1. N. Alikakos, P. Bates, and X. Chen, Convergence of the Cahn-Hilliard equation to the Hele-Shaw model, Arch. Rat. Mech. Anal. 128,165–205 (1994). [The object of study is (3.17) in the scaled form ut + A[e/u — e 1 f (u)] = 0 in a bounded domain. Assume the Mullins-Sekerka problem (3.22) has a classical solution in [0, T].It divides the domain into two subdomains. Then with appropriate initial and boundary conditions, there is a family of solutions (depending on e) converging as e-+0 to +1 in the two subdomains. From Math. Reviews 97b:35174. In this paper, the authors carry out a program of rigorous justification of the connection between the Cahn-Hilliard equation and the Hele-Shaw problem in an appropriate asymptotic limit which corresponds to a sharpening of the transition layer between phase… The Cahn-Hilliard equation, a fourth-order nonlinear partial differential equation, serves as a well-regarded model for the process of phase separation and coarsening in a melted alloy. It has been the subject of much work over the past ten to fifteen years. In particular, formal asymptotic analysis [106] established the connection between the level sets of the solution to the Cahn-Hilliard equation and interfacial motion in the Hele-Shaw (or Mullins-Sekerka) problem. For the time interval in which the Hele-Shaw problem possesses a classical solution, the present paper now makes this formal connection rigorous. To set this result in context, one can compare it to the large body of work devoted to an asymptotic analysis of the (second-order) Allen-Cahn equation—which tends to dissipate the same energy as the Cahn-Hilliard equation but which, unlike the Cahn-Hilliard equation, does not respect conservation of mass. The connection between the level sets of solutions to the Allen-Cahn equation and the problem of motion by mean curvature was made rigorous by Evans, Soner and Souganidis [49], using a weak formulation of the curvature flow via viscosity theory. At roughly the same time, de Mottoni and Schatzman [91] used a detailed spectral analysis to justify the formal asymptotics for the Allen-Cahn equation for as long as the mean curvature flow retained a classical solution. The paper under review is in spirit the Cahn-Hilliard cousin of this last reference. (We should note that more recently Cheri [39] developed a weak formulation of the Hele-Shaw problem using varifolds and carried out the asymptotic connection in this setting as well.) The procedure combines a delicate and nonstandard application of the method of matched asympotic expansions with detailed spectral analysis for the Cahn-Hilliard equation, previously worked out in [4, 37].]
2. N. Alikakos, P. Bates, and G. Fusco, Slow motion for the Cahn-Hilliard equation in one space dimension, J. Diff. Equations 90,81–135 (1991). [See Section 3.4.1.]
3. N. Alikakos, L. Bronsard, and G. Fusco, Slow motion in the gradient theory of phase transitions via energy and spectrum, Calculus of Variations 6, 39–66 (1998).
4. N. Alikakos, G. Fusco, The spectrum of the Cahn-Hilliard operator for generic interface in higher space dimensions, Indiana Mathematics Journal 42,637674 (1993). [See Section 3.4.1.]
5. N. Alikakos, G. Fusco, Slow dynamics for the Cahn-Hilliard equation in higher space dimensions. Part I: Spectral estimates, Comm. in P.D.E. 19, 1397–1447 (1994).