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ISSN 1088-6850(online) ISSN 0002-9947(print)



Geometrical evolution of developed interfaces

Authors: Piero de Mottoni and Michelle Schatzman
Journal: Trans. Amer. Math. Soc. 347 (1995), 1533-1589
MSC: Primary 35B40; Secondary 35A30, 35K57, 58E12
MathSciNet review: 1672406
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Abstract: Consider the reaction-diffusion equation in ${\mathbb {R}^N} \times {\mathbb {R}^ + }:{u_t} - {h^2}\Delta u + \varphi (u) = 0;\varphi$ is the derivative of a bistable potential with wells of equal depth and $h$ is a small parameter. If the initial data has an interface, we give an asymptotic expansion of arbitrarily high order and error estimates valid up to time $O({h^{ - 2}})$. At lowest order, the interface evolves normally, with a velocity proportional to the mean curvature. Soit l’équation de réaction-diffusion dans ${\mathbb {R}^N} \times {\mathbb {R}^ + },\quad {u_t} - {h^2}\Delta u + \varphi (u) = 0$, avec $\varphi$ la dérivée d’un potentiel bistable à puits également profonds et $h$ un petit paramètre. Pour une condition initiale possédant une interface, on donne un développement asymptotique d’ordre arbitrairement élevé, ainsi que des estimations d’erreur valides jusqu’à un temps en $O({h^{ - 2}})$. A l’ordre le plus bas, l’interface évolue normalement, à une vitesse proportionnelle à la courbure moyenne.

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  • N. I. Akhiezer and I. M. Glazman, Theory of linear operators in Hilbert space. Vol. I, Monographs and Studies in Mathematics, vol. 9, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1981. Translated from the third Russian edition by E. R. Dawson; Translation edited by W. N. Everitt. MR 615736
  • Nicholas D. Alikakos and Peter W. Bates, On the singular limit in a phase field model of phase transitions, Ann. Inst. H. Poincaré Anal. Non Linéaire 5 (1988), no. 2, 141–178 (English, with French summary). MR 954469
  • S. Allen and J. Cahn, A microscopic theory for antiphase motion and its application to antiphase domain coarsening, Acta Metall. 27 (1979), 1084-1095. S. Angenent, Parabolic curves on surfaces I. Curves with $p$-integrable curvature, TSR 89-19, Dept. of Math., Univ. of Wisconsin, Madison, November 1988. ---, Parabolic curves on surfaces II. Intersections, blow up and generalized solutions TSR 89-24 Dept. of Math., Univ. of Wisconsin, Madison, January 1988. ---, On the formation of singularities in the curve shortening flow, preprint, Univ. of Wisconsin, Madison, March 1989. S. Angenent and M. E. Gurtin, Multiphase thermomechanics with interfacial structure. $2$. Evolution of an isothermal interface, preprint, 1988.
  • Roberto Benzi, Giovanni Jona-Lasinio, and Alfonso Sutera, Stochastically perturbed Landau-Ginzburg equations, J. Statist. Phys. 55 (1989), no. 3-4, 505–522. MR 1003526, DOI
  • Kenneth A. Brakke, The motion of a surface by its mean curvature, Mathematical Notes, vol. 20, Princeton University Press, Princeton, N.J., 1978. MR 485012
  • L. Bronsard, Reaction-diffusion equations and motion by mean curvature, Ph.D. Thesis, New York Univ., October 1988.
  • Lia Bronsard and Robert V. Kohn, On the slowness of phase boundary motion in one space dimension, Comm. Pure Appl. Math. 43 (1990), no. 8, 983–997. MR 1075075, DOI
  • ---, Motion by mean curvature as the singular limit of Ginzburg-Landau dynamics, preprint, Lefschetz Center for Dynamical Systems, August 1989.
  • Jack Carr, Morton E. Gurtin, and Marshall Slemrod, Structured phase transitions on a finite interval, Arch. Rational Mech. Anal. 86 (1984), no. 4, 317–351. MR 759767, DOI
  • J. Carr and R. L. Pego, Metastable patterns in solutions of ${u_t} = {\varepsilon ^2}{u_{xx}} - f(u)$, Comm. Pure Appl. Math. (in press).
  • G. Caginalp, Mathematical models of phase boundaries, Material instabilities in continuum mechanics (Edinburgh, 1985–1986) Oxford Sci. Publ., Oxford Univ. Press, New York, 1988, pp. 35–52. MR 970516
  • G. Caginalp and P. C. Fife, Elliptic problems involving phase boundaries satisfying a curvature condition, IMA J. Appl. Math. 38 (1987), no. 3, 195–217. MR 983727, DOI
  • Xinfu Chen, Generation and propagation of interfaces for reaction-diffusion equations, J. Differential Equations 96 (1992), no. 1, 116–141. MR 1153311, DOI
  • V. G. Danilov and P. Yu. Subochev, Exact one and two phase wave-like solutions of semilinear parabolic equations, preprint, Steklov Institute, Moscow, 1988.
  • Dennis M. DeTurck, Deforming metrics in the direction of their Ricci tensors, J. Differential Geom. 18 (1983), no. 1, 157–162. MR 697987
  • L. C. Evans, H. M. Soner, and P. E. Souganidis, The Allen-Cahn equation and generalized motion by mean curvature, preprint, 1990. P. C. Fife, Nonlinear diffusive waves, CBMS Conf. at Little Cottonwood Canyon, Utah, 1987, CBMS Conference Series, 1989. P. C. Fife and G. S. Gill, The phase-field description of mushy zones, Proc. Conf. on Nonlinear Partial Differential Equations, Provo, Utah, 1987. P. C. Fife and Ling Hsiao, The generation and propagation of internal layers, Nonlinear Analysis TMA,
  • Paul C. Fife and J. B. McLeod, The approach of solutions of nonlinear diffusion equations to travelling front solutions, Arch. Rational Mech. Anal. 65 (1977), no. 4, 335–361. MR 442480, DOI
  • M. I. Freidlin, Geometric optics approach to reaction-diffusion equations, SIAM J. Appl. Math. 46 (1986), no. 2, 222–232. MR 833475, DOI
  • Avner Friedman, Partial differential equations of parabolic type, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. MR 0181836
  • G. Fusco and J. K. Hale, Slow-motion manifolds, dormant instability, and singular perturbations, J. Dynam. Differential Equations 1 (1989), no. 1, 75–94. MR 1010961, DOI
  • G. Fusco, A geometric approach to the dynamics of ${u_t} = {\varepsilon ^2}{u_{xx}} + f(u)$ for small $\varepsilon$, preprint, June 1989.
  • M. Gage and R. S. Hamilton, The heat equation shrinking convex plane curves, J. Differential Geom. 23 (1986), no. 1, 69–96. MR 840401
  • Peter B. Gilkey, Invariance theory, the heat equation, and the Atiyah-Singer index theorem, Mathematics Lecture Series, vol. 11, Publish or Perish, Inc., Wilmington, DE, 1984. MR 783634
  • Matthew A. Grayson, The heat equation shrinks embedded plane curves to round points, J. Differential Geom. 26 (1987), no. 2, 285–314. MR 906392
  • Morton E. Gurtin, Some results and conjectures in the gradient theory of phase transitions, Metastability and incompletely posed problems (Minneapolis, Minn., 1985) IMA Vol. Math. Appl., vol. 3, Springer, New York, 1987, pp. 135–146. MR 870014, DOI
  • Morton E. Gurtin and Hiroshi Matano, On the structure of equilibrium phase transitions within the gradient theory of fluids, Quart. Appl. Math. 46 (1988), no. 2, 301–317. MR 950604, DOI
  • J. D. Gunton, M. San Miguel, and Paramdeep S. Sahni, The dynamics of first-order phase transitions, Phase transitions and critical phenomena, Vol. 8, Academic Press, London, 1983, pp. 267–482. MR 794319
  • Richard S. Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geometry 17 (1982), no. 2, 255–306. MR 664497
  • Gerhard Huisken, Flow by mean curvature of convex surfaces into spheres, J. Differential Geom. 20 (1984), no. 1, 237–266. MR 772132
  • Jürgen Jost, Nonlinear methods in Riemannian and Kählerian geometry, DMV Seminar, vol. 10, Birkhäuser Verlag, Basel, 1988. MR 925006
  • Ya. I. Kanel’, On the stabilization of solutions of the Cauchy problem for the equations arising in the theory of combustion, Mat. Sb. 59 (1965), 398-413.
  • Tosio Kato and Hiroki Tanabe, On the abstract evolution equation, Osaka Math. J. 14 (1962), 107–133. MR 140954
  • K. Kawasaki and T. Ohta, Kinetic drumhead model of interface I, Progr. Theor. Phys. 67 (1982), 147-163. R. V. Kohn and P. Sternberg, Local minimizers and singular perturbations, Proc. Roy. Soc. Edinburgh (in press).
  • J.-L. Lions, Équations différentielles opérationnelles et problèmes aux limites, Die Grundlehren der mathematischen Wissenschaften, Bd. 111, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1961 (French). MR 0153974
  • Stephan Luckhaus and Luciano Modica, The Gibbs-Thompson relation within the gradient theory of phase transitions, Arch. Rational Mech. Anal. 107 (1989), no. 1, 71–83. MR 1000224, DOI
  • Luciano Modica, The gradient theory of phase transitions and the minimal interface criterion, Arch. Rational Mech. Anal. 98 (1987), no. 2, 123–142. MR 866718, DOI
  • Piero de Mottoni and Michelle Schatzman, Évolution géométrique d’interfaces, C. R. Acad. Sci. Paris Sér. I Math. 309 (1989), no. 7, 453–458 (French, with English summary). MR 1055457
  • ---, Development of interfaces in $N$-dimensional space, Proc. Roy. Soc. Edimburgh 116A (1990), 207-220. J. Neu, Private communication.
  • A. Novick-Cohen, Blow up and growth in the directional solidification of dilute binary alloys, Appl. Anal. 47 (1992), no. 4, 241–257. MR 1307011, DOI
  • E. Presutti, Collective behaviour of interacting particle systems, Proceedings of the 1st World Congress of the Bernoulli Society, Vol. 1 (Tashkent, 1986) VNU Sci. Press, Utrecht, 1987, pp. 395–413. MR 1092379
  • Jacob Rubinstein, Peter Sternberg, and Joseph B. Keller, Fast reaction, slow diffusion, and curve shortening, SIAM J. Appl. Math. 49 (1989), no. 1, 116–133. MR 978829, DOI
  • Barry Simon, On positive eigenvalues of one-body Schrödinger operators, Comm. Pure Appl. Math. 22 (1969), 531–538. MR 247300, DOI
  • H. Spohn, private communication.

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