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Arch Ration Mech Anal
Arch Ration Mech Anal
Archive for Rational Mechanics and Analysis
0003-9527
1432-0673
Springer Berlin Heidelberg Berlin/Heidelberg

2020
10.1007/s00205-024-02020-9
Original Article
Approximation of Classical Two-Phase Flows of Viscous Incompressible Fluids by a Navier–Stokes/Allen–Cahn System
http://orcid.org/0000-0002-6606-4352
Abels Helmut helmut.abels@mathematik.uni-regensburg.de

1
Fischer Julian 2
Moser Maximilian 2
1 https://ror.org/01eezs655 grid.7727.5 0000 0001 2190 5763 Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, Germany
2 https://ror.org/03gnh5541 grid.33565.36 0000 0004 0431 2247 Institute of Science and Technology Austria, Am Campus 1, 3400 Klosterneuburg, Austria
Communicated by K. Ballstadt.

3 9 2024
3 9 2024
2024
248 5 776 11 2023
30 7 2024
© The Author(s) 2024
2024
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We show convergence of the Navier–Stokes/Allen–Cahn system to a classical sharp interface model for the two-phase flow of two viscous incompressible fluids with same viscosities in a smooth bounded domain in two and three space dimensions as long as a smooth solution of the limit system exists. Moreover, we obtain error estimates with the aid of a relative entropy method. Our results hold provided that the mobility mε>0 in the Allen–Cahn equation tends to zero in a subcritical way, i.e., mε=m0εβ for some β∈(0,2) and m0>0. The proof proceeds by showing via a relative entropy argument that the solution to the Navier–Stokes/Allen–Cahn system remains close to the solution of a perturbed version of the two-phase flow problem, augmented by an extra mean curvature flow term mεHΓt in the interface motion. In a second step, it is easy to see that the solution to the perturbed problem is close to the original two-phase flow.

Mathematics Subject Classification

Primary: 76T06
Secondary: 35Q30, 35Q35, 35R35, 76D05, 76D45
http://dx.doi.org/10.13039/100019180 HORIZON EUROPE European Research Council 948819 Universität Regensburg (3161)Open Access funding enabled and organized by Projekt DEAL.

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pmcIntroduction

During its evolution, the interface between two immiscible fluids may undergo topological changes, such as the merging or pinchoff of droplets. Mathematically, when modeling the interface classically as a (d-1)-dimensional manifold, this creates challenges for the analysis and for numerical approximations. Diffuse-interface models circumvent these problems by replacing the sharp interface by a diffuse transition layer of a finite width ε>0, reducing the problem to a set of PDEs posed on the entire domain. However, this procedure comes at the cost of introducing an additional modeling error: For many diffuse-interface models for fluid-fluid interfaces it has remained an open problem to rigorously show convergence to the original sharp-interface model in the limit of vanishing interface width ε→0, even prior to any topology change. In the present work, we prove the convergence of a diffuse-interface approximation for one of the most fundamental macroscopic models for a fluid-fluid interface, the Navier–Stokes equation for two immiscible incompressible fluids separated by a sharp interface with surface tension. To the best of our knowledge, the present work is the first general1 quantitative convergence result for any diffuse-interface approximation of the standard free boundary problem for the interface between two immiscible incompressible viscous fluids.

More specifically, in this contribution we rigorously identify the sharp-interface limit of the following Navier–Stokes/Allen–Cahn system 1.1a ∂tvε+vε·∇vε-Δvε+∇pε=-εdiv(∇φε⊗∇φε)inΩ×(0,T0),

1.1b divvε=0inΩ×(0,T0),

1.1c ∂tφε+vε·∇φε=mεΔφε-1ε2W′(φε)inΩ×(0,T0),

1.1d (vε,φε)|∂Ω=(0,-1)on∂Ω×(0,T0),

1.1e (vε,φε)|t=0=(v0,ε,φ0,ε)inΩ,

in a bounded domain Ω⊆Rd, d=2,3, with smooth boundary. Here vε:Ω×(0,T0)→Rd is the mean velocity of the fluid mixture, pε:Ω×(0,T0)→R its pressure and φε:Ω×(0,T0)→R an order parameter (e.g. the volume fraction difference of the fluids) related to the different phases, where the values φε=±1 describe that only one fluid is present. Moreover, ε>0 is a constant related to the thickness of the diffuse interface and mε>0 is a constant diffusion coefficient, which depends on ε>0. Here W:R→R is a double well-potential, satisfying standard assumptions. More precisely, we assume that W is twice continuously differentiable and we have, for some c>0,W(±1)=0,W(-s)=W(s),W(s)≥cmin{|s-1|2,|s+1|2}

for all s∈R. A standard example is W(s)=c(1-s2)2 for c>0.

This model was introduced by Liu and Shen in [30] to describe a two-phase flow for incompressible fluids with the same viscosity and densities. For simplicity we have set the densities and viscosities to one. The model can be considered as the analogue of the well-know “model H”, cf. [21, 24], if one replaces the convective Cahn-Hilliard equation by a convective Allen–Cahn equation. A first analytic study of the system (1.1a)–(1.1e) was done by Gal and Grasselli [19] in the case of a bounded smooth domain in two space dimensions, where the existence of global and exponential attractors and convergence to stationary solutions was shown. For a more general model with different densities and viscosities Jiang et al. [25] proved the existence of weak solutions globally in time (in two and three space dimensions). Moreover, in the case of two space dimensions they proved global well-posedness in the strong sense and studied the longtime behavior of strong solutions. We refer to Giorgini et al. [20] for a mass-conserving variant of the Navier–Stokes/Allen–Cahn system with different densities and further references.

It is the purpose of this contribution to study the limit (1.1a)–(1.1e) as ε→0 in the case that mε→ε→00 suitably. Then one expects to have convergence to solutions of the classical two-phase Navier–Stokes equation with surface tension: 1.2a ∂tv0±+v0±·∇v0±-Δv0±+∇p0±=0inΩt±,t∈[0,T0],

1.2b divv0±=0inΩt±,t∈[0,T0],

1.2c 〚2Dv0±-p0±I〛nΓt=-σHΓtnΓtonΓt,t∈[0,T0],

1.2d 〚v0±〛=0onΓt,t∈[0,T0],

1.2e VΓt=nΓt·v0±onΓt,t∈[0,T0],

1.2f v0-|∂Ω=0on∂Ω×(0,T0),

1.2g Γ0=Γ0,v0±|t=0=v0,0±inΩ0±.

Here (Γt)t∈[0,T0] is an evolving (d-1)-dimensional submanifold of Ω such that Ω is the disjoint union of two smooth domains Ωt± and Γt as well as ∂Ωt±=Γt for every t∈[0,T0]. Here the absence of a boundary contact of Γt is assumed for all t∈[0,T0]. Moreover, v0±(.,t):Ωt±→Rd and p0±(.,t):Ωt±→R are the velocity and pressure of two fluids filling Ωt+ and Ωt- for every t∈[0,T0], nΓt denotes the interior normal of Γt with respect to Ωt+, HΓt and VΓt denote the mean curvature (sum of principle curvature) and normal velocity, resp., of Γt with respect to the orientation given by nΓt. Furthermore, 〚f〛(s)=limh→0+(f(s+hnΓt(s))-f(s+hnΓt(s))), s∈Γt, denotes the jump of a function f defined in a neighborhood of Γt and Dv0±:=12∇v0±+(∇v0±)⊤ is the symmetric gradient. Finally, σ=∫-112W(s)ds>0 is a surface tension coefficient. For what follows we set that1.3 Γ:=⋃t∈[0,T0]Γt×{t},Ω±:=⋃t∈[0,T0]Ωt±×{t},v0:=∑±v0±χΩ±.

For given vε=v the convective Allen–Cahn equation, i.e., (1.1c)–(1.1d), was discussed formally by the first author in [2] and it was shown formally that the limit system is given by the transport Eqs. (1.2e)–(1.2f) in the case that mε=m0ε for some m0>0. We note that in this case these arguments can be extended to show formally convergence of the full system (1.1a)–(1.1e) to (1.2a)–(1.2f) by combining it with the arguments in [5] for a Navier–Stokes/Cahn-Hilliard system. But in the case that mε=m0εβ for some β>2 nonconvergence was shown in the sense that, in general,φε(x,t)=θ0dΓt(x)ε+O(ε)asε→0,

where dΓt is the signed distance function to Γt, no longer holds and a weak formulation of the right-hand side of (1.1a) does not converge to the mean curvature functional σHΓtnΓtδΓt, which appears in a weak formulation of (1.2a) and (1.2c). Here θ0:R→R is the so-called optimal profile, which is the unique solution of-θ0′′(s)+W′(θ0(s))=0for alls∈R,θ0(s)→s→±∞±1,θ0(0)=0.

Therefore convergence of the full system (1.1a)–(1.1e) cannot be expected in this case. We note that in this case the counterexample given in [6] for a Navier–Stokes/Cahn-Hilliard system in a radially symmetric situation can be adapted to the present Navier–Stokes/Allen–Cahn system. It is the purpose of the present contribution to show convergence of solutions of (1.1a)–(1.1e) to the smooth solution of (1.2a)–(1.2f) on a time interval [0,T0] for which the latter exists in the case of a subcritical scaling of the mobility mε=m0εβ for some β∈(0,2). Moreover, we will derive error estimates with the aid of a relative entropy method. We note that this is the first rigorous convergence result for a vanishing mobility mε→ε→00 in this regime, which includes the natural choice mε=m0ε. Finally, let us remark that the derivation of a similar convergence result using a relative entropy method was attempted in the recent work [26]; however, as the approach of [26] relies on the invalid estimate [26, equation (2.4)], it overlooks the need to devise a careful estimate for the critical interface stretching term that forms the main challenge for our result.

Except for [4], so far only convergence in the case of a non-vanishing mobility mε=m0>0 for all ε>0 has been shown. First this was done in the case of a Stokes/Allen–Cahn system with same viscosities by Abels and Liu [7], then for a Navier–Stokes/Allen–Cahn system with different viscosities in Abels and Fei [3], both in two space dimensions, and by Hensel and Liu [22] for the Navier–Stokes/Allen–Cahn system with same viscosities in two and three space dimensions. We note that the first two results are based on a refined spectral estimate for the linearized Allen–Cahn operator and rigorous asymptotic expansions, while the latter result uses the relative entropy method similarly as for the convergence of the Allen–Cahn equation to the mean curvature flow shown by Fischer, Laux, and Simon [17]. In the case of a non-vanishing mobility the limit system consists of a system, where (1.2e) is replaced by a convective mean curvature flow equationVΓt=nΓt·v0±+m0HΓtonΓt,t∈[0,T0].

In the contribution by Abels, Fei, and Moser [4] convergence of a Navier–Stokes/Allen–Cahn system with different viscosities was shown in the special case of mε=m0ε and two space dimensions using the same method as in [3, 7] refined for this degenerate case. We note that the arguments could be extended to mε=m0εβ for β∈(0,12], but the case β=12 seems to be critical for the estimates in this contribution and new ideas and refinements seem to be needed to treat the cases with β>12 with this method. At this stage, the relative entropy method used in the present contribution appears to be more flexible.

In the following we consider a situation, in which the limit system (1.2a)–(1.2g) is known to possess a unique smooth solution for some T0>0. We note that strong well-posedness of this system was extensively studied starting with the results by Denisova and Solonnikov [14]. Moreover, it was shown by Prüss and Simonett [34] that strong solutions become analytic instantaneously in time. We refer to Köhne, Prüss, and Wilke [27] and the monograph by Prüss and Simonett [35] for results on local well-posedness in an Lp-setting and further references. Finally, we note that global-in-time existence of a notion of weak solutions, called varifold solutions, was shown in [1] and weak-strong uniqueness for these kind of solutions was shown by Fischer and Hensel [16].

In general, there are two main mathematical approaches to the quantitative justification of sharp-interface limits: an approach pioneered by de Mottoni and Schatzman [13] and Chen [12] relies on a matched asymptotic expansion around the sharp-interface limit to obtain an approximate solution to the diffuse interface model; by means of a stability analysis of the linearized operator, it is possible to derive rates of convergence. This approach has recently also been successfully adapted to our Navier–Stokes/Allen–Cahn system with mobility mε=ε1/2 by Fei and the first and the third author [4]. An alternative approach – recently developed by the second author, Laux, and Simon [17] – proceeds via a suitably defined relative entropy. In [17], the relative entropy approach is used to give a short proof of convergence of the Allen–Cahn equation towards mean curvature flow (valid for well-prepared initial data and as long as a classical solution to the latter exists); this result was extended in [23] to interfaces with boundary contact and in [29] to the anisotropic case. The general approach has found numerous further applications: In [18], convergence of the vectorial Allen–Cahn equation with multi-well potential towards multiphase mean curvature flow has been established by the second author and Marveggio; the case of the vectorial Allen–Cahn equation with two-well potential has been considered by Liu [31, 32]. Furthermore, Laux and Liu [28] have obtained the sharp-interface limit for a model for liquid crystals. Hensel and Liu [22] have used the relative entropy approach to study the sharp-interface limit of the Navier–Stokes/Allen–Cahn system (1.1a)–(1.1e) in the regime of nonvanishing mobility, deriving a Navier–Stokes/mean curvature flow system in the limit. In general, a key advantage of the relative entropy approach to sharp interface limits is its robustness, for instance requiring only convergence of the initial energy of solutions to the phase-field model. In contrast, the approach of matched asymptotic expansions may be used to establish an approximation of the diffuse interface model to arbitrary order.

The main result of our contribution is as follows:

Theorem 1.1

(Convergence) Let mε:=m0εβ for ε>0, where m0>0 and β∈(0,2) are fixed, q=2 if d=2, and q=43 if d=3. Moreover, let T0>0 be such that the two-phase Navier–Stokes system with surface tension (1.2a)–(1.2g) has a smooth solution (v0±,p0±,Γ) on [0,T0]. Let (vε,φε) with vε∈L∞(0,T0;Lσ2(Ω))∩L2(0,T0;H01(Ω)d), φε∈L2(0,T0;H2(Ω))∩Wq1(0,T0;L2(Ω)) for ε>0 be energy-dissipating weak solutions to the Navier–Stokes/Allen–Cahn system (1.1a)–(1.1e) on [0,T0] as in Remark 1.2 below for the constant mobility mε and for initial data (v0,ε,φ0,ε) with energy uniformly bounded with respect to ε and satisfying1.4 ∑±∫Ω0±12|v0,ε-v0,0±|2dx+∫Ωε2|∇φ0,ε|2+1εW(φ0,ε)-(ξ·∇ψ0,ε)dx+∫Ω|σχΩ0+-ψ0,ε|min{distΓ0,1}dx≤Cε2mε

for ε>0 sufficiently small, where ψ(r):=∫-1r2W(s)ds and ψ0,ε:=ψ(φ0,ε). Set ψε:=ψ(φε).

Then, for ε>0 small and a.e. T∈[0,T0], it holds that 1.5a ‖(vε-v0)(.,T)‖L2(Ω)+‖σχΩT+-ψε(.,T)‖L1(Ω)≤Cεmε+mε

1.5b ‖∇vε-∇v0‖L2(0,T0;L2(Ω))≤Cεmε+mε

for some C>0 independent of ε>0 and T∈[0,T0]. Finally, there are well-prepared initial data (v0,ε,φ0,ε) for ε>0 small in the sense that (1.4) is satisfied, even with rate ε2.

This result will be a consequence of Corollary 3.2 below.

Let us briefly comment on some aspects of our main theorem. First, note that the choice mε:=ε2/3 leads to the best-possible overall convergence rate O(ε2/3) in (1.5). If one employs the Navier–Stokes/Allen–Cahn system as a numerical approximation of the two-phase flow with sharp interface, this thus suggests the choice of mobility mε∼ε2/3.

Second, our theorem requires a lower bound on the mobility of the form mε≳εβ for some β<2. With a slightly more careful argument, it would in fact be possible to weaken this assumption to mε≳ε2|logε|3. However, in the regime mε≲ε2 one would expect that the Allen–Cahn term no longer suffices to stabilize the Modica-Mortola type profile of the diffuse interface, making it susceptible to stretching or squeezing by drift: To see this heuristically, notice for instance that in this scaling regime the reaction term -mεε2W′(φε) from the Allen–Cahn Eq. (1.1c) no longer suffices to drive the values of φε towards the minima of the potential W in finite time. This possible stretching of the profile by advection in turn is expected to lead to errorneous capillary effects in (1.1a) and hence convergence to a different limiting system; see also [6], in which this has been observed rigorously for certain choices mε=εβ, β>2.

Remark 1.2

Let q=2 if d=2, and q=43 if d=3. We note that weak solutions (vε,φε) with vε∈L∞(0,T0;Lσ2(Ω))∩L2(0,T0;H01(Ω)d), φε∈L2(0,T0;H2(Ω))∩Wq1(0,T0;L2(Ω)) for ε>0 to the Navier–Stokes/Allen–Cahn system (1.1a)–(1.1e) on [0,T0] are understood in the sense of [22, Definition 3] with the difference that we only require φε∈W4/31(0,T0;L2(Ω)) if d=3. In particular, we assume that the energy inequality∫Ω|vε(x,t)|22+ε2|∇φε(x,t)|2+W(φε(x,t))εdx+∫0t∫Ω|∇vε(x,τ)|2+|∂tφε(x,τ)+vε(x,τ)·∇φε(x,τ)|2dxdτ≤∫Ω|v0,ε|22+ε2|∇φ0,ε|2+W(φ0,ε)εdxfor everyt∈[0,T0]

holds true, which is essential for the following arguments. Existence of weak solutions follows from the results in [25]. We note that in the latter contribution the authors do not prove the energy inequality. However, it can be proved in a standard manner using the energy identity for the solutions of the Galerkin system in [25] and lower semi-continuity of norms. We also refer to [20, Theorem 3.1] for the mass-conserved variant of the system.

Let us comment on the novelty of our contribution. As mentioned before, this is the first convergence result for (1.1a)–(1.1e) in the case of a vanishing mobility mε→0, except for the special case mε=m0ε in two space dimensions. We use a relative entropy similar as in [22], which extends the one for the Allen–Cahn equation used in [17] to the coupled Navier–Stokes/Allen–Cahn system. The main step in the proof of convergence consists in showing a suitable estimate for the relative entropy. Parts of the arguments and calculations in our situation follow closely [22], but certain estimates degenerate as mε→0 and some terms become critical. Therefore essential new ingredients are needed.

More precisely, it will turn out that the main challenge in controlling the growth of the relative entropy for mε≪1 consist of controlling terms involving the failure of equipartition of energy such as∫ε2|∂nφε|2-1εW(φε)∂nη~dx.

A naive direct estimate would bound such terms by the square root of the relative entropy, insufficient for a subsequent control of the growth via the Gronwall lemma. Instead, we shall see that by careful integration by parts arguments and an approximation of the diffuse interface by a graph, they may in fact be controlled by the dissipation term from the Allen–Cahn equation and the relative entropy, provided that mε≫ε2.

We deal with these critical terms in Sects. 4.2–4.4. The estimates make use of a suitable parametrization of a suitably chosen level set {φε=b(t)} for some b(t)∈(-12,12) up to an error controlled by the relative entropy using results from [16]. Moreover, it is essential for the construction of the relative entropy that we use sufficiently smooth solutions of a modification of the limit system (1.2a)–(1.2g), where the interface evolution (1.2e) is replaced by the convective mean curvature flow equation with vanishing mobilityVΓt=nΓt·v0±+mεHΓtonΓt,t∈[0,T0].

This is used to obtain a sufficiently good approximation and control of some critical remainder terms. However, it makes the solution depend on mε, ε, respectively, and existence of such solutions together with uniform estimates in ε>0 sufficiently small needs to be shown. We note that existence of strong solutions for a fixed mε>0 locally in time was shown by the first and third author in [8] and in [22, Appendix]. But the existence time might depend on ε. To obtain the uniform bounds for small ε>0 a Hanzawa transformation is used to transform the modified system of the limit system (1.2a)–(1.2g). Then a fixed point argument can be used to obtain strong solutions for mε sufficiently small using suitable uniform bounds for the mε-dependent linearized system in spaces of maximal Lq-regularity.

The structure of the contribution is as follows: in Sect. 2 we define the energy-type functionals, which will be essential for the proof of convergence, and study their coercivity properties. Afterwards the central stability estimate and convergence result is given in Sect. 3. The essential estimates for the relative entropy are done in Sect. 4. The proof of the stability estimate, which implies the convergence result, is given in Sect. 6. Finally, in Sect. 7 existence of strong solutions for the modification of the limit system (1.2a)–(1.2g) for sufficiently small mε>0 is shown and its difference to the solution of the real limit system is estimated by a multiple of mε, cf. Theorem 7.7 below.

Let us finish the introduction with some notation. For example, we denote with Wpk(Ω) for k∈N, 1≤p≤∞ and some domain Ω the Sobolev space with k weak derivatives and integrability exponent p. Moreover, let Hk:=W2k and Lσ2(Ω):={v∈C0∞(Ω)d:divv=0}¯L2(Ω)d, where C0∞(Ω) is the space of smooth functions with compact support in Ω.

Definition and Coercivity Properties of the Energy Functionals

In this section we define the energy/entropy functionals used for our Gronwall argument and show suitable coercivity properties. The definitions are similar to [22], but based on a modification of the limit system as mentioned in the introduction. To this end, we need some notation.

Let (vε,φε) with vε∈L∞(0,T0;Lσ2(Ω))∩L2(0,T0;H01(Ω)d) and φε∈L2(0,T0;H2(Ω))∩Wq1(0,T0;L2(Ω)), where q=2 if d=2 and q=43 if d=3, for ε>0 small be energy-dissipating weak solutions to the Navier–Stokes/Allen–Cahn system (1.1a)–(1.1e) on [0,T0] with constant mobility mε>0 as in Remark 1.2. Moreover, for mε>0 small let (vmε±,pmε±,(Γtε)t∈[0,T0]) be solutions of the two-phase Navier–Stokes equation with surface tension on [0,T0] but with the evolution equation VΓtε=nΓtε·vmε±+mεHΓtε on Γtε instead of (1.2e), cf. (7.1)–(7.7) and Theorem 7.9 in Sect. 7 below. We define that2.1 Γmε:=⋃t∈[0,T0]Γtε×{t},Ωmε,±:=⋃t∈[0,T0]Ωtmε,±×{t}andvmε:=∑±vmε±χΩmε,±.

Let dΓmε be the signed distance function of Γmε and PΓmε be the orthogonal projection on a tubular neighbourhood Γmε(2δ) of Γmε of the width 2δ, where δ>0 is sufficiently small and can be chosen independently of mε>0 sufficiently small, cf. Remark 7.8. Let Γtε(2δ):=Γmε(2δ)∩(Rd×{t}) be the time-slice. On Γmε(2δ) we set that2.2 n:=nΓmε|PΓmεandH:=HΓmε|PΓmε,

where we avoided to add mε in the notation for n and H for convenience, and we used the notation f|PΓmε:=f∘PΓmε for suitable f. Moreover, we denote by Xmε:=(dΓmε,PΓmε,prt)-1 on (-2δ,2δ)×Γmε the coordinate map describing the tubular neighbourhood strip Γmε(2δ) and define the normal derivative and the tangential gradient by2.3 ∂n:=n·∇and∇τ:=(Id-n⊗n)∇,respectively.

On Γmε(2δ) it holds that2.4 ∇=n∂n+∇τ,|∇u|2=|∂nu|2+|∇τu|2for suitableu,

2.5 ∂n=[∂1(.∘Xmε)]∘Xmε-1and∇τ=DxPΓmε[∇Γtε(.∘Xmε)]∘Xmε-1.

Finally, let us recall that2.6 ψ(r):=∫-1r2W(s)ds,σ:=ψ(1)andψε:=ψ(φε)

and define that2.7 nε:=∇φε|∇φε|,if∇φε≠0,s,else,

where s is a fixed unit vector in Rd. Then ψε and nε are defined on Ω×(0,T0) and it holds thatnε|∇φε|=∇φεandnε|∇ψε|=∇ψε.

Relative Entropy

We define the relative entropy as follows for a.e. t∈[0,T0]:2.8 E[vε,φε|vmε,Γmε](t):=∫Ω12|vε-vmε|2(x,t)dx+E[φε|Γmε](t),

2.9 E[φε|Γmε](t):=∫Ωε2|∇φε|2(x,t)+1εW(φε(x,t))-(ξ·∇ψε)(x,t)dx.

Here we chose to introduce a separate notation for the second interface-related part for convenience. Here ξ is an extension (with quadratic cutoff) of the unit normal on Γmε. Note that the interface-related part of the relative entropy – being the same as the relative entropy in [17] – is motivated by the Modica-Mortola trick; as we shall see below, it controls both the error in the equipartition of the diffuse interface energy (2.14) and a tilt-excess type error quantity for the interface normal (2.13). At the same time, the time evolution of the relative entropy (2.9) can be calculated in a straightforward manner using the energy dissipation inequality for the Navier–Stokes/Allen–Cahn equation as well as the phase-field Eq. (1.1c).

For the precise definition of ξ:Ω¯×[0,T0]→Rd and an accompanying B:Ω¯×[0,T0]→Rd (which has the role of an approximate transport and rotational velocity for ξ) we first introduce suitable cutoff functions (analogous to [22, Proof of Theorem 1]). Let η¯:R→[0,1] be smooth and even such that suppη¯⊆[-1,1] and such that it satisfies the quadratic decay2.10 1-Cη¯r2≤η¯(r)≤1-cη¯r2and|η¯′(r)|≤C|r|forr∈[-1,1],

where cη¯,Cη¯,C>0. Moreover, let η~:R→[0,1] be smooth with suppη~⊆[-2,2] and η~=1 on [-1,1]. Finally, we set ηmε:=η¯(dΓmεδ) and η~mε:=η~(dΓmεδ). Then we define2.11 ξ:=nηmεandB:=vmε+mεHnη~mε,

where n=n(mε) and H=H(mε) were defined in (2.2). For important properties of ξ and B we refer to Lemma 2.2 below.

The relative entropy from (2.8) satisfies the following coercivity properties:

Lemma 2.1

(Coercivity Properties of the Relative Entropy) First, the relative entropy provides a control of the velocity error in terms of2.12 ∫Ω12|vε-vmε|2dx≤E[vε,φε|vmε,Γmε].

Moreover, it yields a tilt-excess-type error estimate of the form2.13 ∫Ω(1-nε·ξ)|∇ψε|dx≤E[φε|Γmε].

Additionally, we have some control of the error in the equipartition of the energy in the sense2.14 ∫Ω12ε|∇φε|-1ε2W(φε)2dx≤E[φε|Γmε].

Furthermore, one obtains control of tangential derivatives and for the lack of equipartition of energy in normal direction: for a.e. t∈[0,T0] it holds that2.15 ∫Γtε(2δ)ε2|∇τφε|2(x,t)+12ε∂nφε-1ε2W(φε)2(x,t)dx≤E[φε|Γmε](t),

and one can replace ∂nφε by |∂nφε| in the estimate. Finally, for some C=C(δ)>0 it holds that2.16 ∫Ω|nε-ξ|2+min{dΓmε2,1}ε|∇φε|2+|∇ψε|dx≤CE[φε|Γmε],

2.17 ∫Ωmin{dΓmε,1}+1-nε·ξε|∇φε|2-|∇ψε|dx≤CE[φε|Γmε].

Proof

Up to (2.15) the estimates are analogous to Hensel, Liu [22, Lemma 5]. Note that one only uses the definitions and the elementary estimate |ξ|≤1-cmin{dΓmε2,1}, cf. also Lemma 2.2 below. The new estimate (2.15) is a simple consequence of the properties (2.4) of ∂n and ∇τ. □

Lemma 2.2

(Properties of ξ and B) For (ξ,B)=(ξ,B)(mε) from (2.11) we have Regularity: for some p>d+5 it holds that ξ∈C1([0,T0],C0(Ω¯)d)∩C0([0,T0];Cc2(Ω)d),B∈C0([0,T0];C0,1(Ω)d)∩Lp(0,T0;Wp2(Ω\Γtε)d),∇τ∇B∈L∞Γmε3δ23,

we have uniform bounds with respect to mε and B|∂Ω=0.

Coercivity and consistency: we have 2.18 |ξ|≤1-cmin{dΓmε2,1}inΩ×[0,T],

2.19 ξ=n,∇·ξ=-HonΓmε,

2.20 B-vmεmε·ξ+∇·ξ≤Cmin{dΓmε,1}a.e. inΩ×[0,T],

where (2.19) is a relation for the mean curvature and (2.20) is an approximate equation for interface normal velocity.

Evolution equations for ξ: it holds (where ∇B is defined to be the Jacobian) that 2.21 |(∂t+B·∇)|ξ|2|≤Cmin{dΓmε2,1}a.e.inΩ×[0,T],

2.22 |∂tξ+(B·∇)ξ+(Id-ξ⊗ξ)(∇B)⊤ξ|≤Cmin{dΓmε,1}a.e.inΩ×[0,T],

where (2.21) means that |ξ|2 is approximately transported by the vector field B and (2.22) that ξ is approximately transported and rotated by B.

Proof

The regularity and uniform bounds for ξ are obtained directly from the ones for dΓmε in Theorem 7.9. Moreover, vmε instead of B satisfies the regularity, uniform bounds and the boundary condition stated for B due to Theorem 7.9 together with embeddings and interpolation theory. Moreover, the second term in the definition (2.11) of B is contained in C0([0,T0],Cc2(Ω)2) because of Theorem 7.9 and due to the mε-prefactor and the well-known identities H=(-ΔdΓmε)|PΓmε, n=∇dΓmε and PΓmε=Id-ndΓmε.

The properties (2.18)–(2.19) are clear from the definition and the well-known identities n=∇dΓmε, ∇·n=ΔdΓmε on Γmε(2δ) and ΔdΓmε|Γmε=-H on Γmε. Moreover, a direct calculation givesB-vmεmε·ξ+∇·ξ=Hηmεη~mε+∇·ξ.

The latter vanishes on Γmε, hence (2.20) follows. Moreover,2.23 -∂tdΓmε=VΓmε=vmε·n+mεH=B·∇dΓmεonΓmε

due to (7.6) for mε instead of m, and therefore2.24 (∂t+B·∇)dΓmε=0onΓmε.

We compute |ξ|2=ηmε2 and(∂t+B·∇)|ξ|2=2δηmεη¯′dΓmεδ(∂t+B·∇)dΓmε.

Due to (2.24), η¯′(0)=0 together with a transformation in tubular neighbourhood coordinates and the Taylor Theorem, we obtain (2.21).

Finally, let us prove (2.22). Equation (2.23) yields, by definition,∂tdΓmε+vmε|PΓmε·n+mεH=0onΓmε(2δ).

Differentiating the previous identity implies2.25 ∂tn+∇(vmε|PΓmε)⊤·n+(∇n)⊤vmε|PΓmε+mε∇H=0onΓmε(2δ),

where ∇(vmε|PΓmε)=∇vmε|PΓmε∇PΓmε and it is well-known that ∇PΓmε|Γmε=Id-n⊗n on Γmε. Moreover, on Γmε it holds that∂tξ+(B·∇)ξ+(Id-ξ⊗ξ)(∇B)⊤ξ=∂tn+(vmε+mεHn)·∇n+(Id-n⊗n)∇vmε+mε(n∇H⊤+H∇n)⊤n=∂tn+vmε·∇n+(Id-n⊗n)(∇vmε)⊤n+mε∇H,

where we used (∇n)⊤n=0 and n·∇n=n·∇H=0. Finally, due to n=∇dΓmε on Γmε(2δ) we have ∂xjn=∇nj for j=1,...,d and hence vmε·∇n=(∇n)⊤vmε on Γmε(2δ). Therefore we obtain (2.22) from (2.25). □

Bulk Error Functional

We define the bulk error functional for all t∈[0,T0] by2.26 Ebulk[φε|Γmε](t):=∫ΩσχΩtmε,+-ψε(x,t)ϑ(x,t)dx,

where ϑ:Ω¯×[0,T0]→[0,1] is defined as ϑ:=ϑ¯(dΓmεδ), where ϑ¯:R→[0,1] is smooth withϑ¯>0on(1,∞),ϑ¯<0on(-∞,0)and|ϑ¯|=1onR\[-1,1],

as well as cϑ¯|r|≤|ϑ¯(r)|≤Cϑ¯|r| for all r∈[-1,1] and some cϑ¯,Cϑ¯>0. Note that we use a different sign convention for ϑ¯ as in [22]. Hence ϑ is roughly proportional to the signed distance function of Γmε close to Γmε and appropriately truncated to ±1 outside. The required properties of ϑ will be shown in Lemma 2.4 below.

Let us now prove coercivity properties for the bulk error functional.

Lemma 2.3

(Coercivity Properties of Ebulk) It holds σχΩmε,+-ψεϑ≥0, in particular Ebulk[φε|Γmε]≥0. Moreover,2.27 ∫ΩσχΩmε,+-ψεmin{dΓmε,1}dx+‖σχΩmε,+-ψε‖L1(Ω)2≤CEbulk[φε|Γmε].

Finally, for all c0>0 there exists C=C(c0)>0 such that2.28 ∫Ω|σχΩmε,+-ψε||vε-vmε|dx≤c0∫Ω|∇vε-∇vmε|2dx+CE[vε,φε|vmε,Γmε]+Ebulk[φε|Γmε].

Proof

Because of |φε|≤1 due to the maximum principle, it holds that |ψε|≤σ, by definition. Hence the properties of ϑ¯ yield σχΩmε,+-ψεϑ≥0. Moreover,|σχΩmε,+-ψε|min{dΓmε,1}≤C|σχΩmε,+-ψε||ϑ|=CσχΩmε,+-ψεϑ

and this estimates the first term in (2.27). This yields that the second term in (2.27) is controlled by using the inequality (cf. [17, Proof of Theorem 1])2.29 ∫0δ|g|dr2≤2‖g‖L∞(0,δ)∫0δ|g|(r)rdrfor allg∈L∞(0,δ),

which is derived by dividing the square [0,δ]2 into two triangles and applying Fubini’s theorem.

Finally, (2.28) can be shown analogously to [22, proof of (31)] with (2.27) and elementary estimates, in particular the Gagliardo-Nirenberg inequality in normal direction of Γmε, the Hölder and Young inequality as well as (2.29). □

Lemma 2.4

(Properties of ϑ) For ϑ=ϑ(mε) defined after (2.26) the following properties hold: Regularity: it holds that ϑ∈C1([0,T0],C1(Ω¯))∩C0([0,T0];C3(Ω¯))

and we have uniform bounds with respect to mε.

Coercivity and consistency: we have 2.30 cmin{dΓmε,1}≤|ϑ|≤Cmin{dΓmε,1}for somec,C>0,

2.31 ϑ>0inΩmε,+,ϑ<0inΩmε,-.

Evolution equation: it holds that 2.32 |(∂t+B·∇)ϑ|≤Cmin{dΓmε,1}a.e. onΩ×(0,T0).

Proof

The regularity and uniform bounds for ϑ follow directly from the ones for dΓmε by Theorem 7.9. The estimates (2.30)–(2.31) follow directly from the definition of ϑ and the properties of ϑ¯. Moreover, (2.32) is shown via the chain rule and (∂t+B·∇)dΓmε=0 on Γmε, cf. (2.24). □

Stability Estimate and Convergence Result

In this section we formulate our main results on stability and quantitative convergence for solutions of the Navier–Stokes/Allen–Cahn system (1.1a)–(1.1e) towards solutions of the classical two-phase Navier–Stokes equation (1.2a)–(1.2g) for suitable scalings of the mobility mε.

We obtain the following stability result:

Theorem 3.1

(Stability Estimate) Let mε:=m0εβ>0, where m0>0 and β∈(0,2) are fixed. Let T0>0 be such that the two-phase Navier–Stokes system with surface tension (1.2a)–(1.2g) has a smooth solution (v0±,p0±,Γ) on [0,T0]. Moreover, let (vmε±,pmε±,Γmε) be strong solutions to the modified system (7.1)–(7.7) for ε>0 small, cf. Theorem 7.9 below. Furthermore, let (vε,φε) for ε>0 be energy-dissipating weak solutions to the Navier–Stokes/Allen–Cahn system (1.1a)–(1.1e) on [0,T0] for the constant mobility mε as in Remark 1.2 starting from initial data with energy uniformly bounded with respect to ε. We use the notation from Sect. 2, in particular we define the relative energy functional E[vε,φε|vmε,Γmε] and the bulk error functional Ebulk[φε|Γmε] as in (2.8) and (2.26).

Then for ε>0 small and a.e. T∈[0,T0] it holds that3.1 12‖∇vε-∇vmε‖L2(0,T;L2(Ω))2+E[vε,φε|vmε,Γmε](T)+Ebulk[φε|Γmε](T)≤eC(β,m0)TE[vε,φε|vmε,Γmε](0)+Ebulk[φε|Γmε](0)+Cε2mε.

The proof is done via a Gronwall-type argument in Sect. 6, using coercivity properties for the relative entropy and the bulk error in Sect. 2 as well as preliminary estimates in Sects. 4 and 2.2. As an immediate consequence of Theorem 3.1 we get,

Corollary 3.2

(Convergence Result) Let the assumptions of Theorem 3.1 hold. Moreover, let the initial data satisfy3.2 E[vε,φε|v0,Γ](0)+Ebulk[φε|Γ](0)≤Cε2mε

for ε>0 small. Then for ε>0 small and a.e. T∈[0,T0] it holds that3.3 ‖(vε-vmε)(.,T)‖L2(Ω)+‖σχΩTmε,+-ψε(.,T)‖L1(Ω)≤CeC(β,m0)Tεmε,‖∇vε-∇vmε‖L2(0,T;L2(Ω))≤CeC(β,m0)Tεmε

and for the true limit we obtain3.4 ‖(vε-v0)(.,T)‖L2(Ω)+‖σχΩT+-ψε(.,T)‖L1(Ω)≤CeC(β,m0)Tεmε+mε,‖∇vε-∇v0‖L2(0,T;L2(Ω))≤CeC(β,m0)Tεmε+mε.

Finally, there are well-prepared initial data (v0,ε,c0,ε) for ε>0 small in the sense that (3.2) is satisfied, even with rate ε2.

Proof

Estimate (3.3) is a direct consequence of Theorem 3.1 and the coercivity estimates (2.12) from Lemma 2.1 and (2.27) from Lemma 2.3. Then (3.4) follows from Theorem 7.7. Because of (vmε±,Γmε)(0)=(v0,0±,Γ0), we conclude that E[vε,φε|vmε,Γmε](0)=E[vε,φε|v0,Γ](0) and Ebulk[φε|Γmε](0)=Ebulk[φε|Γ](0). The existence of well-prepared initial data is well-known, cf. [17, Proof of Theorem 1] and [23, Appendix B]. □

Relative Entropy Estimate

Let the assumptions and notation of Sect. 2 be in place. In this section we derive an inequality for the relative entropy E[vε,φε|vmε,Γmε] defined in (2.8) that can be employed later to obtain a Gronwall-type estimate. We use the notation4.1 Hε:=-εΔφε+1εW′(φε).

Preliminary Relative Entropy Inequality

We derive the first important estimate for the relative entropy.

Lemma 4.1

(Relative Entropy Inequality) Let the assumptions and notation of Sect. 2 be valid and Hε be defined as in (4.1). More precisely, let (vmε±,pmε±,(Γtε)t∈[0,T0]) for mε>0 small be solutions of the adjusted two-phase Navier–Stokes equation (7.1)–(7.7) on [0,T0], cf. Theorem 7.9 below. Additionally, let (vε,φε) for ε>0 small be energy dissipating weak solutions to the Navier–Stokes/Allen–Cahn system (1.1a)–(1.1e) on [0,T0] with constant mobility mε>0 as in Remark 1.2. Finally, let Γmε,Ωmε,±,vmε be as in (2.1), σ, ψε, nε be as in (2.6)–(2.7), ξ, B be as in (2.11), E[vε,φε|vmε,Γmε] be as in (2.8). Then for a.e. T∈[0,T0] we obtain 4.2a E[vε,φε|vmε,Γmε](T)≤E[vε,φε|vmε,Γmε](0)-∫0T∫Ω|∇vε-∇vmε|2dxdt-∫0T∫Ωmε2εHε+2W(φε)∇·ξ2dxdt

4.2b -∫0T∫Ωmε2εHε-B-vmεmε·ξε|∇φε|2dxdt

4.2c -∫0T∫Ω(vε-vmε)·((vε-vmε)·∇)vmεdxdt-∫0T∫Ω(σχΩmε,+-ψε)((vε-vmε)·∇)(∇·ξ)dxdt+∫0T∫ΩmεB-vmεmε·ξ+∇·ξ2ε|∇φε|2dxdt

4.2d +∫0T∫Ωmε|∇·ξ|22W(φε)ε-ε|∇φε|2dxdt-∫0T∫Ω1εHε+2W(φε)∇·ξ(vmε-B)·(nε-ξ)ε|∇φε|dxdt

4.2e -∫0T∫Ω(∂tξ+(B·∇)ξ+(Id-ξ⊗ξ)(∇B)⊤ξ)·(nε-ξ)|∇ψε|dxdt-∫0T∫Ω(ξ⊗ξ(∇B)⊤ξ)·(nε-ξ)|∇ψε|dxdt

4.2f -∫0T∫Ω(∂t+B·∇)|ξ|2|∇ψε|dxdt-∫0T∫Ω∇B:(ξ-nε)⊗(ξ-nε)|∇ψε|dxdt+∫0T∫Ω(∇·B)(1-ξ·nε)|∇ψε|dxdt+∫0T∫Ω(∇·B)12ε|∇φε|-1ε2W(φε)2dxdt-∫0T∫Ω(nε⊗nε-ξ⊗ξ):∇B(ε|∇φε|2-|∇ψε|)dxdt-∫0T∫Ωξ⊗ξ:∇B(ε|∇φε|2-|∇ψε|)dxdt.

Proof

Unlike in [22], our choice of B does not have compact support, but due to the boundary condition B|∂Ω=0 by Lemma 2.2 we observe that analogous computations as in the proof of [22, Proposition 6] may be carried out. □

Remark 4.2

The choice of B in (2.11), i.e. B:=vmε+mεHnη~mε with the plateau cutoff η~mε, is natural in order to control the term (4.2c), i.e. ∫0T∫Ωmε|B-vmεmε·ξ+∇·ξ|2ε|∇φε|2dxdt. Note that in Hensel, Liu [22] the projected velocity field (n·vmε|PΓmε)n is used within the definition of B instead. This is not possible here because one would then obtain from (4.2c) a remainder of the form1mε∫0T∫Ωmin{dΓmε2,1}ε|∇φε|2dxdt,

which is only controlled by Cmε∫0TE[φε|Γmε](t)dt due to (2.16). However, with the new choice of B it is not clear anymore how to estimate the last term (4.2f), i.e., ∫0T∫Ωξ⊗ξ:∇B(ε|∇φε|2-|∇ψε|)dxdt.

To this end we rewrite and estimate the term (4.2f) in a novel way in the following Sect. 4.2. The idea is to write ξ⊗ξ:∇B as a normal derivative and use integration by parts in a suitable way. The other terms in the relative entropy estimate from Lemma 4.1 will turn out to be controllable with the choice of B in (2.11), the coercivity properties of the relative entropy and the bulk error functional from Sects. 2.1 and 2.2, cf. Sect. 6 below.

The Remaining Problematic Term

For a.e. t∈[0,T0] consider a 1-Lipschitz-function h=hε(.,t):Γtε→(-δ,δ). This function h will be constructed in Sect. 4.3 below, and it will be used to approximate a suitable level set of φε(.,t). Moreover, the energy outside a strip around the graphΓt,hε:={s+hε(s,t)n(s,t):s∈Γtε}

over Γtε determined by h will be estimated in Sect. 4.4 below. Let us define shifted tubular neighbourhoods for a.e. t∈[0,T0],δ~∈(0,δ]:4.3 Γt,hε(δ~):={x∈Γtε(2δ):dΓmε(x,t)∈hε(PΓmε(x,t),t)+(-δ~,δ~)}.

In order to estimate the problematic term from Remark 4.2, we need the following lemma whose proof is based on integration by parts and will be postponed.

Lemma 4.3

For a.e. t∈[0,T0] let h=hε(.,t):Γtε→(-δ,δ) be a 1-Lipschitz-function. Moreover, let δε∈(0,δ2] for ε>0 small, Γt,hε(δε) be defined as in (4.3) and η~∈Cc0,1(Γt,hε(δε)). Then for a.e. t∈[0,T0] it holds that 4.4a ∫Γt,hε(δε)ε2|∂nφε|2-1εW(φε)∂nη~dx

4.4b =∫Γt,hε(δε)ε2|∂nφε|2+1εW(φε)-2W(φε)∂nφεη~∇·ndx

4.4c +∫Γt,hε(δε)Hε+2W(φε)∇·n∂nφεη~dx

4.4d -∫Γt,hε(δε)ε∇τφε·∇τη~∂nφεdx

4.4e -∫Γt,hε(δε)ε∇τφε·(∇n⊤∇τφε)η~dx

4.4f +∫Γt,hε(δε)ε2|∇τφε|2η~∇·n+∂nη~dx.

We obtain the following important estimate:

Lemma 4.4

For a.e. t∈[0,T0] let h=hε(.,t):Γtε→(-δ,δ) be a 1-Lipschitz-function. Moreover, let δε∈(0,δ2] for ε>0 small and let Γt,hε(δε) be defined as in (4.3). Then for a.e. t∈[0,T0] it holds that 4.5a ∫Ωξ⊗ξ:∇Bε|∇φε|2-|∇ψε||(x,t)dx

4.5b ≤C∫Γtε(2δ)\Γt,hε(δε2)ε|∇φε|2+1εW(φε)|(x,t)dx

4.5c +C∫Γt,hε(δε)εmε|∂nφε|2|dΓmε-h(PΓmε)|2|(x,t)dx

4.5d +mε4ε∫Γt,hε(δε)Hε+2W(φε)∇·ξ2|(x,t)dx

4.5e +C∫Γt,hε(δε)(|h|PΓmε|2+|∇τ(h|PΓmε)|2)ε|∇φε|2+1εW(φε)|(x,t)dx

4.5f +CE[φε|Γmε](t),

where C>0 is independent of ε and t.

Proof

First of all, it suffices to show the estimate with (4.5a) replaced by4.6 ∫Γtε(2δ)ξ⊗ξ:∇Bε2|∂nφε|2-1εW(φε)|(x,t)dx

because of the cutoff for ξ in (2.11), the identityε|∇φε|2-|∇ψε|=ε2|∇φε|2-1εW(φε)+12ε|∇φε|-1ε2W(φε)2,

the coercivity estimate (2.14) and finally the control of the tangential gradient in (2.15) in terms of the relative entropy. In order to rewrite the expression by Lemma 4.3, we use the following idea: for a.e. t∈[0,T0] we write ξ⊗ξ:∇B(.,t) as ∂nη, where η=η(.,t) is defined asη(x):=∫h(PΓmε(x,t))dΓmε(x,t)ξ⊗ξ:∇B(PΓmε(x,t)+rn(PΓmε(x,t),t),t)drforx∈Γtε(2δ).

Moreover, to avoid boundary terms on ∂Γtε(δε) we introduce a smooth α:R→[0,1] with α≡1 on [-12,12] and α≡0 on R\[-34,34] and set for a.e. t∈[0,T0]η~=η~(.,t):=(α~η)(.,t),α~(x,t):=αdΓmε(x,t)-h(PΓmε(x,t))δε,forx∈Γtε(2δ).

Note that Theorem 7.9 and the regularity and uniform bounds in Theorem 2.2 yield that α~(.,t), η~(.,t) and η(.,t) are in C0,1(Ω¯). We rewrite things as follows:4.7 ξ⊗ξ:∇B(.,t)=∂nη=(1-α~)∂nη-∂nα~η+∂nη~.

Because of the definition of α~, it holds that 1-α~=0 on Γt,hε(δε2). Moreover,∂nα~=1δεα′dΓmε(x,t)-h(PΓmε(x,t))δε=0onΓt,hεδε2∪Γtε(2δ)\Γt,hε34δε

and |η|≤Cδε on Γt,hε(δε) for a.e. t∈[0,T0]. Therefore the first two terms on the right hand side of (4.7) yield contributions in (4.6) that can be estimated by (4.5b). Hence we can replace ξ⊗ξ:∇B(.,t) by ∂nη~ in (4.6) for a.e. t∈[0,T0].

For the remaining term4.8 ∫Γt,hε(δε)∂nη~ε2|∂nφε|2-1εW(φε)|(x,t)dx

one can apply integration by parts on Γt,hε(δε) for a.e. t∈[0,T0] and rewrite in a suitable way, cf. Lemma 4.3. Let us estimate the corresponding terms (4.4b)–(4.4f) from Lemma 4.3 now. Note that (4.4b) and (4.4e)–(4.4f) are directly controlled by (4.5f) because of the coercivity properties (2.15) of the relative entropy.

Moreover, (4.4c) with ∇·ξ instead of ∇·n can be estimated by (4.5c) and (4.5d) due to Young’s inequality and |η~|(.,t)≤C|dΓmε-h(PΓmε)|(.,t). Because of the definition of ξ in (2.11) it holds that∇·n=(1-ηmε)∇·n-1δη¯′dΓmεδ+∇·ξ,

where |1-ηmε|≤C|dΓmε|2 and |η¯′dΓmεδ|≤C|dΓmε|. Hence, from exchanging ∇·n with ∇·ξ, we obtain the error term∫Γt,hε(δε)W(φε)|∂nφε|(|dΓmε|2+|h(PΓmε)|2)|(x,t)dx,

which is controlled by (4.5e) and (4.5f) due to the bound W(φε)|∂nφε|≤|∇ψε| and (2.16). Hence (4.4c) is estimated.

Finally, it remains to estimate (4.4d). It holds ∇τη~=∇τα~η+α~∇τη due to η~=α~η, where we have because of (2.5)∇τα~=-1δεα′dΓmε(.,t)-h(PΓmε(.,t))δε∇τ(h(PΓmε))|(.,t),∇τη=-∂nη|(PΓmε(.,t)+h(PΓmε(.,t))n(PΓmε(.,t),t)∇τ(h(PΓmε))|(.,t)+∫h(PΓmε(.,t))dΓmε(.,t)∇τξ⊗ξ:∇B(PΓmε+rn(PΓmε),t)|(.,t)dr.

Due to Lemma 2.2 we obtain|∇τη~|≤C|dΓmε|+|h|PΓmε|+|∇τ(h|PΓmε)||(.,t).

Altogether, (4.4d) is controlled by (4.5e) and (4.5f) due to Young’s inequality, (2.15) and (2.16). This shows Lemma 4.4. □

Proof of Lemma 4.3

First, we use ∂nη~=n·∇η~=∑i=1dni∂xiη~ and integration by parts on Γt,hε(δε) with η~∈Cc0,1(Γt,hε(δε)) for a.e. t∈[0,T0]. This yields∫Γt,hε(δε)ε2|∂nφε|2-1εW(φε)∂nη~dx=-∫Γt,hε(δε)ε2|∂nφε|2-1εW(φε)(∇·n)η~dx-∫Γt,hε(δε)∂nε2|∂nφε|2-1εW(φε)η~dx

for a.e. t∈[0,T0]. Note that ∂n2φε=((n·∇)n)·∇φε+n⊗n:D2φε=n⊗n:D2φε. Therefore-∫Γt,hε(δε)∂nε2|∂nφε|2-1εW(φε)η~dx=-∫Γt,hε(δε)εΔφε-1εW′(φε)∂nφεη~dx+∫Γt,hε(δε)εId-n⊗n:D2φε∂nφεη~dx

for a.e. t∈[0,T0]. In order to rewrite the last term, we compute∇·(Id-n⊗n)∇φε∂nφεη~=-(∇·n)n·∇φε∂nφεη~+(Id-n⊗n):D2φε∂nφεη~+∇τφε·∇(∂nφεη~),

where we used ∇·(Id-n⊗n)=-(∇·n)n. Because of∇(∂nφε)=∇(n·∇)φε=(∇n)⊤∇φε+(n·∇)∇φε

and integration by parts, we obtain, for a.e. t∈[0,T0],∫Γt,hε(δε)εId-n⊗n:D2φε∂nφεη~dx=∫Γt,hε(δε)ε|∂nφε|2(∇·n)η~dx-∫Γt,hε(δε)ε∇τφε·∇η~∂nφεdx-∫Γt,hε(δε)ε∇τφε·(∇n⊤∇φε)η~dx-∫Γt,hε(δε)ε∇τφε·(n·∇)∇φεη~dx.

Note that in (4.4b)–(4.4c) the 2W(φε)-terms cancel and were added for convenience. Moreover, one can directly prove that ∇n⊤∇φε=∇n⊤∇τφε. Hence it is left to show that for a.e. t∈[0,T0]4.9 -∫Γt,hε(δε)ε∇τφε·(n·∇)∇φεη~dx=∫Γt,hε(δε)ε2|∇τφε|2η~∇·n+∂nη~dx.

To this end we use n·∇=∑i=1dni∂xi and integration by parts for ∂xi. This yields-∫Γt,hε(δε)ε∇τφε·(n·∇)∇φεη~dx=-12∫Γt,hε(δε)ε(n·∇)|∇τφε|2η~dx=∫Γt,hε(δε)ε2|∇τφε|2∇·(η~n)dx,

where we used (n·∇)(Id-n⊗n)=0 in the second step. Therefore (4.9) holds and this shows Lemma 4.3. □

Parametrization of the Majority of the Interface

In order for our estimate on the problematic term (4.2f) provided by Lemma 4.4 to work, we need to represent a suitable level set of the phase-field φε as approximately a graph of a small function hε over the surface Γtε, as only then the term (4.5c) becomes controllable in terms of a constant ε2/mε and the relative entropy (see Corollary 4.11 below for details).

In this section we construct for a.e. t∈[0,T0] a 1-Lipschitz-function h=hε(.,t):Γtε→(-δ,δ) that is used to approximate a suitable level set of φε(.,t). To this end, we need the following proposition about local interface errors of a BV-set compared to the strong interface Γtε from Sect. 2.

Proposition 4.5

Let T0>0, δ>0 and Γmε, Ωmε,±, vmε for mε>0 small be as in Sect. 2. Moreover, let n be the extension of the normal from (2.2) and ξ be defined as in (2.11). Let t∈[0,T0] be fixed and χ~∈BV(Rd;{0,1}) be arbitrary. We set χ:=χΩtmε,+.

Let θ:[0,∞)→[0,1] be a smooth cutoff with θ≡0 outside of [0,12] and θ≡1 in [0,14]. We define the local height of the one-sided interface errors ht±=ht,ε±:Γtε→[0,δ2] in ±n-direction asht±(s):=∫0∞(χ-χ~)(s+yn(s,t))θ(yδ)dyfor a.e.s∈Γtε.

Then ht± are BV-functions and we denote the distributional tangential derivative by Dtanht±, by ∇tanht± the density of the absolutely continuous part of Dtanht± with respect to Hd-1 and by Dsht± the singular part. Finally, let G~t:={s+(ht+-ht-)(s)n(s,t):s∈Γtε} denote the graph of ht+-ht-.

Then we have the following estimates with constants independent of χ~ and t∈[0,T0]:4.10 ∫Γtε|ht±|2dHd-1≤C∫Rd|χ-χ~|mindΓtε,1dx

as well as4.11 ∫Γtεmin{|∇tanht±|2,|∇tanht±|}dHd-1+|Dsht±|(Γtε)≤C∫Rd(1-ξ(.,t)·∇χ~|∇χ~|)d|∇χ~|+C∫Rd|χ-χ~|mindΓtε,1dx,

and4.12 ∫Rd\G~t1d|∇χ~|≤C∫Rd(1-ξ(.,t)·∇χ~|∇χ~|)d|∇χ~|+C∫Rd|χ-χ~|mindΓtε,1dx.

Proof

The assertions up to (4.12) can be shown analogously to [16, Proposition 26a–b]. Moreover, (4.12) may be established readily using the arguments for statement c) of [16, Proposition 26]. □

The plan is to apply the previous Proposition 4.5 to a suitable level set of φε(.,t) for a.e. t∈[0,T0]. The latter is selected in the following lemma:

Lemma 4.6

Let t∈[0,T0] be such that φε(.,t)∈H1(Ω). Then the relative entropy E[φε|Γmε](t) from (2.9) is well-defined and finite. Moreover, there exists a level b(t)=bε(t)∈(-12,12) such that the corresponding super-level set Sb(t):={x∈Ω:φε(x,t)>b(t)} is a set of finite perimeter (possibly empty) and satisfies with ξ as in (2.11) and ψ as in (2.6) the estimate∫Rd(1-ξ(.,t)·∇χSb(t)|∇χSb(t)|)d|∇χSb(t)|≤2ψ(12)-ψ(-12)E[φε|Γmε](t).

Proof

Let t be as in the lemma. Then E[φε|Γmε](t) is well-defined and finite because of φε(.,t)∈H1(Ω). Moreover, φε(.,t)∈BV(Ω) yields together with the coarea-formula for BV-functions, cf. Ambrosio, Fusco, Pallara [10, Theorem 3.40] that Sb:={x∈Ω:φε(x,t)>b} is a set of finite perimeter (possibly empty) for a.e. b∈R. Moreover, we use the coarea-formula for BV-functions [10, Theorem 3.40] applied to ψε(.,t)=ψ(φε(.,t))∈H1(Ω) (together with an approximation argument by simple functions) and obtainE[φε|Γmε](t)≥∫Ω|∇ψε|-ξ·∇ψε(.,t)dx=∫Rd1-ξ·∇φε|∇φε|(.,t)|∇ψε(.,t)|dx=∫0σ∫Rd1-ξ·∇φε|∇φε|(.,t)d|∇χSψ-1(r)|dr.

This shows the claim by a contradiction argument. □

We combine Proposition 4.5 and Lemma 4.6 in the following lemma:

Lemma 4.7

Let t∈[0,T0] be fixed such that φε(.,t)∈H1(Ω). Let the level b(t)=bε(t)∈(-12,12) and the super-level set Sb(t) be as in Lemma 4.6. We set χ:=χΩtmε,+. Then there is a 1-Lipschitz function ht=ht,ε:Γtε→(-δ,δ) subject to the estimate4.13 ∫Γtε|ht|2+|∇tanht|2dHd-1≤CE[φε|Γmε](t)+C∫Rd|χ-χSb(t)|mindΓtε,1dx

such that the graph Gt:={s+ht(s)n(s,t):s∈Γtε} approximates the reduced boundary of Sb(t) in the following sense:4.14 ∫Rd\Gt1d|∇χSb(t)|≤CE[φε|Γmε](t)+C∫Rd|χ-χSb(t)|mindΓtε,1dx.

Finally, constants in the estimates (4.13)–(4.14) are independent of φε and t.

Proof

By Lemma 4.6 it holds χ~:=χSb(t)∈BV(Rd;{0,1}). Hence by Proposition 4.5 applied for χ~ there exist BV-functions ht±:Γtε→[0,δ2] such that (4.10)–(4.12) hold. We set h~t:=ht+-ht-. It remains to modify h~ to obtain a 1-Lipschitz function. To do so, we use a standard Lipschitz truncation strategy (see e.g. [9, 15, Section 6.6.3] or [11]): we consider the maximal operator M over Γtε, which can be defined as in [36, Chapter I, §8.1 (ii)] since Hd-1⌊Γtε satisfies the doubling condition with respect to the geodesic balls on Γtε. Then there is some C1>0 such that|u(s1)-u(s2)|≤C1|s1-s2|M|Dtanu|(s1)+M|Dtanu|(s2)forHd-1-a.e.s1,s2∈Γtε

for all u∈BV(Γtε) with some C1>0 independent of u and small mε. More precisely, in the case that Γtε is replaced by Rd-1 this inequality is shown in [11, Lemma 2(c)]. Then the estimate in the present case can be shown by localization. Moreover, because of the continuous dependence on t∈[0,T] and mε∈[0,m0] (cf. Theorem 7.7), the constant can be chosen uniformly in t∈[0,T0], mε∈[0,m0] for ε>0 sufficiently small. Using the precise representative for h~t we defineht(s):=h~t(s)for alls∈At:={s∈Γtε:(M|Dtanh~t|)(s)≤c¯},

where c¯>0 is a small constant to be determined. The so-defined function h is Lipschitz on At with Lipschitz-constant bounded by c1c¯. We extend h to all of Γtε as a Lipschitz function via the standard extension, cf. [10, Proposition 2.12]), i.e. ht(s):=infs~∈At{h~t(s~)+Lip(h~t)|s-s~|},

where the Lipschitz constant stays the same. Hence for c¯ small enough (independent of h~t, ht, t and small mε), we obtain that ht is 1-Lipschitz and bounded by 3δ4. Moreover, using the weak L1-estimate for the maximal operator we obtain4.15 Hd-1Γtε\At≤C∫Γtε|∇tanh~t|dHd-1+|Dsh~t|(Γtε)≤CE[φε|Γmε](t)+C∫Rd|χ-χ~|mindΓtε,1dx,

where we used (4.11) and Lemma 4.6 in the last step. Because of |∇tanht|≤C as well as by (4.11) and the fact that Dtanh~t=∇tanh~t=∇tanht a.e. on At, this establishes the bound∫Γtε|∇tanht|2dHd-1≤CE[φε|Γmε](t)+C∫Rd|χ-χ~|mindΓtε,1dx.

Hence the |ht|2-part of the bound (4.13) is left to prove. The latter follows because of (4.15), the boundedness of ht and (4.11). Finally, (4.14) follows from (4.12) in Proposition 4.5, the estimate (4.15) and Lemma 4.6. □

Corollary 4.8

Let t∈[0,T0] be fixed such that φε(.,t)∈H1(Ω), let b(t)=bε(t),Sb(t) be as in Lemma 4.6 and let ht=ht,ε:Γtε→(-δ,δ) be as in Lemma 4.7. Then, with χ:=χΩtmε,+, it holds that4.16 ∫Γtε(2δ)(|ht|PΓmε|2+|∇τ(ht|PΓmε)|2)ε|∇φε|2+1εW(φε)|(x,t)dx≤CE[φε|Γmε](t)+C∫Rd|χ-χSb(t)|mindΓtε,1dx.

Proof

Using of the identity (2.5) for ∇τ, we can exchange ∇τ(ht|PΓmε) by (∇tanht)|PΓmε in the above estimate. Moreover, since both ht and ∇tanht are uniformly bounded, we can consider the estimate with ∂nφε instead of ∇φε, since tangential derivatives are controlled by (2.15). Then we apply an integral transformation with the tubular neighbourhood coordinates to obtain∫Γtε(2δ)(|ht|PΓmε|2+|(∇tanht)|PΓmε|2)ε|∂nφε|2+1εW(φε)|(x,t)dx=∫Γtε(|ht|2+|∇tanht|2)(s)∫-2δ2δε|∂r(φε|Xmε)|2+1εW(φε|Xmε)|(r,s,t)Jt(r,s)drds,

where the factor Jt=Jt(mε) satisfies for some cJ,CJ>0 independent of t and mεcJ≤Jt(r,s)≤CJfor alls∈Γtε,r∈(-δ,δ).

Let σ=ψ(1) be as in (2.6). For s∈Γtε such that the inner integral with respect to r∈(-2δ,2δ) is less or equal 4σ, the desired estimate follows from Lemma 4.7. For s∈Γtε for which this is not the case, we use that |ψ(φ)|≤σ for all φ∈[-1,1] and therefore∫-2δ2δ∂rψ(φε|Xmε(r,s,t))dr≤2σ.

Hence it follows for such s∈Γtε that∫-2δ2δε|∂r(φε|Xmε)|2+1εW(φε|Xmε)-∂rψ(φε|Xmε)|(r,s,t)Jt(r,s)dr≥cJ∫-2δ2δε|∂r(φε|Xmε)|2+1εW(φε|Xmε)-∂rψ(φε|Xmε)|(r,s,t)dr≥cJ2∫-2δ2δε|∂r(φε|Xmε)|2+1εW(φε|Xmε)|(r,s,t)dr≥cJ2CJ∫-2δ2δε|∂r(φε|Xmε)|2+1εW(φε|Xmε)|(r,s,t)Jt(r,s)dr.

Therefore the contribution in this case can be estimated with (2.15) and by using that ht and ∇tanht are uniformly bounded. □

Finally, in order to control the second term in (4.16) in the end, we need the following lemma:

Lemma 4.9

Let t∈[0,T0] be fixed such that φε(.,t)∈H1(Ω) and let b(t)=bε(t),Sb(t) be as in Lemma 4.6. Then with χ:=χΩtmε,+ and Ebulk[φε|Γmε] from Sect. 2.2 it follows that4.17 ∫Rd|χ-χSb(t)|mindΓtε,1dx≤CEbulk[φε|Γmε](t).

Proof

The integrand is zero on Rd\Ω¯. Moreover, note that (2.27) yields∫Ωσχ-ψε(.,t)min{dΓtε,1}dx≤CEbulk[φε|Γmε](t).

Hence we obtain∫Ωtmε,+σ-ψε(.,t)min{dΓtε,1}dx+∫Ωtmε,-ψε(.,t)min{dΓtε,1}dx≤CEbulk[φε|Γmε](t).

For a.e. x∈Ωtmε,+∩Sb(t) it holds |χ-χSb(t)|(x)=0. For a.e. x∈Ωtmε,+\Sb(t) we have |χ-χSb(t)|(x)=1 and ψε(x,t)≤ψ(b(t))≤ψ(12). Hence |σ-ψε(x,t)|≥|σ-ψ(12)|>0 since b(t)∈(-12,12). This shows the estimate on Ωtmε,+. Moreover, for a.e. x∈Ωtmε,-\Sb(t) we have |χ-χSb(t)|(x)=0. Finally, for a.e. x∈Ωtmε,-∩Sb(t) it holds |χ-χSb(t)|(x)=1 and ψε(x,t)>ψ(b(t))≥ψ(-12)>0. This yields the estimate on Ωtmε,-. □

Estimate of Energy Away from Strip Around Level Set

In this section we estimate the remainder terms (4.5b)–(4.5c) from Lemma 4.4 involving the energy density for φε. Therefore we show the following lemma:

Lemma 4.10

Let t∈[0,T0] be fixed such that φε(.,t)∈H1(Ω) and let b(t)=bε(t),Sb(t) be as in Lemma 4.6. We set χ=χΩtmε,+. Moreover, let h=ht,ε:Γtε→(-δ,δ) be as in Lemma 4.7. We define the shifted tubular neighbourhood Γt,hε(δ~) for δ~∈(0,δ] as in (4.3). For κ>0 fixed and ε>0 small we obtain with some constants c,C>0 independent of φε, h, t and κ∫Γtε(δ)\Γt,hε(κε)ε|∇φε|22+W(φε)εdx≤CE[φε|Γmε](t)+∫Rd|χ-χSb(t)|mindΓtε,1dx+e-cκ.

Proof

Let Gt be the graph of h=ht,ε as in Lemma 4.7. Then Rt:=PΓtε(Gt∩supp|∇χSb(t)|)⊆Γtε satisfiesHd-1(Γtε\Rt)≤CE[φε|Γmε](t)+∫Rd|χ-χSb(t)|mindΓtε,1dx,

where we used that Hd-1((Γtε\Rt)∩PΓtε(supp|∇χSb(t)|)) is controlled by (4.14) and that for the remaining part R~t:=(Γtε\Rt)\PΓtε(supp|∇χSb(t)|) we haveHd-1(R~t)=∫R~t∫δ2δ2δdrdHd-1(s)≤C∫Rd|χ-χSb(t)|mindΓtε,1dx

since s∈R~t implies |χ-χSb(t)|(s+rn(s,t))=1 either for all r∈(-δ,-δ2) or for all r∈(δ2,δ).

Moreover, for s∈Rt it holds that4.18 φε(s+h(s)n(s,t),t)=b(t).

The strategy is to use an ODE inequality argument in normal direction using the control of the relative entropy over the equipartition error. More precisely, by (2.15) we have4.19 ∫Γtε∫-δδ12ε∂rφ~ε-1ε2W(φ~ε)2(.,t)Jt(r,s)drdHd-1(s)≤E[φε|Γmε](t),

where φ~ε:=φε|Xmε(.,t) with Xmε as in Sect. 2 and the factor Jt=Jt(mε) appears due to an integral transformation and satisfies for some cJ,CJ>0 independent of t and mεcJ≤Jt(r,s)≤CJfor alls∈Γtε,r∈(-δ,δ).

With f:=ε∂rφ~ε-1ε2W(φ~ε) it holds that∂rφ~ε=1ε2W(φ~ε)-1εf,

where W(φ)≥c(1-φ)(1+φ)≥c′(1-φ) if φ≥-34. Hence if φ~ε≥-34 we have∂r(1-φ~ε)2=2(1-φ~ε)(-∂rφ~ε)≤-2c′ε(1-φ~ε)2+2εf(1-φ~ε)≤-c′ε(1-φ~ε)2+C1|f|2.

We multiply the preceding inequality by e-cε(r-h(s)) and integrate from h(s) to r. Then we obtain for s∈Rt because of (4.18)4.20 (1-φ~ε)2(s,r)≤e-cε(r-h(s))(1-b(t))2+C1cJ∫-δδ|f(s,r)|2Jt(r,s)dr

for all r∈[h(s),δ) provided that φ~ε(r~,s)≥-34 for all r~∈[h(s),r). We defineSt:=s∈Rt:C1cJ∫-δδ|f(s,r)|2Jt(r,s)dr≤14.

Then, due to b(t)∈(-12,12) and (4.20) we obtain φ~ε(r,s)≥-34 for all r∈[h(s),δ) and s∈St. Analogously one shows φ~ε(r,s)≤34 for all r∈(-δ,h(s)] and s∈St. Moreover, note that because of (4.19) and the definition of f we haveHd-1(Rt\St)≤4cJC1∫Rt∫-δδ|f(s,r)|2Jt(r,s)drdHd-1(s)≤8cJC1E[φε|Γmε](t).

We use an integral transformation to obtain∫Γtε(δ)\Γt,hε(κε)ε2|∇φε|2+W(φε)εdx=∫Γtε∫h(s)+κεδε2|∇φε|2|Xmε+W(φ~ε)εJt(r,s)drdHd-1(s)+∫Γtε∫-δh(s)-κεε2|∇φε|2|Xmε+W(φ~ε)εJt(r,s)drdHd-1(s).

We use the definition (2.9) to derive an estimate. First, for the contribution over Γtε\St we use an integral transformation as above for the last term in (2.9). Since ξ points in normal direction by definition (2.11), the inner integral is uniformly bounded because |ψ|≤σ. Hence the estimates for Hd-1(Γtε\Rt) and Hd-1(Rt\St) yield that the contribution over Γtε\St is suitably estimated by the right hand side of the estimate in the Lemma. For the remaining part over St, we note that, due to integration by parts and since ξ points in normal direction by definition (2.11), we have∫St∫h(s)+κεδξ|Xmε·∇(ψε|Xmε-σ)Jt(r,s)drHd-1(s)≤C∫St∫h(s)+κεδ(ψ(φ~ε)-σ)drdHd-1(s)+∫St(ψ(φ~ε)-σ)Jt(r,s)r=h(s)+kεδdHd-1(s)≤Ce-cκ+‖f‖L2(Γtε(δ))2≤Ce-cκ+E[φε|Γmε](t),

where we have used|ψ(φ~ε(r,s))-σ|≤Ce-cε(r-h(s))+‖f(s,.)‖LJt(.,s)2(-δ,δ)2for allr∈[h(s),δ),s∈St

due to (4.20) and |ψ(r)-σ|≤C(1-r)2 for all r∈R. Analogously, one shows∫St∫-δh(s)-κεξ|Xmε·∇ψε|XmεJt(r,s)drHd-1(s)≤Ce-cκ+E[φε|Γmε](t).

Finally, we have ∫Γtε\St∫(-δ,h(s)-κε)∪(h(s)+κε,δ)ξ·∇ψεdx≤CHd-1(Γtε\St). Combining these bounds, this shows the claim. □

As a corollary of Lemma 4.10, we estimate the remaining term (4.5c).

Corollary 4.11

Let the assumptions and notation of Lemma 4.10 be in place and let C0≥1. Moreover, let C1>0 such that ∫Ωε|∇φε|22+W(φε)εdx≤C1. Then for all ε small with δε:=C0|logε|ε≤δ and for some uniform C>0 we have∫Γt,hε(δε)1mεε2|∇φε|2+1εW(φε)(x,t)dΓtε-h(PΓtε)2dx≤CC03|logε|3ε2mεE[φε|Γmε](t)+∫Rd|χ-χSb(t)|mindΓtε,1dx+CC1ε2mε.

Proof

Let C0≥1 be fixed, then for all ε sufficiently small it holds 2C0|logε|ε≤δ. We fix ε small. Let N∈N be such that C0|logε|≤N≤2C0|logε|. Then δε≤Nε≤δ and∫Γt,hε(Nε)1mεε|∇φε|22+W(φε)ε(x,t)dΓtε-h(PΓtε)2dx≤∑n=2NC∫Γt,hε(nε)\Γt,hε((n-1)ε)1mεε|∇φε|22+W(φε)ε(x,t)dΓtε-h(PΓtε)2dx+C∫Γt,hε(ε)1mεε|∇φε|22+W(φε)ε(x,t)dΓtε-h(PΓtε)2dx≤C∑n=2Nε2mεn2∫Γt,hε(nε)\Γtε((n-1)ε)ε|∇φε|22+W(φε)ε(x,t)dx+CC1ε2mε≤C∑n=2Nε2mεn2E[φε|Γmε](t)+∫Rd|χ-χSb(t)|mindΓtε,1dx+e-c(n-1)+CC1ε2mε≤Cε2mεN3E[φε|Γmε](t)+∫Rd|χ-χSb(t)|mindΓtε,1dx+CC1ε2mε,

where we used Lemma 4.10. This shows the claim. □

Bulk Error Identity

For the bulk error functional Ebulk[φε|Γmε] defined in (2.26) we have the following identity:

Lemma 5.1

(Bulk Error Identity) Let the assumptions of Sect. 2 hold. More precisely, let (vmε±,pmε±,(Γtε)t∈[0,T0]) for mε>0 small be solutions of the adjusted two-phase Navier–Stokes equation (7.1)–(7.7) on [0,T0], cf. Theorem 7.9 below. Moreover, let (vε,φε) for ε>0 small be energy dissipating weak solutions to the Navier–Stokes/Allen–Cahn system (1.1a)–(1.1e) on [0,T0] with constant mobility mε>0 as in Remark 1.2. Finally, let Γmε,Ωmε,±,vmε be as in (2.1), σ, ψε, nε be as in (2.6)–(2.7), ξ, B be as in (2.11), Ebulk[φε|Γmε] and ϑ be as in (2.26) and Hε be defined as in (4.1). Then, for all T∈[0,T0], 5.1a Ebulk[φε|Γmε](T)=Ebulk[φε|Γmε](0)+∫0T∫Ω(σχΩmε,+-ψε)(∂t+B·∇)ϑdxdt

5.1b +∫0T∫Ω(σχΩmε,+-ψε)ϑ∇·Bdxdt

5.1c -∫0T∫Ωϑ(B-vmε)·(nε-ξ)|∇ψε|dxdt

5.1d +∫0T∫Ω(σχΩmε,+-ψε)(vε-vmε)·∇ϑdxdt

5.1e -∫0T∫Ωϑ(B-vmε)·ξ(|∇ψε|-ε|∇φε|2)dxdt

5.1f +mε∫0T∫Ω1εHε-B-vmεmε·ξε|∇φε|ϑε|∇φε|dxdt

5.1g -mε∫0T∫Ωϑ1εHε+2W(φε)∇·ξε|∇φε|-2W(φε)εdxdt

5.1h -mε∫0T∫Ωϑ(∇·ξ)ε|∇φε|-2W(φε)ε2dxdt

5.1i +mε∫0T∫Ωϑε|∇φε|(∇·ξ)ε|∇φε|-2W(φε)εdxdt.

Proof

This can be shown analogously to [22, Lemma 7]. The mε-factors appear here because of the Allen–Cahn part (1.1c) through the equation for ∂tψε. Note that we use a different sign convention for ϑ compared to [22], this just changes the signs in all terms. □

Proof of Theorem 3.1(Stability Estimate)

Proof of Theorem 3.1

Up to (4.2c), (4.2d), (4.2e) and (4.2f) the terms in the estimate for the relative entropy from Lemma 4.1 can be estimated analogously to [22, Proof of Theorem 1]. Here note that ‖∇vε-∇vmε‖L2(0,T;L2(Ω))2 and (4.2a)–(4.2b) are terms with a good sign. At this point, let us estimate a few terms for the convenience of the reader. For example,∫0T∫Ω(σχΩmε,+-ψε)((vε-vmε)·∇)(∇·ξ)dxdt≤C∫0T∫Ω|σχΩmε,+-ψε||vε-vmε|dxdt,

where the latter term can be estimated with (2.28) by using 12‖∇vε-∇vmε‖L2(0,T;L2(Ω))2 for absorption. Moreover, due to (2.21) it holds that∫0T∫Ω(∂t+B·∇)|ξ|2|∇ψε|dxdt≤∫0T∫Ωmin{dΓmε2,1}|∇ψε|dxdt,

which is controlled due to (2.16). Finally, let us estimate∫0T∫Ω(nε⊗nε-ξ⊗ξ):∇B(ε|∇φε|2-|∇ψε|)dxdt≤C∫0T∫Ω1-nε·ξε|∇φε|2-|∇ψε|dxdt,

where we used |(nε⊗nε-ξ⊗ξ):∇B|=|nε⊗(nε-ξ):∇B+(ξ·∇)B·(nε-ξ)|≤|nε-ξ| and |nε-ξ|2=2(1-nε·ξ). Hence the above term is controlled via (2.17).

Let us consider the remaining four terms (4.2c), (4.2d), (4.2e) and (4.2f) for which new estimates are needed. First, the new choice (2.11) of ξ and B compared to [22, (75)–(76)] yields (2.20) and enables us to estimate (4.2c) with (2.16). Moreover, for the estimate of (4.2d) it remains to control∫0T∫Ω1mε(vmε-B)·(nε-ξ)2ε|∇φε|2dxdt

due to absorption with (4.2a). However, we have vmε-B=-mεHnη~mε by definition (2.11) of B, hence the term is controlled by mε∫0TE[vε,φε|vmε,Γmε](t)dt because of (2.16). Additionally, to estimate the term (4.2e) we write (ξ⊗ξ(∇B)⊤ξ)·(nε-ξ)=(ξ⊤(∇B)⊤ξ)ξ·(nε-ξ). Hence because of ξ·(nε-ξ)=nε·ξ-1+1-|ξ|2 and (2.10), the term (4.2e) is controlled via (2.16). Finally, we estimate (4.2f) with Lemma 4.4, where the 1-Lipschitz-function h=hε(.,t) for a.e. t∈[0,T0] is taken from Lemma 4.7 and we consider δε:=C0|logε|ε for some C0≥1 and ε small. It remains to estimate (4.5b)–(4.5e). First, note that (4.5d) is absorbed by (4.2a). Moreover, (4.5e) is controlled via Corollary 4.8 and Lemma 4.9. More precisely, we have∫Γt,hε(δε)(|h|PΓmε|2+|∇τ(h|PΓmε)|2)ε|∇φε|2+1εW(φε)|(x,t)dx≤CE[φε|Γmε](t)+C∫Rd|χ-χSb(t)|mindΓtε,1dx≤C(E[φε|Γmε](t)+Ebulk[φε|Γmε](t)),

where b(t)=bε(t) and Sb(t) are as in Lemma 4.6. Moreover, (4.5b) is suitably estimated by Lemma 4.10 provided that C0≥1 is large enough such that e-cC0|logε|≤ε2. Finally, (4.5c) is controlled by Corollary 4.11 and Lemma 4.9. Altogether, we obtain that12‖∇vε-∇vmε‖L2(0,T;L2(Ω))2+E[vε,φε|vmε,Γmε](T)≤E[vε,φε|vmε,Γmε](0)+Cε2mε+C|logε|3ε2mε+1∫0TE[vε,φε|vmε,Γmε](t)+Ebulk[φε|Γmε](t)dt.

Next, we estimate the terms on the right hand side in the identity for Ebulk[φε|Γmε] from Lemma 5.1. The terms (5.1a)–(5.1b) are controlled by the bulk error itself due to (2.32), (2.30) and (2.27). Moreover, (5.1c) and (5.1e) can be estimated by the relative entropy because of (2.16) and (2.17), respectively. The term (5.1d) is controlled by the bulk functional due to (2.28). Moreover, we estimate (5.1f)–(5.1g) via Young’s inequality in order to absorb one part with the positive terms (4.2b) and (4.2a), respectively. The remainders as well as (5.1h)–(5.1i) are controlled by the relative entropy because of (2.16) and (2.14). Finally, this yieldsEbulk[φε|Γmε](t)≤Ebulk[φε|Γmε](0)+C∫0TE[vε,φε|vmε,Γmε](t)+Ebulk[φε|Γmε](t)dt.

Let mε=m0εβ with m0>0 and β∈(0,2) be as in the theorem. Then it holds |logε|3ε2mε≤Cβm0 for some constant Cβ>0 independent of ε for ε>0 small. Finally, the Gronwall inequality yields Theorem 3.1. □

Existence for the Approximate Two-Phase Flow

In this section we consider the existence of strong solutions for the modified two-phase flow7.1 ∂tvm±+vm±·∇vm±-Δvm±+∇pm±=0inΩtm,±,t∈[0,T0],

7.2 divvm±=0inΩtm,±,t∈[0,T0],

7.3 -〚2Dvm±-pm±I〛nΓtm=σHΓtmnΓtmonΓtm,t∈[0,T0],

7.4 〚vm±〛=0onΓtm,t∈[0,T0],

7.5 VΓtm-nΓtm·vm±=mHΓtmonΓtm,t∈[0,T0],

7.6 vm-|∂Ω=0on∂Ω×(0,T0),

7.7 Γ0m=Γ0,vm±|t=0=v0±inΩ0±,

where T0>0 is such that (1.2a)–(1.2g) possesses a smooth solution and m>0 is sufficiently small. Here, analogously as for Ωt± and Γt, the domain Ω is the disjoint union of two sufficiently smooth domains Ωtm,± and an evolving interface Γtm=∂Ωtm,+ for every t∈[0,T]. Furthermore, nΓtm, VΓtm, and HΓtm denote interior normal (with respect to Ωtm,+), the normal velocity, and the mean curvature of Γtm, respectively. We note that before m=mε>0 depends on ε>0 and mε→ε→00. But in the system above only the value of m>0 enters. Therefore we skip the ε-dependence. The goal is to show that then for every m>0 sufficiently small also (7.1)–(7.7) possesses a strong solution on the same time interval [0,T0], which is close to the solution of (1.2a)–(1.2g) in a certain sense. The idea for the proof is to use the interface Γt of the solution of (1.2a)–(1.2g) for every t∈[0,T0] as a reference surface and to transform the modified system (7.1)–(7.7) with the aid of the Hanzawa transformation with respect to Γt to a perturbed two-phase flow problem in Ωt±, t∈[0,T0]. To show solvability of the transformed system for small m>0 we reduce the system to a fixed-point problem with the aid of the invertibility of the principal part of the linearized system and apply the contraction mapping principle as usual. But to apply this strategy invertibility of the principal part of the linearized system to (7.1)–(7.7) together with uniform estimates in m>0 are essential. These results are obtained in the next subsection.

Throughout this section we will use the notation from [35]. In particular, we note that for a Banach space X, 1≤p≤∞, and T0>0

Analysis of the Linearized System

In this subsection (Γt)t∈[0,T0] is a smooth evolving family of (d-1)-dimensional submanifolds such that Γt=∂Ωt+⊆Ω and Ω is the disjoint union of Γt and two smooth domains Ωt+ and Ωt- for every t∈[0,T0]. Moreover, we assume that there is a continuous X:Ω¯×[0,T0]→Ω¯ such that (X|Ω0±×[0,T0],id):Ω0±¯×[0,T0]→Ω±¯ are smooth and Xt:=X(·,t):Ω0±¯→Ωt±¯ are smooth diffeomorphisms for all t∈[0,T0],

det(DξXt(ξ))=1 for all ξ∈Ω0\Γ0, t∈[0,T0].

Here we use the same notation as in Sect. 2. In particular, we have Xt(Γ0)=Γt for all t∈[0,T0]. In the later applications (Γt)t∈[0,T0] is given by the smooth solution of (1.2a)–(1.2g) and Xt=X(·,t) can be obtained as the solutions ofddtXt(ξ)=v0±(Xt(ξ),t)for allξ∈Ω0±¯,t∈[0,T0],X0(ξ)=ξfor allξ∈Ω0±¯.

Since 〚v0±〛=0, X:Ω0¯×[0,T0]→Ω0¯ is well defined and continuous.

We consider the linearized system7.8 ∂tv±-Δv±+∇p±=finΩt±,t∈[0,T0],

7.9 divv±=ginΩt±,t∈[0,T0],

7.10 -〚2Dv±-p±I〛nΓt=σΔΓthnΓt+aonΓt,t∈[0,T0],

7.11 〚v±〛=0onΓt,t∈[0,T0],

7.12 ∂t∙h=nΓt·v+mΔΓth+bonΓt,t∈[0,T0],

7.13 v0-|∂Ω=0on∂Ω×[0,T0],

7.14 h|t=0=h0,v±|t=0=v0±inΩ0±,

where v±=v|Ω± and ∂t∙h denotes the material time derivative of h:Γ→R defined by∂t∙h=∂th~+∇h~·(∂tXt)∘Xt-1onΓ,

where h~:Γ(δ)→R with h~(x,t)=h(PΓt(x),t) for all x∈Γt(δ),t∈[0,T0] for a sufficiently small δ.

The main result of this subsection is

Theorem 7.1

Let q>d+2, T0∈(0,∞),

m∈(0,1), and Ω,Γt,Ωt±, t∈[0,T0], be as before. Moreover, letf∈Lq(0,T0;Lq(Ω))d,g∈Lq(0,T0;Wq1(Ω\Γt))∩Wq1(0,T0;W˙q-1(Ω)),v0∈Wq2-2q(Ω\Γ0)d,a∈Wq12-12q(0,T0;Lq(Γt))d∩Lq(0,T0;Wq1-1q(Γt))d,h0∈Wq3-2q(Γ0),b∈Wq1-12q(0,T0;Lq(Γt))∩Lq(0,T0;Wq2-1q(Γt))

satisfy the compatibility conditions v0|∂Ω=0, 〚v0±〛=0,

divv0=g|t=0,

-PTΓ0〚2Dv0±〛nΓt=PTΓ0a|t=0,

then there is a unique solution (v,p,h) such thatv∈Wq1(0,T0;Lq(Ω))d∩Lq(0,T0;Wq2(Ω\Γt))d,p∈Lq(0,T0;W˙q1(Ω\Γt)),〚p〛∈Wq12-12q(0,T0;Lq(Γt))∩Lq(0,T0;Wq1-1q(Γt)),h∈Wq2-12q(0,T0;Lq(Γt))∩Wq1(0,T0;Wq2-1q(Γt))∩Lq(0,T0;Wq4-1q(Γt)).

Moreover, there is some C>0 independent of m∈(0,1) and the data (f,g,a,b,v0,h0) such that7.15 ‖v‖Wq1(0,T0;Lq(Ω))+‖v‖Lq(0,T0;Wq2(Ω\Γt))+‖p‖Lq(0,T0;W˙q1(Ω\Γt))+‖〚p〛‖Wq12-12q(0,T0;Lq(Γt))+‖〚p〛‖Lq(0,T0;Wq2-1q(Γt))+‖h‖Wq2-12q(0,T0;Lq(Γt))+‖h‖Wq1(0,T0;Wq1-1q(Γt))+‖h‖Lq(0,T0;Wq3-1q(Γt))+m‖h‖Lq(0,T0;Wq4-1q(Γt))≤C‖f‖Lq(0,T0;Lq(Ω))+‖g‖Lq(0,T0;Wq1(Ω\Γt))+‖a‖Wq12-12q(0,T0;Lq(Γt))∩Lq(0,T0;Wq2-1q(Γt))+‖b‖Lq(0,T0;Wq2-1q(Γt))+‖v0‖Wq2-2q(Ω\Γ0)+‖h0‖Wq4-3q(Γ0)

Here we define Lq(0,T0;Wqm(Ω\Γt)) such that g∈Lq(0,T0;Wqm(Ω\Γt)) if and only if g∘(X,id(0,T0))∈Lq(0,T0;Wqm(Ω\Γ0)) for m∈N0 and analogously for the other function spaces involving Γt.

Case of a Flat Interface

Throughout this section we assume that Ωt±=R±d, Γt=Rd-1×{0}, and Ω=Rd. We follow the strategy of [35, Section 8.2]. To this end we first assume that f=g=a=v0=h0=0, use a spectral shift, and consider, for ω>0,7.16 (∂t+ω)v±-Δv±+∇p±=0inR±d×(0,∞),

7.17 divv±=0inR±d×(0,∞),

7.18 〚2Dv±-p±I〛ed=-σΔRd-1hedonRd-1×{0}×(0,∞),

7.19 〚v±〛=0onRd-1×{0}×(0,∞),

7.20 (∂t+ω)h+vd-mΔRd-1h=bonRd-1×{0}×(0,∞),

7.21 (v,h)|t=0=(0,0),

where v±=v|R±d, p±=p|R±d.

The goal of this subsection is to prove

Theorem 7.2

For any ω>0, 1<q<∞,b∈0Wq1-12q(0,∞;Lq(Rd-1))∩Lq(0,∞;Wq2-1q(Rd-1))

there is a unique solution

satisfying7.22

uniformly in m∈[0,1].

Analogously as in [35] one obtains that (v±,p±,h) solves (7.16)–(7.21) if and only if7.23 (∂t+ω)h+ed·(DN)-10-σΔRd-1h-mΔRd-1h=bonRd-1×{0}×(0,∞),(v,h)|t=0=(0,0),

where DN is the Dirichlet-to-Neumann operator for the two-phase Stokes problem as in [35, Section 8.3.2] and (v±,p±) is determined by (7.16)–(7.19) and v|t=0=0 in dependence on h. More precisely, the Laplace-Fourier transform of u=ed·(DN)-10-σΔRd-1h is given byu^(λ,ξ)=-σ|ξ|22λ/|ξ|+2(λ+|ξ|2)12+|ξ|h^(λ,ξ),

cf. [35, Equation (8.34)], where we note that η1=η2=(λ+|ξ|2)12+|ξ| since ρ1=ρ2=μ1=μ2=1 in our case. Hence the Fourier-Laplace transformation of (7.23) is sm(λ,|ξ|)h^(λ,ξ)=b^(λ,ξ), wheresm(λ,τ)=λ+mτ2+στ22λ/τ+2(λ+τ2)12+τforλ∈C,τ∈C\{0}.

We note that s0(λ,|ξ|) coincides with the symbol s0,0(λ,ξ) studied in [35, Section 8.3.4], for which it was shown|s0,0(λ,ξ)|≤Cη(|λ|+|ξ|)for allλ∈Σπ/2+η,ξ∈Ση∪-Σηn-1,

cf. [35, Equation (8.50)] for every η∈[0,π2). Hence for every η∈[0,π2) there is some Cη such that|sm(λ,τ)|≤Cη(|λ|+|τ|+m|τ|2)for allλ∈Σπ/2+η,τ∈Ση,

uniformly in m∈[0,1]. The essential point is that we have the same kind of lower bound.

Lemma 7.3

For any ω0>0 there is some η>0 and c>0 such that7.24 |sm(λ,τ)|≥c(|λ|+|τ|+m|τ|2)for allλ∈Σπ/2+η,|λ|≥ω0,τ∈Ση,

Proof

First of all, we note thatsm(λ,τ)=λ+mτ2+στk(z)withz=λτ2,τ≠0,

where it was shown in [35, p. 392] that for any ϑ∈[0,π) there is some Cϑ>0 such that|k(z)|≤Cϑ1+|z|for allz∈Σϑ.

Moreover, Rek(z)>0 if Rez≥0.

We first show that (7.24) holds for λ∈Σπ/2-δ,τ∈Ση and any δ∈(0,π/4) and sufficiently small η>0 (depending on δ). To this end we use that for any η>0 there is some Cη>0 such that|z|≤CηRezfor allz∈Σπ/2-η.

Furthermore, observe that2λτ+2(λ+τ2)12+τ∈Σπ/2-δ/2for allλ∈Σπ/2-δ,τ∈Σδ/4

and thereforeστ22λ/τ+2(λ+τ2)12+τ∈Σπ/2-δ/3for allλ∈Σπ/2-δ,τ∈Ση

if η>0 is sufficiently small. Since λ,mτ2∈Σπ/2-δ/3 for all λ∈Σπ/2-δ,τ∈Ση as well, we conclude|sm(λ,τ)|≥Resm(λ,τ)=Reλ+mReτ2+Reστ22λ/τ+2(λ+τ2)12+τ≥cη|λ|+m|τ|2+|τ||k(z)|withz=λτ2

provided η>0 is sufficiently small. Now, if |z|≤1, there is some c>0 such that |k(z)|≥c if Rez≥0 and |z|≤1 and we conclude|sm(λ,τ)|≥cη|λ|+m|τ|2+|τ|if|z|≤1.

On the other hand, if |z|≥1, then |λ|≥|τ|2 and therefore|sm(λ,τ)|≥cη|λ|+m|τ|2+|τ|for allλ∈Σπ/2-δ,|λ|≥ω0,τ∈Ση

uniformly in m∈[0,1] if η>0 is sufficiently small.

Next we consider λ∈Σπ/2+δ\Σπ/2-δ for δ>0 sufficiently small. To this end we note that, since Rek(z)>0 if Rez≥0 and k(z)→z∈Σϑ,|z|→∞0 for every ϑ∈[0,π), we have thatK:=k(z):z∈Σπ/2+η¯⊆{z∈C:Rez>0}

if η>0 is sufficiently small. Moreover, K is compact. Therefore, there is some δ>0 such thatK⊆Σπ/2-3δ.

Moreover, it is easy to prove that there is some Cδ>0 such that|z|+|w|≤Cδ|z+w|for allz∈Σπ/2+δ\Σπ/2-δ,w∈Σδ

provided that δ∈(0,π/4). Hence|sm(λ,τ)|≥cδ|λ|+|mτ2+στk(z)|≥cδ|λ|+mRe(τ2)+σRe(τk(z))≥cδ|λ|+m|τ2|+σ|τ||k(z)|,

and, with the same arguments as before,|sm(λ,τ)|≥cδ|λ|+m|τ2|+|τ|

for all λ∈Σπ/2+δ\Σπ/2-δ, |λ|≥ω0, τ∈Ση uniformly in m∈[0,1] if η>0 is sufficiently small. This finishes the proof. □

Now we can proceed as in [35, Section 8.3.3]. Let Dn12=(-Δx′)12 be the Fourier multiplication operator with symbol |ξ|, ξ∈Rd-1. Then Dn12 possesses an R-bounded functional calculus in Wq2-1q(Rd-1), for any 1<q<∞. Therefore(λ+Dn12+mDn)sm-1(λ,Dn12)

is R-bounded and possesses a bounded H∞-calculus on Σπ/2+η\Bω0(0) for some η>0 and any ω0>0, which is also uniformly bounded in m∈[0,1]. Hence the operator-valued H∞-calculus of G=∂t+ω on 0Hqr(0,∞;Kq(Rd-1)) for ω>0, s,r∈R and K=H,W yields that(∂t+ω+Dn12+mDn)sm-1(∂t+ω,Dn12)∈L0Hqr(0,∞;Kq(Rd-1))

is uniformly bounded in m∈[0,1] for any s,r∈R. Hence we obtain for h=sm-1(λ,Dn12)b and b∈E that h solves (7.23) and satisfies‖ωh‖E+‖∂th‖E+‖∇x′h‖E+m‖∇x′2h‖E≤Cr,s,p‖b‖E

uniformly in m∈[0,1], where E:=Lq(0,∞;Wq2-1q(Rd-1)) or E:=0Hqk(0,∞;Lq(Rd-1)), k=0,1. By real interpolation we obtain that the same is true for b∈E=0Wq1-12q(0,∞;Lq(Rd-1)). This shows existence of a solution as in Theorem 7.2 satisfying (7.22) uniformly in m∈[0,1], where the existence of (v,p) and the corresponding estimates follow from [35, Corollary 8.3.3]. Uniqueness can be shown by a standard duality argument. Hence Theorem 7.2 is proved.

Proof of Theorem 7.1

First we assume that Γt=Γ is independent of t∈[0,T0]. In this case one proves the result by the same localization and perturbation argument and reduction to semi-homogeneous data as in [35, Section 8.2] using the result for a flat interface due to Theorem 7.2. The only difference is related to the new term “mΔΓh” in the evolution equation for h. These extra-terms can be controlled by m‖h‖Lq(0,T0;Wq4-1q) (uniformly in m∈[0,1]), while all other terms can be controlled by ‖h‖Lq(0,T0;Wq3-1q) as in the case m=0. Here we note that in the proof one can reduce to h0=0 by subtracting some h~∈Lq(0,T0;Wq4-1q(Γ))∩Wq1(0,T0;Wq2-1q(Γ)) from h, which exists because ofWq4-3q(Γ)=(Wq2-1q(Γ),Wq4-1q(Γ))1-1q,q

since 1-1q,1-3q∉Z due to q>d+2>3 and by the trace method for real interpolation spaces.

Now, if Γt, t∈[0,T0], are smoothly evolving interfaces as before, we fix an arbitrary t0∈[0,T0] and define Jt0:=[max(t0-δ,0),min(T0,t0+δ)] for δ>0 and X~:Ω¯×Jt0→Ω¯×Jt0 by X~(x,t)=(Xt(Xt0-1(x)),t) for all x∈Ω, t∈Jt0. Then (7.8)–(7.13) with Jt0 instead of [0,T0] is equivalent to a system of the form7.25 ∂tv~±-Δv~±+∇p~±=f~+KfinΩ0±×Jt0,

7.26 divv±=g+KginΩ0±×Jt0,

7.27 -〚2Dv±-p±I〛nΓt=σΔΓthnΓt+a+KaonΓt0×Jt0,

7.28 〚v±〛=0onΓt0×Jt0,

7.29 ∂th=nΓt·v+mΔΓt0h+b+KbonΓt0×Jt0,

7.30 v0-|∂Ω=0on∂Ω×Jt0,

where (Kf,Kg,Ka,Kb) depend linearly on (v±,p±,h) and‖Kf‖Lq(Jt0;Lq(Ω))+‖Kg‖Lq(Jt0;Wq1(Ω\Γt0))+‖Kb‖Lq(Jt0;Wq2-1q(Γt0))+‖Ka‖Wq12-12q(Jt0;Lq(Γt0))∩Lq(Jt0;Wq2-1q(Γt0))≤C(δ)‖v‖Wq1(Jt0;Lq(Ω))+‖v‖Lq(Jt0;Wq2(Ω\Γt0))+‖p‖Lq(Jt0;Wq1(Ω\Γt0))+‖〚p〛‖Wq12-12q(Jt0;Lq(Γt0))+‖〚p〛‖Lq(Jt0;Wq2-1q(Γt0))+‖h‖Lq(Jt0;Wq3-1q(Γt0))+‖h‖Wq1(Jt0;Wq1-1q(Γt0))+m‖h‖Lq(Jt0;Wq4-1q(Γt0)),

wherev~±:=v±∘X~|Ω0±,p±:=p±∘X~|Ω0±,h~:=h∘X~|Γt0,f~:=f∘X~,g~:=g∘X~,a~:=a∘X~|Γt0,b~:=b∘X~|Γt0

and C(δ)→0 as δ→0 since X~→I in Ck(Ω0±×Jt0) as δ→0 for any k∈N. Hence by a standard Neumann series argument (7.25)–(7.30) together with v~|t=t~0=v0∘X~(·,t~0),h~|t=t~0=h0∘X~(·,t~0)|Γt~0, t~0:=max(0,t0-δ), possesses a unique solution (v~±,p~±,h~) for every (f,g,a,b,v0,h0) as in Theorem 7.1 with [0,T0] replaced by Jt0 and the initial time 0 replaced by t~0=max(0,t0-δ) provided δ=δ(t0)>0 is sufficiently small. Transforming back, this yields a unique solution (v±,p±,h) of (7.8)–(7.13) together with v|t=max(0,t0-δ)=v0,h|t=max(0,t0-δ)=h0 for every (f,g,a,b,v0,h0) as in Theorem 7.1 with [0,T0] replaced by Jt0 and the initial time 0 replaced by t~0. Moreover, the Neumann series argument also shows that (7.15) with the same replacements as before holds true uniformly in m∈[0,1].

Since {(t0-δ(t0),t0+δ(t0)):t0∈[0,T0]} is an open covering of [0,T0], there are finitely many 0=t0<t1<…<tN<T0 and δj>0 such that [0,T0]=⋃j=0NJtj and we can solve (7.8)–(7.13) on Jtj=[max(0,tj-δ),min(tj+δ,T0)] together with v~|t=max(0,tj-δj)=v0,h~|t=max(0,tj-δj)=h0 uniquely as before. Hence we can solve (7.8)–(7.14) by solving successively (7.8)–(7.13) on Jtj, j=0,…,N with initial values (v0,h0), (v|t=tj-1,h|t=tj-1) for j=2,…,N. Finally, since (7.15) holds true uniformly in m∈[0,1] on the intervals Jtj, j=0,…,N, one also obtains (7.15) on [0,T0] uniformly in m∈[0,1].

Existence of Solutions for the Transformed System

The idea of the proof is to represent Γtm from the solution of (7.1)–(7.7) for every t∈[0,T0] as a graph over Γt, where Γt, t∈[0,T0], is from the smooth solution of (1.2a)–(1.2g) and to transform (7.1)–(7.7) to a corresponding perturbed system in Ωt±, t∈[0,T0] with the aid of the Hanzawa-Transformation associated to Γt. To this end, let us denote thatΓt(3δ)={x∈Rd:|dΓt(x)|<3δ},t∈[0,T0],

wheredΓt(x)=dist(x,Γt)ifx∈Ωt+,-dist(x,Γt)else

is the signed distance to Γt. Since (Γt)t∈[0,T0] are smoothly evolving, compact, and [0,T0] is compact, there is some δ>0 such that for every x∈Γt(3δ), t∈[0,T0] there is a unique closest point PΓt(x)∈Γt andΓ(3δ):=⋃t∈[0,T0]Γt(3δ)×{t}∋(x,t)↦(dΓt,PΓt)∈Rd+1

is smooth. Moreover, we choose δ>0 so small that Γ(3δ)⊆Ω×[0,T0].

Furthermore, for a given continuous “height function” h:Γ→R letθh:Γ→Rn:x↦x+h(x,t)nΓt(x).

Here Γ is defined as in (1.3). Then θh is injective provided that ‖h‖C0(Γ)<δ. Moreover, we define the Hanzawa transformation associated to Γ as7.31 Θh(x,t)=x+χ(dΓt(x)/δ)h(PΓt(x),t)nΓt(PΓt(x)),

where χ∈C∞(R) such that χ(s)=1 for |s|≤2δ and χ(s)=0 for |s|>23 as well as |χ′(s)|≤4 for all s∈R, and ‖h‖C0(Γ)<δ. Then Θh(.,t):Ω→Ω is a smooth diffeomorphism for every t∈[0,T0], cf. e.g. [35, Chapter 1, Section 3.2]. Hence for any h:Γ→R that is continuously differentiable with ‖h‖C0(Γ)<δ we have that (Γth)t∈[0,T0]:=(θh(Γt,t))t∈[0,T0] is an oriented, compact evolving C1-manifold, such that Γth is a C2-manifold for every t∈[0,T0] if h(·,t):Γt→R is twice continuously differentiable.

In the following we look for a solution of (7.1)–(7.7) such that Γtm=Γth for all t∈[0,T0] a sufficiently regular h:Γ→R with ‖h‖C0(Γ)<δ. Then (vm±,pm±,(Γtm)t∈[0,T0]) solves (7.1)–(7.7) if and only ifv±(x,t):=vm±(Θh(x,t),t),p±(x,t)=pm±(Θh(x,t),t)forx∈Ωt±,t∈[0,T0],h(x,t):=hm(PΓt(θh(x,t)),t)forx∈Γt,t∈[0,T0]

solves the transformed system7.32 ∂tv±-Δv±+∇p±=a±(h;Dx)(v±,p±)+∂tΘh·∇hv±-v±·∇hv±inΩ±,

7.33 divv±=tr((I-A(h))∇v±)=:g(h)v±inΩ±,

7.34 〚v〛=0onΓ,

7.35 〚2Dv±-p±I〛nΓt=t(h;Dx)(v,p)+σK(h)nΓth∘θthonΓ,

7.36 v-|∂Ω=0on∂Ω×(0,T0),

7.37 ∂t∙h-nΓt·v=mK(h)+(nΓth∘θth-nΓt)·vonΓ,

7.38 v|t=0=v0onΩ0±,

wherea±(h;Dx)(v±,p±)=divh(∇hv±)-Δv±+(∇-∇h)p±,∇h=A(h)∇,divhu=tr(∇hu),A(h)=DxΘh-T,t(h,Dx)(v,p)=[[(nΓ0-nh)·(2μ±Dv-pI)+2nh·sym(∇v-∇hv)]],nh=A(h)nΓ0|A(h)nΓ0|,K(h)=HΓth∘θth.

For the fixed-point argument we introduce the solution spaceEm(T0):=E1(T0)×E2(T0)×E3(T0)×E4,m(T0),E(T0):=E1(T0)×E2(T0)×E3(T0)×E4(T0),E1(T0):=0Wq1(0,T0;Lq(Ω))d∩Lq(0,T0;Wq2(Ω\Γt))d,E2(T0):=Lq(0,T0;W˙q1(Ω\Γt)),E3(T0):=0Wq12-12q(0,T0;Lq(Γt))∩Lq(0,T0;Wq1-1q(Γt)),E4,m(T0):=Wq2-12q(0,T0;Lq(Γt))∩0Wq1(0,T0;Wq2-1q(Γt))∩Lq(0,T0;Wq4-1q(Γt)),E4(T0):=Wq2-12q(0,T0;Lq(Γt))∩0Wq1(0,T0;Wq2-1q(Γt))∩Lq(0,T0;Wq3-1q(Γt)),

where E4,m(T0) is normed by‖h‖E4,m(T0):=‖h‖E4(T0)+m‖h‖Lq(0,T0;Wq4-1q(Γt))+m‖h‖Wq1-12q(0,T0;Wq2(Γt))+m‖h|t=0‖Wq4-3q(Γ0)‖h‖E4(T0):=‖h‖Wq2-1q(0,T0;Lq(Γt))+‖h‖Wq1(0,T0;Wq2-1q(Γt))+‖h‖Lq(0,T0;Wq3-1q(Γt)+‖h|t=0‖Wq3-3q(Γ0)

and E1(T0), E2(T0) are normed in a standard manner. We note that in comparison to [27] the conditions w|t=0=0, h|t=0=0 are already included in the definition of Em(T0). Moreover, we define the space for the right-hand side asF(T0):=F1(T0)×F2(T0)×F3(T0)×F4(T0)F1(T0):=Lq(0,T0;Lq(Ω))d,F2(T0):=Lq(0,T0;Wq1(Ω\Γt))∩Wq1(0,T;W˙q-1(Ω)),F3(T0):=Wq12-12q(0,T0;Lq(Γt))d∩Lq(0,T;Wq1-1q(Γt))d,F4(T0):=Wq1-12q(0,T0;Lq(Γt))∩Lq(0,T;Wq2-1q(Γt))

normed in the standard manner.

Then (7.32)–(7.37) can be written as7.39 Lz+mL¯z=N(z)+mN¯(z),

for z=(v,p,〚p〛,h)∈E(T0) with ‖h‖C0(Γ)<δ, whereLz=(L1(v,p),L2v,L3(v,π,h),L4(v,h)),N(z)=(N1(v,p),N2(v,h),N3(v,π,h),N4(v,h))

for z=(v,p,π,h) with ‖h‖C0(Γ)<δ are defined as in [27, Section 4]. More precisely,L1(v,p):=∂tv-Δv+∇p,L2v:=divv,L3(v,p,h):=-〚2Dv±〛nΓt-πnΓt-(ΔΓth)nΓtL4(v,h):=∂t∙h-nΓt·v+v0·∇Γth

andN1(v,p,h):=F(h,v)∇v+M4(h):∇2v+M1(h)∇p,N2(v,h):=M1(h):∇v,N3(v,h):=Gτ(h)∇v+(Gν(h)∇v+Gγ(h))nΓtN4(v,h):=([M0(h)-I]∇Γth)·v+(v-v0)·∇Γth,

where F,M1,…,M4 are defined as in [27, Section 2], with the only difference that the time-independent reference surface Σ is replaced by the smoothly evolving reference surface Γt, t∈[0,T0] and ∂th is replaced by ∂t∙h. Moreover,L¯z=(0,0,0,ΔΓth),N¯z=(0,0,0,K(h)-ΔΓth)

for z=(v,p,π,h). Let us note thatLz0=N(z0),

for z0=(v0,p0,0), where v0|Ω±=v0±, p0|Ω±=p0± is the solution of the limit system (1.2a)–(1.2g). Therefore (7.39) is equivalent to7.40 Lmw:=Lw+mL¯w-DN(z0)w=N(w+z0)-N(z0)-DN(z0)w+mN¯(w+z0)=:Nm(w)

for w=(u,π,〚π〛,h) with u=v-v0, π=p-p0, whereDN(z0)w=v0·∇w+w·∇v0+(DM1(0)h+DM2(0)h+DM3(0)h)∇v0(DM1(0)h):∇v0(DGτ(0)h)∇v0+((DGν(0)h)∇v0)nΓt0

since L¯z0=0, M0(0)=I, Mj(0)=0 for j=1,…,4, Gτ(0)=Gν(0)=Gγ(0)=0, and DGγ(0)h=σ(DHΓt(0)-DHΓt(0))h=0. Hence DN(z0) is a linear operator of lower order with respect to w compared to L+mL¯.

As in [27, Proposition 3] we have

Proposition 7.4

Let q>d+2. Then N∈Cω(U,F(T)) and N¯∈Cω(Um,F(T)), whereU:={(u,π,r,h)∈E(T):‖h‖E4(T)<r0},Um=U∩Em(T)

for r0>0 sufficiently small. Moreover, for every r1∈(0,r0) there is some C>0 such that7.41 ‖mN¯(w1)-mN¯(w2)‖F(T)≤CR‖w1-w2‖Em(T)

for all w1,w2∈Um with ‖wj‖Em(T)≤R, j=1,2, where R∈(0,r1].

Proof

The statement for N is proved in the same way as in [27, Proposition 3], where the only difference is that the reference surface is time-dependent. But because of the compactness of [0,T0] all relevant norms are uniformly controlled. For the statement on N¯ we use the quasilinear structure of K(h), i.e.,K(h)=C0(h):∇t2h+C1(h),

where Cj(h) depend only on h and ∇Γth and are analytic in (h,∇Γth) (pointwise), cf. e.g. [35, Section 2.2.5]. Moreover, due to [33, Lemma 6.1] F4(T0) is a Banach algebra and m‖∇t2h‖F4(T0)≤C‖w‖Em(T0) uniformly in m∈(0,1]. From this N¯∈Cω(Um,F(T0)) easily follows and one obtains (7.41) in a straight forward manner. □

The central technical results are

Proposition 7.5

Let r0 be as in Proposition 7.4. There are some CN>0, R0∈(0,r0) such that for every R∈(0,R0], m∈(0,1] we have7.42 ‖Nm(w1)-Nm(w2)‖F(T0)≤CNR‖w1-w2‖Em(T0)

for all w1,w2∈Em(T0) with ‖wj‖Em(T0)≤R for j=1,2.

Proof

Let R0∈(0,1] be at least so small that R0<r0. Then, by the definition of Nm,Nm(w1)-Nm(w2)=N(w1+z0)-N(w2+z0)-DN(z0)(w1-w2)+mN¯(w1+z0)-N¯(w2+z0)

for all wj∈U, j=1,2. Now using the power series expansion for N(wj+z0) and N¯(wj+z0), we obtain for R0 sufficiently small‖Nm(w1)-Nm(w2)‖F(T0)≤C‖w1-w2‖E(T0)2+CR‖w1-w2‖Em(T0)≤CR‖w1-w3‖Em(T0)

for all wj∈Um with ‖wj‖Em(T0)≤R, j=1,2. □

Proposition 7.6

Let Lm be defined as in (7.40). Then there are some m1∈(0,1], CL>0 such that Lm:Em(T0)→F(T0) is invertible and‖Lm-1‖L(F(T0),Em(T0))≤CLfor allm∈(0,m1].

Proof

First of all L+mL¯:Em(T)→F(T) is invertible for all m∈(0,m1], m1∈(0,1] sufficiently small, and all T∈(0,T0] because of Theorem 7.1. Moreover, there is some CL′>0 such that‖(L+mL¯)-1‖L(F(T),Em(T))≤CL′for allm∈(0,m1]andT∈(0,T0].

UsingE4(T)↪0Wq1(0,T;Wq2-1q(Γt))∩Wr1(0,T;Wq1(Γt))↪C1-1q[0,T];Wq2-1q(Γt)

uniformly in T∈(0,T0] for some r>q and the smoothness of v0 one can show‖v0·∇w+w·∇v0+(DM1(0)h+DM2(0)h+DM3(0)h)∇v0‖Lq(Ω×(0,T))≤CT1q‖h‖L∞(0,T;Wq2-1q(Γt))≤C′T1q‖h‖E4(T)

and‖(DM1(0)h):∇v0‖F2(T)≤C‖(DM1(0)h):∇v0‖Wq1(Ω)×(0,T))≤CT1q-1r‖h‖E4(T),‖(DGτ(0)h)∇v0+((DGν(0)h)∇v0)nΓt‖F3(T)≤CTα‖h‖E4(T)

uniformly in T∈(0,T0] and h∈E4(T) for some α>0 by straightforward estimates. Hence‖DN(z0)(L+mL¯)-1‖L(F(T))≤CTαfor allm∈(0,m1]andT∈(0,T0]

for some α>0. Hence by a Neumann series argument Lm is invertible and‖Lm-1‖L(F(T),Em(T))≤CLfor allm∈(0,m1]

provided that T∈(0,T∗] for some T∗>0 sufficiently small. Finally, the invertibility of Lm and the uniform estimate on the time interval (0,T0) can be shown by the same arguments as in the end of the proof of Theorem 7.1 by dividing (0,T0) in finitely many intervals (0,T1), (T1,T2),…,(TN-1,TN) of length less than T∗ and solving the system iteratively on the intervals. □

Now letRm:=2mCL‖N¯(z0)‖F(T)

and choose m0∈(0,m1] so small thatCLCN(Rm0+m0)≤12andRm0≤R0.

Then we have for every m∈(0,m0]‖Lm-1Nm(w1)-Nm(w2)‖Em(T)≤CLCN(Rm+m)‖w1-w2‖Em(T)≤12‖w1-w2‖Em(T)

for all w1,w2∈Em(T) with ‖wj‖Em(T)≤Rm for j=1,2 and‖Lm-1Nm(w)‖Em(T)≤‖Lm-1Nm(w)-Nm(0)‖Em(T)+CL‖Nm(0)‖Em(T)≤Rm2+mCL‖N¯(z0)‖F(T)≤Rm

for all w∈Em(T) with ‖w‖Em(T)≤Rm. Hence Lm-1Nm is a contraction on BRm(0)¯ in Em(T) for all m∈(0,m0] and we obtain for every m∈(0,m0] a unique wm∈BRm(0)¯ such that7.43 wm=Lm-1(Nmwm).

In summary, we obtain

Theorem 7.7

There is some m0>0 such that for every m∈(0,m0] the transformed system (7.32)–(7.38) possesses a solution (v,p,〚p〛,h)∈Em(T0), which satisfies‖(v-v0,p-p0,〚p-p0〛,h)‖Em(T0)≤Cm

for some C>0 independent of m∈(0,m0].

Transforming back with Θh-1 finally yields a solution (vm±,pm±,(Γtm)t∈[0,T0]) of (7.1)–(7.7).

Remark 7.8

Since E4,m(T0)↪C0([0,T0];C2(Γt)) uniformly in m∈(0,1), one can show in a straight forward manner that there are m1∈(0,m0] and δ>0 such that the signed distance function dΓtm and the orthogonal projection PΓtm on Γt for every t∈[0,T0] are well-defined and smooth in Γm(2δ)={(x,t)∈Ω×[0,T0]:|dist(x,Γtm)|<2δ} for every m∈(0,m1]. Moreover,Γm(2δ)⊆Γ(3δ)for allm∈(0,m1]

and δ>0 can be chosen (as before) such that the signed distance function dΓt and the orthogonal projection PΓt on Γt for every t∈[0,T0] are smooth in Γ(3δ).

Uniform Regularity

The goal of this section is to prove

Theorem 7.9

Let q>d+5. There is some m0>0 such that for every m∈(0,m0] (7.1)–(7.6) possesses a solution (vm±,pm±,Γm) with Γm⊆Γ(δ), dΓtm∈Ct1Cx1∩Ct0Cx3 in Γ(δ)¯ and vm∈Lq(0,T;Wq1(Ω)d)∩Lq(0,T;Wq2(Ω\Γtm))∩Wq1(0,T;Lσq(Ω)) and ∇τ∇vm∈L∞(Ω×(0,T)), ∇τpm∈Lq(0,T;Wq1(Ω\Γt)) with uniformly bounded norms with respect to m∈(0,m0].

Proof

We use the so-called parameter-trick to obtain that the tangential derivative of the solution belongs locally to the same function space as the solution itself (together with uniform bounds). To this end we follow the arguments in [35, Section 9.4.2] with modifications to our time dependent reference manifold Γt. Let (t0,x0)∈Γ be arbitrary and X~:Ω¯×Jt0→Ω¯ with X~(x,t)=Xt(X0-1(x)) for all x∈Ω, t∈Jt0:=[max(t0-δ,0),min(T0,t0+δ)] as in Section 7.3. Moreover, let φ:B3R(0)⊆Rd-1→Rd be a local parametrization of Γt0 with φ(0)=x0. Furthermore, for t∈Jt0 let ϕt:B3R(0)×(-2δ0,2δ0)→Rd be defined byϕt(s,ρ)=X~(φ(s),t)+ρnΓtfor all(s,ρ)∈B3R(0)×(-2δ0,2δ0).

Then PΓt(ϕt(s,ρ))=X~(φ(s),t). We define the truncated shift τξ:Jt0×B3R(0)×(-2δ0,2δ0)→B3R(0)×(-2δ0,2δ0)τξ(t,s,ρ)=(s+ξη(t)χ0(s)ζ0(ρ),ρ)for all(t,s,ρ)∈Jt0×B3R(0)×(-2δ0,2δ0),

where ξ∈Br(0)⊆Rd-1, 0<r≤R, χ0∈C0∞(Rd-1) with 0≤χ0≤1, suppχ0⊆B2R(0) and χ0(s)=1 if |s|≤R, ζ0∈C0∞(R) with suppζ0⊆[-5δ/2,5δ/2] and ζ0(ρ)=1 if |ρ|≤2δ0, η∈C0∞(R) with η(t)=0 if t∈[t0-δ/2,t0+δ/2] and suppη⊆(t0-δ,t0+δ). Finally, we defineτξ(t,x)=ϕt(τξ(t,ϕt-1(x)))for all(t,x)∈U:=⋃t∈Jt0{t}×Ut,Ut:=ϕtB3R(0)×(-3δ02,3δ02).

and τξ(t,x)=(t,x) for all t∈Jt0, x∈Ω\Ut and t∈[0,T]\Jt0, x∈Ω. Note that PΓt(τξ(t,x))=τξ(t,PΓt(x)) for all x∈Ut, t∈Jt0 and thereforeh∘(id,PΓ·)∘τξ(t,x)=h(t,PΓt(τξ(t,x)))=(h(t,·)∘τξ)∘(id,PΓ·)(t,x)

for all (t,x)∈U and thusΘh(τξ(t,x),t)=Θh∘(id,τξ)(x,t)for all(t,x)∈U

since dΓt∘τξ(t,·)=dΓt on Ut and χ∘(dΓt/δ)=0 on Γt(2δ)\Ut for all t∈Jt0.

Altogether we observe that for r>0 sufficiently small τξ(t,·):Ω¯→Ω¯ is a smooth diffeomorphism, which depends smoothly on ξ∈Br(0) and t∈[0,T]. We denote by Tξ:Em(T)→Em(T) the operator obtained by pointwise composition with τξ. Let Gm:BR(0)¯⊆Em(T)→BR(0)¯ with Gm(w):=w-Lm-1(Nmw) and wm be as in (7.43). Then0=TξGm(wm)=TξGm(Tξ-1Tξwm)for allξ∈Br(0).

Therefore we define Gm:Br(0)×BR(0)⊆Rd-1×Em(T)→Em byGm(ξ,w):=TξGm(Tξ-1w).

Then as in [35, Section 9.4.2] one observes that Gm is continuously differentiable and ∂ξjGm(0,wm)∈Em(T) is bounded with respect to m∈(0,1]. Furthermore wm,ξ:=Tξwm solvesGm(ξ,wm,ξ)=0for allξ∈Br(ξ).

In particular, Gm(0,wm)=0 andDwGm(0,wm)=DGm(wm)=I-Lm-1DNm(wm):Em(T)→Em(T)

is invertible since ‖Lm-1DNm(wm)‖L(Em(T))≤12 due to Proposition 7.5 for m∈(0,m1] with m1>0 sufficiently small as before. Furthermore, ‖DGm(wm)-1‖L(Em(T))≤2 uniformly in m∈(0,m1]. Now the Implicit Function Theorem yields that, if r>0 is sufficiently small, there are some r′>0 and continuously differentiable Φm:Br(0)→Br′(wm)⊆Em(T) such that Br′(wm)⊆BR(0) and Gm(ξ,Φm(ξ))=0 for all ξ∈Br(0).

If Gm(ξ,u)=0 for some u∈BR(0)⊆Em(T) and ξ∈Br(0), then u=Φm(ξ).

Hence Φm(ξ)=wm,ξ for all ξ∈Br(0). To obtain uniform boundedness of ∂ξjwm,ξ|ξ=0 we use that∂ξjwm,ξ|ξ=0=-DwGm(0,wm)-1∂ξjGm(0,wm)=-DGm(wm)-1∂ξjGm(0,wm),

where ∂ξjGm(0,wm)∈Em(T) and DwGm(wm)-1 are uniformly bounded in m∈(0,m1] by the same observations before. Since ∂ξju∘τξ|ξ=0=∂τju in a neighborhood of x0, where τj(x)=∂pjX~(φ(p),t), j=1,…,d-1, (with x=ϕt(p,q)) form a basis of TxΓt, the statement of the theorem follows. □

Acknowledgements

J. Fischer and M. Moser have received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 948819)

Funding

Open Access funding enabled and organized by Projekt DEAL.

Data Availibility

There is no associated data to the manuscript.

1 The only previous work in this direction [4] only explicitly covers the case of the scaling regime mε=ε1/2 and works under substantially stronger assumptions on the initial data.

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