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Quasi-stationary stefan problem with values on the front depending on its geometry

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Abstract

The problem presented below is a singular-limit problem of the extension of the Cahn-Hilliard model obtained via introducing the asymmetry of the surface tension tensor under one of the truncations (approximations) of the inner energy [2, 5–8, 10, 12, 13].

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References

  1. G. I. Barenblat, V. M. Entov, and V. M. Rizhik, Behavior of Fluids and Gases in Porous Media [in Russian], Nedra, Moscow (1984).

    Google Scholar 

  2. V. T. Borisov, “On mechanism of normal crystal growth,” Dokl. Akad. Nauk SSSR, 151, 1311–1314 (1963).

    Google Scholar 

  3. J. W. Cahn and J. E. Hilliard, “Free energy of a non-uniform system, Part I: Interfacial free energy,” J. Chemical Physics, 28, No. 1, 258–267 (1958).

    Article  Google Scholar 

  4. W. Dreyer and W. H. Muller, “A study of the coarsening in tin/lead solders,” Int. J. Solids Structures, 37, 3841–3871 (2000).

    Article  MATH  Google Scholar 

  5. M. Fabbri and V. R. Voller, “The phase-field method in the sharp-interface limit: A comparison between model potentials,” J. Comput. Phys., 130, 256–265 (1997).

    Article  MATH  Google Scholar 

  6. G. Fix, “Phase field models for free boundary problems,” In: Free Boundary Problems: Theory and Application, Pittman, London (1983), pp. 580–589.

  7. P. C. Hohenberg and B. I. Halperin, “Theory of dynamic critical phenomena,” Rev. Mod. Phys., 49, 435–479 (1977).

    Article  Google Scholar 

  8. V. I. Mazhukin, A. A. Samarskii, O. Kastel’yanos, and A. V. Shapranov, “Dynamical adaptation method for nonstandard problems with large gradients,” Mat. Model., 5, 33–56 (1993).

    MathSciNet  Google Scholar 

  9. A. M. Meiermanov, Stefan Problem [in Russian], Nauka, Novosibirsk (1986).

    Google Scholar 

  10. O. Penrose and P. Fife, “On the relation between the standard phase-field model and a “thermodynamically consistent” phase-field model,” Physica D, 69, 107–113 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  11. E. V. Radkevich and M. Zakharchenko, “Asymptotic solyution of extended Cahn–Hilliard model,” In: Contemporary Mathematics and Its Applications [in Russian], 2, Institute of Cybernetics, Tbilisi (2003).

  12. V. A. Solonnikov and E. V. Frolova, “On fulfillment of quasi-stationary approximation for Stefan problem,” Zap. Nauchn. Sem. LOMI, 348, 209–253 (2007).

    Google Scholar 

  13. A. R. Umantsev, “Plane front motion under crystallization,” Kristallografiya, 30, 153–160 (1985).

    Google Scholar 

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Authors and Affiliations

  1. All-Russian Institute for Scientific and Technical Information, Russian Academy of Sciences, Usievicha 20, Moscow, Russia

    N. Yu. Selivanova

  2. M. V. Lomonosov Moscow State University, Moscow, Russia

    M. V. Shamolin

Authors
  1. N. Yu. Selivanova
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  2. M. V. Shamolin
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Correspondence to N. Yu. Selivanova.

Additional information

Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 78, Partial Differential Equations and Optimal Control, 2012.

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Selivanova, N.Y., Shamolin, M.V. Quasi-stationary stefan problem with values on the front depending on its geometry. J Math Sci 189, 301–310 (2013). https://doi.org/10.1007/s10958-013-1187-y

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  • DOI: https://doi.org/10.1007/s10958-013-1187-y

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