SHARP-INTERFACE LIMITS OF THE CAHN-HILLIARD EQUATION WITH DEGENERATE MOBILITY

摘要

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.