On the viscous Allen-Cahn and Cahn-Hilliard systems with Willmore regularization

Ahmad Makki

Applications of Mathematics (2016)

  • Volume: 61, Issue: 6, page 685-725
  • ISSN: 0862-7940

Abstract

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We consider the viscous Allen-Cahn and Cahn-Hilliard models with an additional term called the nonlinear Willmore regularization. First, we are interested in the well-posedness of these two models. Furthermore, we prove that both models possess a global attractor. In addition, as far as the viscous Allen-Cahn equation is concerned, we construct a robust family of exponential attractors, i.e. attractors which are continuous with respect to the perturbation parameter. Finally, we give some numerical simulations which show the effects of the viscosity term on the anisotropic and isotropic Cahn-Hilliard equation.

How to cite

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Makki, Ahmad. "On the viscous Allen-Cahn and Cahn-Hilliard systems with Willmore regularization." Applications of Mathematics 61.6 (2016): 685-725. <http://eudml.org/doc/287531>.

@article{Makki2016,
abstract = {We consider the viscous Allen-Cahn and Cahn-Hilliard models with an additional term called the nonlinear Willmore regularization. First, we are interested in the well-posedness of these two models. Furthermore, we prove that both models possess a global attractor. In addition, as far as the viscous Allen-Cahn equation is concerned, we construct a robust family of exponential attractors, i.e. attractors which are continuous with respect to the perturbation parameter. Finally, we give some numerical simulations which show the effects of the viscosity term on the anisotropic and isotropic Cahn-Hilliard equation.},
author = {Makki, Ahmad},
journal = {Applications of Mathematics},
keywords = {viscous Cahn-Hilliard equation; viscous Allen-Cahn equation; Willmore regularization; well-posedness of models; global attractor; robust exponential attractors; anisotropy; simulations; viscous Cahn-Hilliard equation; viscous Allen-Cahn equation; Willmore regularization; well-posedness of models; global attractor; robust exponential attractors; anisotropy; simulations},
language = {eng},
number = {6},
pages = {685-725},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On the viscous Allen-Cahn and Cahn-Hilliard systems with Willmore regularization},
url = {http://eudml.org/doc/287531},
volume = {61},
year = {2016},
}

TY - JOUR
AU - Makki, Ahmad
TI - On the viscous Allen-Cahn and Cahn-Hilliard systems with Willmore regularization
JO - Applications of Mathematics
PY - 2016
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 61
IS - 6
SP - 685
EP - 725
AB - We consider the viscous Allen-Cahn and Cahn-Hilliard models with an additional term called the nonlinear Willmore regularization. First, we are interested in the well-posedness of these two models. Furthermore, we prove that both models possess a global attractor. In addition, as far as the viscous Allen-Cahn equation is concerned, we construct a robust family of exponential attractors, i.e. attractors which are continuous with respect to the perturbation parameter. Finally, we give some numerical simulations which show the effects of the viscosity term on the anisotropic and isotropic Cahn-Hilliard equation.
LA - eng
KW - viscous Cahn-Hilliard equation; viscous Allen-Cahn equation; Willmore regularization; well-posedness of models; global attractor; robust exponential attractors; anisotropy; simulations; viscous Cahn-Hilliard equation; viscous Allen-Cahn equation; Willmore regularization; well-posedness of models; global attractor; robust exponential attractors; anisotropy; simulations
UR - http://eudml.org/doc/287531
ER -

References

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