(Non-)Convergence of Solutions of the Convective Allen-Cahn Equation

We consider the sharp interface limit of a convective Allen-Cahn equation, which can be part of a Navier-Stokes/Allen-Cahn system, for different scalings of the mobility $m_\varepsilon=m_0\varepsilon^\theta$ as $\varepsilon\to 0$. In the case $\theta>2$ we show a (non-)convergence result in the sense that the concentrations converge to the solution of a transport equation, but they do not behave like a rescaled optimal profile in normal direction to the interface as in the case $\theta=0$. Moreover, we show that an associated mean curvature functional does not converge the corresponding functional for the sharp interface. Finally, we discuss the convergence in the case $\theta=0,1$ by the method of formally matched asymptotics.


Introduction
In this contribution we consider the so-called sharp interface limit, i.e., the limit ε → 0, of the convective Allen-Cahn equation Here v : Ω × [0, T ) → R d is a given smooth divergence free velocity field with n · v| ∂Ω = 0 and c ε : Ω × [0, T ) → R is an order parameter, which will be close to the "pure states" ±1 for small ε > 0. Here f = F ′ , where F : R → R is a suitable double well potential with globale minima ±1, e.g. F (c) = (1−c 2 ) 2 . c ε can describe the concentration difference of two different phases in the case of phase transitions, where the total mass of each phase is not necessarily conserved. Moreover, Ω ⊆ R d is assumed to be a bounded domain with smooth boundary, m ε is a (constant) mobility coefficient and ε > 0 is a parameter that is proportional to the "thickness" of the diffuse interface {x ∈ Ω : |c ε (x, t)| < 1 − δ} for δ ∈ (0, 1). The convective Allen-Cahn equation (1) is part of the following diffuse interface model for the two-phase flow of two incompressible Newtonian, partly miscible fluids with phase transition div v ε = 0, in Ω × (0, T ), where v ε : Ω × [0, T ) → R d is the velocity of the mixture, Dv ε = 1 2 (∇v ε + (∇v ε ) T ), p ε : Ω × [0, T ) → R is the pressure, and ν(c ε ) > 0 is the viscosity of the mixture. This model can be considered as a model for a two-phase flow with phase transition or an approximation of a classical sharp interface model for a two-phase flow of incompressible fluids with surface tension. Here the densities of the two separate fluids are assumed to be the same. A derivation of this model in a more general form with variable densities can be found in Jiang et al. [10]. We refer to Gal and Grasselli [6] for the existence of weak solutions and results on the longtime behavior of solutions for this model and to Giorgini et al. [7] for analytic results for a volume preserving variant with different densities. Mathematically, this system arises if one replaces the Cahn-Hilliard equation in the well-known "model H", cf. e.g. [1,8], by an Allen-Cahn equation.
With the aid of formally matched asymptotic expansions one can formally show that solutions of this system converge to solutions of the following free boundary value problem div when m ε = m 0 > 0 and [ when m ε = m 0 ε, m 0 > 0. We will discuss this formal result in the appendix in more detail, cf. Remark 2 below. Here ν ± > 0 are viscosity constants, Ω ± (t) ⊂ Ω are open and disjoint such that ∂Ω − (t) = Γ t = ∂Ω + (t) ∩ Ω, n Γt denotes the outer normal of ∂Ω − (t) and the normal velocity and the mean curvature of Γ t are denoted by V Γt and H Γt , respectively, taken with respect to n Γt . Furthermore, [ . ] Γt denotes the jump of a quantity across the interface in the direction of n Γt , i.e., In the case ν + = ν − and that the Navier-Stokes equation is replaced by a (quasi-stationary) Stokes system Liu and the author proved rigorously in [4] that the convergence holds true in the first case m ε = m 0 > 0 for sufficiently small times and for well-prepared initial data. More precisely, it was shown that in a neighborhood of Γ t (even with O(ε 2 )), where d Γt is the signed distance function to Γ t and h ε are correction terms, which are uniformly bounded in ε ∈ (0, 1), and θ 0 : R → R is the so-called optimal profile that is determined by This form is important in order to obtain in the limit ε → 0 the Young-Laplace law (15), cf. e.g. [2,Section 4]. It is the goal of the present contribution to show that in the case m ε = m 0 ε θ with θ > 2 the solutions of the convective Allen-Cahn equation (1)-(2) do not have the form (17) in general. Moreover, we will show that the functional does not converge to the mean curvature functional We note that H ε is the weak formulation of the right-hand side of (4), which should converge to a weak formulation of the right-hand side of (15). Therefore there is no hope that solutions of the full system (4)-(6) converge to solutions of the corresponding limit system with (15) as ε → 0 in the case that m ε = m 0 ε θ , θ > 2. We note that this effect was first observed for the corresponding Navier-Stokes/Cahn-Hilliard system by Schaubeck and the author in [5] in the case θ > 3. These results are also contained in the PhDthesis of Schaubeck [11]. It is not difficult to show that H ε , ϕ converges to (19) if (17) holds true in a sufficiently strong sense. Moreover, in the case θ > 3 non-convergence of the Navier-Stokes/Cahn-Hilliard system in the case of radial symmetry and an inflow boundary condition was shown by Lengeler and the author in [3,Section 4]. We note that the latter counter example can be adapted to the present case of a Navier-Stokes/Allen-Cahn equation in the case θ > 2.
The structure of this contribution is as follows: In Section 2 we summarize some preliminaries and notation. Afterwards we prove the nonconvergence result in Section 3. Finally, in Section 4 we discuss briefly the sharp interface limit of the convective Allen-Cahn equation in the case m ε = m 0 ε θ with θ = 0, 1.

Preliminaries and Notation
We denote We assume that Ω ⊂ R d is a bounded domain with smooth boundary ∂Ω. Furthermore, we define Ω T = Ω × (0, T ) and ∂ T Ω = ∂Ω × (0, T ) for T > 0. Moreover, n ∂Ω denotes the exterior unit normal on ∂Ω. For a hypersurface Γ t ⊂ Ω, t ∈ [0, T ], without boundary such that Γ t = ∂Ω − (t) for a domain Ω − (t) ⊂⊂ Ω, the interior domain is denoted by Ω − (t) and the exterior domain by Ω + (t) := Ω\(Ω − (t)∪ Γ t ), i.e., Γ t separates Ω into an interior and an exterior domain. n Γt is the exterior unit normal on ∂Ω − (t) = Γ t . The mean curvature of Γ t with respect to n Γt is denoted by H Γt . In the following d Γt is the signed distance function to Γ t chosen such that d Γt < 0 in Ω − (t) and d Γt > 0 in Ω + (t). By this convention we obtain ∇d Γt = n Γt on Γ t . Moreover, we define The "double-well" potential F : R → R is a smooth function taking its global minimum 0 at ±1. For its derivative f (c) = F ′ (c) we assume for all u ∈ (−1, 1). In equation (1) the given velocity field satisfies v ∈ C 0 b ([0, T ]; C 4 b (Ω)) d with div v = 0 and v · n ∂Ω = 0 on ∂Ω and the mobility constant m ε has the form m ε = m 0 ε θ for some θ ≥ 0 and m 0 > 0. In equation (3) we choose the special initial value where we determine the constant δ > 0 later. Here ζ ∈ C ∞ 0 (R) is a cut-off function such that and θ 0 is the unique solution to (18). This choice of the initial value is natural in view of (17).

Nonconvergence Result
Our main result is: where the evolving hypersurface Γ t , t ∈ [0, T ], is the solution of the evolution equation where V Γt is the normal velocity of Γ t . Moreover, it holds To prove the theorem we follow the same strategy as in [5]: First we construct a family of approximate solutions {c ε A } 0<ε≤1 . Afterwards we estimate the difference ∇(c ε − c ε A ), which will enable us to prove the assertion of the theorem. We start with the observation that Γ t := X t (Γ 0 ) is the solution to the evolution equation.
We will need the following result: ) the following properties hold: We refer to [5,Lemma 4] for the proof. As mentioned in Section 2, let θ 0 be the solution to (18) and let ζ be a cut-off function as in (22). Then we define Then we have c ε A (., 0) = c ε (., 0) since e(., 0) = d Γ 0 and Furthermore, we define the approximate mean curvature functional by for all ϕ ∈ D(Ω) d . Then we have: Lemma 3. Let c ε A be defined as above. Then there exists some constant C > 0 independent of ε and ε 0 ∈ (0, 1] such that the estimates hold for all t ∈ [0, T ] and ε ∈ (0, ε 0 ).
We refer to [5,Lemma 5] for the proof. Now we are able to prove the central lemma for the proof of Theorem 1.

Lemma 4.
Let c A ε be defined as above and let c ε be the unique solution to (1)-(2) with initial condition (21). Then, for θ ≥ 2, there exists some constant C > 0 independent of ε and ε 0 > 0 such that Proof. First of all, we note that c ε (x, t), c ε A (x, t) ∈ [−1, 1] for all x ∈ Ω, t ∈ (0, T ). For c ε A this follows from the construction and for c ε by the maximum principle.
We denote by u = c ε − c ε A the difference between exact and approximate solution, which solves We multiply the difference of the differential equations for c ε and c ε A by u and integrate the resulting equation over Ω. Then we get for all t ∈ (0, T ) where we have used u = 0 on ∂Ω as well as div v = 0 in Ω. By Hölder's and Young's inequalities we obtain since θ ≥ 2 for some C 1 > 0 independent of ε and t ∈ [0, T ]. Hence the Gronwall inequality implies sup 0≤t≤T u 2 L 2 (Ω) + ε θ ∇u 2 L 2 ((0,T )×Ω) ≤ Cε 2θ−3 for some C = C(T ) > 0 independent of ε. Therefore the lemma is proved.
Now we can show that H ε − H ε A converges to 0 as ε goes to zero. Lemma 5. Let H ε and H ε A be defined as above and let θ > 2. Then it holds Proof. The proof is almost the same as in [5,Lemma 6]. But we include it for the convenience of the reader since the argument is central for our main

Because of Lemma 3 and Lemma 4, we have
Using Lemma 4 we conclude for some constant C = C(ϕ) > 0 and for all ε small enough. Since θ > 2, the assertion follows.
We refer to [5,Lemma 8] for the proof.
Proof of Theorem 1: The first assertion of the theorem immediately follows by Lemma 5 and 6. The second assertion is a consequence of Lemma 3 and Lemma 4 since θ > 3.

Formal Asymptotics
In this section we will use the method of formally matched asymptotic expansions to identify the sharp interface limit of the convective Allen-Cahn equation (1)-(2) in the cases m ε = m 0 ε θ for θ = 0, 1 and some m 0 > 0. We follow similar arguments as in [2,Section 4]. In particular we assume that there are smoothly evolving hypersurfaces Γ t , t ∈ (0, T ), such that Γ t = ∂Ω − (t), and we have the following expansions: Outer expansion: "Away from Γ t " we assume that c ε has an expansion of the form: for every x ∈ Ω ± (t).
Inner expansion: In a neighborhood Γ t (δ), δ > 0, of Γ t c ε has an expansion of the form: Matching condition: Moreover, all functions in the expansions above are assumed to be sufficiently smooth.
In the following we will use the expansions above and the matching conditions, insert them into the convective Allen-Cahn equation (1) and equate all terms of same order in order to determine the leading parts in the inner and outer expansions formally.
If k = 1, the corresponding term is of order O(1) and we do not use this information. Moreover, we will not determine c ± 1 and c 1 in this case.

Inner Expansion
In Γ t (δ) we use the inner expansion in (1) in order to determine the leading coefficients c 0 (ρ, s, t) and, in the case k = 0, c 1 (ρ, s, t), where s := s(x) :=