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Multiphase thermomechanics with interfacial structure. II: Evolution of an isothermal interface. (English) Zbl 0723.73017
Among the results of M. E. Gurtin’s paper [see this Zbl 0723.73016] on the thermodynamics of two-phase rigid continua based on Gibbs’s notion of sharp phase-interface endowed with energy, entropy, and superficial force we remained those regarding the exact and approximate free boundary conditions at the interface and a hierarchy of free boundary problems. These problem are extremely difficult to solve because of nonlinearities inherent in the free boundary conditions. In the case of perfrect conductors, i.e. materials with infinite thermal conductivity, the temperature is constant and the corresponding free boundary problem reduces to a single set of evolution equations for interfaces.
The goal of this paper is the theory of rigid perfect conductors with interfaces that evolve as curves in $${\mathbb{R}}2$$. The paper contains a lot of results of great interest regarding the evolution of an isothermal interface. To form an idea of the problems treated in this paper we present its main sections. I. The thermodynamics of evolving curves, II. Smooth interfacial motions, III. Interfacial motions with corners, and IV. Nonsmooth interfacial energies.

##### MSC:
 74A15 Thermodynamics in solid mechanics 80A17 Thermodynamics of continua 35R35 Free boundary problems for PDEs
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 [1] [1901]Wulff, G., Zur Frage der Geschwindigkeit des Wachsthums und der Auflösung der Krystallflächen, Zeit. Krystall. Min.34, 449-530. [2] [1944]Dinghas, A., Über einen geometrischen Satz von Wulff für die Gleichgewichts-form von Krystallen, Zeit. Krystall.105, 304-314. · Zbl 0028.43001 [3] [1951a]Herring, C., Surface tension as a motivation for sintering,The Physics of Powder Metallurgy (ed.W. E. Kingston) McGraw-Hill, New York. [4] [1951b]Herring, C., Some theorems on the free energies of crystal surfaces, Phys. Rev.82, 87-93. · Zbl 0042.23201 · doi:10.1103/PhysRev.82.87 [5] [1958]Frank, F. C., On the kinematic theory of crystal growth and dissolution processes,Growth and Perfection of Crystals (eds.R. H. Doremus, B. W. Roberts &D. Turnbull) John Wiley, New York. [6] [1963]Frank, F. C., The geometrical thermodynamics of surfaces, Metal Surfaces: Structure, Energetics, and Kinetics, Am. Soc. Metals, Metals Park, Ohio. [7] [1963]Gjostein, N. A., Adsorption and surface energy (II): thermal faceting from minimization of surface energy, Act. Metall.11, 969-977. · doi:10.1016/0001-6160(63)90066-X [8] [1967]Protter, M., &H. Weinberger,Maximum Principles in Differential Equations, Prentice Hall, New York. · Zbl 0153.13602 [9] [1974]Cahn, J. W., &D. W. Hoffman, A vector thermodynamics for anisotropic surfaces ? 2. curved and faceted surfaces, Act. Metall.22, 1205-1214. · doi:10.1016/0001-6160(74)90134-5 [10] [1978]Brakke, K. A.,The Motion of a Surface by its Mean Curvature, Princeton University Press. · Zbl 0386.53047 [11] [1978]Taylor, J. E., Crystalline variational problems, Bull. Am. Math. Soc.84, 568-588. · Zbl 0392.49022 · doi:10.1090/S0002-9904-1978-14499-1 [12] [1979]Allen, S. M., &J. W. Cahn, A macroscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Act. Metall.27, 1085-1098. · doi:10.1016/0001-6160(79)90196-2 [13] [1984]Gage, M., Curve shortening makes convex curves circular, Invent. Math.76, 76, 357-364. · Zbl 0542.53004 · doi:10.1007/BF01388602 [14] [1985]Sethian, J. A., Curvature and the evolution of fronts, Comm. Math. Phys.101, 487-499. · Zbl 0619.76087 · doi:10.1007/BF01210742 [15] [1986]Abresch, U., &J. Langer, The normalized curve shortening flow and homothetic solutions. J. Diff. Geom.23, 175-196. · Zbl 0592.53002 [16] [1986]Gage, M., &R. S. Hamilton, The heat equation shrinking convex plane curves, J. Diff. Geom.23, 69-95. · Zbl 0621.53001 [17] [1986]Gage, M., Oh an area preserving evolution equation for plane curves, Contemporary Math.51, 51-62. [18] [1986g]Gurtin, M. E., On the two-phase Stefan problem with interfacial energy and entropy, Arch. Rational Mech. Anal.96, 199-241. · Zbl 0654.73008 [19] [1987]Grayson, M. A., The heat equation shrinks embedded plane curves to round points, J. Diff. Geom.26, 285-314. · Zbl 0667.53001 [20] [1987]Huisken, G., Deforming hypersurfaces of the sphere by their mean curvature, Math. Z.198, 138-146. · Zbl 0626.53039 [21] [1987]Maris, H. J., & A. A. Andreev, The surface of crystalline helium 4, Phys. Today, February, 25-30. [22] [1987]Osher, S., &J. A. Sethian, Fronts propagating with curvature dependent speed: algorithms based on Hamilton-Jacobi formulations, CAM Report 87-12, Dept. Math., U. California, Los Angeles. · Zbl 0659.65132 [23] [1987]Rubinstein, J., Sternberg, P., & J. B. Keller, Fast reaction, slow diffusion, and curve shortening, Forthcoming. · Zbl 0701.35012 [24] [1988]Angenent, S., Forthcoming. [25] [1988g]Gurtin, M. E., Multiphase thermomechanics with interfacial structure. 1. Heat conduction and the capillary balance law. Arch. Rational Mech. Anal.104, 195-221. · Zbl 0723.73016 · doi:10.1007/BF00281354 [26] [1988gg]Gurtin, M. E., Multiphase thermomechanics with interfacial structure. Toward a nonequilibrium thermomechanics of two phase materials, Arch. Rational Mech. Anal.100, 275-312. · Zbl 0673.73007 [27] [1988]Fonseca, I., Interfacial energy and the Maxwell rule, Res. Rept. 88-18, Dept. Math., Carnegie Mellon, Pittsburgh. [28] [1989]Angenent, S., & M. E. Gurtin, Forthcoming.
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