EXISTENCE OF SOLUTIONS FOR A ONE-DIMENSIONAL ALLEN-CAHN EQUATION

Alain Miranville

doi:10.11948/2013019

2013 Volume 3 Issue 3

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Alain Miranville. EXISTENCE OF SOLUTIONS FOR A ONE-DIMENSIONAL ALLEN-CAHN EQUATION[J]. Journal of Applied Analysis & Computation, 2013, 3(3): 265-277. doi: 10.11948/2013019

Citation:

Alain Miranville. EXISTENCE OF SOLUTIONS FOR A ONE-DIMENSIONAL ALLEN-CAHN EQUATION[J]. Journal of Applied Analysis & Computation, 2013, 3(3): 265-277. doi: 10.11948/2013019

EXISTENCE OF SOLUTIONS FOR A ONE-DIMENSIONAL ALLEN-CAHN EQUATION

Alain Miranville

Université de Poitiers, Laboratoire de Mathématiques et Applications, UMR CNRS 7348-SP2MI, 86962 Chasseneuil Futuroscope Cedex, France

Abstract

Abstract

Our aim in this paper is to prove the existence and uniqueness of solutions for a one-dimensional Allen-Cahn type equation based on a modification of the Ginzburg-Landau free energy proposed in[10]. In particular, the free energy contains an additional term called Willmore regularization and takes into account anisotropy effects.
- Allen-Cahn equation /
- Willmore regularization /
- anisotropy effects /
- well-posedness
MSC: 35B45;35K55

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References

[1]	S.M. Allen and J.W. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metall., 27(1979),1085-1095. Google Scholar
[2]	F. Chen and J. Shen, Efficient energy stable schemes with spectral discretization in space for anisotropic Cahn-Hilliard systems, Commun. Comput. Phys., 13(2013), 1189-1208. Google Scholar
[3]	Y. Giga, T. Ohtsuka and R. Schätzle, On a uniform approximation of motion by anisotropic curvature by the Allen-Cahn equations, Interfaces Free Bound., 8(2006), 317-348. Google Scholar
[4]	A. Miranville, Asymptotic behavior of a sixth-order Cahn-Hilliard system, Central Europ. J. Math., to appear. Google Scholar
[5]	A. Miranville and R. Quintanilla, A generalization of the Allen-Cahn equation, submitted. Google Scholar
[6]	A. Novick-Cohen, On the viscous Cahn-Hilliard equation, in Material instabilities in continuum and related problems, J.M. Ball ed., Oxford University Press Oxford, 1988, 329-342. Google Scholar
[7]	K. Promislow and H. Zhang, Critical points of functionalized Lagrangians, Discrete Cont. Dyn. System, 33(2013), 1231-1246. Google Scholar
[8]	J.E. Taylor and J.W. Cahn, Diffuse interfaces with sharp corners and facets:phase-field models with strongly anisotropic surfaces, Phys. D, 112(1998), 381-411. Google Scholar
[9]	R. Temam, Infinite-dimensional dynamical systems in mechanics and physics, Second edition, Applied Mathematical Sciences, Springer-Verlag New York, 68(1997). Google Scholar
[10]	S. Torabi, J. Lowengrub, A. Voigt and S. Wise, A new phase-field model for strongly anisotropic systems, Proc. R. Soc. A, 465(2009), 1337-1359. Google Scholar
[11]	S.M. Wise, C. Wang and J.S. Lowengrub, Solving the regularized, strongly anisotropic Cahn-Hilliard equation by an adaptative nonlinear multigrid method, J. Comput. Phys., 226(2007), 414-446. Google Scholar