The phenomenon of spontaneous isothermal phase separation in a binary alloy is described mathematically by the Cahn-Hilliard equation. It is named after John W. Cahn and John E. Hilliard, who proposed a new way to model the free energy of systems of nonuniform composition in a well-known article from 1958 [1].

The resulting nonlinear partial differential equation (PDE) naturally asks for a rigorous mathematical analysis. A lot of research has been carried out on its well-posedness, properties of solutions and the asymptotic behaviour of the dynamical system. While the Cahn-Hilliard equation on a fixed, stationary domain is quite well-understood with respect to these aspects and from the numerical as well as the theoretical point of view, this has not yet been the case when the PDE is considered on a domain that is allowed to evolve over time.

The manuscript “Cahn–Hilliard equations on an evolving surface” [2] by Diogo Caetano and Charles Elliott (both University of Warwick) which has recently been published in European Journal of Applied Mathematics (EJAM) progresses in this direction as it treats the mathematical properties of the problem in the general case of time-dependent domains. Including this important case, according to the authors, allows a more accurate and realistic modelling of dynamic phenomena such as surface dissolution of binary alloys, cell motility or movements of elastic membranes.

The new article by Caetano and Elliott extends the existing work on Cahn-Hilliard equations in that well-posedness and regularity results for the dynamic case are proven. Under the assumption that the surface is sufficiently smooth and that its deformation is described by a diffeomorphism that preserves the regularity, they carry out their analysis for three different important choices of the homogeneous free energy potential. Relevant models arising from a thermodynamical mean-field model are the logarithmic potential and a typical regular approximation of the latter in the form of a W-shaped quartic polynomial. Further, a singular double-obstacle potential is considered.

For the smooth potential, the authors prove well-posedness for any initial data analogous to the static-domain case by an evolving space Galerkin method. Extra regularity of the weak solution is shown under additional conditions on the surface and its movement. The regular potential, however, does not produce physically meaningful solutions but only possibly approximations under certain assumptions, hence the authors then proceed by proving well-posedness in the case of the two singular potentials. For both the logarithmic and the double-obstacle form, the interplay between the evolving domain and the behaviour of the PDE requires additional initial conditions which ensure that the mean of a solution remains bounded by 1.

Caetano and Elliott further make an effort to unify their approach with other, slightly different derivations of the dynamical system and apply their previous arguments to the latter. They finish their work by remarking how to extend the work from surfaces in space both to curves in two dimensions and to hypersurfaces in higher dimensions.

[1]: Cahn, J. and Hilliard, J.: Free Energy of a Nonuniform System. I. Interfacial Free Energy; The Journal of Chemical Physics, 28, 258 (1958);

[2]: Caetano, D. and Elliott, C.: Cahn–Hilliard equations on an evolving surface; European Journal of Applied Mathematics, 32(5), 937-1000