The Cahn–Hilliard Equation: Recent Advances and Applications

The front matter includes the title page, series page, copyright page, in memorium page, TOC, and preface.

The Cahn–Hilliard system

is usually rewritten, equivalently, as the fourth-order-in-space parabolic equation

which is precisely the equation known as the Cahn–Hilliard equation. It was proposed by J.W. Cahn¹ and J.E. Hilliard in 1958 (see [81]). These equations play an essential role in materials science and describe important qualitative features of two-phase systems related to phase separation processes, assuming isotropy and a constant temperature. This can be observed, e.g., when a binary alloy (e.g., aluminum/zinc (see [467]) or iron/chromium (see [364, 365, 366])) is cooled down sufficiently. We then observe a partial nucleation (i.e., the appearance of nuclides in the material) or a total nucleation, known as spinodal decomposition: the initially homogeneous material quickly becomes inhomogeneous, resulting in a very finely dispersed microstructure. In a second stage, which occurs at a slower time scale, these microstructures coarsen (hence the term “coarsening”). See, e.g., YouTube, https://www.youtube.com/watch?v=wWXS52OFo7w, for an animation. Such phenomena play an essential role in the mechanical properties of the material, e.g., strength, hardness, fracture, toughness, and ductility. We refer the reader to, e.g., [79, 81, 328, 333, 357, 358, 415, 417] for more details.

In this chapter, we give preliminary materials (linear operators, compactness results, inequalities, global attractors) which will be useful in the succeeding chapters. In what follows, Ω is a bounded and regular (as regular as needed) domain of ℝⁿ, n = 1, 2, or 3.

We consider the following initial and boundary value problem, in a bounded and regular domain Ω ⊂ ℝⁿ, n = 1, 2, 3, with boundary Γ:

Note that we have set all physical parameters equal to one (we will also do so, at least for most of the parameters, in the following chapters).

4.1 Setting of the problem

We consider the initial and boundary value problem

As far as the nonlinear term f is concerned, we assume that it is logarithmic:

It thus immediately follows that f is odd and

where c₀ = θ_c > 0. Furthermore, we can write

The function f also enjoys the following properties.

We consider the initial and boundary value problem

where ∆_Γ denotes the Laplace–Beltrami operator.

In this chapter we consider the initial and boundary value problem

known as the Cahn–Hilliard–Oono equation (see Chapter 1 for more details).

Image inpainting consists of filling in parts of an image or a video by using information from the surrounding area; it can thus be seen as a type of interpolation. Its applications include restoration of antique paintings by museum artists, removing scratches from old photographs, altering scenes in photographs, and restoring motion pictures.

We consider the equation

proposed in [318] to model cells which move, proliferate, and interact via diffusion and adhesion in wound healing and tumor growth (we can also think, for a two-dimensional extension of the model, of the clustering of malignant brain tumor cells). Here, u is the local density of cells, q is the adhesion parameter, λ is the proliferation rate, and F is the local free energy. Furthermore,

where J corresponds to the interatomic interactions (e.g., the coupling strength in the Ising model); k_B is Boltzmann's constant; and T is the absolute temperature, assumed constant. Finally, F is a double-well potential of the form

where a and b are taken as

q_cr > 0 and q > q_cr (this corresponds to the unstable region; note, however, that, as noted in [318], even in the case q < q_cr, when the proliferation term is switched on, the initially homogeneous state can become inhomogeneous, leading to phase separation and clustering). Furthermore, the positive function c(q) is such that

so that

In this final chapter, we briefly discuss several other important variants and generalizations of the Cahn–Hilliard equation.

The back matter includes bibliography, index, and back cover.