Sharp-interface limits of the Cahn-Hilliard equation with degenerate mobility. (English) Zbl 1343.35129
Summary: 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 C. M. Elliott and H. Garcke [SIAM J. Math. Anal. 27, No. 2, 404–423 (1996; Zbl 0856.35071)]), 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.
MSC:
| 35K35 | Initial-boundary value problems for higher-order parabolic equations |
| 35Q35 | PDEs in connection with fluid mechanics |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 74N20 | Dynamics of phase boundaries in solids |
| 76M45 | Asymptotic methods, singular perturbations applied to problems in fluid mechanics |
| 76E17 | Interfacial stability and instability in hydrodynamic stability |
| 82C26 | Dynamic and nonequilibrium phase transitions (general) in statistical mechanics |
Keywords:
Cahn-Hilliard equation; degenerate mobility; sharp-interface limit; surface diffusion; matched asymptotics; singular perturbation methodsCitations:
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