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Higher-Order Anisotropic Caginalp Phase-Field Systems

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Abstract

Our aim in this paper was to study the well-posedness and the dissipativity of higher-order anisotropic phase-field systems. More precisely, we prove the existence and uniqueness of solutions and the existence of the global attractor.

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References

  1. Agmon, S.: Lectures on elliptic boundary value problems, Mathematical Studies. Van Nostrand, New York (1965)

  2. Agmon S., Douglis A., Nirenberg L.: Estimates near the boundary for solutions of elliptic partial differential equations, I . Commun. Pure Appl. Math. 12, 623–727 (1959)

    Article  MathSciNet  MATH  Google Scholar 

  3. Agmon S., Douglis A., Nirenberg L.: Estimates near the boundary for solutions of elliptic partial differential equations, II. Commun. Pure Appl. Math. 17, 35–92 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  4. Aizicovici S., Feireisl E.: Long-time stabilization of solutions to a phase-field model with memory. J. Evol. Eqs. 1, 69–84 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  5. Aizicovici S., Feireisl E., Issard-Roch F.: Long-time convergence of solutions to a phase-field system. Math. Methods Appl. Sci. 24, 277–287 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  6. Brochet D., Chen X., Hilhorst D.: Finite dimensional exponential attractors for the phase-field model. Appl. Anal. 49, 197–212 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  7. Caginalp G.: An analysis of a phase field model of a free boundary. Arch. Ration. Mech. Anal. 92, 205–245 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  8. Caginalp G., Esenturk E.: Anisotropic phase field equations of arbitrary order. Discrete Contin. Dyn. Syst. S 4, 311–350 (2011)

    MathSciNet  MATH  Google Scholar 

  9. Cahn J.W., Hilliard J.E.: Free energy of a nonuniform system I. Interfacial free energy. J. Chem. Phys. 2, 258–267 (1958)

    Article  Google Scholar 

  10. Chen X., Caginalp G., Esenturk E.: Interface conditions for a phase field model with anisotropic and non-local interactions. Arch. Ration. Mech. Anal. 202, 349–372 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  11. Cherfils L., Miranville A.: Some results on the asymptotic behavior of the Caginalp system with singular potentials. Adv. Math. Sci. Appl. 17, 107–129 (2007)

    MathSciNet  MATH  Google Scholar 

  12. Cherfils L., Miranville A.: On the Caginalp system with dynamic boundary conditions and singular potentials. Appl. Math. 54, 89–115 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  13. Cherfils, L., Miranville, A., Peng, S.: Higher-order models in phase separation. J. Appl. Anal. Comput. (2016, to appear)

  14. Chill R., Fašangovà E., Prüss J.: Convergence to steady states of solutions of the Cahn-Hilliard equation with dynamic boundary conditions. Math. Nachr. 279, 1448–1462 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  15. Conti M., Gatti S., Miranville A.: A generalization of the Caginalp phase-field system with Neumann boundary conditions. Nonlinear Anal. 87, 11–21 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  16. Gal C.G., Grasselli M.: The nonisothermal Allen-Cahn equation with dynamic boundary conditions. Discrete Contin. Dyn. Syst. A 22, 1009–1040 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  17. Gatti, S., Miranville, A.: Asymptotic behavior of a phase-field system with dynamic boundary conditions. In: Favini, A., Lorenzi, A., (eds.) Differential equations: inverse and direct problems (Proceedings of the workshop “Evolution Equations: Inverse and Direct Problems”, Cortona, June 21–25, 2004), A series of Lecture notes in pure and applied mathematics, vol. 251, pp. 149–170. Chapman & Hall (2006)

  18. Giacomin G., Lebowitz J.L.: Phase segregation dynamics in particle systems with long range interaction I. Macroscopic limits. J. Stat. Phys. 87, 37–61 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  19. Giacomin G., Lebowitz J.L.: Phase segregation dynamics in particle systems with long range interaction II. Interface motion. SIAM J. Appl. Math. 58, 1707–1729 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  20. Grasselli M., Miranville A., Pata V., Zelik S.: Well-posedness and long time behavior of a parabolic-hyperbolic phase-field system with singular potentials. Math. Nachr. 280, 1475–1509 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  21. Grasselli M., Miranville A., Schimperna G.: The Caginalp phase-field system with coupled dynamic boundary conditions and singular potentials. Discrete Contin. Dyn. Syst. 28, 67–98 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  22. Grasselli M., Petzeltová H., Schimperna G.: Long time behavior of solutions to the Caginalp system with singular potential. Z. Anal. Anwend. 25, 51–72 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  23. Grasselli M., Pata V.: Existence of a universal attractor for a fully hyperbolic phase-field system. J. Evol. Eqs. 4, 27–51 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  24. Kobayashi R.: Modelling and numerical simulations of dendritic crystal growth. Phys. D 63, 410–423 (1993)

    Article  MATH  Google Scholar 

  25. Miranville A.: Some mathematical models in phase transition. Discrete Contin. Dyn. Syst. Ser. S 7, 271–306 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  26. Miranville A., Zelik S.: Robust exponential attractors for singularly perturbed phase-field type equations. Electron. J. Diff. Eqn. 2002, 1–28 (2002)

    MathSciNet  MATH  Google Scholar 

  27. Miranville, A., Zelik, S.: Attractors for dissipative partial differential equations in bounded and unbounded domains. In: Dafermos, C.M., Pokorny, M., (eds.) Handbook of differential equations, evolutionary partial differential equations, vol. 4, pp. 103–200. Elsevier, Amsterdam (2008)

  28. Taylor J.E.: Mean curvature and weighted mean curvature. Acta Metall. Mater. 40, 1475–1495 (1992)

    Article  Google Scholar 

  29. Temam, R.: Infinite-dimensional dynamical systems in mechanics and physics, Applied Mathematical Sciences, vol. 68, 2nd edn. Springer-Verlag, New York (1997)

  30. Wheeler A.A., McFadden G.B.: On the notion of \({\xi }\)-vector and stress tensor for a general class of anisotropic diffuse interface models. Proc. R. Soc. Lond. Ser. A 453, 1611–1630 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  31. Zhang Z.: Asymptotic behavior of solutions to the phase-field equations with Neumann boundary conditions. Commun. Pure Appl. Anal. 4, 683–693 (2005)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Laboratoire de Mathématiques et Applications, Université de Poitiers, UMR CNRS 7348-SP2MI, Boulevard Marie et Pierre Curie-Téléport 2, 86962, Chasseneuil Futuroscope Cedex, France

    Alain Miranville

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  1. Alain Miranville
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Miranville, A. Higher-Order Anisotropic Caginalp Phase-Field Systems. Mediterr. J. Math. 13, 4519–4535 (2016). https://doi.org/10.1007/s00009-016-0760-2

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  • DOI: https://doi.org/10.1007/s00009-016-0760-2

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