Higher-order Generalized Cahn–hilliard Equations

Our aim in this paper is to study higher-order (in space) anisotropic generalized Cahn–Hilliard models. In particular, we obtain well-posedness results, as well as the existence of the global attractor. Such models can have applications in biology, image processing, etc. We also give numerical simulations which illustrate the effects of the higher-order terms on the anisotropy.


Introduction
The Cahn-Hilliard equation, plays an essential role in materials science and describes important qualitative features of two-phase systems related with phase separation processes, assuming isotropy and a constant temperature.This can be observed, e.g., when a binary alloy is cooled down sufficiently.One then observes a partial nucleation (i.e., the apparition of nuclides in the material) or a total nucleation, the so-called spinodal decomposition: the material quickly becomes inhomogeneous, forming a fine-grained structure in which each of the two components appears more or less alternatively.In a second stage, which is called coarsening, occurs at a slower time scale and is less understood, these microstructures coarsen.Such phenomena play an essential role in the mechanical properties of the material, e.g., strength.We refer the reader to, e.g., [8,9,16,20,29,30,32,33,38,39] for more details.
Here, u is the order parameter (e.g., a density of atoms) and f is the derivative of a double-well potential F. A thermodynamically relevant potential F is the following logarithmic function which follows from a mean-field model: although such a function is very often approximated by regular ones, typically, i.e., f (s) = s 3 − s. (1.5) Now, it is interesting to note that the Cahn-Hilliard equation and some of its variants are also relevant in other phenomena than phase separation.We can mention, for instance, population dynamics (see [18]), tumor growth (see [4] and [26]), bacterial films (see [27]), thin films (see [41] and [44]), image processing (see [5,6,10,12,19]) and even the rings of Saturn (see [45]) and the clustering of mussels (see [31]).
In particular, several such phenomena can be modeled by the following generalized Cahn-Hilliard equation: We studied in [35] and [36] (see also [4,12,17,21]) this equation.The Cahn-Hilliard equation is based on the so-called Ginzburg-Landau free energy, where Ω is the domain occupied by the system (we assume here that it is a bounded and regular domain of R n , n = 1, 2 or 3, with boundary Γ).In particular, in (1.7), the term |∇u| 2 models short-ranged interactions.It is however interesting to note that such a term is obtained by truncation of higher-order ones (see [9]); it can also be seen as a first-order approximation of a nonlocal term accounting for long-ranged interactions (see [22] and [23]).G. Caginalp and E. Esenturk recently proposed in [7] (see also [11]) higher-order phasefield models in order to account for anisotropic interfaces (see also [28,42,47] for other approaches which, however, do not provide an explicit way to compute the anisotropy).More precisely, these authors proposed the following modified free energy, in which we omit the temperature: where, for α = (k 1 , . . ., and, for α = (0, . . ., 0), (we agree that D (0,...,0) v = v).The corresponding higher-order Cahn-Hilliard equation then reads We studied in [13] and [14] the corresponding isotropic model which reads where The anisotropic model (1.9) is treated in [15].Our aim in this paper is to study the higher-order generalized Cahn-Hilliard model In particular, we study the well-posedness and the regularity of solutions.We also prove the dissipativity of the corresponding solution operators, as well as the existence of the global attractor.We finally give numerical simulations which show the effects of the higher-order terms on the anisotropy.

Setting of the problem
We consider the following initial and boundary value problem, for k ∈ N, k ≥ 2 (the case k = 1 can be treated as in [35]): ) We assume that and we introduce the elliptic operator A k defined by where H −k (Ω) is the topological dual of H k 0 (Ω).Furthermore, ((•, •)) denotes the usual L 2scalar product, with associated norm • .More generally, we denote by • X the norm on the Banach space X; we also set • −1 = (−∆) − 1 2 • , where (−∆) −1 denotes the inverse minus Laplace operator associated with Dirichlet boundary conditions.We can note that is bilinear, symmetric, continuous and coercive, so that is indeed well defined.It then follows from elliptic regularity results for linear elliptic operators of order 2k (see [1][2][3]) that A k is a strictly positive, selfadjoint and unbounded linear operator with compact inverse, with domain where, for v ∈ D(A k ), We further note that D A We finally note that (see, e.g., [43]) Similarly, we can define the linear operator which is a strictly positive, selfadjoint and unbounded linear operator with compact inverse, with domain where, for v ∈ D(A k ), We finally consider the operator Ãk = (−∆) −1 A k , where Ãk : note that, as −∆ and A k commute, then the same holds for (−∆) −1 and A k , so that Ãk = A k (−∆) −1 .
We have the following lemma.
Lemma 2.1.The operator Ãk is a strictly positive, selfadjoint and unbounded linear operator with compact inverse, with domain where, for v ∈ D( Ãk ), Proof.We first note that Ãk clearly is linear and unbounded.Then, since (−∆) −1 and A k commute, it easily follows that Ãk is selfadjoint.
Next, the domain of Ãk is defined by , it follows from the elliptic regularity results of [1], [2] and [3] and recalling that k ≥ 2, we deduce that Ãk has compact inverse.
We now note that, considering the spectral properties of −∆ and A k (see, e.g., [43]) and recalling that these two operators commute, −∆ and A k have a spectral basis formed of common eigenvectors.This yields that, ∀s 1 , s 2 ∈ R, (−∆) s 1 and A s 2 k commute.Having this, we see that Finally, as far as the equivalences of norms are concerned, we can note that, for instance, the norm Ã 1 Having this, we rewrite (2.1) as where As far as the nonlinear term f is concerned, we assume that where In particular, the usual cubic nonlinear term f (s) = s 3 − s satisfies these assumptions.
Furthermore, as far as the function g is concerned, we assume that ) where h ≥ 0 is continuous and satisfies (i) Cahn-Hilliard-Oono equation (see [34], [40] and [46]).In that case, This function was proposed in [40] in order to account for long-ranged (i.e., nonlocal) interactions, but also to simplify numerical simulations.
(ii) Proliferation term.In that case, This function was proposed in [26] in view of biological applications and, more precisely, to model wound healing and tumor growth (in one space dimension) and the clustering of brain tumor cells (in two space dimensions); see also [4] for other quadratic functions.
(iii) Fidelity term.In that case, where χ denotes the indicator function.This function was proposed in [5] and [6] in view of applications to image inpainting.Here, ϕ is a given (damaged) image and D is the inpainting (i.e., damaged) region.Furthermore, the fidelity term g(x, u) is added in order to keep the solution close to the image outside the inpainting region.The idea in this model is to solve the equation up to steady state to obtain an inpainted (i.e., restored) version u(x) of ϕ(x).
Throughout the paper, the same letters c, c and c denote (generally positive) constants which may vary from line to line.Similarly, the same letters Q and Q denote (positive) monotone increasing and continuous (with respect to each argument) functions which may vary from line to line.

A priori estimates
c > 0, t ≥ 0, r > 0 given, where the continuous and monotone increasing function Q is of the form Q(s) = cse c s .
Proof.The estimates below will be formal, but they can easily be justified within, e.g., a standard Galerkin scheme.We multiply (2.6) by (−∆) −1 ∂u ∂t and integrate over Ω and by parts.This gives where k [u] is not necessarily nonnegative).This yields, owing to (2.12) and (2.14), We can note that, owing to the interpolation inequality there holds This yields, employing (2.10), noting that, owing to Young's inequality, We then multiply (2.6) by (−∆) −1 u and have, owing to (2.9), (2.12), (2.13) and the interpolation inequality (3.5), hence, proceeding as above and employing, in particular, (2.10), Summing δ 1 times (3.4) and (3.8),where δ 1 > 0 is small enough, we obtain a differential inequality of the form where satisfies, owing to (3.6), Estimates (3.1)-(3.2) then follow from (3.9)-(3.10)and Gronwall's lemma.Multiplying next (2.6)by Ãk u, we find, owing to (2.12) and the interpolation inequality (3.5), It follows from the continuity of f , F and h, the continuous embedding H k (Ω) ⊂ C(Ω) (recall that k ≥ 2) and (3.1) that Summing (3.9) and (3.13), we have a differential inequality of the form where We now multiply (2.6) by ∂u ∂t and obtain, noting that f is of class C 2 , so that and proceeding as above, where Summing finally (3.14) and (3.16), we find a differential inequality of the form where k [u] satisfies, proceeding as above, In particular, it follows from (3.17)-(3.18) that We then rewrite (2.6) as an elliptic equation, for t > 0 fixed, Multiplying (3.20) by A k u, we have, owing to (2.12) and the interpolation inequality (3.5), hence, proceeding as above (employing, in particular, (3.12)), In a next step, we differentiate (2.6) with respect to time and obtain We multiply (3.23) by (−∆) −1 ∂u ∂t and find, owing to (2.8), (2.15), the interpolation inequality (3.5) and the continuous embedding which yields, employing the interpolation inequality and proceeding as above (note that l is continuous), the differential inequality In particular, this yields, owing to (3.2) and employing the uniform Gronwall's lemma (see, e.g., [43]), ∂u ∂t  (3.3), gives an H 2kestimate on u, for all times.This is however not satisfactory, in particular, in view of the study of attractors.Remark 3.3.We assume that, for simplicity, g(x, s) = g(s) and we further assume that f is of class C k+1 and g is of class C k−1 .Multiplying (2.6) by Ãk ∂u ∂t , we have ) and owing to (3.19), Combining (3.28) with (3.17), it follows from (3.18) and the interpolation inequality (3.5) that so that, owing to (3.3),

The dissipative semigroup
We first give the definition of a weak solution to (2.1)-(2.3).
Definition 4.1.We assume that u 0 ∈ L 2 (Ω).A weak solution to (2.1)-(2.3) is a function u such that, for any given T > 0, in the sense of distributions.
We have the following theorem.
(i) We assume that u 0 ∈ H k 0 (Ω).Then, (2.1)-(2.3)possesses a unique weak solution u such that, Proof.The proofs of existence and regularity in (i), (ii) and (iii) follow from the a priori estimates derived in the previous section and, e.g., a standard Galerkin scheme.Indeed, we can note that, since the operators −∆, A k , A k and Ãk are linear, selfadjoint and strictly positive operators with compact inverse which commute, they have a spectral basis formed of common eigenvectors.We then take this spectral basis as Galerkin basis, so that all the a priori estimates derived in the previous section are justified within the Galerkin scheme.
Here, we have used the fact that, owing to (2.15) and (3.1), It follows from (4.4) and Gronwall's lemma that hence the uniqueness, as well as the continuous dependence with respect to the initial data in the H −1 -norm.
It follows from Theorem 4.2 that we can define the family of solving operators where Φ = H k 0 (Ω).This family of solving operators forms a semigroup which is continuous with respect to the H −1 -topology.Finally, it follows from (3.1) that we have the following theorem.

Remark 4.4.
(i) Actually, it follows from (3.3) that we have a bounded absorbing set B 1 which is compact in Φ and bounded in H 2k (Ω).This yields the existence of the global attractor A which is compact in Φ and bounded in H 2k (Ω).
(ii) We recall that the global attractor A is the smallest (for the inclusion) compact set of the phase space which is invariant by the flow (i.e., S(t)A = A, ∀t ≥ 0) and attracts all bounded sets of initial data as time goes to infinity; it thus appears as a suitable object in view of the study of the asymptotic behavior of the system.We refer the reader to, e.g., [37] and [43] for more details and discussions on this.
(iii) We can also prove, based on standard arguments (see, e.g., [37] and [43]) that A has finite dimension, in the sense of covering dimensions such as the Hausdorff and the fractal dimensions.The finite-dimensionality means, very roughly speaking, that, even though the initial phase space has infinite dimension, the reduced dynamics can be described by a finite number of parameters (we refer the interested reader to, e.g., [37] and [43] for discussions on this subject).
Remark 4.5.In the numerical simulations given in the next section below, the equations will be endowed with periodic boundary conditions.From a mathematical point of view, these boundary conditions are much more delicate to handle, since we have to estimate the spatial average of the order parameter u = 1 Vol(Ω) Ω u dx (see [12], [16] and [21]).When g ≡ 0, this is straightforward, since we have the conservation of mass, namely, However, when g does not vanish, we are not able to estimate this quantity in general.

Numerical simulations
We give in this section several numerical simulations in order to illustrate the effects of the higher-order terms on the anisotropy.The computations presented below are performed with the software FreeFem++ (see [24]), for k = 2.We also take Ω bi-dimensional and rectangular.Finally, the system is associated with periodic boundary conditions.The problem can be written as, for k = 2, where ε > 0 is introduced to take into account the diffuse interface thickness.Setting we have the variational formulation: find (u, w, p, q) ∈ H 1 per (Ω) 4 such that where the test functions v 1 , v 2 , v 3 , v 4 all belong to H 1 per (Ω).The mesh is obtained by dividing Ω into 149 2 rectangles, each rectangle being divided along the same diagonal into two triangles.The computations in Fig. 5.2, 5.3, 5.4 are based on a P 1 finite element method for the space discretization, while we used a P 2 finite element The initial conditions u