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Solutions of an advected phase field system with low regularity velocity

Authors: Bianca Morelli Rodolfo Calsavara and José Luiz Boldrini
Journal: Proc. Amer. Math. Soc. 141 (2013), 943-958
MSC (2010): Primary 35K40, 80A22; Secondary 47H10, 76R10
Published electronically: July 24, 2012
MathSciNet review: 3003687
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Abstract: We present a result on existence of solutions for a system of highly nonlinear partial differential equations related to a phase field model for non-isothermal solidification/melting processes in the case of two possible crystallization states and flow of the molten material.

The flow is incompressible with a velocity which is assumed to be given, but with low regularity. We prove the existence of solutions for the associated system and also give estimates for the temperature and the phase fields related to each of the crystallization states in terms of the low regularity norms of the given flow velocity.

These results constitute a fundamental step in the proof of the existence of solutions of a complete model for solidification obtained by coupling the present equations with a singular Navier-Stokes system for the flow velocity. The analysis of this complete model is done in a forthcoming article.

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  • 1. R. Adams, Sobolev Spaces. Academic Press, 1975. MR 0450957 (56:9247)
  • 2. J. L. Boldrini, B. M. C. Caretta, E. Fernández-Cara, Analysis of a two-phase field model for the solidification of an alloy, J. Math. Anal. Appl. 357 (2009), 25-44. MR 2526803 (2010e:35138)
  • 3. J.L. Boldrini, G. Planas, Weak solutions of a phase-field model for phase change of an alloy with thermal properties, Math. Meth. in Appl. Sci. 25, no. 14 (2002), 1177-1193. MR 1925439 (2003i:35279)
  • 4. J.L. Boldrini, G. Planas, A tridimensional phase-field model with convection for phase-change of an alloy, Disc. and Cont. Dyn. Syst. 13, no. 2 (2005), 429-450. MR 2152398 (2006d:35105)
  • 5. J.L. Boldrini, C.L.D. Vaz, Existence and regularity of solutions of a phase field model for solidification with convection of pure materials in two dimensions, Elect. J. Diff. Equations 2003, no. 109 (2003), 1-25. MR 2011582 (2005b:35302)
  • 6. G. Caginalp, J. Jones, A derivation and analysis of phase field models of thermal alloys, Annal. Phys. 237 (1995), 66-107.
  • 7. G. Caginalp, Phase field computations of single-needle crystals, crystal growth and motion by mean curvature, SIAM J. Sci. Comput. 15 (1) (1994), 106-126. MR 1257157 (94k:65122)
  • 8. G. Caginalp, Stefan and Hele-Shaw type models as asymptotic limits of the phase-field equations, Phys. Rev. A 39 (11) (1989), 5887-5896. MR 998924 (90c:80004)
  • 9. G. Caginalp, An analysis of a phase field model of a free boundary, Arch. Rat. Mech. Anal. 92 (1986), 205-245. MR 816623 (87c:80011)
  • 10. B.M.R. Calsavara, J.L. Boldrini, On a system coupling the two-crystallizations Allen-Cahn equations and a singular Navier-Stokes system, in preparation (2011).
  • 11. A. Friedman, Partial Differential Equations of Parabolic Type. Prentice Hall, 1964. MR 0181836 (31:6062)
  • 12. K. Hoffman, L. Jiang, Optimal control of a phase field model for solidification, Numer. Funct. Anal. and Optimiz. 13 (1992), 11-27. MR 1163315 (93h:35218)
  • 13. P. Krejči, J. Sprekels, U. Stefanelli, Phase-field models with hysteresis in one-dimensional thermoviscoplasticity, SIAM J. Math. Anal. 34, no. 2 (2002), 409-434. MR 1951781 (2005d:74031)
  • 14. O. A. Ladyzenskaja, V. A. Solonnikov, N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs, vol. 23, Amer. Math. Soc., Providence, RI, 1968. MR 0241822 (39:3159b)
  • 15. J. L. Lions, Control of Singular Distributed Systems, Gauthier-Villars, 1983. MR 0712486 (85c:93002)
  • 16. V. P. Mikhaylov, Partial Differential Equations. Mir Publishers, Moscow, 1978. MR 498162 (80b:35002)
  • 17. G. Planas, J. L. Boldrini, A bidimensional phase-field model with convection for change phase of an alloy, J. Math. Anal. and Appl. 303 (2005), 669-687. MR 2122569 (2005k:35407)
  • 18. G. Planas, Existence of solutions to a phase-field model with phase-dependent heat absorption, Elect. J. Diff. Equations 2007, no. 28 (2007), 1-12. MR 2299582 (2007k:35240)
  • 19. P. Sprekels, S. Zheng, Global existence and asymptotic behaviour for a nonlocal phase-field model for non-isothermal phase transitions, J. Math. Anal. 279 (2003), 97-110. MR 1970493 (2004c:45015)
  • 20. J. P. Simon, Compact sets in the space $ L^{p}(0,T;B)$, Annali di Matematica Pura ed Applicata, serie quarta, tomo 146 (1987), 65-96. MR 916688 (89c:46055)
  • 21. I. Steinbach, F. Pezzolla, B. Nestler, M. Seesselberg, R. Prieler, G.J. Schimitz, J.L.L. Rezende, A phase field concept for multiphase systems. Physica D 94 (1996), 135-147.
  • 22. I. Steinbach, F. Pezzolla, A generalized field method for multiphase transformations using interface fields, Physica D 134 (1999), 385-393. MR 1725913 (2000h:80008)
  • 23. C.L.D. Vaz, J.L. Boldrini, A semidiscretization scheme for a phase-field type model for solidification, Portugaliae Mathematica 63, no. 3 (2006), 261-292. MR 2254930 (2007e:35303)

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Additional Information

Bianca Morelli Rodolfo Calsavara
Affiliation: University of Campinas – UNICAMP, School of Applied Sciences, Pedro Zaccaria Street, 1300, CEP 13484-350, Limeira, SP, Brazil

José Luiz Boldrini
Affiliation: University of Campinas – UNICAMP, IMECC, Sergio Buarque de Holanda Street, 651, CEP 13083-859, Campinas, SP, Brazil

Keywords: Parabolic partial differential equations, fixed point, solidification, phase field, advection, convection.
Received by editor(s): July 27, 2011
Published electronically: July 24, 2012
Additional Notes: The first author was supported in part by FAPESP, Brazil, Grant 2010/10087-8
The second author was supported in part by CNPq, Brazil, Grant 307833/2008-9
Communicated by: Walter Craig
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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