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Transactions of the American Mathematical Society

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Compactness properties for families of quasistationary solutions of some evolution equations


Author: Giuseppe Savaré
Journal: Trans. Amer. Math. Soc. 354 (2002), 3703-3722
MSC (2000): Primary 47J25, 80A22; Secondary 37L05, 47J35, 49J45
DOI: https://doi.org/10.1090/S0002-9947-02-03035-0
Published electronically: May 7, 2002
MathSciNet review: 1911517
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Abstract: The following typical problem occurs in passing to the limit in some phase field models: for two sequences of space-time dependent functions $\{\theta_n\}, \{{\raise.3ex\hbox{$\chi$ }}_n\}$ (representing, e.g., suitable approximations of the temperature and the phase variable) we know that the sum $\theta_n + {\raise.3ex\hbox{$\chi$ }}_n$ converges in some $L^p$-space as $n\uparrow+\infty$ and that the time integrals of a suitable ``space'' functional evaluated on $\theta_n, {\raise.3ex\hbox{$\chi$ }}_n$ are uniformly bounded with respect to $n$. Can we deduce that $\theta_n$ and ${\raise.3ex\hbox{$\chi$ }}_n$ converge separately? LUCKHAUS (1990) gave a positive answer to this question in the framework of the two-phase Stefan problem with Gibbs-Thompson law for the melting temperature. PLOTNIKOV (1993) proposed an abstract result employing the original idea of Luckhaus and arguments of compactness and reflexivity type. We present a general setting for this and other related problems, providing necessary and sufficient conditions for their solvability: these conditions rely on general topological and coercivity properties of the functionals and the norms involved, and do not require reflexivity.


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  • [BBGT88] C. Baiocchi, G. Buttazzo, F. Gastaldi, and F. Tomarelli, General existence theorems for unilateral problems in continuum mechanics, Arch. Rational Mech. Anal. 100 (1988), no. 2, 149-189. MR 88k:73014
  • [CV77] C. Castaing and M. Valadier, Convex analysis and measurable multifunctions, Springer, Berlin-New York, 1977. MR 57:7169
  • [Dal93] G. Dal Maso, An introduction to ${\Gamma}$-convergence, Progress in Nonlinear Differential Equations and Their Applications, vol. 8, Birkhäuser, Boston, 1993. MR 94a:49001
  • [Gag61] E. Gagliardo, A unified structure in various families of function spaces. Compactness and closure theorems, Proc. Internat. Sympos. Linear Spaces (Jerusalem, 1960), Jerusalem Academic Press, Jerusalem, 1961, pp. 237-241. MR 24:A2836
  • [Giu84] E. Giusti, Minimal surfaces and functions of bounded variation, Birkhäuser Verlag, Basel, 1984. MR 87a:58041
  • [HS98a] K.-H. Hoffmann and V. N. Starovoitov, Phase transitions of liquid-liquid type with convection, Adv. Math. Sci. Appl. 8 (1998), no. 1, 185-198. MR 99d:35186
  • [HS98b] K.-H. Hoffmann and V. N. Starovoitov, The Stefan problem with surface tension and convection in Stokes fluid, Adv. Math. Sci. Appl. 8 (1998), no. 1, 173-183. MR 99d:35185
  • [LM72] J.-L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications. Vol. I-II, Springer-Verlag, New York, 1972, Die Grundlehren der mathematischen Wissenschaften, Band 182. MR 50:2670; MR 50:2671
  • [LP64] J.-L. Lions and J. Peetre, Sur une classe d'espaces d'interpolation, Inst. Hautes Études Sci. Publ. Math. no. 19, (1964), 5-68. MR 29:2627
  • [Luc90] S. Luckhaus, Solutions for the two-phase Stefan problem with the Gibbs-Thomson Law for the melting temperature, European Journal of Applied Mathematics 1 (1990), 101-111. MR 92i:80004
  • [Mod86] L. Modica, Gradient theory of phase transitions and minimal interface criterion, Arch. Rational Mech. Anal. 98 (1986), 123-142. MR 88f:76038
  • [PS93] P.I. Plotnikov and V.N. Starovoitov, The Stefan problem with surface tension as the limit of a phase field model, Differential Equations 29 (1993), 395-404. MR 94f:35155
  • [Rud87] W. Rudin, Real and complex analysis, third ed., McGraw-Hill Book Co., New York, 1987. MR 88k:00002
  • [Sch00] R. Schätzle, The quasistationary phase field equations with Neumann boundary conditions, J. Differential Equations 162 (2000), no. 2, 473-503. MR 2001b:35143
  • [Vis96] A. Visintin, Models of phase transitions, Progress in Nonlinear Differential Equations and Their Applications, vol. 28, Birkhäuser, Boston, 1996. MR 98a:80006
  • [Yos80] K. Yosida, Functional analysis, sixth ed., Springer-Verlag, Berlin, 1980, Die Grundlehren der Mathematischen Wissenschaften, Band 123. MR 82i:46002

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Additional Information

Giuseppe Savaré
Affiliation: Dipartimento di Matematica “F. Casorati”, Università di Pavia. Via Ferrata, 1. I-27100 Pavia, Italy
Email: savare@ian.pv.cnr.it

DOI: https://doi.org/10.1090/S0002-9947-02-03035-0
Keywords: Quasistationary solutions of evolution equations, phase-field models, Stefan problem with Gibbs-Thomson law, compactness methods
Received by editor(s): October 3, 2001
Published electronically: May 7, 2002
Additional Notes: This work was partially supported by the M.U.R.S.T. (Italy) through national research project funds, and by the Institute of Numerical Analysis of the C.N.R., Pavia, Italy
Article copyright: © Copyright 2002 American Mathematical Society

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