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Structured phase transitions on a finite interval. (English) Zbl 0564.76075
Consider a mass M of a compressible van der Waals fluid, i.e. a fluid whose free energy per unit volume E depends on density u and its gradient u’ as \(E=W(u)+\epsilon^ 2u'{}^ 2\) and W’(u) has the well-known undulating dependence on u. Consider the fluid contained in a, say, cylindrical vessel of length 2L. Consider density distributions u(x) depending only on the axial variable x and respecting the constraint of conservation of mass. Then among these distributions there are only two (specular) minimizers of the total free energy and they are monotonic (single-interface solutions), provided that the average density M/2L is contained within the Maxwell interval and \(\epsilon\) is sufficiently small.
The proof of this theorem of uniqueness (modulo reversals) is difficult and long (it extends well over 20 pages); it is achieved by transforming the associated Euler problem (with constraint) into a set of two integral equations, the study of which is delicate because of singularities present.
The result is an important contribution to the study of phase transitions within a finite domain.
Reviewer: G.Capriz

76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
49J27 Existence theories for problems in abstract spaces
Full Text: DOI
[1] [1893] van der Waals, J. D., The thermodynamic theory of capillarity under the hypothesis of a continuous variation of density (in Dutch), Verhandel. Konink. Akad. Weten. Amsterdam (Sec. 1) Vol. 1, No. 8. · JFM 24.0967.02
[2] [1958] Cahn, J. W., & J. E. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy, J. Chem. Physics 28, 258-267. · doi:10.1063/1.1744102
[3] [1962] Toupin, R. A., Elastic materials with couple-stresses, Arch. Rational Mech. Anal. 11, 385-414. · Zbl 0112.16805 · doi:10.1007/BF00253945
[4] [1966] Morrey, C. B., Multiple Integrals in the Calculus of Variations, Springer-Verlag, Berlin Heidelberg New York. · Zbl 0142.38701
[5] [1969] Hale, J. K., Ordinary Differential Equations, Wiley Interscience, New York. · Zbl 0186.40901
[6] [1971] Langer, J. S., Theory of spinodal decomposition in alloys, Ann. Phys. 65, 53-87. · doi:10.1016/0003-4916(71)90162-X
[7] [1973] Antman, S. S., Nonuniqueness of equilibrium states for bars in tension, J. Math. Anal. Appl. 44, 333-349. · Zbl 0267.73031 · doi:10.1016/0022-247X(73)90063-2
[8] [1974] Antman, S. S., Qualitative theory of the ordinary differential equations of nonlinear elasticity, Mechanics Today, 1972; Pergamon, New York, 58-101.
[9] [1975] Ericksen, J. L., Equilibrium of bars, J. Elasticity 5, 191-201. · Zbl 0324.73067 · doi:10.1007/BF00126984
[10] [1977] Antman, S. S., & E. R. Carbone, Shear and necking instabilities in nonlinear elasticity, J. Elast. 7, 125-151. · Zbl 0356.73048 · doi:10.1007/BF00041087
[11] [1977] Jackiw, R., Quantum meaning of classical field theory, Rev. Modern Phys. 49, 681-706. · doi:10.1103/RevModPhys.49.681
[12] [1978] Arnold, V., Mathematical Methods of Classical Mechanics, Springer-Verlag, Berlin Heidelberg New York. · Zbl 0386.70001
[13] [1979] Rowlinson, J. S., Translation of J. D. van der Waals ?The thermodynamic theory of capillarity under the hypothesis of a continuous variation of density?, J. Stat. Phys. 20, 197-244. · Zbl 1245.82006 · doi:10.1007/BF01011513
[14] [1982] Davis, H. T., & L. E. Scriven, Stress and structure in fluid interfaces, Advances Chem. Phys. 49, 681-706.
[15] [1983] Aifantis, E. C. & J. B. Serrin, The mechanical theory of fluid interfaces and Maxwell’s rule, J. Colloidal Interface Sci. 96, 517-529. · doi:10.1016/0021-9797(83)90053-X
[16] [1983] Aifantis, E. C. & J. B. Serrin, Equilibrium solutions in the mechanical theory of fluid microstuctures, J. Colloidal Interface Sci. 96, 530-547. · doi:10.1016/0021-9797(83)90054-1
[17] [1983] Carr, J., M. E. Gurtin, & M. Slemrod, One-dimensional structured phase transformations under prescribed loads, Technical Report 2559, Mathematics Research Center, University of Wisconsin, Madison. To appear in J. Elast. · Zbl 0581.73020
[18] [1983] Coleman, B. D., Necking and drawing in polymeric fibers under tension, Arch. Rational Mech. Anal. 83, 115-137. · Zbl 0535.73016 · doi:10.1007/BF00282158
[19] [1984] Novick-Cohen, A., & L. A. Segel, Nonlinear aspects of the Cahn-Hilliard equation, Physica 10D, 277-298.
[20] [1984] Novick-Cohen, A., The nonlinear Cahn-Hilliard equation: Transition from spinodal decomposition to nucleation behavior. To appear.
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