The existence of travelling waves for phase field equations and convergence to sharp interface models in the singular limit
Authors:
G. Caginalp and Y. Nishiura
Journal:
Quart. Appl. Math. 49 (1991), 147-162
MSC:
Primary 35K60; Secondary 35B05, 80A22
DOI:
https://doi.org/10.1090/qam/1096237
MathSciNet review:
MR1096237
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Additional Information
- 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, Stefan and Hele-Shaw type models as asymptotic limits of the phase-field equations, Phys. Rev. A (3) 39 (1989), no. 11, 5887–5896. MR 998924, DOI https://doi.org/10.1103/PhysRevA.39.5887
- L. I. Rubenšteĭn, The Stefan problem, American Mathematical Society, Providence, R.I., 1971. Translated from the Russian by A. D. Solomon; Translations of Mathematical Monographs, Vol. 27. MR 0351348
- 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 https://doi.org/10.1093/imamat/38.3.195
- Robert V. Kohn and Peter Sternberg, Local minimisers and singular perturbations, Proc. Roy. Soc. Edinburgh Sect. A 111 (1989), no. 1-2, 69–84. MR 985990, DOI https://doi.org/10.1017/S0308210500025026
- 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
- 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 https://doi.org/10.1007/BF00251427
- J. N. Dewynne, S. D. Howison, J. R. Ockendon, and Wei Qing Xie, Asymptotic behavior of solutions to the Stefan problem with a kinetic condition at the free boundary, J. Austral. Math. Soc. Ser. B 31 (1989), no. 1, 81–96. MR 1002093, DOI https://doi.org/10.1017/S0334270000006494
- J. Chadam and G. Caginalp, Stability of interfaces with velocity correction term, Rocky Mountain J. Math. 21 (1991), no. 2, 617–629. Current directions in nonlinear partial differential equations (Provo, UT, 1987). MR 1121530, DOI https://doi.org/10.1216/rmjm/1181072956
- Shui Nee Chow and Jack K. Hale, Methods of bifurcation theory, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 251, Springer-Verlag, New York-Berlin, 1982. MR 660633
- Shui Nee Chow, Jack K. Hale, and John Mallet-Paret, An example of bifurcation to homoclinic orbits, J. Differential Equations 37 (1980), no. 3, 351–373. MR 589997, DOI https://doi.org/10.1016/0022-0396%2880%2990104-7
- Yasumasa Nishiura and Hiroshi Fujii, Stability of singularly perturbed solutions to systems of reaction-diffusion equations, SIAM J. Math. Anal. 18 (1987), no. 6, 1726–1770. Translated in J. Soviet Math. 45 (1989), no. 3, 1205–1218. MR 911661, DOI https://doi.org/10.1137/0518124
- J. W. Wilder, Traveling wave solutions for interfaces arising from phase boundaries based on a phase field model, Quart. Appl. Math. 49 (1991), no. 2, 333–350. MR 1106396, DOI https://doi.org/10.1090/qam/1106396
H. Fujii, Y. Nishiura, M. Mimura, and R. Kobayashi, Existence of curved fronts for the phase field model, In preparation
- 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 https://doi.org/10.1007/BF00250432
G. Caginalp, Mathematical models of phase boundaries, Material Instabilities in Continuum Problems and Related Mathematical Problems (Ed. J. Ball), Heriot-Watt Symposium 1985–1986, Oxford Publ. (1988)
G. Caginalp, Stefan and Hele-Shaw type models as asymptotic limits of the phase field equations, Phys. Rev. A 39, 5887–5896 (1989)
L. I. Rubinstein, The Stefan problem, Transl. Math. Monographs, Vol. 27, American Mathematical Society, Providence, Rhode Island, 1971
G. Caginalp and P. C. Fife, Elliptic problems involving phase boundaries satisfying a curvature condition, IMA J. Appl. Math. 38, 195–217 (1987)
R. Kohn and P. Sternberg, Local minimizers and singular perturbations, Proc. Roy. Soc. Edinburgh, Sect. A 111, 69–84 (1989)
N. Alikakos and P. Bates, On the singular limit in a phase field model of a phase transition, Ann. Inst. H. Poincaré Non Linéaire 5, 141–178 (1988)
S. Luckhaus and L. Modica, The Gibbs-Thompson relation within the gradient theory of phase transitions, Arch. Rational Mech. Anal. 107, 71–83 (1989)
J. N. Dewynne, S. D. Howison, J. R. Ockendon, and W. Xie, Asymptotic behavior of solutions to the Stefan problem with a kinetic condition at the free boundary, J. Austral. Math. Soc. Ser. B 31, 81–96 (1989)
G. Caginalp and J. Chadam, Stability of interfaces with velocity correction term, to appear in Rocky Mountain J. Math.
S.-N. Chow and J. Hale, Methods of Bifurcation Theory, Springer, Berlin, 1982
S.-N. Chow, J. Hale, and J. Mallet-Paret, An example of bifurcation to homeoclinic orbits, J. Differential Equations 37, 351–373 (1980)
Y. Nishiura and H. Fujii, Stability of singularly perturbed solutions to systems of reaction-diffusion equations, SIAM J. Math. Anal. 18, 1726–1770 (1987)
J. W. Wilder, Travelling wave solutions for interfaces arising from phase boundaries based on a phase field model, Rensselaer Polytechnic Inst. preprint
H. Fujii, Y. Nishiura, M. Mimura, and R. Kobayashi, Existence of curved fronts for the phase field model, In preparation
P. C. Fife and J. B. McLeod, The approach of solutions of nonlinear diffusion equations to travelling front solutions, Arch. Rational Mech. Anal. 65, 335–361 (1977)
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© Copyright 1991
American Mathematical Society