). On the Caginalp for a Conserved Phase-Field with Two Temperatures. American Journal of Applied Mathematics, 10(5), 205-211. https://doi.org/10.11648/j.ajam.20221005.12

    Copy | Download

    ACS Style

    Narcisse Batangouna; Cyr Séraphin Ngamouyih Moussata; Urbain Cyriaque Mavoungou. On the Caginalp for a Conserved Phase-Field with Two Temperatures. Am. J. Appl. Math. 2022, 10(5), 205-211. doi: 10.11648/j.ajam.20221005.12

    Copy | Download

    AMA Style

    Narcisse Batangouna, Cyr Séraphin Ngamouyih Moussata, Urbain Cyriaque Mavoungou. On the Caginalp for a Conserved Phase-Field with Two Temperatures. Am J Appl Math. 2022;10(5):205-211. doi: 10.11648/j.ajam.20221005.12

    Copy | Download

  • @article{10.11648/j.ajam.20221005.12,
      author = {Narcisse Batangouna and Cyr Séraphin Ngamouyih Moussata and Urbain Cyriaque Mavoungou},
      title = {On the Caginalp for a Conserved Phase-Field with Two Temperatures},
      journal = {American Journal of Applied Mathematics},
      volume = {10},
      number = {5},
      pages = {205-211},
      doi = {10.11648/j.ajam.20221005.12},
      url = {https://doi.org/10.11648/j.ajam.20221005.12},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20221005.12},
      abstract = {The general theme of this article is the theorical study of phase field systems, more precisely that of Caginalp. This work is motivated by their immense applications in many physical fields, industriels… The Caginalp problem gives the authors a formulation based on the fact that the phases separated by an unknown regular interface, which evolves in a regular way. The authors’ aim in this paper is to study on Caginalp for a conserved Phase-field with two temperatures. The authors have worked on the existence and uniqueness of the Caginalp phase field in a conservative version. Moreover, the authors have also used Dirichlet type boundary conditions with a regular potential; existence and uniqueness are analyzed by means of absorbing bounded sets. The authors build the solution of the conservative problem on the estimates which lead authors to treat the problem well to arrive at the result. These equations are known as the conserved phase-field based on type II heat conduction and two temperatures. The authors consider a regular potential, more precisely a polynomial with edge conditions of Dirichlet type. More precisely, the authors prove the existence and uniqueness of solutions.},
     year = {2022}
    }
    

    Copy | Download

  • TY  - JOUR
    T1  - On the Caginalp for a Conserved Phase-Field with Two Temperatures
    AU  - Narcisse Batangouna
    AU  - Cyr Séraphin Ngamouyih Moussata
    AU  - Urbain Cyriaque Mavoungou
    Y1  - 2022/10/24
    PY  - 2022
    N1  - https://doi.org/10.11648/j.ajam.20221005.12
    DO  - 10.11648/j.ajam.20221005.12
    T2  - American Journal of Applied Mathematics
    JF  - American Journal of Applied Mathematics
    JO  - American Journal of Applied Mathematics
    SP  - 205
    EP  - 211
    PB  - Science Publishing Group
    SN  - 2330-006X
    UR  - https://doi.org/10.11648/j.ajam.20221005.12
    AB  - The general theme of this article is the theorical study of phase field systems, more precisely that of Caginalp. This work is motivated by their immense applications in many physical fields, industriels… The Caginalp problem gives the authors a formulation based on the fact that the phases separated by an unknown regular interface, which evolves in a regular way. The authors’ aim in this paper is to study on Caginalp for a conserved Phase-field with two temperatures. The authors have worked on the existence and uniqueness of the Caginalp phase field in a conservative version. Moreover, the authors have also used Dirichlet type boundary conditions with a regular potential; existence and uniqueness are analyzed by means of absorbing bounded sets. The authors build the solution of the conservative problem on the estimates which lead authors to treat the problem well to arrive at the result. These equations are known as the conserved phase-field based on type II heat conduction and two temperatures. The authors consider a regular potential, more precisely a polynomial with edge conditions of Dirichlet type. More precisely, the authors prove the existence and uniqueness of solutions.
    VL  - 10
    IS  - 5
    ER  - 

    Copy | Download

Author Information
  • Functional Analysis Laboratory and Partial, Differential Equations Sciences and Technologies Faculty, University Marien Ngouabi, Brazzaville, Congo

  • Functional Analysis Laboratory and Partial, Differential Equations Sciences and Technologies Faculty, University Marien Ngouabi, Brazzaville, Congo

  • Functional Analysis Laboratory and Partial, Differential Equations Sciences and Technologies Faculty, University Marien Ngouabi, Brazzaville, Congo

  • Sections