The existence of travelling waves for phase field equations and convergence to sharp interface models in the singular limit

@article{Caginalp1991TheEO,
  title={The existence of travelling waves for phase field equations and convergence to sharp interface models in the singular limit},
  author={Gunduz Caginalp and Yasumasa Nishiura},
  journal={Quarterly of Applied Mathematics},
  year={1991},
  volume={49},
  pages={147-162}
}
Title The Existence of traveling waves for phase field equations and convergence to sharp interface models in the singular limit Author(s) Caginalp, G.; Nishiura, Yasumasa Citation Quarterly of Applied Mathematics, 49(1): 147-162 Issue Date 1991-03 Doc URL http://hdl.handle.net/2115/39998 Rights First published in Quarterly of Applied Mathematics in volume 49, number 1, published by Brown University. Type article File Information nishiura-83.pdf 

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References

SHOWING 1-10 OF 20 REFERENCES
Traveling wave solutions for interfaces arising from phase boundaries based on a phase field model
This work presents an analysis of the traveling wave solutions resulting from the phase field model which has been proposed for solidification. It is shown that solutions only exist for very
Asymptotic behavior of solutions to the Stefan problem with a kinetic condition at the free boundary
Abstract We study the large time behaviour of the free boundary for a one-phase Stefan problem with supercooling and a kinetic condition u = −ε|⋅ṡ| at the free boundary x = s(t). The problem is posed
Stefan and Hele-Shaw type models as asymptotic limits of the phase-field equations.
  • Caginalp
  • Physics, Medicine
    Physical review. A, General physics
  • 1989
TLDR
L'echelonnement des parametres physiques dans le domaine microscopique conduit a des modeles marcroscopiques distincts, avec des differences critiques de la dynamique du modele du champ de phase.
The approach of solutions of nonlinear diffusion equations to travelling front solutions
AbstractThe paper is concerned with the asymptotic behavior as t → ∞ of solutions u(x, t) of the equation ut—uxx—∞;(u)=O, x∈(—∞, ∞) , in the case ∞(0)=∞(1)=0, ∞′(0)<0, ∞′(1)<0. Commonly, a travelling
The Gibbs-Thompson relation within the gradient theory of phase transitions
This paper discusses the asymptotic behavior as ɛ → 0+ of the chemical potentials λɛ associated with solutions of variational problems within the Van der Waals-Cahn-Hilliard theory of phase
Stability of singularly perturbed solutions to systems of reaction-diffusion equations
Stability theorem is presented for large amplitude singularly perturbed solutions (SPS) of reactiondiffusion systems on a finite interval. Spectral analysis shows that there exists a unique real
Local minimisers and singular perturbations
Synopsis We construct local minimisers to certain variational problems. The method is quite general and relies on the theory of Γ-convergence. The approach is demonstrated through the model problem
The Stefan Problem
...
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