×

zbMATH — the first resource for mathematics

Convergence of the Cahn-Hilliard equation to the Hele-Shaw model. (English) Zbl 0828.35105
Summary: We prove that level surfaces of solutions to the Cahn-Hilliard equation tend to solutions of the Hele-Shaw problem under the assumption that classical solutions of the latter exist. The method is based on a new matched asymptotic expansion for solutions, a spectral analysis for linearized operators, and an estimate for the difference between the true solutions and certain approximate ones.

MSC:
35Q35 PDEs in connection with fluid mechanics
35R35 Free boundary problems for PDEs
76D99 Incompressible viscous fluids
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] N. D. Alikakos, P. W. Bates & G. Fusco, Slow motion for the Cahn-Hilliard equation in one space dimension, J. Diff. Eqns. 90 (1991), 81-135. · Zbl 0753.35042
[2] S. Allen & J. W. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta. Metall., 27 (1979), 1084-1095.
[3] N. D. Alikakos & G. Fusco, Equilibrium and dynamics of bubbles for the Cahn-Hilliard equation, preprint. · Zbl 0938.35565
[4] N. D. Alikakos & G. Fusco, Slow dynamics for the Cahn-Hilliard equation in higher space dimensions, Part I: Spectral estimates, to appear, Comm. Partial Diff. Eqns., Part II: The motion of bubbles. preprint.
[5] N. D. Alikakos & G. Fusco, The spectrum of the Cahn-Hilliard operator for generic interface in higher space dimensions, Indiana University Math. J. 41 (1993), 637-674. · Zbl 0798.35123
[6] P. W. Bates & P. C. Fife, Spectral comparison principles for the Cahn-Hilliard and phase-field equations, and time scales for coarsening, Phys. D 43 (1990), 335-348. · Zbl 0706.58074
[7] P. W. Bates & P. C. Fife, The dynamics of nucleation for the Cahn-Hilliard equation, SIAM J. Appl. Math., 53 (1993), 990-1008. · Zbl 0788.35061
[8] P. W. Bates & P.-J. Xun, Metastable patterns for the Cahn-Hilliard equation, Parts I and II, to appear in J. Diff. Eqns. · Zbl 0805.35046
[9] L. Bronsard & D. Hilhorst, On the slow dynamics for the Cahn-Hilliard equation in one space dimension, Proc. R. Soc. London A 439 (1992), 669-682. · Zbl 0777.35007
[10] L. Bronsard & R. V. Kohn, On the slowness of the phase boundary motion in one space dimension, Comm. Pure Appl. Math. 43 (1990), 983-997. · Zbl 0761.35044
[11] L. Bronsard & R. V. Kohn, Motion by mean curvature as the singular limit of Ginzburg-Landau dynamics, J. Diff. Eqns. 90 (1991), 211-237. · Zbl 0735.35072
[12] G. Caginalp, The dynamics of a conserved phase field system: Stefan-like, Hele-Shaw and Cahn-Hilliard models as asymptotic limits, IMA J. Appl. Math. 44 (1989), 77-94. · Zbl 0712.35114
[13] J. W. Cahn, On the spinodal decomposition, Acta Metall. 9 (1961), 795-801.
[14] J. W. Cahn & J. E. Hilliard, Free energy of a nonuniform system I. Interfacial free energy, J. Chem. Phys. 28 (1958), 258-267.
[15] J. Carr, M. Gurtin & M. Slemrod, Structured phase transitions on a finite interval, Arch. Rational Mech. Anal. 86 (1984), 317-351. · Zbl 0564.76075
[16] J. Carr & R. Pego, Very slow phase separation in one dimension, Lecture Notes in Physics 344 (M. Rascle, ed.), Springer-Verlag, 216-226 (1989). · Zbl 0991.35515
[17] J. Carr & R. Pego, Invariant manifolds for metastable patterns in \(u_t = \varepsilon ^2 u_{xx} - f(u)\) , Proc. Roy. Soc. Edinburgh 116 (1990), 133-160. · Zbl 0738.35023
[18] P. Constantin & M. Pugh, Global solutions for small data to the Hele-Shaw problem, preprint. · Zbl 0808.35104
[19] Xinfu Chen, Hele-Shaw problem and area-preserving, curve shorting motion, Arch. Rational Mech. Anal. 123 (1993), 117-151. · Zbl 0780.35117
[20] Xinfu Chen, Spectrums for the Allen-Cahn, Cahn-Hilliard, and phase-field equations for generic interface, Comm. Partial Diff Eqns. 19 (1994), 1371-1395. · Zbl 0811.35098
[21] Xinfu Chen, Generation and propagation of interface in reaction-diffusion equations, J. Diff. Eqns. 96 (1992), 116-141. · Zbl 0765.35024
[22] Xinfu Chen & C. M. Elliott, Asymptotics for a parabolic double obstacle problem, Proc. Roy. Soc. Lond. A, 444 (1994), 429-445. · Zbl 0814.35044
[23] P. de Mottoni & M. Schatzman, Evolution géométrique d’interfaces, C. R. Acad. Sci. Sér. I Math 309 (1989), 453-458. · Zbl 0698.35078
[24] P. de Mottoni & M. Schatzman, Geometrical evolution of developed interfaces, to appear, Trans. Amer. Math. Soc. · Zbl 0797.35077
[25] J. Duchon & R. Robert, Évolution d’une interface par capillarité et diffusion de volume I. Existence locale en temps, Ann. Inst. H. Poincaré, Analyse non linéaire 1 (1984), 361-378. · Zbl 0572.35051
[26] C. M. Elliott & D. A. French, Numerical studies of the Cahn-Hilliard equation for phase separation, IMA J. Appl. Math. 38 (1987), 97-128. · Zbl 0632.65113
[27] C. M. Elliott & S. Zheng, On the Cahn-Hilliard equation, Arch. Rational Mech. Anal. 96 (1986), 339-357. · Zbl 0624.35048
[28] L. C. Evans, H. M. Soner & P. E. Souganidis, Phase transitions and generalized motion by mean curvature, Comm. Pure Appl. Math. 45 (1992), 1097-1123. · Zbl 0801.35045
[29] P. C. Fife, Dynamical Aspects of the Cahn-Hilliard Equations, Barret Lectures, University of Tennessee, Spring, 1991.
[30] P. C. Fife & L. Hsiao, The generation and propagation of internal layers, Nonlinear Anal. TMA 70 (1988), 31-46. · Zbl 0685.35055
[31] G. Fusco, A geometric approach to the dynamics of \(u_t = \varepsilon ^2 u_{xx} - f(u)\) for small ?, Lecture Notes in Physics 359 (K. Kirchgässner, ed.), Springer-Verlag, 53-73 (1990).
[32] G. Fusco & J. K. Hale, Slow-motion manifolds, dormant instability, and singular perturbations, J. Dynamics Differential Equations 1 (1989), 75-94. · Zbl 0684.34055
[33] M. Gurtin & H. Matano, On the structure of equilibrium phase transitions within the gradient theory of fluids, Quart. Appl. Math. 46 (1988), 301-317. · Zbl 0665.76120
[34] C. P. Grant, Stow motion in one dimensional Cahn-Morral systems, to appear, SIAM J. Math. Anal, 1994.
[35] D. Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics 840, Springer-Verlag, New York, 1981. · Zbl 0456.35001
[36] T. Ilmanen, Convergence of the Allen-Cahn equation to Brakke’s motion by mean curvature, preprint. · Zbl 0784.53035
[37] O. A. Lady?enskaja, V. A. Solonnikov & N. N. Ural’ceva, Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs, vol. 23, Amer. Math. Soc, Providence (1968).
[38] S. Luckhaus & L. Modica, The Gibbs-Thomson relation within the gradient theory of phase transitions, Arch. Rational Mech. Anal. 107 (1989), 71-83. · Zbl 0681.49012
[39] L. Modica, The gradient theory of phase transitions and the minimal interface condition, Arch. Rational Mech. Anal. 98 (1986), 123-142. · Zbl 0616.76004
[40] R. L. Pego, Front migration in the nonlinear Cahn-Hilliard equation, Proc. Roy. Soc. London A 422 (1989), 261-278. · Zbl 0701.35159
[41] J. Rubinstein, P. Sternberg & J. B. Keller, Fast reaction, slow diffusion and curve shortening, SIAM J. Appl. Math. 49 (1989), 116-133. · Zbl 0701.35012
[42] P. Sternberg, The effect of a singular perturbation on non-convex variational problems, Arch. Rational Mech. Anal. 101 (1988), 209-260. · Zbl 0647.49021
[43] B. Stoth, The Stefan Problem coupled with the Gibbs-Thomson law as singular limit of the phase-field equations in the radial case, preprint.
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.