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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
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##### 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.
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