Sharp Interface Limit for a Navier-Stokes/Allen-Cahn System in the Case of a Vanishing Mobility

We consider the sharp interface limit of a Navier-Stokes/Allen Cahn equation in a bounded smooth domain in two space dimensions, in the case of vanishing mobility $m_\varepsilon=\sqrt{\varepsilon}$, where the small parameter $\varepsilon>0$ related to the thickness of the diffuse interface is sent to zero. For well-prepared initial data and sufficiently small times, we rigorously prove convergence to the classical two-phase Navier-Stokes system with surface tension. The idea of the proof is to use asymptotic expansions to construct an approximate solution and to estimate the difference of the exact and approximate solutions with a spectral estimate for the (at the approximate solution) linearized Allen-Cahn operator. In the calculations we use a fractional order ansatz and new ansatz terms in higher orders leading to a suitable $\varepsilon$-scaled and coupled model problem. Moreover, we apply the novel idea of introducing $\varepsilon$-dependent coordinates.


Introduction and Main Result
Two-phase flows of macroscopically immiscible fluids is an important research area with many applications.There are two important model categories: sharp interface models and diffuse interface models.For sharp interface models the interface separating the fluids is assumed to be a hypersurface.These models usually consist of an evolution law for the hypersurface, coupled to equations in the bulk domains and on the interface.Solutions often develop singularities in finite time, in particular when the interface changes its topology.In contrast, diffuse interface models use a typically smooth order parameter (e.g. the density or volume fraction of the two fluids) that distinguishes the bulk domains and inbetween typically has steep gradients in a small transition zone (also called diffuse interface), which is proportional to a small parameter, e.g.ε > 0. In applications the diffuse interface can be interpreted as microscopically small mixing region of the fluids.Quantities defined on the hypersurface in sharp interface models typically have a diffuse analogue that is defined in the diffuse interface.An important example is the relation of surface tension and capillary stress tensor, see Anderson, McFadden, Wheeler [9].Diffuse interface models may be more suited to describe phenomena acting on length scales related to the interface thickness, e.g.interface thicking phenomena, complicated contact angle behaviour and topology changes, cf.[9].Moreover, topology changes typically are no problem from an analytical or numerical point of view in contrast to sharp interface models.However, both model types are usually derived from physical principles or observations and can be used to model the same situations in applications.This motivates to study the connection between diffuse and sharp interface models by sending the small parameter ε (related to the thickness of the diffuse interface) to zero.Such limits are known as "sharp interface limits".
Moreover, HΓ t is the (mean) curvature and VΓ t is the normal velocity of Γt with respect to nΓ t .Furthermore, (Γ 0 , v ± 0,0 ) are suitable initial data.For the following we denote Γt × {t}.
Then there is a unique solution θ0 : R → R of (1.13), which is monotone.Moreover, for every m ∈ N0, there is some Cm > 0 such that |∂ m ρ (θ0(ρ) ∓ 1)| ≤ Cme −α|ρ| for all ρ ∈ R with ρ ≷ 0, where α = min( f ′′ (−1), f ′′ (1)).Since f is assumed to be even, θ0 is odd and θ ′ 0 is even.Strong solutions for the problem (1.6)-(1.12)have been studied extensively in the literature starting with the results by Denisova and Solonnikov [14].For further references we refer to Köhne, Prüss, and Wilke [24] and the monograph by Prüss and Simonett [30], where in particular local well-posedness in an L p -setting is shown.Existence of a notion of weak solutions, called varifold-solutions, globally in time was shown in [1].Weak-strong uniqueness for these kind of solutions was shown by Hensel and Fischer [16].
Let us now comment on the choice of vanishing mobility mε = √ ε → 0 in (1.4).In [2] a nonconvergence result was shown for a convective Allen-Cahn equation for a mobility mε = m0ε α , where m0 > 0 is a constant and α > 2, and formal asymptotic calculations were carried out for the case α = 0, 1. Hence for constant mobility mε = m0 the formal limit is a transport equation coupled to mean curvature flow, whereas for the case mε = m0ε the formal limit is a pure transport equation, cf.(1.11) above.It is possible to adapt the formal calculations to the case of mobilities mε = m0ε α for all exponents α ∈ [0, 1] (with the same limit system for α ∈ (0, 1]), the expansions just become more tedious and lengthy due to the fractional order ansatz.In Abels, Fei [3] the case of mε = 1, α = 0 was studied and rigorous convergence to a two-phase Navier-Stokes system coupled with mean curvature flow was shown as expected from the formal asymptotic expansions as long as a smooth solution of the limit system exists.However, for this limit system there is no conservation of mass and hence it could be considered physically less relevant compared to the classical two-phase Navier-Stokes system with surface tension, where one has pure transport of the interface.This clearly motivates the study of the case of vanishing mobility mε for ε → 0. To the best of our knowledge there is no rigorous convergence result in the case of vanishing mobility in the literature.The choice of mε = √ ε and α = 1 2 for our result is motivated as follows: for the arguments in [3], the exponent α = 1 2 is critical in a heuristic sense by calculating the orders for α = 1 and by assuming some linear depencence on α.The cases α ∈ (0, 1  2 ] should work formally with the strategy in [3], but we decided to simply choose α = 1  2 , in particular in order to have simpler asymptotic expansions with just √ ε-spacing in the sums.We note that in a joint-work with Fischer the first and third author show convergence for more general scalings of mε > 0 using the relative entropy method. In this work the convergence is obtained in weaker norms (and assuming same viscosities for simplicity), but it also holds for three space dimensions, see [4].
Our strategy to prove the sharp interface limit is via rigorous asymptotic expansions.The method goes back to de Mottoni and Schatzman [13] who first applied it to prove the rigorous sharp interface limit for the Allen-Cahn equation.The strategy works as follows: it is assumed that there exists a smooth solution to the limit sharp interface problem locally in time (usually this is no restriction).Then in the first step, one rigorously constructs an approximate solution to the diffuse interface system via rigorous asymptotic expansions based on the evolving hypersurface that is part of the solution to the limit problem.In the second step, one estimates the difference between the exact and approximate solution with the aid of a spectral estimate for a linear operator depending on the diffuse interface equation and the approximate solution.Comparison principles are not needed for the method and one even obtains the typical profile of solutions across the diffuse interface.The strategy was applied to many other sharp interface limits as well, see Moser [28] for a list of results.Let us just mention the famous result by Alikakos, Bates, Chen [8] for the Cahn-Hilliard equation, Abels, Liu [5] for a Stokes/Allen-Cahn system, Abels, Marquardt [6,7] for a Stokes/Cahn-Hilliard system, and the recent result Abels, Fei [3] for the Navier-Stokes/Allen Cahn system with constant mobility.
In general, rigorous results for sharp interface limits can be grouped into results concerning strong solutions for the limit system, in particular before singularities appear, and global time results using some weak notion for the sharp interface system.As described above, our result relies on the existence of a smooth solution for the limit system and assumes sufficiently small times.Another important strategy for sharp interface limits using strong solutions is the relative entropy method, see Fischer, Laux, Simon [17] where the convergence of the Allen-
Let us comment on the novelty of our contribution.We use a similar strategy as in Abels, Fei [3].Compared to [3], we consider the case of vanishing mobility mε = √ ε in (1.4), leading to the classical two-phase Navier-Stokes system with surface tension (1.6)-(1.12) in the sharp interface limit instead of the coupling with mean curvature flow in (1.11) obtained in [3].Some remarks on the choice of the scaling for the mobility were included before.Note that our choice turns out to be critical for the arguments we use, and therefore we need to take time small in our result compared to [3].Moreover, we need fractional order expansions with √ εspacing in the terms, cf.Section 3 below.Additionally, note that in [3] a new type of ansatz in higher orders was introduced based on a linearization idea that simplified the previous works [5,6,7].However, a direct modification with uncoupled equations for the higher order ansatz terms as in [3] does not lead to suitable estimates and hence is not enough to close the argument in our case.Therefore we modify this type of ansatz and obtain as model problem a coupled system (and another uncoupled problem in higher order) with suitable scaling in ε, see Section 2.3 and Section 4 below.Moreover, we even have a term at order O( √ ε) in the expansion of the distance function which leads to problems when applying spectral estimates within standard tubular neighbourhood coordinates.Therefore we use the novel idea of working with ε-dependent coordinates, in particular as framework for the spectral estimates, cf.Section 2.1 and Section 2.4 below.
Finally, let us summarize the structure of the paper.Section 2 contains the required preliminaries, i.e., ε-dependent coordinates, estimates of remainder terms, the (coupled and uncoupled) model problems with scalings in ε as well as spectral estimates based on the ε-scaled coordinates.The asymptotic expansion is done in Section 3, where the novelty lies in the expansion in integer powers of √ ε instead of integer powers of ε.The sophisticated higher order ansatz terms and remainder estimates are the content of Section 4. Finally, the main result is proven in Section 5, where a major part is the control of the error in the velocities in Section 5.1.

Preliminaries
Throughout the manuscript N denotes the set of natural numbers (without 0) and and X be a Banach space.Then we denote with L p (Ω; X) and W m p (Ω; X) the standard Lebesgue and Sobolev spaces.In the case X = R we write L p (Ω) and W m p (Ω), respectively.Moreover, if Ω has finite measure, we define for 1 ≤ q ≤ ∞ and k ∈ N0
This motivates to define for suitable ψ Moreover for all (x, t) ∈ Γ(3δ) \ Γ, ∂na(x, t) for all (x, t) ∈ Γ is well-defined, smooth in normal direction and tangential regularity is conserved.In particular ã is smooth provided that a is smooth.This can be shown with a Taylor expansion in dΓ.
A similar statement, based on a Taylor expansion in normal direction for Sobolev functions, is given by the following lemma and will be useful to estimate remainder terms.
For the following let h : T 1 × [0, T0] → R be sufficiently smooth.Then we have for all (x, t) ∈ Γ(3δ), where r ∈ (−3δ, 3δ) and s ∈ T 1 are determined by x = Xε(r, s, t).Therefore we define for every sufficiently smoth h : We note that coefficients of the differences , be open sets and U := t∈[0,T ] Ut.Then we define for s ≥ 0
We remark that R k,α and R 0 k,α are closed under multiplication and R k,α ⊂ R k−1,α .
In the same way we estimate for all ε ∈ (0, ε1] and t ∈ [0, T0], which shows the second estimate.
Proof: We use that since div ϕ = 0. To treat the remaining integral, we use integration by parts twice to get for all ε ∈ (0, ε1), t ∈ [0, T0] for some uniformly bounded Qε, Rε, where Now using g ∈ S(R) and (2.15) we obtain This finishes the proof.
We equip XT with the norm and the operator norm of the embedding is uniformly bounded in T .
Proof of Theorem 2.8: Existence of a unique solution follows by standard results on linear parabolic equations.Therefore we only need to prove the uniform estimates.
First we consider the case r = 0. Then testing (2.18) with h and integrating with respect to t we obtain .
In order to prove (2.21) in the case r = 0 we test (2.18) with −κ∂ 2 s h and obtain Hence integration in time, (2.20) with r = 0 and Young's inequality finally yield (2.21).
In the case r = 1 we use again that h = ∂sh solves (2.22).Testing this equation with −κ∂ 2 s h yields in the same way as before Therefore integration in time, (2.20) with r = 1 and Young's inequality yield (2.21) in the case r = 1.Finally, the case r ∈ (0, 1) follows again by interpolation.
For the construction of the approximate solutions we will essentially use solution to the following linearized system: together with where for all s ∈ T 1 , t ∈ (0, T ).We note that by chain rule for every sufficiently smooth hε : More precisely, (2.23)-(2.25)are understood in the following weak sense: and almost every t ∈ (0, T ), where as usual (2.31) + wε )) follows from the standard theory of abstract parabolic evolution equation for the Gelfand triple )). Hence it only remains to show the uniform estimates.
Proof of (2.31): First of all we can reduce to the case u = 0 simply by replacing w by w − u in the equations, where w0 is replaced by w0 − u|t=0 and f has to be replaced by f defined by ν ± Du : Dϕ dx for all ϕ ∈ V (Ω) and t ∈ (0, T ).Adding u afterwards to wε yields the desired solution.Since one also obtains (2.31).Now let u ≡ 0. Choosing ϕ = wε in (2.30) and testing (2.28) with |∂sXε(0, s, t)|∆Γ ε hε we obtain for some smooth and uniformly bounded ãε, bε, cε, dε : where we note that Hence Young's and Gronwall's inequality yield the desired estimate (2.31).
Proof of (2.32): Now assume additionally that f ∈ L 2 (0, T ; L 2 (Ω) 2 ), w0 ∈ V (Ω) and h0 ∈ H with the assumed regularity for u.)The estimate of hε follows directly from (2.21) for r = 1 2 and (2.31) using the equation (2.28).Hence )) follows from standard estimates for the two-phase Stokes system, cf.e.g.[30] for the case that the interface Γ ε t is independent of t ∈ (0, T ).The result in the present case that Γ ε t evolves smoothly with respect to t can be shown by the same perturbation argument as in the proof of Theorem A.1 in the appendix.

Spectral Estimate
For the spectral estimate in ε-coordinates as in Section 2.1 let ε1 > 0 be as in Theorem 2.1 and assume that (2.6) hold true.Moreover, we consider the rescaled variable ρε(x, t) := dε(x, t) ε for (x, t) ∈ Γ(3δ). (2.33) Finally, we assume the following structure of the approximate solution: let where µ ∈ [1, 2) and pε : We set The following spectral estimate will be a key ingredient for the proof of convergence.
Proof of Lemma 2.11: First, due to (2.4) and the definition (2.36) of c A ε , we obtain that for ε > 0 small it holds ). Therefore let us first consider the integral over Γε( 3δ 2 ): we can transform it into (dε, Sε)coordinates and get where we have set ψε := ψ • Xε and Jε is defined in (2.5).Via the chain rule we have the following transformation identity: (2.37) Therefore the asymptotics (2.6) together with Young's inequality yields for ε small Altogether we obtain The last term on the right hand side of (2.38) can be treated by well-known scaling and perturbation arguments as well as the spectral properties of differential operators on the real line similar to Chen [12].More precisely, except for the dependency of Jε on ε (which is not a problem because it is uniform in ε) the abstract 1D-spectral estimates in Moser [29, Section 5.1.3]are applicable after rescaling and yield the desired estimate.This shows the spectral estimate in Lemma 2.11.
Here note that βε from Corollary 2.13 is bounded uniformly for ε small and in order to obtain the estimate one has to take care of the Ψε-term from Corollary 2.13.However, this can be done by using the estimates in Corollary 2.13 and a rescaling argument.We note that for every ε > 0 the norm .Vε is equivalent to the standard norm in H 1 ( 3δ 2 )) (with ε-dependent constants).For the estimates of some critical remainder terms the choice of this norm will be essential.To estimate such remainder terms the following lemma will be used.
Remark 2.16 If we define dual norm the lemma states that

Formally Matched Asymptotics
In this section we will discuss the construction of the approximate solutions except some higher order terms, which will be added in the next section.In comparision with previous works the main difference is that we obtain an expansion in terms of integer powers of ε in Ω × (0, T0) by replacing pε by pε + 1 2 |∇cε| 2 .

The Inner Expansion
Close to the interface Γ we introduce a stretched variable for ε ∈ (0, 1), where dε : Γ(3δ which has to be understood as |∇dε(x, t)| 2 = 1 + O(ε N+ 1 2 ) for any N ∈ 1 2 N0 similary as before.Formally, dε is the signed distance function to Γ ε , which is the 0-level set of cε.Moreover, we assume the asymptotic expansion dε(x, t) ≈ understood in the same way as before, where d0(x, t) = dΓ t (x) for all (x, t) ∈ Γ(3δ).Here and in the following we assume already that (v ± 0 , p ± 0 , Γ) is a smooth solution of (1.6)-(1.12),although these equations can also be derived throughout the formal expansion.Since for the asymptotic expansion as ε → 0 only small values of ε > 0 matter, we may assume that for some M0 > 0.Moreover, we choose η : R → Then we have by integration by parts since ν ′ (θ0) is even by the assumptions on ν ′ .Furthermore, we define Remark 3.2 In the following we will insert terms W ± η ε,± in the equation to ensure some matching conditions.We have to make sure that these terms vanish if ρ = dε(x,t) ε .Because of (3.13), we have for ρ In the same way one shows this in Ω − .
Furthermore, we assume that we have the expansions ĉε(ρ, x, t) ≈ understood in the same way as before.Actually, in the expansion it turns out that ĉ0 = θ0 and ĉ 1 2 = ĉ1 = 0. To simplify the following presentation we already assume ĉ 1 2 = ĉ1 = 0.As usual we normalize ĉk such that ĉk (0, x, t) = 0 for all (x, t) ∈ Γ(3δ), k ≥ 0. (3.25) In order to match the inner and outer expansions, we require that for all k the so-called inner-outer matching conditions sup where ϕ = ĉk , vk with k ≥ 0 and pk with k ≥ −1 hold for constants α, C > 0 and all ρ > 0, m, n, l ≥ 0. For the non-linear terms Φ = ν, f ′ we use the expansion , . . ., ĉk−1 ) Using this expansion in (3.12) we obtain Hence, in order to satisfy (3.12) (up to higher order terms in ε) in Γ(3δ) we choose Furthermore, we choose d 1/2 such that which will ensure that (3.21) is well-defined and we can choose ĉ1 = 0, cf.(3.36) below.In what follows (see Corollary 3.7 below) we will find that v0, v 1 2 • ∇dΓ and φ0 will be independent of ρ on Γ.
To proceed we use that for since ĉε, dε and their derivatives have a corresponding expansion.Matching the O(ε 0 )-terms in the Allen-Cahn equation (3.19), we find ĉ0(ρ, x, t) = θ0(ρ) for all ρ ∈ R, (x, t) ∈ Γ(3δ). (3.31) Matching the O(ε 0 )-terms in the transformed momentum equation (3.17) and the divergence equation (3.18), we derive the following ordinary differential equations in ρ: where the right-hand side vanishes for the choice p−1(ρ) 2 )-order terms in the Allen-Cahn equation (3.19), we find 2 )-order terms in the transformed momentum equation and the divergence equation, we obtain the following ordinary differential equations with respect to ρ: Furthermore comparing the O(ε)-order terms in the Allen-Cahn equation (3.19), we have which again justifies the choice ĉ1 = 0. Then matching the O(ε)-order terms in the transformed momentum equation and the divergence equation, we obtain the following ordinary differential equations and Similarly, comparing the O(ε k )-order terms for k ≥ 3 2 in the transformed momentum equation (3.17), the divergence equation (3.18) and the Allen-Cahn equation (3.19), we obtain the following ordinary differential equations and where depend on the terms up to k − 3 2 order and converge exponentially to zero as |ρ| → ∞ because of the choice of W ± k and div v ± k = 0 for all k ∈ 1 2 N0.

Existence of Expansion Terms
The following two lemmas are used to solve the ordinary differential equations with respect to ρ and can be found in [3, Lemma A.2 and Lemma A.3].
Lemma 3.3 Let U ⊂ R n be an open subset and let A : R × U → R, (ρ, x) → A(ρ, x) be given and smooth.Assume that there exists A ± (x) such that the decay property A(±ρ, x) − A ± (x) = O(e −αρ ) as ρ → ∞ is fulfilled.Then for every x ∈ U the system has a smooth and bounded solution if and only if In addition, if the solution exists, then it is unique and satisfies for all x ∈ U ∂ ℓ ρ w(±ρ, x) +
Solving p− 1 : Γ(3δ) → R 2 .Because of the matching conditions we get as well as and Solving the O(ε)-order terms: To proceed we give the following proposition, which can be found in [3,Proposition A.5]. ) and ν is defined as in (3.14).
We need to point out that the first equalities in (3.59)-(3.61)hold not only on Γ but also in Γ(3δ).Moreover, because of (1.9) • ∇dΓ and φ0 are independent of ρ on Γ.Moreover, we can rewrite the evolution law (3.30)for d 1/2 as  (x, t) on Γ.
Together with (3.62) this leads to This implies the statement.
Remark 3.8 We note that solvability of (3.64) together with a system for v 1 2 is given by Theorem A.1 in the appendix and will be discussed later.

Solving the Higher Order Terms
Determining ĉ 3 2 , the evolution law of d1 on Γ, and g0: The compatibility condition (3.43) for (3.76) is equivalent to where and D 1 2 depends on the terms up to order 1 2 , which were determined before.Here we have used the definition of v0 and R θ ′ (ρ)(η(ρ) − 1 2 ) dρ = 0 since η − 1 2 is odd.On Γ the latter equation is satisfied if and only if d1 solves the evolution equation In order to satisfy the compatibility condition on Γ(3δ) \ Γ we define : First of all, we rewrite (3.41) as where S 1,k− 3 2 depends on lower order terms which are known by the induction hypothesis.Then the compatibility condition (3.43) for (3.80) is equivalent to where and Determining vk , k ≥ 3 2 , the jump conditions on Γ and l k−1 : Firstly, the compatibility condition (3.47) for (3.39) is equivalent to ). (3.84) Here S 2,k− 3 2 depends on low order terms, which were determined before.If it is satisfied, the solution to (3.39) is given by vk where V k−1 consists of terms up to k − 1 order.By taking ρ → +∞ in (3.85) and the matching conditions for vk we obtain where we have used u0 • n|Γ = 0 due to Proposition 3.6, and on Γ.Since by the induction hypothesis one assumes that the compatibility condition (3.84) for k − 1 instead of k is already satisfied, one obtains where V−1 ≡ 0. Inserting this we can rewrite (3.84) as where Here we have used and In order to satisfy the matching conditions for vk we define u k by satisfies the inner-outer matching conditions (3.26).
In order to determine pk−1 we multiply (3.39) by ∇dΓ and use (3.40).This yields where A k−1 and V k−1 consist of some terms up to k − 1 order.Thus which satisfies the inner-outer matching conditions (3.26).In summary we have: Lemma 3.10 (The k-th order terms) Let k ≥ 1 2 and all functions with negative index be supposed to be zero.Then there are smooth functions , d k , pk−1 , p ± k , which are bounded on their respective domains, such that for the k-th order the outer equations (3.4), (3.5), (3.6), the inner equations (3.39), (3.40), (3.41), the inner-outer matching conditions (3.26) are satisfied.Moreover, where Proof: The lemma is proved by mathematical induction with respect to k ∈ 1 2 N, where the beginning of the induction is given by Lemma 3.9.In Theorem A.1 in the appendix we will show solvability of the system (3.97)-(3.103),which will be smooth due to Remark A.2.In the induction hypothesis we assume that are known and satisfy the statements of the lemma with i instead of k for all 0 ≤ i ≤ k − 1 2 .Then we obtain the terms for i = k by the following four steps: Step 3: Since pk−1 is determined, S 1,k−1 is known on Γ and we can determine (v ± k , p ± k , d k ) as solution of (3.97)-(3.103),cf.Theorem A.1 in the appendix.
Step 4: Using that (v ± k , p ± k ) are known, u k is now determined uniquely by (3.93) and we can determine vk by (3.39) on Γ(3δ) uniquely.
Step 5: Using that d k is determined on Γ, one can integrate (3.29) in normal direction to determine d k uniquely on Γ(3δ).
Finally, we note that (v k , pk−1 , ĉk ) satisfy the matching conditions on Γ(3δ) by construction, in particular because of the choice of u k .

Summary of the Construction
The result of this section can be summarized as follows: Then there are smooth (c in A , ṽin A , pin A ) defined in Γ(3δ) and smooth (c ± A , ṽ± A , p± A ) defined on Ω × [0, T0] such that: 1. Inner expansion: In Γ(3δ) we have
Proof: We define as well as aε(t) = k∈ 1 2 N 0 ,k≤N+2 ε k a k (t).From the construction one can verify the statements of Theorem 3.11 in the same way as e.g. in [7,Section 4].Remark 3.12 We note that dA defined in (3.109) satisfies with respect to C k (Γ(3δ)) for every k ∈ N by the construction (3.29).
Since by the construction we only have to consider the terms from the inner expansion.

The Leading Error in the Velocity
For the following let (cA, vA, pA) and (cA, ṽA, pA) are given as in Section 4, where (cA, vA, pA) still depends on the choice of u, which will be chosen in the following, but (cA, ṽA, pA) are independent of u.Moreover, we define w := vε − vA.Hence we obtain for some q : Ω × [0, T0] → R.Here u = cε − cA, a ⊗ s b = a ⊗ b + b ⊗ a and Rε, Gε are as in Theorem 4.1.
The main goal of this subsection is to obtain the following bound for the error w in the velocity, which again is by a factor ε 1 4 worse than the corresponding result in [3]: Theorem 5.1 Let M > 0, cA, w be as in (5.1) and u satisfy (5.2) for some R > 0 and τ = Tε ∈ (0, T0] and N ≥ 3. Then there are some C(R, M ) > 0, C0(R) independent of ε ∈ (0, ε1), where ε1 is as in Theorem 2.1, and T ∈ (0, Tε] and w = w1 −w0, where w1 (5.7) As in [3] this yields a non-linear evolution equation with a globally Lipschitz nonlinearity for u, which can be solved in the same manner as in [5, Proof of Lemma 4.2].
Proof of Theorem 5.1: The proof is a variant of the proof of [3,Theorem 4.1].But there are several careful modifications necessary because of the different powers in the estimates (5.2) in the present case and the new ε-dependent coordinates (dε, Sε), which are only approximatively orthogonal.
The most important step is to show To this end we decompose Ω into Ω\Γ ε t ( 3δ 2 ) and Γ ε t ( 3δ 2 ) and split the integrals accordingly.Then the proof of (5.11) will consist of three parts.

Proof of the Main Result Theorem 1.1
In order to estimate the error due to linearization of ε − L 2 is decreased by 1  8 , which cause the loss of 1 8 in the power of ε in the present case compared to [3, Proposition 4.3].
In the following the proof is similar to [3,Section 4.2].But because of the different powers of ε in the estimates, some terms are critical compared to [3] and we have to choose additionally T > 0 sufficiently small to finally control all terms.
Finally, (1.18) follows from w = vε − vA and Theorem 5.1, in particular (5.5), and the remaining two conclusions in Theorem 1.1 are a consequence of the constructions of cA and vA.This finishes the proof of Theorem 1.1.