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On nonnegative solutions for the Functionalized Cahn-Hilliard equation with degenerate mobility

The Functionalized Cahn-Hilliard equation has been proposed as a model for the interfacial energy of phase-separated mixtures of amphiphilic molecules. We study the existence of a nonnegative weak solutions of a gradient flow of the Functionalized Cahn-Hilliard equation subject to a degenerate mobility M(u) that is zero for u<=0. Assuming the initial data u0(x) is positive, we construct a weak solution as the limit of solutions corresponding to nondegenerate mobilities and verify that it satisfies an energy dissipation inequality.


Introduction
The Functionalized Cahn-Hilliard (FCH) free energy was introduced in [12]. It is an extension of the model of Gompper and Goos [13], proposed to describe the free energy of microemulsions of amphiphilic molecules and solvent. Amphiphilic molecules are formed by chemically bonding two components whose individual interactions with the solvent are energetically favorable and unfavorable, respectively. When blended with the solvent, amphiphilic molecules have a propensity to phase separate, forming amphiphilic rich domains that are thin, generically the thickness of two molecules, in at least one direction. For a binary mixture with composition described by u on Ω ⊂ R d , the FCH free energy takes the form where Ω is a bounded domain in R d with the boundary ∂Ω. The FCH equation, which is the gradient flow of the FCH energy functional, is written as where T > 0 is a given number. This equation is often subject to periodic or zero-flux boundary conditions on ∂Ω. We also prescribe initial values u(x, 0) = u 0 (x) for all x ∈ Ω, where u 0 ∈ H 2 (Ω) is a given function. The function µ is the chemical potential defined by the first variational derivative of the FCH energy functional (1). The diffusion mobility M : R → [0, ∞) is nonnegative and continuous. The double well potential W : R → R is smooth enough and has two unequal depth local minima at 0 and b + > 0 for which W (0) = 0 > W (b + ), and W ′ has exactly three zeroes at 0, b + , and b 0 ∈ (0, b + ). The parameter η > 0 characterizes key structural properties of the amphiphilic molecules. The function u is the order parameter, representing the relative volume fraction of amphiphilic materials, with u ≡ 0 being pure solvent and u ≡ b + being pure amphiphile. Highly amphiphilic lipids have long hydrophobic tails. The energy of low concentrations of highly amphiphilic materials grows exponentially with the tail length [2]. This leads to models in which the concavity of the left well, W ′′ (0), is large [3]. Indeed, arguing formally, a perturbation v of a u ≡ 0 distribution satisfies the linear diffusion equation To prevent spuriously high diffusivity at low concentrations, it is natural to take the mobility of lipids so that the product M (0)(W ′′ (0)) 2 remains bounded. To compensate for the high energy of dispersed amphiphilic molecules requires mobilities that are zero or asymptotically zero. In this paper we fix the low-density energy and establish the existence and energy dissipation of weak solutions with vanishing mobility. In [8] it was proved that for there exists a weak solution for this class of degenerate FCH equation. Here 0 < m < ∞ if the spatial dimension d ≤ 4, and 0 < m The degenerate mobility (6) is an appropriate choice for the degenerate Cahn-Hilliard (CH) equation, which models phase separations in composite materials [4,5,6,7,15]. It has been shown by numerical simulations and by asymptotic analysis that for d ≥ 2 the degenerate mobility (6) does not guarantee the weak solution for the degenerate CH to remain positive, even if the initial value is positive [4,5,6,15]. This is a consequence of the Gibbs-Thomson effect (see also [1,14,16,17] for discussions on the role of degenerate diffusions, and [18] for the one-dimensional case). The situation is different for the degenerate FCH equation, since there is no Gibbs-Thomson effect [9,10]. Instead, a more important feature we need to guarantee is that the relative volume fraction of amphiphilic materials u remains nonnegative. It is the purpose of this paper to show that, with a specific form of degenerate mobility there exists a nonnegative weak solution to the FCH equation (2)-(4) when the initial data u 0 (x) is positive for all x ∈ Ω. Since the FCH equation is a gradient flow for the FCH functional, it is natural to expect the weak solution to satisfy an energy dissipation inequality for F(u). In [8] this energy dissipation property for F was stated informally and without a proof.
In this paper we establish that the weak solutions dissipate the FCH energy.

Main result
In this paper, we assume the dimension d = 1, 2, 3 and the set Ω = (0, 2π) d , and consider the periodic boundary condition on the boundary ∂Ω. We choose the mobility M (u) to be (7), which is degenerate at u = 0. The degeneracy of mobility at u = 0 presents the technical difficulties. We also assume that W ∈ C 4 (R) and there exist positive constants C 1 , C 2 , C 3 , C 4 such that for all z ∈ R, for a constant 1 < p < ∞ if d = 1, 2, and 1 < p One example of such a potential W is with p = 2. Under these assumptions, along with the cut-off degenerate mobility (7), we will prove that the FCH equation (2)-(4) has a nonnegative weak solution that is not zero everywhere in Ω T , assuming that the initial data u 0 (x) is positive in Ω. Our analysis follows the same strategy as in [7,8,11] and involves two steps. The first step is to approximate the degenerate mobility M (u) by a non-degenerate mobility M θ (u) defined for θ ∈ (0, 1) by The positive lower bound of M θ (u) allows us to find a sufficiently regular weak solution to (2)-(4) with the positive mobility M θ (u).
The estimate (16) in part (v) of Theorem 1.1 is essential. It is the key to prove the existence of a nonnegative weak solution to the equation (2)-(4) with the degenerate mobility (7).
The second step is to consider the limit of u θ as θ → 0. The limiting function u of u θ does exist and, in the weak sense, solves the FCH equation (2)-(4) with the mobility M (u) defined by (7). It can be interpreted that u As for the set where u does not have enough regularity, that set is contained in the set where M (u) is degenerate and another set of Lebesgue measure zero. Moreover, if the initial data u 0 (x) is positive in Ω, we obtain a nonnegative weak solution to the equation (2)-(4) that is not constantly zero in Ω T . Theorem 1.2. Let u 0 ∈ H 2 (Ω). With the potential W (u) satisfying (8)- (12) and the mobility M (u) defined by (7), for any given constant T > 0, there exists a function u that satisfies the following conditions: . u can be considered as a weak solution for the FCH equation (2)- (4) in the following weak sense: (a) Let P be the set where M (u) is not degenerate, that is, There exists a set B ⊂ Ω T with |Ω T \B| = 0 and a function ζ : where ω = −∆u + W ′ (u). (c) In addition, for any t ≥ 0, the following energy inequality holds:

Notation
In this paper, we use C to denote a generic positive constant that may depend on d, T, Ω, η, u 0 and C j (j = 1, 2, 3, 4) but nothing else, in particular not on θ. We also use C θ to denote a generic positive constant that may depend on d, T, Ω, η, u 0 , C j (j = 1, 2, 3, 4) and θ.
This paper is organized as follows. In Section 2 we prove Theorem 1.1 using the Galerkin method. In Section 3 we prove Theorem 1.2 that establishes the existence of a weak solution to the equation (2)-(4) with degenerate mobility.

Weak solution for the positive mobility case
In this section we prove Theorem 1.1. The proof for the existence of a weak solution u θ can be found in Section 3 in [8], which is based on Galerkin approximations. We just sketch the idea of that proof, as well as state the convergences and estimates that are necessary for later parts. The main purpose of this section is to prove the energy inequality (15), and the estimate (16) when the initial data u 0 (x) is positive in Ω.

Galerkin approximation and Weak solution
Let {φ j : j = 1, 2, ...} be the normalized eigenfunctions, in the sense that φ j L 2 (Ω) = 1, of the eigenvalue problem −∆u = λu in Ω subject to periodic boundary condition on ∂Ω. The eigenfunctions φ j are orthogonal in the H 2 (Ω) and L 2 (Ω) scalar product. Without loss of generality, we assume that We consider the Galerkin approximation for the equation (2)- (4): where ω N = −∆u N + W ′ (u N ). This gives an initial value problem for a system of ordinary differential equations for c N 1 , ..., c N N : Since the right hand side of (22) depends continuously on c N 1 , ..., c N N , the initial value problem (22)-(24) has a local solution. From the Subsections 2.2 and 3.1 in [8], we have the following estimates for u N and µ N .
The estimates in Lemma 2.1 give the uniform bound for c N 1 , ..., c N N . Therefore a global solution for the initial value problem (22)-(24) exists. With the specific form (13) of the positive mobility M θ (u), we obtain the following bound for ∇µ N .
Combining (48) and (49) we have So we can find a subsequence of {u N } (not relabeled) that satisfies (39)-(50). Then by taking the limit as N → ∞ in (39), we get the energy inequality (15).

Weak solutions for the degenerate mobility case
In this section we prove the main theorem that is Theorem 1.2. We now consider the FCH equation (2)-(4) with the degenerate mobility M (u) defined by (7). The proof for the existence of a weak solution u and a function ζ that satisfy (17) is similar to that in Section 4.4 in [8]. Again, here we just sketch the idea of that proof, as well as state the convergences and estimates that are necessary for later parts. In this section we provide a more detailed proof for the relation between u and ζ, as well as prove the energy in equality (18). Moreover, we prove the existence of a nonnegative solution that is not constantly zero in Ω T when the initial data u 0 is positive in Ω.

The relation between ζ and u
Since u ∈ L ∞ (0, T ; H 2 (Ω)), the function ω := −∆u+W ′ (u) is completely defined, and ω ∈ L ∞ (0, T ; L 2 (Ω)). The desired relation between ζ and u is But the terms ∇ω and ∇∆ω are only defined in the sense of distributions and may not even be functions. So we need some higher regularity conditions on u.
Claim. For any open set U ⊂ Ω such that ∇∆ 2 u ∈ L q (U T ) for some q > 1 that may depend on U , where U T = U × (0, T ), we have By the equation (71) in [8], we have Since ∇∆ 2 u ∈ L q (U T ) and where q > 1 and 0 < α < 1/2, we have for the values of q and α indicated. By (65), there exists a subsequence of {ω k,N k } ∞ k=1 (not relabeled) such that ω k,N k ⇀ ω weakly in L 2 (0, T ; L 2 (U )).
Since |Ω T \B| = 0 and M (u) = 0 in Ω T \P , the value of ζ outside of (B ∩ P ) ∪Ω T does not contribute to the integral on the right hand side of (70), so we may just let ζ = 0 outside of (B ∩ P ) ∪Ω T .

Nonnegative weak solution with positive initial data
Assume, in addition, that the initial data u 0 (x) > 0 for all x ∈ Ω. By (16), there exists a constant C independent on {θ i } ∞ i=1 ⊂ (0, 1) such that for each i = 1, 2, ..., By the convergence in (67) and (68), and since u is continuous in Ω T , passing to the limit as i → ∞ in (81) yields u(x, t) ≥ 0 for all (x, t) ∈ Ω T . Furthermore, since u 0 > 0 in Ω, by the continuity of u in Ω T , u is not constantly zero in Ω T . This completes the proof of Theorem 1.2.