×

zbMATH — the first resource for mathematics

Spectrum for the Allen-Cahn, Cahn-Hilliard, and phase-field equations for generic interfaces. (English) Zbl 0811.35098
The Allen-Cahn, Cahn-Hilliard, and phase-field equations are a set of PDE which describe the phase transition phenomena in a solidification process. The analysis of the stability of the solution at time zero results in an eigenvalue problem, and the purpose of the paper is to estimate the lower bound of the latter, given some mathematical conditions. The author first reviews some of the results of de Mottoni and Schatzman and then he characterizes the eigenfunctions for the Allen- Cahn operator. The main theorem is proven, and finally some other related theorems are stated.

MSC:
35Q35 PDEs in connection with fluid mechanics
35Q55 NLS equations (nonlinear Schrödinger equations)
80A22 Stefan problems, phase changes, etc.
35R35 Free boundary problems for PDEs
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Alikakos N. D., Ann. Inst. H. Poincaré 5 pp 1– (1988)
[2] N. D. Alikakos, P. W. Bates, & X. Chen, The convergence of solutions of the Cahn-Hillard equation to the solution of Hele-Shaw model. preprint. · Zbl 0828.35105
[3] DOI: 10.1016/0022-0396(91)90163-4 · Zbl 0753.35042
[4] N. D. Alikakos & G. Fusco, Equilibrium and Dynamics of Bubbles for the Cahn-Hilliard equation, preprint. · Zbl 0938.35565
[5] N. D. Alikakos & G. Fusco, Slow Dynamics for the Cahn-Hilliard equation on high space dimensions, Parts I & II, preprint · Zbl 0906.35049
[6] DOI: 10.1512/iumj.1993.42.42028 · Zbl 0798.35123
[7] Allen S., Acta. Metall. 27 pp 1084– (1979)
[8] DOI: 10.1016/0167-2789(90)90141-B · Zbl 0706.58074
[9] P. W. Bates & P. J. Xun, Metastable patterns for the Cahn-Hilliard equations, Parts I & II, to appear in J. Diff. Eqns. · Zbl 0805.35046
[10] Brakke K. A., The Motion of a Surface by its Mean Curvature (1978) · Zbl 0386.53047
[11] DOI: 10.1002/cpa.3160430804 · Zbl 0761.35044
[12] DOI: 10.1016/0022-0396(91)90147-2 · Zbl 0735.35072
[13] DOI: 10.1098/rspa.1992.0176 · Zbl 0777.35007
[14] DOI: 10.1007/BF00254827 · Zbl 0608.35080
[15] DOI: 10.1093/imamat/44.1.77 · Zbl 0712.35114
[16] DOI: 10.1103/PhysRevA.39.5887 · Zbl 1027.80505
[17] Caginalp G., IMA Volume of Mathematics and Its Applications 43 pp 1– (1992)
[18] G. Caginalp & X. Chen, Convergence of solutions of the phase-field equations to solutions of the sharp interface model, in preparation. · Zbl 0930.35024
[19] DOI: 10.1093/imamat/38.3.195 · Zbl 0645.35101
[20] Caginalp G., Quart. Appl. Math. 49 pp 147– (1991)
[21] DOI: 10.1016/0001-6160(61)90182-1
[22] DOI: 10.1063/1.1744102
[23] DOI: 10.1007/BF00280031 · Zbl 0564.76075
[24] Carr J., Lecture Notes in Physics 344 pp 216– (1989)
[25] Carr J., Poc. Roy. Soc. Edinburgh 116 pp 133– (1990)
[26] DOI: 10.1016/0022-0396(92)90146-E · Zbl 0765.35024
[27] G. Caginalp & X. Chen, Convergence of solutions of the phase-field equations to solutions of the sharp interface model, in preparation. X. Chen & C. M. Elliott, Asymptotics for a parabolic double obstacle problem, preprint. · Zbl 0930.35024
[28] Chen Y. G., J. Diff. Geom. 33 pp 749– (1991)
[29] X. Chen & C. M. Elliott, Asymptotics for a parabolic double obstacle problem, preprint. T. Ilmanen, Convergence of the Allen-Cahn equation to Brakke’s motion, preprint.
[30] de Mottoni P., Proc. Roy. Soc. Edinburgh Sect. A 116 pp 207– (1990) · Zbl 0725.35009
[31] de Mottoni, P. and Schatzman, M. 1989.Evolution géométrique d’interfaces, I Math Vol. 309, 453–458. C. R. Acad. Sci. · Zbl 0698.35078
[32] P. de Mottoni & M. Schatzman, Geometrical Evolution of developed interface, to appear in Trans. Amer. Math. Soc. · Zbl 0797.35077
[33] L. C. Evans, H. M. Soner, & P. E. Souganidis, The Allen-Cahn equation and the generalized motion by mean curvature, preprint. · Zbl 0801.35045
[34] Evans L. C., J. Diff. Geom. 33 pp 635– (1991)
[35] Fife P. C., CCMS–NSF Regional Conf. Ser. in Appl. Math. (1988)
[36] Fife P. C., Nonlinear Anal. TMA 70 pp 31– (1988)
[37] Fusco G., Lecture Notes in Physics 359 pp 53– (1990)
[38] DOI: 10.1007/BF01048791 · Zbl 0684.34055
[39] Gage M., J. Diff. Geom. 23 pp 69– (1986)
[40] Grayson M., J. Diff. Geom. 26 pp 285– (1987)
[41] Gurtin M., Quart. Appl. Math. 46 pp 301– (1988) · Zbl 0665.76120
[42] DOI: 10.1103/PhysRevB.10.139
[43] Hamilton R. S., J. Diff. Geom. 17 pp 255– (1982)
[44] Huisken G., J. Diff. Geom/ 20 pp 237– (1984)
[45] Langer J. S., Directions in Condensed Matter physics pp 164– (1986)
[46] DOI: 10.1007/BF00251427 · Zbl 0681.49012
[47] Modica L., Arch. Rat. Mech. Anal. 98 pp 123– (1986)
[48] Pego, R. L. 1989.Front migration in the nonlinear Cahn-Hilliard equation, A Vol. 422, 261–278. Proc. Roy. Soc. · Zbl 0701.35159
[49] DOI: 10.1137/0149007 · Zbl 0701.35012
[50] H. M. Soner, Motion of a set by the curvature of its boundary, to apper in J. Diff. Eqns. · Zbl 0769.35070
[51] DOI: 10.1007/BF00253122 · Zbl 0647.49021
[52] B. Stoth, A model with sharp interface as limit of phase-field equations in one space dimension, preprint · Zbl 0876.35133
[53] 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.