zbMATH — the first resource for mathematics

Abstract approach to evolution equations of phase-field type and applications. (English) Zbl 0978.35075
In this extended paper the author is concerned with the nonlinear parabolic equations that relate to the phase-field models of heat transfer in anisotropic fluids with phase transitions. An abstract approach is chosen, and general results on existence and uniqueness are proven.
In the second part of the article (Section 5) the results are applied to two physical situations: firstly, a heat diffusion problem, showing how existence and uniqueness results can be obtained from the previously proven theory and, secondly, a far more interesting problem, dealing with the heat and phase-field diffusion problem between two adjoining regions filled with fluids of possibly different physical characteristics. Finally, suggestions for possible expansions are included together with an open question.

35Q72 Other PDE from mechanics (MSC2000)
35K90 Abstract parabolic equations
Full Text: DOI
[1] Baiocchi, C., Sulle equazioni differenziali astratte lineari del primo e del secondo ordine negli spazi di Hilbert, Ann. mat. pura appl. (4), 76, 233-304, (1967) · Zbl 0153.17202
[2] Barbu, V., Nonlinear semigroups and differential equations in Banach spaces, (1976), Noordhoff Leyden
[3] V. Barbu, P. Colli, G. Gilardi, and, M. Grasselli, Existence, uniqueness, and longtime behaviour for a nonlinear Volterra integrodifferential equation, Differential Integral Equations, in press. · Zbl 0981.45006
[4] Bergh, J.; Löfström, J., Interpolation spaces: an introduction, (1976), Springer-Verlag Berlin/Heidelberg · Zbl 0344.46071
[5] Brézis, H., Opérateurs maximaux monotones et Sémi-groupes de contractions dans LES espaces de Hilbert, North-holland math. studies, 5, (1973), North-Holland Amsterdam · Zbl 0252.47055
[6] Brézis, H., Intégrales convexes dans LES espaces de Sobolev, Israel J. math., 13, 9-23, (1972)
[7] Caginalp, G., An analysis of a phase field model of a free boundary, Arch. rational mech. anal., 92, 205-245, (1986) · Zbl 0608.35080
[8] Caginalp, G.; Fife, P.C., Dynamics of layered interfaces arising from phase boundaries, SIAM J. appl. math., 48, 506-518, (1988)
[9] Carroll, R.W.; Showalter, R.E., Singular and degenerate Cauchy problems, (1976), Academic Press New York
[10] Colli, P.; Gilardi, G.; Grasselli, M., Global smooth solution to the standard phase-field model with memory, Adv. differential equations, 2, 453-486, (1997) · Zbl 1023.45500
[11] Colli, P.; Gilardi, G.; Grasselli, M., Well-posedness of the weak formulation for the phase-field model with memory, Adv. differential equations, 2, 487-508, (1997) · Zbl 1023.45501
[12] Damlamian, A.; Kenmochi, N., Evolution equations generated by subdifferentials in the dual space of H1(ω), Discrete contin. dynam. systems, 5, 269-278, (1999) · Zbl 0954.35098
[13] Damlamian, A.; Kenmochi, N.; Sato, N., Subdifferential operator approach to a class of nonlinear systems for Stefan problems with phase relaxation, Nonlinear anal., 23, 115-142, (1994) · Zbl 0820.35143
[14] Ekeland, I.; Temam, R., Analyse convexe et problèmes variationnels, (1974), Gauthier-VillarsDunod Paris
[15] Fix, G.J., Phase field models for free boundary problems, (), 580-589
[16] Landau, L.D.; Lifshitz, E.M., Statistical physics, (1958), Addison-Wesley Reading · Zbl 0080.19702
[17] Lions, J.L., Quelques Méthodes de Résolution des problèmes aux limites non linéaires, (1969), Gauthier-VillarsDunod Paris · Zbl 0189.40603
[18] Lions, J.L.; Magenes, E., Non-homogeneous boundary value problems and applications, (1972), Springer-Verlag Berlin · Zbl 0223.35039
[19] Penrose, O.; Fife, P.C., Thermodynamically consistent models of phase-field type for the kinetics of phase transitions, Phys. D, 43, 44-62, (1990) · Zbl 0709.76001
[20] Rockafellar, R.T., Integrals which are convex functionals, Pacific J. math., 24, 525-539, (1968) · Zbl 0159.43804
[21] Savaré, G.; Visintin, A., Variational convergence of nonlinear diffusion equations: applications to concentrated capacity problems with change of phase, Atti accad. naz. lincei, cl. sci. fis. mat. natur. rend. lincei (9) mat. appl., 8, 49-89, (1997) · Zbl 0888.35139
[22] Schimperna, G., Weak solution to a phase-field transmission problem in a concentrated capacity, Math. methods appl. sci., 22, 1235-1254, (1999) · Zbl 0933.35198
[23] G. Schimperna, Singular limit of a transmission problem for the parabolic phase-field model, Appl. Math, in press. · Zbl 1058.35041
[24] Simon, J., Compact sets in the space L^p(0, T; B), Ann. mat. pura appl. (4), 146, 65-96, (1987) · Zbl 0629.46031
[25] Visintin, A., Stefan problem with phase relaxation, IMA J. appl. math., 34, 225-245, (1985) · Zbl 0585.35053
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.