(Non-)Convergence of Solutions of the Convective Allen-Cahn Equation
aa r X i v : . [ m a t h . A P ] F e b (Non-)Convergence of Solutions of the ConvectiveAllen-Cahn Equation Helmut Abels ∗ February 22, 2021
Dedicated to Prof. Hideo Kozono on the occasion of his 60th birthday.
Abstract
We consider the sharp interface limit of a convective Allen-Cahnequation, which can be part of a Navier-Stokes/Allen-Cahn system,for different scalings of the mobility m ε = m ε θ as ε →
0. In thecase θ > θ = 0. Moreover, we show that an asso-ciated mean curvature functional does not converge the correspondingfunctional for the sharp interface. Finally, we discuss the convergencein the case θ = 0 , Key words:
Two-phase flow, diffuse interface model, Allen-Cahnequation, sharp interface limit
AMS-Classification:
In this contribution we consider the so-called sharp interface limit, i.e., thelimit ε →
0, of the convective Allen-Cahn equation ∂ t c ε + v · ∇ c ε = m ε (cid:0) ∆ c ε − ε − f ( c ε ) (cid:1) in Ω × (0 , T ) , (1) c ε | ∂ Ω = − ∂ Ω × (0 , T ) , (2) c ε | t =0 = c ε in Ω . (3)Here v : Ω × [0 , T ) → R d is a given smooth divergence free velocity fieldwith n · v | ∂ Ω = 0 and c ε : Ω × [0 , T ) → R is an order parameter, whichwill be close to the “pure states” ± ε >
0. Here f = F ′ , where ∗ Faculty of Mathematics, University of Regensburg, 93040 Regensburg, Germany, e-mail: [email protected] : R → R is a suitable double well potential with globale minima ±
1, e.g. F ( c ) = (1 − c ) . c ε can describe the concentration difference of two differentphases in the case of phase transitions, where the total mass of each phaseis not necessarily conserved. Moreover, Ω ⊆ R d is assumed to be a boundeddomain with smooth boundary, m ε is a (constant) mobility coefficient and ε > { x ∈ Ω : | c ε ( x, t ) | < − δ } for δ ∈ (0 , ∂ t v ε + v ε · ∇ v ε − div( ν ( c ε ) D v ε ) + ∇ p ε = − ε div( ∇ c ε ⊗ ∇ c ε ) , (4)div v ε = 0 , (5) ∂ t c ε + v ε · ∇ c ε = m ε (cid:0) ∆ c ε − ε − f ( c ε ) (cid:1) (6)in Ω × (0 , T ), where v ε : Ω × [0 , T ) → R d is the velocity of the mixture, D v ε = ( ∇ v ε + ( ∇ v ε ) T ), p ε : Ω × [0 , T ) → R is the pressure, and ν ( c ε ) > ∂ t v + v · ∇ v − div( ν ± D v ) + ∇ p = 0 in Ω ± ( t ) , t ∈ (0 , T ) , (7)div v = 0 in Ω ± ( t ) , t ∈ (0 , T ) , (8)[ v ] Γ t = 0 on Γ t , t ∈ (0 , T ) , (9) − (cid:2) n Γ t · ( ν ± D v − p Id) (cid:3) Γ t = σH Γ t n Γ t on Γ t , t ∈ (0 , T ) , (10) V Γ t − n Γ t · v = m H Γ t on Γ t , t ∈ (0 , T ) , (11)2hen m ε = m > ∂ t v + v · ∇ v − div( ν ± D v ) + ∇ p = 0 in Ω ± ( t ) , t ∈ (0 , T ) , (12)div v = 0 in Ω ± ( t ) , t ∈ (0 , T ) , (13)[ v ] Γ t = 0 on Γ t , t ∈ (0 , T ) , (14) − (cid:2) n Γ t · ( ν ± D v − p Id) (cid:3) Γ t = σH Γ t n Γ t on Γ t , t ∈ (0 , T ) , (15) V Γ t − n Γ t · v = 0 on Γ t , t ∈ (0 , T ) , (16)when m ε = m ε , m >
0. We will discuss this formal result in the appendixin more detail, cf. Remark 2 below. Here ν ± > ± ( t ) ⊂ Ω are open and disjoint such that ∂ Ω − ( t ) = Γ t = ∂ Ω + ( t ) ∩ Ω, n Γ t denotes the outer normal of ∂ Ω − ( t ) and the normal velocity and themean curvature of Γ t are denoted by V Γ t and H Γ t , respectively, taken withrespect to n Γ t . Furthermore, [ . ] Γ t denotes the jump of a quantity acrossthe interface in the direction of n Γ t , i.e., [ f ] Γ t ( x ) = lim h → ( f ( x + h n Γ t ) − f ( x − h n Γ t )) for x ∈ Γ t .In the case ν + = ν − and that the Navier-Stokes equation is replacedby a (quasi-stationary) Stokes system Liu and the author proved rigorouslyin [4] that the convergence holds true in the first case m ε = m > t c ε ( x, t ) = θ (cid:18) d Γ t ( x ) − εh ε ( x, t ) ε (cid:19) + O ( ε ) (17)(even with O ( ε )), where d Γ t is the signed distance function to Γ t and h ε arecorrection terms, which are uniformly bounded in ε ∈ (0 , θ : R → R is the so-called optimal profile that is determined by − θ ′′ + f ( θ ) = 0 in R , θ (0) = 0 , lim z →±∞ θ ( z ) = ± . (18)This form is important in order to obtain in the limit ε → m ε = m ε θ with θ > h H ε , ϕ i := ε Z Ω ∇ c ε ⊗ ∇ c ε : ∇ ϕ dx does not converge to the mean curvature functional2 σ Z Γ t n Γ t ⊗ n Γ t : ∇ ϕ d H d − = − σ Z Γ t H Γ t n Γ t · ϕ d H d − (19)3or all ϕ ∈ C ∞ ,σ (Ω) = (cid:8) f ∈ C ∞ (Ω) d : div f = 0 (cid:9) , where σ = 12 Z R (cid:0) θ ′ ( z ) (cid:1) dz. 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) convergeto solutions of the corresponding limit system with (15) as ε → m ε = m ε θ , θ >
2. We note that this effect was first observed for thecorresponding Navier-Stokes/Cahn-Hilliard system by Schaubeck and theauthor in [5] in the case θ >
3. These results are also contained in the PhD-thesis of Schaubeck [11]. It is not difficult to show that h H ε , ϕ i convergesto (19) if (17) holds true in a sufficiently strong sense. Moreover, in thecase θ > θ > m ε = m ε θ with θ = 0 , We denote a ⊗ b = ( a i b j ) di,j =1 for a, b ∈ R d and A : B = P di,j =1 A ij B ij for A, B ∈ R d × d . We assume that Ω ⊂ R d is a bounded domain withsmooth 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 thatΓ t = ∂ Ω − ( t ) for a domain Ω − ( t ) ⊂⊂ Ω, the interior domain is denoted byΩ − ( t ) and the exterior domain by Ω + ( t ) := Ω \ (Ω − ( t ) ∪ Γ t ), i.e., Γ t separatesΩ into an interior and an exterior domain. n Γ t is the exterior unit normalon ∂ Ω − ( t ) = Γ t . The mean curvature of Γ t with respect to n Γ t is denotedby H Γ t . In the following d Γ t is the signed distance function to Γ t chosensuch that d Γ t < − ( t ) and d Γ t > + ( t ). By this convention weobtain ∇ d Γ t = n Γ t on Γ t . Moreover, we define Q ± := { ( x, t ) ∈ Ω T : d ( x, t ) ≷ } . F : R → R is a smooth function taking itsglobal minimum 0 at ±
1. For its derivative f ( c ) = F ′ ( c ) we assume f ( ±
1) = 0 , f ′ ( ± > , Z u − f ( s ) ds = Z u f ( s ) ds > u ∈ ( − , v ∈ C b ([0 , T ]; C b (Ω)) d with div v = 0 and v · n ∂ Ω = 0 on ∂ Ω and the mobilityconstant m ε has the form m ε = m ε θ for some θ ≥ m >
0. Inequation (3) we choose the special initial value c ε | t =0 = ζ (cid:16) d Γ0 δ (cid:17) θ (cid:16) d Γ0 ε (cid:17) + (cid:16) − ζ (cid:16) d Γ0 δ (cid:17)(cid:17) (cid:16) χ { d Γ0 ≥ } − (cid:17) in Ω , (21)where we determine the constant δ > ζ ∈ C ∞ ( R ) is a cut-offfunction such that ζ ( z ) = 1 if | z | < , ζ ( z ) = 0 if | z | > , zζ ′ ( z ) ≤ R , (22)and θ is the unique solution to (18). This choice of the initial value isnatural in view of (17). Our main result is:
Theorem 1.
Let Ω ⊂ R d be a bounded domain with smooth boundary ∂ Ω , Γ a smooth hypersurface such that Γ = ∂ Ω − for a domain Ω − ⊂⊂ Ω and let c ε be the solution to the convective Allen-Cahn equation (1)-(2) with initialcondition (21). Then for every T > and for all ϕ ∈ C ∞ ([0 , T ]; D (Ω) d ) wehave Z T h H ε , ϕ i dt → ε → σ Z T Z Γ t (cid:12)(cid:12) ∇ ( d Γ ( X − t )) (cid:12)(cid:12) n Γ t ⊗ n Γ t : ∇ ϕ d H d − dt , where the evolving hypersurface Γ t , t ∈ [0 , T ] , is the solution of the evolutionequation V Γ t ( x ) = n Γ t ( x, t ) · v ( x, t ) for x ∈ Γ t , t ∈ (0 , T ] , Γ(0) = Γ , where V Γ t is the normal velocity of Γ t . Moreover, it holds (cid:13)(cid:13) c ε − (2 χ Q + − (cid:13)(cid:13) L (Ω T ) = O ( ε ) as ε → . Remark 1.
In general (cid:12)(cid:12) ∇ ( d Γ ( X − t )) (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) DX − Tt ∇ d Γ ◦ X − t (cid:12)(cid:12)(cid:12) = 1 , we re-fer to [5, Remark 1] for a proof. This shows that the weak formulation of H ε does not converge to the weak formulation of the right-hand side of theYoung-Laplace law (15) in general.
5o prove the theorem we follow the same strategy as in [5]: First we con-struct a family of approximate solutions { c εA } <ε ≤ . Afterwards we estimatethe difference ∇ ( c ε − c εA ), which will enable us to prove the assertion of thetheorem. We start with the observation that Γ t := X t (Γ ) is the solution tothe evolution equation. Lemma 1.
Let Γ ⊂ Ω be a given smooth hypersurface such that Γ = ∂ Ω − for a domain Ω − ⊂⊂ Ω . Then the evolving hypersurface Γ t := X t (Γ ) ⊂ Ω , t ∈ [0 , T ] , is the solution to the problem V Γ t = n Γ t · v on Γ t , t ∈ (0 , T ) , Γ(0) = Γ . We refer to [5, Lemma 3] for the proof.For the following let P Γ t ( x ) be the orthogonal projection of x onto Γ t .Then there exists a constant δ > t ( δ ) := { x ∈ Ω : | d Γ t ( x )) | < δ } ⊂ Ω and τ t : Γ t ( δ ) → ( − δ, δ ) × Γ t defined by τ t ( x ) = ( d Γ t ( x ) , P Γ t ( x )) is a smoothdiffeomorphism, cf. e.g. [9, Kapitel 4.6].We will need the following result: Lemma 2.
For e : S t ∈ [0 ,T ] X t (Γ ( δ )) ×{ t } → R defined by e ( x, t ) := d Γ ( X − t ( x )) the following properties hold:1. ddt e ( x, t ) = − v ( x, t ) · ∇ e ( x, t ) for all ( x, t ) ∈ S t ∈ [0 ,T ] X t (Γ ( δ )) × { t } .2. e ( x, t ) is a level set function for Γ t , i.e., e ( x, t ) = 0 if and only if x ∈ Γ t . We refer to [5, Lemma 4] for the proof.As mentioned in Section 2, let θ be the solution to (18) and let ζ be acut-off function as in (22). Then we define c εA ( x, t ) := ± Q ± ∩ S t ∈ [0 ,T ] X t (Ω \ Γ ( δ )) × { t } ,ζ (cid:0) eδ (cid:1) θ (cid:0) eε (cid:1) ± (1 − ζ (cid:0) eδ (cid:1) ) in Q ± ∩ S t ∈ [0 ,T ] X t (Γ ( δ ) \ Γ (cid:0) δ (cid:1) ) × { t } ,θ (cid:0) eε (cid:1) in S t ∈ [0 ,T ] X t (Γ (cid:0) δ (cid:1) ) × { t } . Then we have c εA ( .,
0) = c ε ( .,
0) since e ( .,
0) = d Γ and ∂ t c εA + v · ∇ c εA = 0 in Ω T since ∂ t e + v · ∇ e = 0. Moreover, by the construction c εA | ∂ Ω = 0 on ∂ Ω . Furthermore, we define the approximate mean curvature functional by h H εA , ϕ i = ε Z Ω ∇ c εA ⊗ ∇ c εA : ∇ ϕ dx . for all ϕ ∈ D (Ω) d . Then we have: 6 emma 3. Let c εA be defined as above. Then there exists some constant C > independent of ε and ε ∈ (0 , such that the estimates k ∆ c εA ( ., t ) k L (Ω) ≤ Cε − , (23) k∇ c εA ( ., t ) k L (Ω) ≤ Cε − , (24) k f ( c εA ( ., t )) k L (Ω) ≤ Cε , (25) (cid:13)(cid:13) c εA ( ., t ) − (2 χ Q + ( ., t ) − (cid:13)(cid:13) L (Ω) ≤ Cε (26) hold for all t ∈ [0 , T ] and ε ∈ (0 , ε ) . 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 Aε be defined as above and let c ε be the unique solutionto (1) - (2) with initial condition (21) . Then, for θ ≥ , there exists someconstant C > independent of ε and ε > such that ε k∇ ( c ε − c εA ) k L (Ω T ) ≤ Cε θ − and (27) k c ε − c εA k L ∞ (0 ,T ; L (Ω)) ≤ Cε θ − (28) for all ε ∈ (0 , ε ] .Proof. First of all, we note that c ε ( x, t ) , c εA ( x, t ) ∈ [ − ,
1] for all x ∈ Ω, t ∈ (0 , T ). For c εA this follows from the construction and for c ε by themaximum principle.We denote by u = c ε − c εA the difference between exact and approximatesolution, which solves ∂ t c εA + v · ∇ c εA = 0 in Ω T . We multiply the difference of the differential equations for c ε and c εA by u and integrate the resulting equation over Ω. Then we get for all t ∈ (0 , T )0 = Z Ω u h ∂ t u + v · ∇ u − m ε θ ∆ u − m ε θ ∆ c εA + m ε θ − f ( c ε ) u i dx = Z Ω (cid:16) ∂ t | u | − v · ∇ | u | + m ε θ |∇ u | (cid:17) dx + Z Ω (cid:16) m ε θ ∇ u · ∇ c εA − m ε θ − f ( c ε ) u (cid:17) dx = 12 ddt Z Ω | u | dx + m ε θ Z Ω |∇ u | dx + Z Ω (cid:16) m ε θ ∇ u · ∇ c εA + m ε θ − uf ( c ε ) (cid:17) dx , u = 0 on ∂ Ω as well as div v = 0 in Ω. By H¨older’s andYoung’s inequalities we obtain12 ddt Z Ω | u | dx + m ε θ Z Ω |∇ u | dx ≤ Cε θ k∇ c εA k L (Ω) + ε θ − (cid:12)(cid:12)(cid:12)(cid:12)Z Ω f ( c ε ) u dx (cid:12)(cid:12)(cid:12)(cid:12) (29)for all ε ∈ (0 , ε ), where (cid:12)(cid:12)(cid:12)(cid:12)Z Ω f ( c ε ) u dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)Z Ω f ( c εA ) u dx (cid:12)(cid:12)(cid:12)(cid:12) + C k u k L (Ω) ≤ k f ( c εA ) k L (Ω) k u k L (Ω) + C k u k L (Ω) ≤ Cε k u k L (Ω) + C k u k L (Ω) (30)since f ′ is Lipschitz continuous on [ − , ddt Z Ω | u | dx + m ε θ Z Ω |∇ u | dx ≤ C (cid:16) k u k L (Ω) + ε θ − + ε θ − k u k L (Ω) (cid:17) ≤ C (cid:16) k u k L (Ω) + ε θ − (cid:17) since θ ≥ C > ε and t ∈ [0 , T ]. Hence theGronwall inequality impliessup ≤ t ≤ T k u k L (Ω) + ε θ k∇ u k L ((0 ,T ) × Ω) ≤ Cε θ − for some C = C ( T ) > ε . Therefore the lemma is proved. (cid:3) Now we can show that H ε − H εA converges to 0 as ε goes to zero. Lemma 5.
Let H ε and H εA be defined as above and let θ > . Then it holds (cid:12)(cid:12)(cid:12)(cid:12)Z T h H ε − H εA , ϕ i dt (cid:12)(cid:12)(cid:12)(cid:12) → ε → , for all ϕ ∈ C ∞ ([0 , T ]; D (Ω) d ) .Proof. The proof is almost the same as in [5, Lemma 6]. But we include itfor the convenience of the reader since the argument is central for our mainresult. Let ϕ ∈ C ∞ ([0 , T ]; D (Ω) d ) and set u = c ε − c εA . Then ε (cid:12)(cid:12)(cid:12)(cid:12)Z Ω T ( ∇ c ε ⊗ ∇ c ε − ∇ c εA ⊗ ∇ c εA ) : ∇ ϕ dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε (cid:12)(cid:12)(cid:12)(cid:12)Z Ω T ( ∇ c ε ⊗ ∇ u ) : ∇ ϕ dx (cid:12)(cid:12)(cid:12)(cid:12) + ε (cid:12)(cid:12)(cid:12)(cid:12)Z Ω T ( ∇ u ⊗ ∇ c εA ) : ∇ ϕ dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε k∇ ϕ k L ∞ (Ω T ) k∇ u k L (Ω T ) (cid:16) k∇ c ε k L (Ω T ) + k∇ c εA k L (Ω T ) (cid:17) . k∇ c ε k L (Ω T ) ≤ k∇ c εA k L (Ω T ) + k∇ u k L (Ω T ) ≤ C (cid:16) ε − + ε θ − (cid:17) . Using Lemma 4 we conclude (cid:12)(cid:12)(cid:12)(cid:12)Z T h H ε − H εA , ϕ i dt (cid:12)(cid:12)(cid:12)(cid:12) ≤ Cε θ − (cid:16) ε θ − (cid:17) for some constant C = C ( ϕ ) > ε small enough. Since θ > (cid:3) Lemma 6.
Let H εA and c εA be defined as above. Then it holds for all ϕ ∈D (Ω) d and t ∈ [0 , T ] h H εA , ϕ i → ε → σ Z Γ t (cid:12)(cid:12) ∇ ( d Γ ( X − t )) (cid:12)(cid:12) n Γ t ⊗ n Γ t : ∇ ϕ d H d − . We refer to [5, Lemma 8] for the proof.
Proof of Theorem 1:
The first assertion of the theorem immediatelyfollows by Lemma 5 and 6. The second assertion is a consequence of Lemma3 and Lemma 4 since θ > (cid:3) In this section we will use the method of formally matched asymptotic ex-pansions to identify the sharp interface limit of the convective Allen-Cahnequation (1)-(2) in the cases m ε = m ε θ for θ = 0 , m > t , t ∈ (0 , T ), such thatΓ t = ∂ Ω − ( t ), and we have the following expansions: Outer expansion: “Away from Γ t ” we assume that c ε has an expansion ofthe form: c ε ( x, t ) = ∞ X k =0 ε k c ± k ( x, t ) for every x ∈ Ω ± ( t ) . Inner expansion:
In a neighborhood Γ t ( δ ), δ >
0, of Γ t c ε has an expansionof the form: c ε ( x, t ) = ∞ X k =0 ε k c k ( d Γ t ε , P Γ t ( x ) , t ) for all x ∈ Γ t ( δ ) . atching condition: lim z →±∞ c k ( z, x, t ) = c ± k ( x, t ) for all x ∈ Γ t , k = 0 , , lim z →±∞ ∂ z c ( z, x, t ) = 0 for all x ∈ Γ t . Moreover, all functions in the expansions above are assumed to be sufficientlysmooth.In the following we will use the expansions above and the matchingconditions, insert them into the convective Allen-Cahn equation (1) andequate all terms of same order in order to determine the leading parts in theinner and outer expansions formally.
First we use a power series expansion of c ε due to the outer expansion. Then f ′ ( c ε ( x, t )) = f ′ ( c ± ( x, t )) c ± ( x, t ) + εf ′′ ( c ± ( x, t )) c ± ( x, t ) + O ( ε )and we obtain from (1)1 ε − k f ′ ( c ± ( x, t )) + 1 ε − k f ′′ ( c ± ( x, t )) c ± ( x, t ) + O (1) = 0for all x ∈ Ω ± ( t ). This yieldsi) At order ε − k we obtain f ′ ( c ± ( x, t )) = 0. Thus c ± ( x, t ) ∈ (cid:8) ± , (cid:9) .Here we exclude the case c ± ( x, t ) = 0 since 0 is unstable and defineΩ ± ( t ) such that c ± ( x, t ) = ± x ∈ Ω ± ( t ) . ii) If k = 0, we obtain at order ε that f ′′ ( c ( x, t )) c ± ( x, t ) = 0. Since f ′′ ( ± >
0, we conclude c ± ( x, t ) = 0 for all x ∈ Ω ± ( t ) . If k = 1, the corresponding term is of order O (1) and we do not usethis information. Moreover, we will not determine c ± and c in thiscase. In Γ t ( δ ) we use the inner expansion in (1) in order to determine the leadingcoefficients c ( ρ, s, t ) and, in the case k = 0, c ( ρ, s, t ), where s := s ( x ) :=10 Γ t ( x ). To this end we use v · ∇ c j ( ρ, s, t ) = 1 ε v · ∇ d Γ t ( ρ, s, t ) + O (1) , ∆ c j ( ρ, s, t ) = 1 ε ( ∂ ρ c j ) ( ρ, s, t ) + 1 ε ( ∂ ρ c j ) ( ρ, s, t ) ∆ d Γ t ( x ) + O (1) ,∂ t c j ( ρ, s, t ) = 1 ε ( ∂ ρ c j ) ( ρ, s, t ) ∂ t d Γ t ( x ) + O (1)on Γ t , where ρ = d Γ t ( x,t ) ε and ∇ d Γ t = n Γ t , ∆ d Γ t = − H Γ t , ∂ t d Γ t = − V Γ t on Γ t . Hence inserting the inner expansion in (1) and equating terms of the sameorder yields for all x ∈ Γ t : m (cid:2) − ∂ ρ c ( ρ, s, t ) + f ′ ( c ( ρ, s, t )) (cid:3) · ε + m (cid:2) − ∂ ρ c ( ρ, s, t ) + f ′′ ( c ( ρ, s, t )) c ( ρ, s, t ) (cid:3) · ε + [ − ∂ ρ c ( ρ, s, t )( V Γ t − n Γ t · v − m H Γ t )] · ε = O (1)in the case k = 0 and (cid:2) m (cid:0) − ∂ ρ c ( ρ, s, t ) + f ′ ( c ( ρ, s, t )) (cid:1) − ( ∂ ρ c )( ρ, s, t )( V Γ t − n Γ t · v ) (cid:3) · ε = O (1)in the case k = 1. For the following we distinguish the cases k = 0 , Case k = 0 : The O ( ε )-terms yield − ∂ ρ c ( ρ, s, t ) + f ′ ( c ( ρ, s, t )) = 0 for all ρ ∈ R , s ∈ Γ t , t ∈ [0 , T ] . Because of the matching condition, we obtainlim ρ →±∞ c ( ρ, s, t ) = c ± ( s, t ) = ± s ∈ Γ t , t ∈ [0 , T ] . In order to obtain that Γ t approximates the zero-level set of c ε ( x, t ) = c ( d Γ t ε , s ( x ) , t ) + O ( ε ) sufficiently well, we obtain c (0 , s, t ) = 0. Hence c ( ρ, x, t ) = θ ( ρ ) for all x ∈ Γ t , ρ ∈ R . Furthermore, the O ( ε )-terms yield m (cid:0) − ∂ ρ c ( ρ, x, t ) + f ′′ ( θ ( ρ )) c ( ρ, x, t ) (cid:1) = θ ′ ( ρ )( V Γ t − n Γ t · v − m H Γ t ) =: g ( ρ )Since θ ′ is in the kernel of the differential operator − ∂ ρ + f ′′ ( θ ), this ODEhas a bounded solution if and only if Z R g ( ρ ) θ ′ ( ρ ) dρ = 0 , (31)11hich is equivalent to V Γ t − n Γ t · v = H Γ t on Γ t . Now the matching condition yields c ( ρ, x, t ) → ρ →±∞ c ± ≡
0. Hence c ≡ c ε ( x, t ) = θ (cid:18) d Γ t ( x ) ε (cid:19) + O ( ε )close to Γ t . Case k = 1 : The O ( ε )-terms yield m (cid:0) − ∂ ρ c ( ρ, s, t ) + f ′ ( c ( ρ, s, t )) (cid:1) − ∂ ρ c ( ρ, s, t )( V Γ t ( s ) − n Γ t ( s ) · v ( s, t )) = 0 (32)for all s ∈ Γ t . Testing with ∂ ρ c ( ρ, x, t ) yields0 = Z R | ∂ ρ c ( ρ, s, t ) | dρ ( V Γ t ( s ) − n Γ t · v ( s, t ))since Z R ∂ ρ (cid:18) | ∂ ρ c ( ρ, s, t ) | f ( c ( ρ, s, t )) (cid:19) dρ = 0because of the matching condition for ∂ ρ c . Because of c ( ρ, s, t ) → ρ →±∞ ± ∂ ρ c does not vanish and we obtain V Γ t = n Γ t · v on Γ t . Moreover, we obtain from (32) − ∂ ρ c ( ρ, s, t ) + f ′ ( c ( ρ, s, t )) = 0 for all s ∈ Γ t , ρ ∈ R . Hence we can conclude as in the case k = 0 that c ( ρ, s, t ) = θ ( ρ ) for all ρ ∈ R and s ∈ Γ t , t ∈ [0 , T ]. Remark 2.
The formal calculations show that c ε should have an expansionof the form (17) in the case θ = 0 , . This is important to obtain (15) in thelimit. Actually, using c ( ρ, s, t ) = θ ( ρ ) one can easily modify the results in[2, Section 4] to show formally convergence of the Navier-Stokes/Allen-Cahnsystem (4) - (6) to (7) - (11) in the case θ = 0 and (12) - (16) in the case θ =1 . A rigorous justification of this convergence under suitable assumptionsremains open. eferences [1] Abels, H.: On a diffuse interface model for two-phase flows of viscous,incompressible fluids with matched densities. Arch. Rat. Mech. Anal. (2), 463–506 (2009)[2] Abels, H., Garcke, H., Gr¨un, G.: Thermodynamically consistent, frameindifferent diffuse interface models for incompressible two-phase flowswith different densities. Math. Models Methods Appl. Sci. (3),1150013 (40 pages) (2012)[3] Abels, H., Lengeler, D.: On sharp interface limits for dif-fuse interface models for two-phase flows. Interfaces FreeBound. (3), 395–418 (2014). DOI 10.4171/IFB/324. URL https://doi.org/10.4171/IFB/324 [4] Abels, H., Liu, Y.: Sharp interface limit for a Stokes/Allen-Cahn system. Arch. Ration. Mech. Anal. (1), 417–502 (2018). DOI 10.1007/s00205-018-1220-x. URL https://doi.org/10.1007/s00205-018-1220-x [5] Abels, H., Schaubeck, S.: Nonconvergence of the capillary stress func-tional for solutions of the convective Cahn-Hilliard equation. In: Math-ematical fluid dynamics, present and future, Springer Proc. Math. Stat. ,vol. 183, pp. 3–23. Springer, Tokyo (2016)[6] Gal, C.G., Grasselli, M.: Longtime behavior for a model of ho-mogeneous incompressible two-phase flows. Discrete Contin. Dyn.Syst. (1), 1–39 (2010). DOI 10.3934/dcds.2010.28.1. URL http://dx.doi.org/10.3934/dcds.2010.28.1 [7] Giorgini, A., Grasselli, M., Wu, H.: Diffuse interface models for incom-pressible binary fluids and the mass-conserving allen-cahn approxima-tion. Preprint, arXiv:2005.07236 (2020)[8] Gurtin, M.E., Polignone, D., Vi˜nals, J.: Two-phase binary fluids andimmiscible fluids described by an order parameter. Math. Models Meth-ods Appl. Sci. (6), 815–831 (1996)[9] Hildebrandt, S.: Analysis 2. Springer-Lehrbuch. [Springer Textbook].Springer, Berlin (2003)[10] Jiang, J., Li, Y., Liu, C.: Two-phase incompressible flows with vari-able density: an energetic variational approach. Discrete Contin. Dyn.Syst. (6), 3243–3284 (2017). DOI 10.3934/dcds.2017138. URL https://doi.org/10.3934/dcds.2017138https://doi.org/10.3934/dcds.2017138