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Sharp-InterfaceLimits of theCahn--HilliardEquation withDegenerate Mobility Alpha Albert LeeAndreas MünchEndre Süli2016 год

Sharp-Interface Limits of the Cahn--Hilliard Equation with Degenerate Mobility
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Аннотация: In this work, sharp-interface limits for the degenerate Cahn--Hilliard equation with a polynomial double-well free energy and a mobility that vanishes at the minima of the double well are derived. For the choice of a quadratic mobility, the leading order sharp-interface motion is not governed by pure surface diffusion, as has been previously claimed in the literature, but contains a contribution from nonlinear, porous-medium-type bulk diffusion at the same order. Our analysis reveals that there are two subcases: One, where the solution for the order parameter is bounded between the minima (proven to exist for the first mobility by Elliott and Garcke [SIAM J. Math. Anal., 27 (1996), pp. 404--423]), and one where it converges to the classical stationary solution of the Cahn--Hilliard equation. Consistent treatment of the bulk diffusion requires the matching of exponentially large and small terms in combination with multiple inner layers. Moreover, the leading order sharp-interface motion depends sensitively on the choice of mobility. The asymptotic analysis shows that, for example, with a biquadratic mobility, the leading order sharp-interface motion is driven only by surface diffusion. The sharp-interface models are corroborated by comparing relaxation rates of perturbations to a radially symmetric stationary state with those obtained by the phase field model.
Год издания: 2016
Авторы: Alpha Albert Lee, Andreas Münch, Endre Süli
Издательство: Society for Industrial and Applied Mathematics
Источник: SIAM Journal on Applied Mathematics
Ключевые слова: Solidification and crystal growth phenomena, Fluid Dynamics and Thin Films, nanoparticles nucleation surface interactions
Другие ссылки: SIAM Journal on Applied Mathematics (HTML)
Oxford University Research Archive (ORA) (University of Oxford) (PDF)
Oxford University Research Archive (ORA) (University of Oxford) (HTML)
arXiv (Cornell University) (PDF)
arXiv (Cornell University) (HTML)