zbMATH — the first resource for mathematics

Well-posedness and regularity for a parabolic-hyperbolic Penrose-Fife phase field system. (English) Zbl 1099.35021
Summary: This work is concerned with the study of an initial boundary value problem for a non-conserved phase field system arising from the Penrose-Fife approach to the kinetics of phase transitions. The system couples a nonlinear parabolic equation for the absolute temperature with a nonlinear hyperbolic equation for the phase variable \(\chi \), which is characterized by the presence of an inertial term multiplied by a small positive coefficient \(\mu \). This feature is the main consequence of supposing that the response of \(\chi \) to the generalized force (which is the functional derivative of a free energy potential and arises as a consequence of the tendency of the free energy to decay towards a minimum) is subject to delay. We first obtain well-posedness for the resulting initial-boundary value problem in which the heat flux law contains a special function of the absolute temperature \(\vartheta \), i.e. \(\alpha (\vartheta )\sim \vartheta -1/\vartheta \). Then we prove convergence of any family of weak solutions of the parabolic-hyperbolic model to a weak solution of the standard Penrose-Fife model as \(\mu \searrow 0\). However, the main novelty of this paper consists in proving some regularity results on solutions of the parabolic-hyperbolic system (including also estimates of Moser type) that could be useful for the study of the longterm dynamics.
35G25 Initial value problems for nonlinear higher-order PDEs
35B45 A priori estimates in context of PDEs
35B65 Smoothness and regularity of solutions to PDEs
80A22 Stefan problems, phase changes, etc.
35D10 Regularity of generalized solutions of PDE (MSC2000)
Full Text: DOI EuDML
[1] N. D. Alikakos: L p -bounds of solutions of reaction-diffusion equations. Comm. Partial Differential Equations 4 (1979), 827–868. · Zbl 0421.35009 · doi:10.1080/03605307908820113
[2] H. Brezis: Operateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. North-Holland Math. Studies 5. North-Holland, Amsterdam, 1973.
[3] M. Brokate, J. Sprekels: Hysteresis and Phase Transitions. Appl. Math. Sci. Vol. 121. Springer-Verlag, New York, 1996. · Zbl 0951.74002
[4] G. Caginalp: An analysis of a phase field model of a free boundary. Arch. Rational Mech. Anal. 92 (1986), 205–245. · Zbl 0608.35080 · doi:10.1007/BF00254827
[5] P. Colli, G. Gilardi, and M. Grasselli: Well-posedness of the weak formulation for the phase field model with memory. Adv. Differential Equations 3 (1997), 487–508. · Zbl 1023.45501
[6] P. Colli, M. Grasselli, and A. Ito: On a parabolic-hyperbolic Penrose-Fife phase-field system. Electron. J. Differential Equations 100 (2002), electronic; [Erratum, Electron. J. Differential Equations 100 (2002), erratum, (electronic)]. · Zbl 1011.35038
[7] P. Colli, Ph. Laurencot: Weak solutions to the Penrose-Fife phase field model for a class of admissible heat ux laws. Phys. D 111 (1998), 311–334. · Zbl 0929.35062 · doi:10.1016/S0167-2789(97)80018-8
[8] P. Colli, Ph. Laurencot, and J. Sprekels: Global solution to the Penrose-Fife phase field model with special heat ux law. In: Variation of Domains and Free-Boundary Problems in Solid Mechanics (Paris 1997) (P. Argoul, M. Fremond, Q. S. Nguyen, eds.). Kluwer, Dordrecht, 1997; Solid Mech. Appl. 66 (1999), 181–188.
[9] P. C. Fife, O. Penrose: Interfacial dynamics for thermodynamically consistent phase field models with nonconserved order parameter. Electron. J. Differential Equations 16 (1995), electronic. · Zbl 0851.35059
[10] A. Friedman: Partial Differential Equations. Holt-Rinehart-Winston, New York, 1969. · Zbl 0224.35002
[11] P. Galenko: Phase field model with relaxation of the diffusion flux in nonequilibrium solidification of a binary system. Phys. Lett. A 287 (2001), 190–197. · Zbl 01638760 · doi:10.1016/S0375-9601(01)00489-3
[12] G. Gilardi: Teoremi di regolarita per la soluzione di un’equazione differenziale astratta lineare del secondo ordine. Ist. Lombardo Accad. Sci. Lett. Rend. A 106 (1972), 641–675. · Zbl 0298.34057
[13] C. Giorgi, M. Grasselli, and V. Pata: Uniform attractors for a phase field model with memory and quadratic nonlinearity. Indiana Univ. Math. J. 48 (1999), 1395–1445. · Zbl 0940.35037 · doi:10.1512/iumj.1999.48.1793
[14] M. Grasselli, V. Pata: Existence of a universal attractor for a parabolic-hyperbolic phase field system. Adv. Math. Sci. Appl. 13 (2003), 443–459. · Zbl 1057.37068
[15] M. Grasselli, V. Pata: Existence of a universal attractor for a fully hyperbolic phase-field system. J. Evol. Equ. 4 (2004), 27–51. · Zbl 1063.35038 · doi:10.1007/s00028-003-0074-2
[16] M. Grasselli, V. Pata: Asymptotic behaviour of a parabolic-hyperbolic system. Commun. Pure Appl. Anal 3 (2004), 849–881. · Zbl 1079.35022 · doi:10.3934/cpaa.2004.3.849
[17] M. Grasselli, H. G. Rotstein: Hyperbolic phase field dynamics with memory. J. Math. Anal. Appl. 261 (2001), 205–230. · Zbl 0988.35117 · doi:10.1006/jmaa.2001.7492
[18] N. Kenmochi, M. Kubo: Weak solutions of nonlinear systems for non-isothermal phase transitions. Adv. Math. Sci. Appl. 9 (1999), 499–521. · Zbl 0930.35037
[19] Ph. Laurencot: Solutions to a Penrose-Fife model of phase field type. J. Math. Anal. Appl. 185 (1994), 262–274. · Zbl 0819.35159 · doi:10.1006/jmaa.1994.1247
[20] J. L. Lions: Quelques methodes de resolution des problemes aux limites non lineaires. Dunod, Gauthier-Villars, Paris, 1969.
[21] O. Penrose, P. C. Fife: Thermodynamically consistent models of phase field type for the kinetics of phase transitions. Phys. D 43 (1990), 44–62. · Zbl 0709.76001 · doi:10.1016/0167-2789(90)90015-H
[22] O. Penrose, P. C. Fife: On the relation between the standard phase field model and a ” thermodynamically consistent” phase field model. Phys. D 69 (1993), 107–113. · Zbl 0799.76084 · doi:10.1016/0167-2789(93)90183-2
[23] E. Rocca, G. Schimperna: Uniform attractor for some singular phase transition systems. Physica D 192 (2004), 279–307. · Zbl 1062.82015 · doi:10.1016/j.physd.2004.01.024
[24] E. Rocca, G. Schimperna: Global attractor for a parabolic-hyperbolic Penrose-Fife phase field system. Discrete Contin. Dyn. Syst. (Special Volume). To appear. · Zbl 1116.35028
[25] H. G. Rotstein, S. Brandon, A. Novick-Cohen, and A. A. Nepomnyashchy: Phase field equations with memory: the hyperbolic case. SIAM J. Appl. Math. 62 (2001), 264–282. · Zbl 0990.80007 · doi:10.1137/S0036139900369102
[26] J. Simon: Compact sets in the space L p (0; T; B). Ann. Mat. Pura Appl. 146 (1987), 65–96. · Zbl 0629.46031 · doi:10.1007/BF01762360
[27] R. Temam: Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Springer-Verlag, New York, 1997. · Zbl 0871.35001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.