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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References | Similar Articles | Additional Information

**[1]**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) MR**970516****[2]**G. Caginalp,*Stefan and Hele-Shaw type models as asymptotic limits of the phase field equations*, Phys. Rev. A**39**, 5887-5896 (1989) MR**998924****[3]**L. I. Rubinstein,*The Stefan problem*, Transl. Math. Monographs, Vol. 27, American Mathematical Society, Providence, Rhode Island, 1971 MR**0351348****[4]**G. Caginalp and P. C. Fife,*Elliptic problems involving phase boundaries satisfying a curvature condition*, IMA J. Appl. Math.**38**, 195-217 (1987) MR**983727****[5]**R. Kohn and P. Sternberg,*Local minimizers and singular perturbations*, Proc. Roy. Soc. Edinburgh, Sect. A**111**, 69-84 (1989) MR**985990****[6]**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) MR**954469****[7]**S. Luckhaus and L. Modica,*The Gibbs-Thompson relation within the gradient theory of phase transitions*, Arch. Rational Mech. Anal.**107**, 71-83 (1989) MR**1000224****[8]**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) MR**1002093****[9]**G. Caginalp and J. Chadam,*Stability of interfaces with velocity correction term*, to appear in Rocky Mountain J. Math. MR**1121530****[10]**S.-N. Chow and J. Hale,*Methods of Bifurcation Theory*, Springer, Berlin, 1982 MR**660633****[11]**S.-N. Chow, J. Hale, and J. Mallet-Paret,*An example of bifurcation to homeoclinic orbits*, J. Differential Equations**37**, 351-373 (1980) MR**589997****[12]**Y. Nishiura and H. Fujii,*Stability of singularly perturbed solutions to systems of reaction-diffusion equations*, SIAM J. Math. Anal.**18**, 1726-1770 (1987) MR**911661****[13]**J. W. Wilder,*Travelling wave solutions for interfaces arising from phase boundaries based on a phase field model*, Rensselaer Polytechnic Inst. preprint MR**1106396****[14]**H. Fujii, Y. Nishiura, M. Mimura, and R. Kobayashi,*Existence of curved fronts for the phase field model*, In preparation**[15]**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) MR**0442480**

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Additional Information

DOI:
https://doi.org/10.1090/qam/1096237

Article copyright:
© Copyright 1991
American Mathematical Society