1 Introduction

The dynamics of two immiscible, viscous fluids like oil and water or polymer blends is a fascinating and difficult topic because in general the interface, which separates both fluids, develops singularities in finite time, e.g. because of pinch off of droplets or collisions. In classical sharp interface models it is assumed that both fluids are separated by a (two-dimensional) surface. In so-called diffuse interface models a partial mixing of the fluids in a thin interfacial region is taken into account, cf. Fig. 9.1.

Fig. 9.1
figure 1

Partial mixing in an interfacial region

Full size image

Moreover, mixing and phase separation because of diffusion of molecules is modeled as well.

The purpose of this contribution is to discuss recent rigorous mathematical results regarding the limit of diffuse interface models to sharp interface models as the parameter ɛ > 0, which is proportional to the “thickness” of the diffuse interface model, tends to zero. Actually, this is a subtle problem and the results depend significantly on the scaling of a mobility coefficient m ɛ in the system as ɛ → 0, as will be shown below.

A fundamental and well accepted diffuse interface model for the flow of two macroscopically immiscible viscous, incompressible Newtonian fluids with same densities is the so-called model H, cf. Hohenberg and Halperin [19] or Gurtin et al. [18]. This model leads to a system of Navier-Stokes/Cahn-Hilliard type:

$$\displaystyle{ \rho \partial _{t}\mathbf{v}_{\varepsilon } +\rho \mathbf{v}_{\varepsilon } \cdot \nabla \mathbf{v}_{\varepsilon } -\mathop{\mathrm{div}}\nolimits (2\nu (c_{\varepsilon })D\mathbf{v}_{\varepsilon }) + \nabla p_{\varepsilon } = -\varepsilon \mathop{\mathrm{div}}\nolimits (\nabla c_{\varepsilon } \otimes \nabla c_{\varepsilon })\quad \rm{in}\ Q_{T}, }$$
(9.1)
$$\displaystyle{ \mathop{\mathrm{div}}\nolimits \mathbf{v}_{\varepsilon } = 0\qquad \qquad \quad \qquad \text{in}\ Q_{T}, }$$
(9.2)
$$\displaystyle{ \partial _{t}c_{\varepsilon } + \mathbf{v}_{\varepsilon } \cdot \nabla c_{\varepsilon } = m_{\varepsilon }\varDelta \mu _{\varepsilon }\qquad \qquad \quad \qquad \text{in}\ Q_{T}, }$$
(9.3)
$$\displaystyle{ \mu _{\varepsilon } = -\varepsilon \varDelta c_{\varepsilon } + \varepsilon ^{-1}f'(c_{\varepsilon })\qquad \qquad \quad \qquad \text{in}\ Q_{ T}, }$$
(9.4)
$$\displaystyle{ (\mathbf{v}_{\varepsilon },c_{\varepsilon })\vert _{t=0} = (\mathbf{v}_{0,\varepsilon },c_{0,\varepsilon })\qquad \qquad \quad \qquad \text{in}\ \varOmega. }$$
(9.5)

where Q T = Ω × (0, T) and (a ⊗ b) i, j = a i b j for all i, j = 1, …, d. Here v ɛ and p ɛ are the mean velocity and pressure of the fluid mixture, \(D\mathbf{v}_{\varepsilon } = \frac{1} {2}(\nabla \mathbf{v}_{\varepsilon } + \nabla \mathbf{v}_{\varepsilon }^{T})\), c ɛ is an order parameter related to the concentrations of the fluids, which will be the concentration difference in the following. Furthermore, μ ɛ is the chemical potential of the mixture, ν is the viscosity of the fluid mixture, ρ is the density of the fluids, which is assumed to be constant. Lastly, \(f: \mathbb{R} \rightarrow \mathbb{R}\) is a suitable (homogeneous) free energy density of double well shape, which will be specified below, and m ɛ > 0 is a (constant) mobility coefficient related to the strength of the diffusion in the mixture. In the following m ɛ will depend on ɛ > 0 and it turns out that the choice of the scaling influences the results fundamentally. The system has to be closed by suitable boundary conditions if ∂Ω ≠ ∅. One standard choice are no-slip boundary conditions for v ɛ and Neumann boundary conditions for c ɛ and μ ɛ , i.e.,

$$\displaystyle{ (\mathbf{v}_{\varepsilon },\mathbf{n}_{\partial \varOmega } \cdot \nabla c_{\varepsilon },\mathbf{n}_{\partial \varOmega } \cdot \nabla \mu _{\varepsilon })\vert _{\partial \varOmega } = 0\qquad \text{on}\ \partial \varOmega \times (0,T), }$$
(9.6)

where n ∂Ω denotes the exterior normal of ∂Ω. Moreover, \(\varOmega \subseteq \mathbb{R}^{d}\) will be a suitable domain, which will be specified in the following in dependence on the result we are discussing. Since the boundary conditions on ∂Ω will not play an important role, we will not specify it in the following for simplicity of the presentation. Finally, the results hold true for any T ∈ (0, ∞) if not stated differently.

We note that sufficiently smooth solutions of (9.1)– (9.5) satisfy the energy inequality

$$\displaystyle{ \begin{array}{ll} &E_{\varepsilon }(c_{\varepsilon }(t)) +\int _{\varOmega }\tfrac{\rho \vert \mathbf{v}_{\varepsilon }(t)\vert ^{2}} {2} \,dx \\ &\ +\int _{ 0}^{t}\int _{\varOmega }\left (2\nu (c_{\varepsilon })\vert D\mathbf{v}_{\varepsilon }\vert ^{2} + m_{\varepsilon }\vert \nabla \mu _{\varepsilon }\vert ^{2}\right )\,dx\,dt \leq E_{\varepsilon }(c_{0,\varepsilon }) +\int _{\varOmega }\tfrac{\rho \vert \mathbf{v}_{0,\varepsilon }\vert ^{2}} {2} \,dx \end{array} }$$
(9.7)

for all t ∈ (0, T), where

$$\displaystyle{ E_{\varepsilon }(c) =\int _{\varOmega }\left (\varepsilon \frac{\vert \nabla c\vert ^{2}} {2} + \frac{f(c)} {\varepsilon } \right )\,dx }$$

is the free energy of the fluid mixture. This energy inequality is fundamental for the analysis of weak and strong solutions for fixed ɛ > 0, cf. e.g. A. [2]. It also provides a limited control of (v ɛ , c ɛ , μ ɛ ) as ɛ → 0 as will be discussed in Sect. 9.3.

So far the sharp interface limits of diffuse interface models in fluid mechanics as (9.1)– (9.4) were mainly discussed with the method of formally matched asymptotics. There it is assumed that the quantities possess suitable power series expansions close to and away from the interface and suitable matching conditions between both expansions hold. With this method it was shown by A., Garcke and Grün [8] that there are (at least) two possible sharp interface limits of (9.1)– (9.5), which can both be formulated as the following system:

$$\displaystyle{ \rho \partial _{t}\mathbf{v} +\rho \mathbf{v} \cdot \nabla \mathbf{v} -\mathop{\mathrm{div}}\nolimits \mathbf{T}(\mathbf{v},p) = 0\quad \qquad \qquad \text{in}\ \varOmega ^{\pm }(t),t \in (0,T), }$$
(9.8)
$$\displaystyle{ \mathop{\mathrm{div}}\nolimits \mathbf{v} = 0\ \qquad \qquad \quad \text{in}\ \varOmega ^{\pm }(t),t \in (0,T), }$$
(9.9)
$$\displaystyle{ [\mathbf{v}] = 0\ \qquad \qquad \quad \text{on}\ \varGamma _{t},t \in (0,T), }$$
(9.10)
$$\displaystyle{ -[\mathbf{n}_{\varGamma _{t}} \cdot \mathbf{T}(\mathbf{v},p)] =\sigma H_{\varGamma _{t}}\mathbf{n}_{\varGamma _{t}}\ \qquad \qquad \quad \text{on}\ \varGamma _{t},t \in (0,T), }$$
(9.11)
$$\displaystyle{ \mathbf{v}\vert _{t=0} = \mathbf{v}_{0}\ \qquad \qquad \quad \text{in}\ \varOmega, }$$
(9.12)
$$\displaystyle{ V _{\varGamma _{t}} -\mathbf{n}_{\varGamma _{t}} \cdot \mathbf{v}\vert _{\varGamma _{t}} = -\tfrac{\tilde{m}} {2} [\mathbf{n}_{\varGamma _{t}} \cdot \nabla \mu ]\quad \text{on}\ \varGamma _{t},t \in (0,T), }$$
(9.13)
$$\displaystyle{ 2\mu =\sigma H_{\varGamma _{t}}\quad \qquad \qquad \text{on}\ \varGamma _{t},t \in (0,T), }$$
(9.14)
$$\displaystyle{ \varDelta \mu = 0\ \qquad \qquad \quad \text{in}\ \varOmega ^{\pm }(t),t \in (0,T), }$$
(9.15)
$$\displaystyle{ \varOmega ^{+}(0) =\varOmega _{ 0}^{+}.\ }$$
(9.16)

Here Ω is the disjoint union of Ω +(t), Ω −(t), and Γ t = ∂Ω +(t), \(\mathbf{n}_{\varGamma _{ t}}\) is the interior normal of Γ t with respect to Ω +(t), \(V _{\varGamma _{t}},H_{\varGamma _{t}}\) are the normal velocity, mean curvature of the interface, respectively, and T(v, p) = 2ν ± D v − p I in Ω ±(t) is the stress tensor in each fluid phase. Furthermore, \([u](x) =\lim _{h\rightarrow 0+}u(x + h\mathbf{n}_{\varGamma _{t}}(x)) - u(x - h\mathbf{n}_{\varGamma _{t}}(x))\), x ∈ Γ t , denotes the jump of a quantity u at the interface Γ t , \(\sigma =\int _{ -1}^{1}\sqrt{2f(s)}\,ds> 0\) is a (constant) surface tension coefficient, depending only on f, which will be specified below, \(\tilde{m} = 0\) if m ɛ = m 0 ɛ for some m 0 > 0 and \(\tilde{m} = m_{0}\) if m ɛ = m 0. Actually in the latter contribution a more general model for the case of fluids with different densities was considered.

We note that in the case \(\tilde{m} = 0\) (9.14)– (9.15) and the chemical potential μ can be disregarded and solutions of (9.1)– (9.5) converge to solutions of the classical sharp interface model for a two-phase flow of viscous, incompressible fluids with pure transport of the interface Γ t in time. That is

$$\displaystyle{ V _{\varGamma _{t}} = \mathbf{n} \cdot \mathbf{v}\vert _{\varGamma _{t}}\qquad \text{on}\ \varGamma _{t},t \in (0,T), }$$

Moreover, we have the well-known Young-Laplace equation (9.11) relating the jump of the normal component of the stress tensor to surface tension forces. We note that this convergence was formally verified by Starovoitov [29] in the case m ɛ = m 0 ɛ and Lowengrub and Truskinovsky [23] in the case m ɛ = m 0 ɛ 2. A more detailed analysis can be found in [8] and an overview in [17, Sect. 4].

In the case \(\tilde{m}> 0\) the system is a new sharp interface model for a two-phase flow of viscous, incompressible fluids. In this case the dynamics of the interface Γ t is given by a Mullins-Sekerka equation with additional convection term, cf. (9.13). Here m 0 is related to the strength of diffusion in the system and non-local effects like Ostwald ripening are present in the system as for the separate Mullins-Sekerka and Cahn-Hilliard system.

Here and in the following we will not consider the case that the interface Γ t intersects the boundary ∂Ω. Therefore we will not discuss situations where contact angle conditions at ∂Ω ∩Γ t occur. Throughout this contribution we will not specify the precise (technical) assumptions on f for each result discussed. Typical assumptions are that f is sufficiently smooth, even, non-negative, f(c) = 0 if and only if c = ±1, and satisfies f″(±1) > 0 together with suitable growth conditions. We refer to the corresponding publications for the precise assumptions for each result discussed in the following. A standard example is \(f(c) = \frac{1} {8}(1 - c^{2})^{2}\), \(c \in \mathbb{R}\). Moreover, the so-called optimal profile will play an important role for the following, which is the unique solution of

$$\displaystyle\begin{array}{rcl} -\theta _{0}^{{\prime\prime}}(x) + f^{{\prime}}(\theta _{ 0}(x))& =& 0\quad \text{for all}\ x \in \mathbb{R}, \\ \theta _{0}(0)& =& 0,\ \theta _{0}(x) \rightarrow _{x\rightarrow \pm \infty }\pm 1.{}\end{array}$$
(9.17)

If \(f(c) = \frac{1} {8}(1 - c^{2})^{2}\), \(c \in \mathbb{R}\), then \(\theta _{0}(x) =\tanh (\tfrac{x} {2} )\) for all \(x \in \mathbb{R}\). In general θ 0 has the same qualitative behaviour as in the latter case. In particular, θ 0(x) → x → ±∞ ± 1 and θ 0′(x), θ 0″(x) → x → ±∞ 0 exponentially, cf. e.g. [28, Lemma 2.6.1]. In the following we will use that

$$\displaystyle{ \sigma =\int _{ -\infty }^{\infty }\sqrt{2f(\theta _{ 0}(x))}\theta _{0}'(x)\,dx =\int _{ -\infty }^{\infty }\theta _{ 0}'(x)^{2}\,dx }$$

because of \(f(\theta _{0}(x)) = \tfrac{\theta _{0}'(x)^{2}} {2}\), which follows from integration of (9.17) multiplied by θ 0′(x).

The structure of this contribution is as follows: In Sect. 9.2 we will discuss rigorous results on convergence and non-convergence of the convective Cahn-Hilliard equation (9.3)– (9.4) as ɛ → 0 for a given velocity v, which indicate what kind of results can be expected for the full coupled system (9.1)– (9.4). In Sect. 9.3 we discuss mathematical results on convergence of weak solutions of the model H to so-called varifold solutions of (9.8)– (9.16) for large times. A disadvantage of these results is that uniqueness of varifold solutions is unknown, convergence is only obtained for a suitable subsequence and no convergence rates can be shown. Therefore a goal is to prove convergence results of the full sequence together with convergence rates at least for sufficiently small times when the limit system possesses a smooth solution. In Sect. 9.4 we present a first convergence result of this kind, when the Navier-Stokes/Cahn-Hilliard system (9.1)– (9.4) is replaced by a Stokes/Allen-Cahn system. In this case the limit system is a Stokes system coupled to a mean curvature flow equation with an additional convection term. Finally, in Sect. 9.5 we discuss works in progress and further perspectives.

2 Sharp Interface Limit of the Convective Cahn-Hilliard Equation

In this section we discuss the sharp interface limit of the convective Cahn-Hilliard equation

$$\displaystyle\begin{array}{rcl} \partial _{t}c_{\varepsilon } + \mathbf{v} \cdot \nabla c_{\varepsilon }& =& m_{\varepsilon }\varDelta \mu _{\varepsilon },{}\end{array}$$
(9.18)
$$\displaystyle\begin{array}{rcl} \mu _{\varepsilon }& =& -\varepsilon \varDelta c_{\varepsilon } + \varepsilon ^{-1}f'(c_{\varepsilon }).{}\end{array}$$
(9.19)

Here v is assumed to be a given, sufficiently smooth velocity field such that \(\mathop{\mathrm{div}}\nolimits \mathbf{v} = 0\) and n ⋅ v | ∂Ω = 0. More precisely, we assume that m ɛ = m 0 ɛ k for some k ≥ 0, m 0 > 0, and pose the following:

Question:

For which k do solutions c ɛ of  (9.18)– (9.19) (with suitably well prepared initial data) converge to a solution of

$$\displaystyle{ 2\partial _{t}\chi _{\varOmega ^{+}(t)} + 2\mathbf{v} \cdot \nabla \chi _{\varOmega ^{+}(t)} =\tilde{ m}\varDelta \mu \quad \mathit{\text{in}}\ \mathcal{D}'(\varOmega \times (0,T)), }$$
(9.20)
$$\displaystyle{ 2\mu =\sigma H_{\varGamma _{t}}\quad \mathit{\text{on}}\ \partial \varOmega ^{+}(t) }$$
(9.21)

for some \(\tilde{m} \geq 0\) together with

$$\displaystyle{ \varepsilon \int _{0}^{T}\int _{ \varOmega }\nabla c_{\varepsilon } \otimes \nabla c_{\varepsilon }: \nabla \boldsymbol{\varphi }\,dx\,dt \rightarrow _{\varepsilon \rightarrow 0}\sigma \int _{0}^{T}\int _{ \varGamma _{t}}\mathbf{n}_{\varGamma _{t}} \otimes \mathbf{n}_{\varGamma _{t}}: \nabla \boldsymbol{\varphi }\,d\mathcal{H}^{d-1}\,dt }$$
(9.22)

for all \(\boldsymbol{\varphi }\in C_{0}^{\infty }(\varOmega \times (0,T))^{d}\) with \(\mathop{\mathrm{div}}\nolimits \boldsymbol{\varphi } = 0\) ?Here \(\mathcal{H}^{d-1}\) denotes the (d − 1)-dimensional Hausdorff measure. We note that \(\mathcal{H}^{d-1}(M)\) is the area of M if d = 3 and \(M \subseteq \mathbb{R}^{3}\) is a smooth and compact surface. Moreover, we remark that

$$\displaystyle{ \int _{\varGamma _{t}}\mathbf{n}_{\varGamma _{t}} \otimes \mathbf{n}_{\varGamma _{t}}: \nabla \boldsymbol{\varphi }\,d\mathcal{H}^{d-1} = -\int _{\varGamma _{ t}}(\mathbf{I} -\mathbf{n}_{\varGamma _{t}} \otimes \mathbf{n}_{\varGamma _{t}}): \nabla \boldsymbol{\varphi }\,d\mathcal{H}^{d-1} =\int _{\varGamma _{ t}}H_{\varGamma _{t}}\mathbf{n}_{\varGamma _{t}} \cdot \boldsymbol{\varphi }\, d\mathcal{H}^{d-1} }$$

since \(\mathop{\mathrm{div}}\nolimits \boldsymbol{\varphi } = 0\) provided Γ t is sufficiently regular. We will call

$$\displaystyle{ \boldsymbol{\varphi }\mapsto -\int _{\varGamma _{t}}(\mathbf{I} -\mathbf{n}_{\varGamma _{t}} \otimes \mathbf{n}_{\varGamma _{t}}): \nabla \boldsymbol{\varphi }\,d\mathcal{H}^{d-1} }$$
(9.23)

the mean curvature functional in the following. It can be used for weak formulations of the sharp interface models.

The motivation for (9.22) comes from the final goal to pass to the limit in the coupled system (9.1)– (9.5), where the left-hand side of (9.22) is a weak formulation of the right-hand side of (9.1), which should converge to a weak formulation of the right-hand side of the Young-Lapace law (9.11).

Surprisingly, the answer to the question above is negative if k > 3, i.e., (9.22) does not hold in that case. But in the case k = 0, 1 it is possible to prove that (9.22) and (9.20)– (9.21) hold for k = 0 with \(\tilde{m} = m_{0}\) and for k = 1 with \(\tilde{m} = 0\). We will first explain the negative result.

In order to prove the negative result we first considered the case “k = ∞”, which corresponds to choosing m ɛ = 0 for all ɛ ∈ (0, 1). Then (9.18) reduces to the transport equation, which can be solved by the method of characteristics. We will denote by c ɛ ∞ its solution with initial value c 0,ɛ . Hence

$$\displaystyle{ c_{\varepsilon }^{\infty }(x,t) = c_{ 0,\varepsilon }\left (X_{t}^{-1}(x)\right )\qquad \text{for all}\ x \in \varOmega,t \in (0,T), }$$

where X t (ξ) is the solution of

$$\displaystyle{ \tfrac{d} {dt}X_{t}(\xi ) = \mathbf{v}(X_{t}(\xi ),t),\quad \left.X_{t}(\xi )\right \vert _{t=0} =\xi \, }$$

for all ξ ∈ Ω. Then one can show that

$$\displaystyle{ \varepsilon \int _{0}^{T}\int _{ \varOmega }\nabla c_{\varepsilon }^{\infty }\otimes \nabla c_{\varepsilon }^{\infty }: \nabla \boldsymbol{\varphi }\,dx\,dt \rightarrow _{\varepsilon \rightarrow 0}\sigma \int _{0}^{T}\int _{ \varGamma _{t}}\kappa \mathbf{n}_{\varGamma _{t}} \otimes \mathbf{n}_{\varGamma _{t}}: \nabla \boldsymbol{\varphi }\,d\mathcal{H}^{d-1}\,dt }$$

for all \(\boldsymbol{\varphi }\in C_{0}^{\infty }(\varOmega \times (0,T))^{d}\), where κ is determined explicitly by X t and κ ≢ 1 in general, cf. [28, Theorem 5.2.2] or [6]. Hence (9.22) does not hold in general. To get a more precise description we assume for simplicity that \(\varOmega = \mathbb{R}^{d}\), \(\varGamma _{0} = \mathbb{R}^{d-1} \times \{ 0\}\), ν ≡ const. , and \(c_{0,\varepsilon }(x_{d}) =\theta _{0}(\tfrac{x_{d}} {\varepsilon } )\). Then one has

$$\displaystyle\begin{array}{rcl} \int _{\varOmega }\nabla c_{\varepsilon }^{\infty }\otimes \nabla c_{\varepsilon }^{\infty }: \nabla \boldsymbol{\varphi }\,dx& = & \tfrac{1} {\varepsilon } \int _{\mathbb{R}^{d}}\left (\theta _{0}^{{\prime}}\!\left (\tfrac{x_{d}} {\varepsilon } \right )\right )^{2}Ae_{ d} \otimes Ae_{d}: \nabla \varphi \circ X_{t}(x)\,dx {}\\ & \rightarrow _{\varepsilon \rightarrow 0}& \sigma \int _{\mathbb{R}^{d-1}}Ae_{d} \otimes Ae_{d}: \nabla \varphi \circ X_{t}(x)\,dx {}\\ & = & \sigma \int _{\varGamma _{t}}\mathop{\underbrace{\left \vert Ae_{d}\right \vert \circ X_{t}^{-1}}}\limits _{=\kappa }\mathbf{n}_{\varGamma _{t}} \otimes \mathbf{n}_{\varGamma _{t}}: \nabla \varphi \,d\mathcal{H}^{d-1} {}\\ \end{array}$$

for every t ∈ (0, T), where \(\sigma =\int _{\mathbb{R}}\vert \theta _{0}^{{\prime}}(s)\vert ^{2}\,ds\),

$$\displaystyle{ A = (DX_{t}^{-1})^{T} = \mbox{ cof}DX_{ t},\quad Ae_{d} = \left \vert Ae_{d}\right \vert \mathbf{n}_{\varGamma _{t}}, }$$

Γ t = X t (Γ 0), | Ae d | is the surface element of Γ t , and κ = | Ae d | ∘ X t −1. Choosing a velocity field v such that X t increases or decreases the length locally, one can easily obtain that κ ≢ 1.

In the case k > 3 one gets the same result since one is able to prove that

$$\displaystyle{ \varepsilon \|\nabla (c_{\varepsilon } - c_{\varepsilon }^{\infty })\|_{ L^{2}(\varOmega \times (0,T))}^{2} \rightarrow _{\varepsilon \rightarrow 0}0 }$$
(9.24)

with the aid of the energy method applied to the equation for the difference c ɛ − c ɛ ∞, cf. [28, Lemma 5.2.7]. Here c ɛ is the solution of (9.18)– (9.19) and c ɛ ∞ is the solution of the limit case “k = ∞” as above. Hence one obtains as before

$$\displaystyle{ \varepsilon \int _{0}^{T}\int _{ \varOmega }\nabla c_{\varepsilon } \otimes \nabla c_{\varepsilon }: \nabla \boldsymbol{\varphi }\,dx\,dt \rightarrow _{\varepsilon \rightarrow 0}\sigma \int _{0}^{T}\int _{ \varGamma _{t}}\kappa \mathbf{n} \otimes \mathbf{n}: \nabla \boldsymbol{\varphi }\,d\mathcal{H}^{d-1}\,dt, }$$

where κ ≢ 1 in general as in the case k = ∞. The reason that (9.22) does not hold in the case k > 3 in general is that the convection in the equation caused by v ⋅ ∇c dominates the diffusion related to the term m ɛ Δμ ɛ . More precisely, in that case one does not have the relation

$$\displaystyle{ c_{\varepsilon }(x,t) =\theta _{0}\left (\frac{\mathop{\mathrm{sdist}}\nolimits (x,\varGamma _{t})} {\varepsilon } \right ) +\mathcal{ O}(\varepsilon )\quad \text{as}\ \varepsilon \rightarrow 0, }$$
(9.25)

where \(\mathop{\mathrm{sdist}}\nolimits (x,\varGamma _{t})\) is the signed distance of x to Γ t , cf. (9.42) below. This relation holds true in the case k = 0, 1. If the initial values satisfy \(c_{0,\varepsilon } \approx \theta _{0}(\tfrac{\mathop{\mathrm{sdist}}\nolimits (x,\varGamma _{0})} {\varepsilon } )\), then

$$\displaystyle{ c_{\varepsilon }(x,t) \approx \theta _{0}\left (\frac{\mathop{\mathrm{sdist}}\nolimits (X_{t}^{-1}(x),\varGamma _{0})} {\varepsilon } \right )\neq \theta _{0}\left (\frac{\mathop{\mathrm{sdist}}\nolimits (x,\varGamma _{t})} {\varepsilon } \right ) +\mathcal{ O}(\varepsilon ) }$$

in general.

On the other-hand in the case k = 0 one can adapt the construction of suitable approximative solutions in [9] to include the convection term v ⋅ ∇c. Then the approximative solution satisfies (9.25) and one can prove that the difference of approximative and exact solution of (9.18)– (9.19) converges to zero in a sufficiently strong norm provided that the limit system (9.20)– (9.21) possesses a sufficiently smooth solution, which is true at least for small times and sufficiently regular initial surfaces. Using (9.25) one is able to prove (9.22) in the case k = 0.

In the case k = 1 it is possible to modify the approach of [9] to prove that solutions of (9.18)– (9.19) converge to (9.20) with \(\tilde{m} = 0\) and that (9.25) as well as (9.22) hold. We refer to [28, Chap. 6] for the details.

We note that the result is consistent with the scaling m ɛ = m 0 ɛ k with k ∈ [1, 2) proposed and used by Jacqmin [20] for numerical simulations.

3 Sharp Interface Limit for the Navier-Stokes/Cahn-Hilliard System

3.1 A Counter Example for Too Fast Decreasing Mobility

As before we have considered a scaling of the mobility m ɛ = m 0 ɛ k for some m 0 > 0, k ≥ 0 for the full system (9.1)– (9.5). Then the question arises: For which k do solutions of the system converge to solutions of (9.8)– (9.16)? Based on the experience for the convective Cahn-Hilliard system we were able to construct a radial symmetric solution of (9.1)– (9.5), which does not converge to a solution of (9.8)– (9.16) if k > 3. On the other hand, if k ∈ [0, 1), we were able to prove convergence for the full system in the sense of varifold solutions due to Chen [13] for the Cahn-Hilliard equation.

More precisely, for the case k > 3 we consider (9.1)– (9.5) in the exterior domain \(\varOmega =\{ x \in \mathbb{R}^{d}: \vert x\vert> 1\}\) with the “inflow” boundary conditions

$$\displaystyle{ \begin{array}{llll} \mathbf{v}_{\varepsilon }\vert _{\partial \varOmega }& = a \tfrac{x} {\vert x\vert } \equiv a\mathbf{e}_{r}&\qquad \ \ &\text{on}\ \partial \varOmega \times (0,T), \\ c_{\varepsilon }\vert _{\partial \varOmega } &= 1 &&\text{on}\ \partial \varOmega \times (0,T), \end{array} }$$

where a > 0. Moreover, we considered solutions of the form v ɛ (x, t) = u ɛ (r, t)e r , \(c_{\varepsilon }(x,t) =\tilde{ c}_{\varepsilon }(r,t)\), where \(r = \vert x\vert,\mathbf{e}_{r} = \tfrac{x} {\vert x\vert }\) such that \(c_{\varepsilon }\vert _{t=0} = c_{0,\varepsilon } =\theta (\tfrac{r-r_{0}} {\varepsilon } )\) with r 0 > 1 and

$$\displaystyle{ \theta (s) = \left \{\begin{array}{@{}l@{\quad }l@{}} 1 \quad &\text{if}\ s \leq -1,\\ -1\quad &\text{if} \ s \geq 1. \end{array} \right. }$$

I.e., c 0,ɛ describes a spherical droplet of radius r 0 > 1 with a diffuse interface of thickness 2ɛ, cf. Fig. 9.2.

Fig. 9.2
figure 2

Radial symmetric flow with inflow

Full size image

Again we first considered the case k = ∞, i.e., m ɛ ≡ 0, where the solution can be calculated explicitly. Since v ɛ (x, t) = u ɛ (r, t)e r , (9.2) reduces to ∂ r (r d−1 u ɛ (r, t)) = 0. Therefore

$$\displaystyle{ u_{\varepsilon }(r,t) = \frac{a} {r^{d-1}}\qquad \text{for all}\ r> 1,t> 0 }$$

because of the boundary condition for v ɛ . Hence (9.3) (with m ɛ ≡ 0) reduces to

$$\displaystyle{ \partial _{t}c_{\varepsilon }(r,t) + \frac{a} {r^{d-1}}\partial _{r}c_{\varepsilon }(r,t) = 0. }$$

Together with the initial value the solution is given by

$$\displaystyle{ c_{\varepsilon }^{\infty }(x,t) =\theta \left (\frac{\root{d}\of{r^{d} - dat} - r_{ 0}} {\varepsilon } \right )\qquad \text{with}\ r = \vert x\vert. }$$

In Fig. 9.3 one can observe that the gradient in the diffuse interface regions increases as time increases and the diffuse interface thickness decreases. This results in an “increase in the surface tension coefficient” as follows: Using the explicit formulas for v ɛ and c ɛ ∞, one can determine p ɛ ∞ uniquely (up to a constant) by (9.1). Then one verifies that

$$\displaystyle{ \begin{array}{llll} c_{\varepsilon }^{\infty }& \rightarrow _{\varepsilon \rightarrow 0}2\chi _{B_{R(t)}(0)} - 1&\qquad \ &\text{for every}\ x \in \varOmega \setminus \partial B_{R(t)}(0),t \in (0,T), \\ p_{\varepsilon }^{\infty }& \rightarrow _{\varepsilon \rightarrow 0}p &\qquad &\text{for every}\ x \in \varOmega \setminus \partial B_{R(t)}(0),t \in (0,T),\end{array} }$$

where \(R(t) = \root{d}\of{r_{0}^{d} + dat}\) and

$$\displaystyle{ [\,p] = \kappa (t)\sigma H\qquad \text{on}\ \varGamma _{t} = \partial B_{R(t)}(0) }$$
(9.26)

with 1 < κ(t) → t → ∞ ∞ and \(\sigma =\int _{\mathbb{R}}\vert \theta _{0}^{{\prime}}(s)\vert ^{2}\,ds\) as before. In particular (v, p) do not satisfy (9.11) since [D v] = 0. As in the case of the convective Cahn-Hilliard equation (9.18)– (9.19), one can prove in the case k > 3 that (9.24) holds. This yields finally the same result as in the case k = ∞. In particular (9.26) holds with the same κ(t). We refer to [3] for the details.

Fig. 9.3
figure 3

Plot of the solution in radial direction for t = 0, 1, 2, 3, 4 (from left to right) with a = 1, ɛ = 0. 4, r 0 = 2, d = 2

Full size image

3.2 Convergence to Varifold Solutions

For the following we assume that c 0,ɛ , v 0,ɛ , ɛ ∈ (0, 1) are initial values such that

$$\displaystyle{ \sup _{\varepsilon \in (0,1)}\left (E_{\varepsilon }(c_{0,\varepsilon }) +\int _{\varOmega }\tfrac{\rho \vert \mathbf{v}_{0,\varepsilon }\vert ^{2}} {2} \,dx\right ) <\infty. }$$

Then (9.7) yields that

$$\displaystyle{ E_{\varepsilon }(c_{\varepsilon }(t)) +\int _{\varOmega }\tfrac{\rho \vert \mathbf{v}_{\varepsilon }(t)\vert ^{2}} {2} \,dx }$$

is uniformly bounded with respect to t ∈ (0, T) and ɛ ∈ (0, 1) as well. Moreover, we note that by Modica and Mortola [26] or Modica [25], we have

$$\displaystyle{ E_{\varepsilon } \rightarrow _{\varepsilon \rightarrow 0}\mathcal{P}\qquad \text{w.r.t.}\ L^{1}\text{-}\varGamma \text{-convergence}, }$$
(9.27)

where

$$\displaystyle{ \mathcal{P}(u) = \left \{\begin{array}{@{}l@{\quad }l@{}} \sigma \,\mathcal{H}^{d-1}(\partial ^{{\ast}}E)\quad &\text{if}\ u = -1 + 2\chi _{ E}\ \text{and}\ E\ \text{has finite perimeter}, \\ +\infty \quad &\text{else}. \end{array} \right. }$$

Here E is a set of finite perimeter (in Ω) if and only if the distribution ∇χ E coincides with a vector-valued Radon measure and ∂ ∗ E is the reduced boundary of a set of finite perimeter, cf. e.g. [16]. We remark that for sets of finite perimeter we have \(\vert \nabla \chi _{E}\vert = \mathcal{H}^{d-1}\lfloor \partial ^{{\ast}}E\), where | ∇χ E | denotes the total variation measure of ∇χ E , and we have that the Radon-Nikodym derivative \(\frac{\nabla \chi _{E}} {\vert \nabla \chi _{E}\vert }\) coincides with a measure theoretic interior normal of ∂ ∗ E. For the following we denote by \(\mathcal{M}(\varOmega )\) the set of all signed Radon measures on Ω and recall that \(\mathcal{M}(\varOmega ) = C_{c}(\varOmega )'\) by the Riesz representation theorem. Here C c k(Ω), \(k \in \mathbb{N}_{0}\), denotes the set of all k-times continuously differentiable functions \(f: \varOmega \rightarrow \mathbb{R}\) with compact support and C c (Ω) = C c 0(Ω). Furthermore,

$$\displaystyle{ BV (\varOmega ) =\{\, f \in L^{1}(\varOmega ): \nabla f \in \mathcal{ M}(\varOmega )^{d}\}, }$$

where L p(Ω), 1 ≤ p ≤ ∞, denotes the standard Lebesgue space with respect to the Lebesgue measure and L p(0, T; X) denotes its X-valued variant for Ω = (0, T), where X is a Banach space. H 1(Ω) denotes the L 2-Sobolev space of first order.

The proof of the following convergence result as ɛ → 0 is based on ideas of the proof of the latter results by Modica and Mortola, similar to Chen [13]. First of all, using the uniform boundedness of E ɛ (c ɛ (t)), f″(±1) > 0 and a suitable growth condition for f″ one obtains that there is some C > 0 such that

$$\displaystyle{ \int _{\varOmega }(\vert c_{\varepsilon }(t)\vert - 1)^{2}\,dx \leq C\varepsilon R }$$
(9.28)

for all t ∈ (0, T), ɛ ∈ (0, 1). Hence there is a subsequence \((c_{\varepsilon _{k}})_{k\in \mathbb{N}}\), which will for simplicity be again denoted by (c ɛ ) ɛ ∈ (0, 1), such that

$$\displaystyle{ c_{\varepsilon }(x,t) \rightarrow _{\varepsilon \rightarrow 0} \pm 1\qquad \text{for almost all }x \in \varOmega,t \in (0,T). }$$
(9.29)

Moreover, if we define

$$\displaystyle{ W(c) =\int _{ -1}^{c}\sqrt{2\tilde{f}(s)}\,ds,\ \text{ where }\tilde{f}(s) =\min (f(s),1 + \vert s\vert ^{2}), }$$

and

$$\displaystyle{ w_{\varepsilon }(x,t) = W(c_{\varepsilon }(x,t)), }$$

then (w ɛ ) ɛ ∈ (0, 1) are uniformly bounded in L ∞(0, T; BV (Ω)) since

$$\displaystyle{ \int _{\varOmega }\vert \nabla w_{\varepsilon }(x,t)\vert \,dx =\int _{\varOmega }\sqrt{2\tilde{f}(c_{\varepsilon }(x, t))}\vert \nabla c_{\varepsilon }(x,t)\vert \,dx \leq E_{\varepsilon }(c_{\varepsilon }(t)) }$$
(9.30)

is uniformly bounded with respect to t ∈ (0, T), ɛ ∈ (0, 1). Hence there is some w ∈ L w∗ ∞(0, T; BV (Ω)) such that

$$\displaystyle{ w_{\varepsilon } \rightharpoonup _{ \varepsilon \rightarrow 0}^{{\ast}}w\qquad \text{in }L_{ w{\ast}}^{\infty }(0,T;BV (\varOmega )) }$$

for a suitable subsequence. Here L w∗ ∞(0, T; X′) is the space of all weakly-∗ measurable and essentially bounded functions u: (0, T) → X′ for a Banach space X. Moreover, using that BV (Ω) is compactly embedded into L 1(Ω), w ɛ → ɛ → 0 w in L 1(Ω) for every t ∈ (0, T) (and a suitable subsequence). Because of (9.29), we have w(x, t) ∈ {0, σ}. Hence there is a measurable set E such that w = σχ E and we define c = −1 + 2χ E . Together with suitable estimates for differences in time one can show, cf. [3, Lemmas 3.4 and 3.6]:

Lemma 1

There exists a subsequence and a measurable set E ⊂ Ω× [0, T] such that, as ɛ → 0,

$$\displaystyle\begin{array}{rcl} w_{\varepsilon } \rightarrow \sigma \chi _{E}& & \quad \mathit{\text{ a.e. in }}\varOmega \times (0,T)\mathit{\text{ and in }}C^{\frac{1} {9} }([0,T];L^{1}(\varOmega )) {}\\ c_{\varepsilon } \rightarrow -1 + 2\chi _{E}& & \quad \mathit{\text{ a.e. in }}\varOmega \times (0,T)\mathit{\text{ and in }}C^{\frac{1} {9} }([0,T];L^{2}(\varOmega )) {}\\ \end{array}$$

Moreover, \(\chi _{E} \in L_{w{\ast}}^{\infty }(0,T;BV (\varOmega )) \cap C^{\frac{1} {4} }([0,T];L^{1}(\varOmega ))\) and for all t ∈ [0, T] we have | E t | = | E 0 | .

Here C α([0, T]; X), α ∈ (0, 1), is the set of all f: [0, T] → X that are Hölder continuous with exponent α.

With aid of the last considerations and a suitable one-sided estimate of the so-called discrepancy measure \(\xi _{\varepsilon }:= \varepsilon \frac{\vert \nabla c_{\varepsilon }\vert ^{2}} {2} -\frac{f(c_{\varepsilon })} {\varepsilon }\) due to [13, Theorem 3.6], the following result was obtained:

Theorem 1

Let (v ɛ , c ɛ , μ ɛ )0 < ɛ ≤ 1 be weak solutions of  (9.1)– (9.4) with m ɛ → m 0 ≥ 0 such that lim ɛ → 0 ɛm ɛ −1 = 0. Then for a suitable subsequence

$$\displaystyle{ \begin{array}{rlll} (\mathbf{v}_{\varepsilon },m(\varepsilon )\mu _{\varepsilon })& \rightharpoonup _{\varepsilon \rightarrow 0}(\mathbf{v},m_{0}\mu ) &\quad &\mathit{\text{in }}L^{2}(0,T;H^{1}(\varOmega )) \\ c_{\varepsilon }& \rightarrow _{\varepsilon \rightarrow 0} - 1 + 2\chi _{E}&\quad &\mathit{\text{in }}C^{\frac{1} {9} }([0,T];L^{2}(\varOmega ))\mathit{\text{ and a.e.}} \end{array} }$$

where χ E ∈ L ∞(0, T; BV (Ω)) and

$$\displaystyle{ \begin{array}{rlll} \partial _{t}(\rho \mathbf{v}) +\mathop{ \mathrm{div}}\nolimits (\rho \mathbf{v} \otimes \mathbf{v}) -\mathop{\mathrm{div}}\nolimits (2\nu (\chi _{E})D\mathbf{v}) + \nabla q& =\delta V && \\ \partial _{t}\chi _{E} + v \cdot \nabla \chi _{E}& = \tfrac{m_{0}} {2} \varDelta \mu & \qquad & \\ \mathit{\text{If}}\ m_{0}> 0: - 2\mu \cdot \nabla \chi _{E}& =\sigma \delta V && \end{array} }$$

in the sense of distributions on Ω × (0, T), where \(V: (0,T) \rightarrow \mathcal{ M}(\varOmega \times G_{d-1})\) is bounded and weak-∗ measurable, G d−1 is the space of (d − 1)-dimensional linear subspaces of \(\mathbb{R}^{d}\) , and

$$\displaystyle{ \langle \delta V (t),\boldsymbol{\psi }\rangle =\int _{\varOmega \times G_{d-1}}(\mathbf{I} -\mathbf{s} \otimes \mathbf{s}): \nabla \boldsymbol{\psi }(x)\,dV (t)(x,\mathbf{s}) }$$

for all \(\boldsymbol{\psi }\in C_{c}^{1}(\varOmega )^{d}\) , cf. Chen [ 13 ].

Here elements in \(\mathcal{M}(\varOmega \times G_{d-1})\) are called (general) varifolds. Therefore (v, χ E , μ, V ) is called varifold solution. The result was proved by A. and Röger [5, Appendix] in the case m ɛ ≡ m 0 > 0 and by A. and Lengeler [3] for the assumptions above even for a model with different densities.

Since every (d − 1)-dimensional hyperplane \(U \subseteq \mathbb{R}^{d}\) is uniquely determined by its normal (up to orientation), we can identify G d−1 with \(\mathbb{S}^{d-1}/ \sim\), where p 0 ∼ p 1 for \(\mathbf{p}_{0},\mathbf{p}_{1} \in \mathbb{S}^{d-1}\) if and only if p 0 = ±p 1 and \(\mathbb{S}^{d-1}\) is the unit sphere in \(\mathbb{R}^{d}\). Then by disintegration V (t) can be decomposed in a non-negative measure \(\vert V _{t}\vert \in \mathcal{ M}(\varOmega )\) and a family of probability measures \(V _{t,x} \in \mathcal{ M}(\mathbb{S}^{d-1}/\{ \pm \mathbf{p}\})\), x ∈ Ω, t ∈ (0, T) such that

$$\displaystyle{ \langle V (t),\psi \rangle =\int _{\varOmega }\int _{\mathbb{S}^{d-1}}\psi (x,\mathbf{p})\,dV _{t,x}(\mathbf{p})\,d\vert V _{t}\vert (x)\quad \text{for all}\ \psi \in C_{c}(\varOmega \times \mathbb{S}^{d-1}), }$$

cf. [10, Theorem 2.28]. Here | V t | corresponds to the measure of “area of the interface” and V t, x can be considered as a probability for the “normal at the interface”. If Γ(t) ⊆ Ω, t ∈ (0, T) is a smoothly evolving family of hypersurfaces, then the canonically associated varifold V (t) associated to it is given by

$$\displaystyle{ \langle V (t),\psi \rangle =\int _{\varGamma _{t}}\psi (x,[\mathbf{n}_{\varGamma _{t}}(x)])\,d\mathcal{H}^{d-1}(x)\qquad \text{for all }\psi \in C_{ c}(\varOmega \times \mathbb{S}^{d-1}/\{ \pm \mathbf{p}\}). }$$

Then

$$\displaystyle{ \langle \delta V (t),\boldsymbol{\psi }\rangle =\int _{\varGamma _{t}}\mathop{\underbrace{ (\mathbf{I} -\mathbf{n}_{\varGamma _{t}} \otimes \mathbf{n}_{\varGamma _{t}}): \nabla \psi }}\limits _{=\mathop{\mathrm{div}}\nolimits _{\boldsymbol{\tau }}\boldsymbol{\psi }}\,d\mathcal{H}^{d-1} = -\int _{\varGamma _{ t}}H_{\varGamma _{t}}\mathbf{n}_{\varGamma _{t}} \cdot \boldsymbol{\psi }\, d\mathcal{H}^{d-1} }$$

for all \(\boldsymbol{\psi }\in C_{c}^{1}(\varOmega )^{d}\) because of Gauß’ theorem on hypersurfaces for non-tangential vectorfields. Hence δV (t) generalizes the mean curvature functional (9.23).

Since the normal \(\mathbf{n}_{\varGamma _{t}}\) is replaced by a probability distribution for the normal, varifold solutions are similar to measure-valued solutions, cf. e.g. [24]. We note that in the case m 0 > 0 existence of weak solutions to the Navier-Stokes/Mullins-Sekerka system (9.8)– (9.16) was proved by A. and Röger in [5]. In the definition of weak solutions a suitable formulation of (9.23) instead of the first variation of a (general) varifold V (t) is used. These weak solutions are also varifold solutions as in Theorem 1. But it is unknown whether the converse statement is true and convergence to weak solutions in the sharp interface limit ɛ → 0 is an open problem even in the case of the Cahn-Hilliard equation, i.e., (9.3)– (9.4) with v ɛ ≡ 0. The convergence to varifold solutions was shown by Chen [13]. Finally, we remark that in the case m 0 = 0 even existence of weak solutions to the limit system (9.8)– (9.16) is an open problem. We refer to [1] or [17, section on “Weak Solutions and Diffuse Interface Models for Incompressible Two-Phase Flows”] for a further discussion.

Finally, we remark that every sufficiently smooth solution of (9.8)–(9.16) satisfies

$$\displaystyle{ \frac{d} {dt} \frac{1} {2}\int _{\varOmega }\rho \,\vert \mathbf{v}\vert ^{2}\,dx +\sigma \frac{d} {dt}\mathcal{H}^{d-1}(\varGamma _{ t}) = -\int _{\varOmega }2\nu (c)\vert D\mathbf{v}\vert ^{2}\,dx -\int _{\varOmega }m_{ 0}\vert \nabla \mu \vert ^{2}\,dx, }$$
(9.31)

where \(c(t,x) = -1 + 2\chi _{\varOmega ^{+}(t)}(x)\). In view of (9.27) this is consistent with (9.7).

4 Sharp Interface Limit of a Stokes/Allen-Cahn System

In this section we discuss a first rigorous convergence result with convergence rates in strong norms for the sharp interface limit ɛ → 0 in the case of a two-phase flow in fluid mechanics, which is comparable to results known for single phase field models like the Allen-Cahn equation, which is due to De Mottoni and Schatzman [15], or to the Cahn-Hilliard equation, which is due to Alikakos et al. [9].

More precisely, we consider the asymptotic limit ɛ → 0 of the following system:

$$\displaystyle{ -\varDelta \mathbf{v}_{\varepsilon } + \nabla p_{\varepsilon } = -\varepsilon \mathop{\mathrm{div}}\nolimits (\nabla c_{\varepsilon } \otimes \nabla c_{\varepsilon })\quad \ \text{in}\ \varOmega \times (0,T_{0}), }$$
(9.32)
$$\displaystyle{ \mathop{\mathrm{div}}\nolimits \mathbf{v}_{\varepsilon } = 0\qquad \quad \qquad \ \ \text{in}\ \varOmega \times (0,T_{0}), }$$
(9.33)
$$\displaystyle{ \partial _{t}c_{\varepsilon } + \mathbf{v}_{\varepsilon } \cdot \nabla c_{\varepsilon } =\varDelta c_{\varepsilon } -\tfrac{1} {\varepsilon ^{2}} f'(c_{\varepsilon })\quad \quad \quad \text{in}\ \varOmega \times (0,T_{0}), }$$
(9.34)
$$\displaystyle{ \mathbf{v}_{\varepsilon }\vert _{\partial \varOmega } = 0,\quad c_{\varepsilon }\vert _{\partial \varOmega } = -1,\qquad \text{on }\partial \varOmega \times (0,T_{0}), }$$
(9.35)
$$\displaystyle{ c_{\varepsilon }\vert _{t=0} = c_{0,\varepsilon }\qquad \quad \qquad \ \ \text{in }\varOmega, }$$
(9.36)

where \(\varOmega \subseteq \mathbb{R}^{2}\) is bounded domain with smooth boundary, and for suitable “well-prepared” initial data c 0,ɛ .

Similar to (9.1)– (9.4), every sufficiently smooth solution of (9.32)– (9.36) satisfies the energy identity

$$\displaystyle{ E_{\varepsilon }(c_{\varepsilon }(t)) +\int _{ 0}^{t}\int _{ \varOmega }\left (\vert \nabla \mathbf{v}_{\varepsilon }\vert ^{2} + \frac{1} {\varepsilon } \vert \mu _{\varepsilon }\vert ^{2}\right )\,dx\,d\tau = E_{\varepsilon }(c_{ 0,\varepsilon }) }$$
(9.37)

for all t ∈ (0, T 0), where \(\mu _{\varepsilon } = -\varepsilon \varDelta c_{\varepsilon } + \frac{1} {\varepsilon } f'(c_{\varepsilon })\) and

$$\displaystyle{ E_{\varepsilon }(c_{\varepsilon }(t)) =\int _{\varOmega }\varepsilon \frac{\vert \nabla c_{\varepsilon }(x,t)\vert ^{2}} {2} \,dx +\int _{\varOmega }\frac{f(c_{\varepsilon }(x,t))} {\varepsilon } \,dx }$$

as before. The sharp interface limit of (9.32)– (9.36) is the system

$$\displaystyle{ -\varDelta \mathbf{v} + \nabla p = 0\quad \qquad \ \ \text{in }\varOmega ^{\pm }(t),t \in [0,T_{ 0}], }$$
(9.38)
$$\displaystyle{ \mathop{\mathrm{div}}\nolimits \mathbf{v} = 0\quad \qquad \ \ \text{in }\varOmega ^{\pm }(t),t \in [0,T_{ 0}], }$$
(9.39)
$$\displaystyle{ -[2D\mathbf{v} - p\mathbf{I}]\mathbf{n}_{\varGamma _{t}} =\sigma H_{\varGamma _{t}}\mathbf{n}_{\varGamma _{t}}\qquad \text{on }\varGamma _{t},t \in [0,T_{0}], }$$
(9.40)
$$\displaystyle{ V _{\varGamma _{t}} -\mathbf{n}_{\varGamma _{t}} \cdot \mathbf{v}\vert _{\varGamma _{t}} = H_{\varGamma _{t}}\quad \qquad \ \ \text{on }\varGamma _{t},t \in [0,T_{0}], }$$
(9.41)

where Ω is the disjoint union of Ω +(t), Ω −(t), Γ t for every t ∈ [0, T 0], Ω ±(t) are smooth domains, Γ t = ∂Ω +(t) and \(\sigma =\int _{\mathbb{R}}\theta _{0}'(x)^{2}\,dx\) as before.

We note that, if the material time derivative ∂ t v ɛ + v ɛ ⋅ ∇v ɛ is added to the left-hand side of (9.32) (i.e., the Navier-Stokes equations are considered), the system (9.32)– (9.36) was suggested by Liu and Shen in [21] as an alternative approximation of a classical sharp interface model for a two-phase flow of viscous, incompressible, Newtonian fluids. It has advantages for numerical simulations since the Allen-Cahn equation is of second order and not of fourth order as the Cahn-Hilliard equation. A disadvantage of it is that the total mass ∫ Ω c ɛ (x, t) dx is in general not preserved in time for solutions of (9.32)– (9.36). This property is desirable if the model is used to approximate a two-phase flow without phase transitions. But (9.32)– (9.36) can be considered as a simplified model for a two-phase flow with phase transitions. For such flows one can obtain systems of Navier-Stokes/Allen-Cahn type, cf. e.g. Blesgen [11].

Existence of weak solutions of the sharp interface model (9.38)– (9.41) was proved by Liu et al. [22] if the Stokes equation on the right-hand side is replaced by a modified Navier-Stokes equation for a shear thickening non-Newtonian fluid of power-law type. To this end they used a Galerkin approximation by a corresponding Navier-Stokes/Allen-Cahn system. Then they pass to the limit in the Galerkin approximation and ɛ → 0 simultaneously. But they do not perform a sharp interface limit separately.

In the following we assume that (v, p, Γ) is a smooth solution of (9.38)– (9.41) for some T 0 > 0, where \((\varGamma _{t})_{t\in [0,T_{0}]}\) is a family of smoothly evolving compact, non-selfintersecting, closed curves in Ω. More precisely, we assume that

$$\displaystyle{ \varGamma:=\bigcup _{t\in [0,T_{0}]}\varGamma _{t} \times \{ t\} }$$

is a smooth two-dimensional submanifold of \(\varOmega \times \mathbb{R}\) (with boundary), and \(\mathbf{v}\vert _{\varOmega ^{\pm }} \in C^{\infty }(\overline{\varOmega ^{\pm }})^{2}\), \(p\vert _{\varOmega ^{\pm }} \in C^{\infty }(\overline{\varOmega ^{\pm }})\), where

$$\displaystyle{ \varOmega ^{\pm } =\bigcup _{ t\in [0,T_{0}]}\varOmega ^{\pm }(t) \times \{ t\}. }$$

Moreover, let (v ɛ , p ɛ , c ɛ ) be the (classical) solution of (9.32)– (9.36) with smooth initial values \(c_{0,\varepsilon }: \varOmega \rightarrow \mathbb{R}\), which will be specified in the main result below.

We note that the mean curvature and the Stokes equation in (9.38)– (9.41) are coupled by terms of lower order, which is also the case for the Navier-Stokes/Mullins-Sekerka system (9.8)– (9.16). Hence existence of a local strong solution of the system can be obtained by adapting the strategy in [7], where this result was proven for (9.8)– (9.16). This was carried out by Moser in [27] in the case that the Stokes system (9.38)– (9.39) is replaced by the instationary Navier-Stokes system. By standard arguments from the regularity theory of parabolic equations and the Stokes system, one can prove that the solution is indeed smooth for smooth initial values.

For the statement of our main result we need tubular neighborhoods of Γ t . For δ > 0 and t ∈ [0, T 0] we define

$$\displaystyle{ \varGamma _{t}(\delta ):=\{ y \in \varOmega:\mathop{ \mathrm{dist}}\nolimits (y,\varGamma _{t}) <\delta \},\quad \varGamma (\delta ) =\bigcup _{t\in [0,T_{0}]}\varGamma _{t}(\delta ) \times \{ t\}. }$$

Moreover, we define the signed distance function

$$\displaystyle{ d_{\varGamma }(x,t):=\mathop{ \mathrm{sdist}}\nolimits (\varGamma _{t},x) = \left \{\begin{array}{@{}l@{\quad }l@{}} \mathop{\mathrm{dist}}\nolimits (\varOmega ^{-}(t),x) \quad &\text{if }x\not\in \varOmega ^{-}(t) \\ -\mathop{\mathrm{dist}}\nolimits (\varOmega ^{+}(t),x)\quad &\text{if }x \in \varOmega ^{-}(t) \end{array} \right. }$$
(9.42)

for all x ∈ Ω, t ∈ [0, T 0]. Since Γ is smooth and compact, there is some δ > 0 sufficiently small such that \(d_{\varGamma }: \varGamma (3\delta ) \rightarrow \mathbb{R}\) is smooth.

The main result of A. and L. [4] is:

Theorem 2

Let (v, Γ) be a smooth solution of  (9.38)– (9.41) for some T 0 ∈ (0, ∞) and let

$$\displaystyle{ \begin{array}{ll} c_{A,0}^{0}(x)& =\zeta (d_{\varGamma _{0}}(x))\theta _{0}\left (\frac{d_{\varGamma _{0}}(x)} {\varepsilon } \right ) + (1 -\zeta (d_{\varGamma _{0}}(x)))\left (\chi _{\varOmega ^{+}(0)}(x) -\chi _{\varOmega ^{-}(0)}(x)\right ) \end{array} }$$

for all x ∈ Ω, where \(d_{\varGamma _{0}} = d_{\varGamma }\vert _{t=0}\) is the signed distance function to Γ 0 and \(\zeta \in C^{\infty }(\mathbb{R})\) such that

$$\displaystyle{ \zeta (s) = 1,\ \mathit{\text{if}}\ \vert s\vert \leq \delta;\ \zeta (s) = 0,\ \mathit{\text{if}}\ \vert s\vert \geq 2\delta;\ 0 \leq s\zeta '(s) \leq 4\ \mathit{\text{if}}\ \delta \leq \vert s\vert \leq 2\delta. }$$
(9.43)

Moreover, let \(c_{0,\varepsilon }: \varOmega \rightarrow \mathbb{R}\) , 0 < ɛ ≤ 1, be smooth such that

$$\displaystyle{ \|c_{0,\varepsilon } - c_{A,0}^{0}\|_{ L^{2}(\varOmega )} \leq C\varepsilon ^{2+\frac{1} {2} }\qquad \mathit{\text{for all }}\varepsilon \in (0,1] }$$
(9.44)

and some C > 0, \(\sup _{0<\varepsilon \leq 1}\|c_{0,\varepsilon }\|_{L^{\infty }(\varOmega )} <\infty\) and (v ɛ , c ɛ ) the corresponding solutions of  (9.32)– (9.36). Then there are some ɛ 0 ∈ (0, 1], R > 0, T ∈ (0, T 0], and \(c_{A}: \varOmega \times [0,T_{0}] \rightarrow \mathbb{R}\) , \(\mathbf{v}_{A}: \varOmega \times [0,T_{0}] \rightarrow \mathbb{R}^{2}\) (depending on ɛ) such that

$$\displaystyle{ \sup _{0\leq t\leq T}\|c_{\varepsilon }(t) - c_{A}(t)\|_{L^{2}(\varOmega )} +\| \nabla (c_{\varepsilon } - c_{A})\|_{L^{2}(\varOmega \times (0,T)\setminus \varGamma (\delta ))} \leq R\varepsilon ^{2+\frac{1} {2} } }$$
(9.45a)
$$\displaystyle{ \left \|\left (\nabla _{\boldsymbol{\tau }}(c_{\varepsilon } - c_{A}),\varepsilon \partial _{\mathbf{n}}(c_{\varepsilon } - c_{A})\right )\right \|_{L^{2}(\varOmega \times (0,T)\cap \varGamma (2\delta ))} \leq R\varepsilon ^{2+\frac{1} {2} } }$$
(9.45b)

and

$$\displaystyle{ \|\mathbf{v}_{\varepsilon } -\mathbf{v}_{A}\|_{L^{2}(0,T;H^{1}(\varOmega ))} \leq R\varepsilon ^{2} }$$
(9.46)

hold true for all ɛ ∈ (0, ɛ 0]. Moreover,

$$\displaystyle{ \lim _{\varepsilon \rightarrow 0}c_{A} = \pm 1\qquad \mathit{\text{uniformly on compact subsets in }}\varOmega ^{\pm }. }$$

and

$$\displaystyle{ \mathbf{v}_{A} = \mathbf{v} + O(\varepsilon )\qquad \mathit{\text{in }}L^{\infty }(\varOmega \times (0,T))\mathit{\text{ as }}\varepsilon \rightarrow 0. }$$

Here c A is constructed such that

$$\displaystyle{ c_{A}(x,t) =\mathop{\underbrace{ \theta _{0}\left (\frac{d_{\varGamma }(x,t)} {\varepsilon } - h_{\varepsilon }(S(x,t),t)\right )}}\limits _{=c_{A,0}} + O(\varepsilon ^{2}),\quad \text{where }h_{\varepsilon } = h_{ 1} + \varepsilon h_{2,\varepsilon }, }$$

in Γ(δ) for some suitable \(h_{1},h_{2,\varepsilon }: \mathbb{T}^{1} \times [0,T_{0}] \rightarrow \mathbb{R}\), where \(S(\cdot,t): \varGamma _{t}(3\delta ) \rightarrow \mathbb{T}^{1}\) is the pull-back of a suitable parametrization of Γ t for every t ∈ [0, T 0] and \(\mathbb{T}^{1}\cong \mathbb{S}^{1} =\{ x \in \mathbb{R}^{2}: x_{1}^{2} + x_{2}^{2} = 1\}\) is the one dimensional torus, which corresponds to [0, 2π] if 0 and 2π are identified. Here h ɛ is introduced in order to obtain that the zero-level set of c A, 0 approximates the zero-level set of c ɛ sufficiently well. More precise information on c A can be found in [4, Sect. 4].

Here 2 is the basic convergence order (w.r.t. the L ∞(Ω)-norm). The order of convergence differs by \(\frac{1} {2}\) if the L 2-norm is considered in space and by \(-\frac{1} {2}\) if L 2-norm of the gradient is considered. Here the additional \(\frac{1} {2}\) in the power of ɛ is natural since e.g. a simple change of variables yields

$$\displaystyle{ \|\theta _{0}(\tfrac{\cdot } {\varepsilon } ) - (\chi _{[0,\infty )} -\chi _{(-\infty,0)})\|_{L^{2}(\mathbb{R})} = M\varepsilon ^{\frac{1} {2} }. }$$

Here θ 0(s) → s → ±∞ ± 1 exponentially is used. Moreover, from the L 2-estimates one can obtain an L ∞-estimate in normal direction with the aid of the interpolation inequality

$$\displaystyle{ \|u\|_{L^{\infty }(-2\delta,2\delta )} \leq C_{\delta }\|u\|_{L^{2}(-2\delta,2\delta )}^{\frac{1} {2} }\|u\|_{H^{1}(-2\delta,2\delta )}^{\frac{1} {2} }\quad \text{for all }u \in H^{1}(-2\delta,2\delta ). }$$

Hence (9.45) yields a control of c ɛ − c A of the order ɛ 2 in the L ∞-norm in normal direction and L 2-norms in the tangential direction.

For the proof of our main results we will follow the same basic strategy, which was already successfully used in [15] for the Allen-Cahn equation, in [9] for the Cahn-Hilliard equation, and in [14] for the mass-preserving Allen-Cahn equation. Following this strategy the proof consists of two parts. In the first part a suitable approximative solution for (9.32)– (9.36) upto an error term of a certain order in ɛ is constructed, which will be denoted by (c A , v A ) in the following. In the second step the error of the approximative (c A , v A ) and the exact solutions (c ɛ , v ɛ ) is estimated with the aid of a suitable estimate for the linearized Allen-Cahn operator \(\mathcal{L}_{\varepsilon }\), where

$$\displaystyle{ \mathcal{L}_{\varepsilon }u = -\varDelta u + \frac{1} {\varepsilon ^{2}} f^{{\prime\prime}}(c_{ A})u\qquad \text{for all }u \in H^{2}(\varOmega ) }$$

is used, cf. (9.52) below.

But in order to adapt this strategy to the present system several new difficulties, which are mainly related to the coupling of the Allen-Cahn and the Stokes system, have to be overcome. More precisely, in order to estimate the difference u: = c ɛ − c A a suitable estimate of the convection term v ɛ ⋅ ∇c ɛ is needed. To this end it is essential how this term is approximated in the equation of c A . More precisely, c A is constructed such that the following result holds true, cf. [4, Theorem 1.3]:

Theorem 3

Let the assumption of Theorem  2 be satisfied and R ≥ 1. Then for every ɛ ∈ (0, 1) there are

$$\displaystyle{ \mathbf{v}_{A},\mathbf{w}_{1},\mathbf{w}_{2}: \varOmega \times [0,T_{0}] \rightarrow \mathbb{R}^{2},\quad c_{ A}: \varOmega \times [0,T_{0}] \rightarrow \mathbb{R},\quad r_{A}: \varOmega \times [0,T_{0}] \rightarrow \mathbb{R} }$$

(depending on ɛ ∈ (0, 1]) such that v ɛ = v A + ɛ 2 w 1 + ɛ 2 w 2 and

$$\displaystyle{ \partial _{t}c_{A} + (\mathbf{v}_{A} + \varepsilon ^{2}\mathbf{w}_{ 2}) \cdot \nabla c_{A} + \varepsilon ^{2}\mathbf{w}_{ 1}\vert _{\varGamma }\cdot \nabla c_{A} =\varDelta c_{A} -\frac{f^{{\prime}}(c_{A})} {\varepsilon ^{2}} + r_{A} }$$
(9.47)

in Ω × [0, T 0]. Moreover, there are some ɛ 0 > 0, T 1 > 0 and M R : (0, 1] × (0, T 0] → (0, ∞), which is increasing with respect to both variables, such that M R (ɛ, T) →(ɛ, T) → 00 and, if

$$\displaystyle{ \sup _{0\leq t\leq T_{\varepsilon }}\|c_{\varepsilon }(t) - c_{A}(t)\|_{L^{2}(\varOmega )} +\| \nabla (c_{\varepsilon } - c_{A})\|_{L^{2}(\varOmega \times (0,T_{\varepsilon })\setminus \varGamma (\delta ))} \leq R\varepsilon ^{2+\frac{1} {2} }, }$$
(9.48a)
$$\displaystyle{ \left \|\left (\nabla _{\boldsymbol{\tau }}(c_{\varepsilon } - c_{A}),\varepsilon \partial _{\mathbf{n}}(c_{\varepsilon } - c_{A})\right )\right \|_{L^{2}(\varOmega \times (0,T_{\varepsilon })\cap \varGamma (2\delta ))} \leq R\varepsilon ^{2+\frac{1} {2} } }$$
(9.48b)

holds true for some T ɛ ∈ (0, T 0], ɛ 0 ∈ (0, 1], and all ɛ ∈ (0, ɛ 0], then

$$\displaystyle{ \int _{0}^{T}\left \vert \int _{\varOmega }r_{ A}(x,t)(c_{\varepsilon }(x,t) - c_{A}(x,t))\,dx\right \vert \,dt \leq M_{R}(\varepsilon,T)\varepsilon ^{2(2+\frac{1} {2} )},\quad }$$
(9.49)

for all T ∈ (0, min(T ɛ , T 1)), ɛ ∈ (0, ɛ 0].

Here \(\nabla _{\boldsymbol{\tau }}\) denotes the tangential gradient and w 1 is the leading part of the error \(\mathbf{w} = \frac{\mathbf{v}_{\varepsilon }-\mathbf{v}_{A}} {\varepsilon ^{2}}\) and \(\mathbf{w}_{1}\vert _{\varGamma }(x,t) = \mathbf{w}_{1}(P_{\varGamma _{t}}(x),t)\) for x ∈ Γ t (2δ), where \(P_{\varGamma _{t}}\) denotes the orthogonal projection onto Γ t . Moreover, v A is a suitable approximation of the solution of

$$\displaystyle{ \begin{array}{rlll} -\varDelta \mathbf{v} + \nabla p& = -\varepsilon \mathop{\mathrm{div}}\nolimits (\nabla c_{A,0} \otimes \nabla c_{A,0})&\quad &\text{in }\varOmega,\text{for almost all }t \in (0,T_{0}), \\ \mathop{\mathrm{div}}\nolimits \mathbf{v}& = 0 &\quad &\text{in }\varOmega,\text{for almost all }t \in (0,T_{0}).\end{array} }$$

We note that in (9.47) ɛ 2 w 2 ⋅ ∇c A could also be omitted since it is of the same order as r A . But the presence of the term ɛ 2 w 1 | Γ ⋅ ∇c A is essential for the error estimates. Theorem 3 is proved with the aid of finitely many terms of an expansion in ɛ > 0 close to Γ t (the inner expansion) and away from Γ t (the outer expansion), using the method of formally matched asymptotics. It is shown rigorously that all terms in the expansion are well-defined, which yields that c A is well-defined, and that the estimates above hold true.

Sketch of the proof of Theorem 2

Because of the regularity of c ɛ and c A for every fixed ɛ and the assumption on the initial data, for every ɛ > 0 (9.48) holds true for some T ɛ ∈ (0, T 0]. The essential step in the proof of Theorem 2 is to show that for sufficiently small ɛ > 0 there is some T ∈ (0, T 0) independent of ɛ such that (9.48) hold for T ɛ ≥ T. To this end the equation for u = c ɛ − c A is considered, which can be written as

$$\displaystyle{ \partial _{t}u + \mathbf{v}_{\varepsilon } \cdot \nabla u +\mathcal{ L}_{\varepsilon }u = -r_{\varepsilon }(c_{\varepsilon },c_{A}) - r_{A} +\mathcal{ R} }$$
(9.50)

where

$$\displaystyle\begin{array}{rcl} r_{\varepsilon }(c_{\varepsilon },c_{A})& =& \frac{1} {\varepsilon ^{2}} \left (f^{{\prime}}(c_{\varepsilon }) - f^{{\prime}}(c_{ A}) - f^{{\prime\prime}}(c_{ A})(c_{\varepsilon } - c_{A})\right ), {}\\ \mathcal{R}& =& -\varepsilon ^{2}\mathbf{w}_{ 1} \cdot \nabla c_{A} + \varepsilon ^{2}\mathbf{w}_{ 1}\vert _{\varGamma }\cdot \nabla c_{A}. {}\\ \end{array}$$

Here r ɛ (c ɛ , c A ) is the error due to a linearization of \(\tfrac{1} {\varepsilon ^{2}} f'(c_{\varepsilon })\), r A is the error in the equation for the approximation c A , cf. (9.47), and \(\mathcal{R}\) is the error in the approximation of the convection term v ɛ ⋅ ∇c ɛ . Multiplying (9.50) with u and integrating yield

$$\displaystyle\begin{array}{rcl} & & \frac{1} {2}\|u(t)\|_{L^{2}(\varOmega )}^{2} +\int _{ 0}^{t}\int _{ \varOmega }\left (\vert \nabla u\vert ^{2} + \frac{f^{{\prime\prime}}(c_{ A})} {\varepsilon ^{2}} u^{2}\right )\,dx\,ds \\ & & \leq \int _{0}^{t}\int _{ \varOmega }\vert r_{\varepsilon }(c_{\varepsilon },c_{A})u\vert \,dx\,ds +\int _{ 0}^{t}\left \vert \int _{\varOmega }r_{ A}\ u\,dx\right \vert \,ds +\int _{ 0}^{t}\left \vert \int _{\varOmega }\mathcal{R}\ u\,dx\right \vert \,ds + \frac{1} {2}\|u(0)\|_{L^{2}(\varOmega )}^{2}{}\end{array}$$
(9.51)

for all t ∈ [0, T ɛ ]. Here

$$\displaystyle{ \int _{\varOmega }\left (\vert \nabla u\vert ^{2} + \frac{f^{{\prime\prime}}(c_{ A})} {\varepsilon ^{2}} u^{2}\right )\,dx =\langle \mathcal{ L}_{\varepsilon }u,u\rangle. }$$

Next the following estimate is used: There are some C, ɛ 1 > 0 such that for all ψ ∈ H 1(Ω), ɛ ∈ (0, ɛ 1], and t ∈ [0, T 0]

$$\displaystyle\begin{array}{rcl} & & \int _{\varOmega }\left (\vert \nabla \psi (x)\vert ^{2} + \varepsilon ^{-2}f^{{\prime\prime}}(c_{ A}(x,t))\psi ^{2}(x)\right )\,dx \\ & & \quad \geq -C\int _{\varOmega }\psi ^{2}\,dx +\int _{\varOmega \setminus \varGamma _{ t}(\delta )}\vert \nabla \psi \vert ^{2}\,dx +\int _{\varGamma _{ t}(2\delta )}\vert \nabla _{\tau }\psi \vert ^{2}\,dx.{}\end{array}$$
(9.52)

This is called a spectral estimate since it implies that the L 2-spectrum of \(\mathcal{L}_{\varepsilon }\) is bounded below by \(-\sqrt{C}\) independent of ɛ ∈ (0, ɛ 1). The estimate holds true if c A is of suitable form, which is in particular true for c A in Theorem 3. The estimate is a refinement of [12, Theorem 2.3], cf. [4, Theorem 2.12]. Now using the latter spectral estimate, the assumption on the initial data (9.44), and (9.49) from Theorem 3, we obtain

$$\displaystyle\begin{array}{rcl} & & \sup _{0\leq s\leq t}\|u(s)\|_{L^{2}(\varOmega )}^{2} +\| \nabla _{\boldsymbol{\tau }}u\|_{ L^{2}(\varOmega \times (0,t)\cap \varGamma (2\delta ))}^{2} +\| \nabla u\|_{ L^{2}(\varOmega \times (0,t)\setminus \varGamma (\delta ))}^{2} \\ & & \leq C\int _{0}^{t}\|u(s)\|_{ L^{2}(\varOmega )}^{2}\,ds +\int _{ 0}^{t}\int _{ \varOmega }\vert r_{\varepsilon }(c_{\varepsilon },c_{A})u\vert \,dx\,ds +\int _{ 0}^{t}\left \vert \int _{\varOmega }\mathcal{R}\ u\,dx\right \vert \,ds \\ & & \quad + \left (\frac{R^{2}} {4} + M_{R}(\varepsilon,t)\right )\varepsilon ^{2(2+\frac{1} {2} )} {}\end{array}$$
(9.53)

for all t ∈ [0, T ɛ ] and some C > 0 independent of ɛ, T ɛ . Using suitable interpolation inequalities in normal direction one can estimate the error due to linearization by

$$\displaystyle{ \int _{0}^{T}\int _{ \varOmega }\vert r_{\varepsilon }(c_{\varepsilon },c_{A})u\vert \,dx\,dt \leq C(R,T_{0})\varepsilon ^{3\cdot 2+1} }$$

as long as (9.48) is valid, cf. [4, Lemma 5.4] for the details. Moreover, using the structure of the leading part of c A and \(\mathop{\mathrm{div}}\nolimits \mathbf{w}_{1} = 0\) one can prove for the error in the convection term

$$\displaystyle{ \int _{0}^{t}\left \vert \int _{\varOmega }\mathcal{R}\ u\,dx\right \vert \,ds \leq C(R,T_{\varepsilon },\varepsilon )\varepsilon ^{2(2+\frac{1} {2} )}, }$$

where C(R, T, ɛ) → 0 as (T, ɛ) → 0, cf. [4, Lemma 5.3]. Together with Gronwall’s inequality we obtain

$$\displaystyle\begin{array}{rcl} & & \sup _{0\leq t\leq T_{\varepsilon }}\|u(t)\|_{L^{2}(\varOmega )}^{2} +\| \nabla _{\boldsymbol{\tau }}u\|_{ L^{2}(\varOmega \times (0,t)\cap \varGamma (2\delta ))}^{2} {}\\ & & \quad +\| \nabla u\|_{L^{2}(\varOmega \times (0,t)\setminus \varGamma (\delta ))}^{2} \leq C(R,T_{\varepsilon },\varepsilon )\varepsilon ^{2(2+\frac{1} {2} )} {}\\ \end{array}$$

for all t ∈ [0, T ɛ ], where C(R, T, ɛ) → 0 as (T, ɛ) → 0. Moreover, putting in (9.51) the integral of \(\frac{1} {\varepsilon ^{2}} f^{{\prime\prime}}(c_{A})\) on the right-hand side and using the previous estimates, one can add \(\varepsilon ^{2}\|\partial _{\mathbf{n}}u\|_{L^{2}(\varOmega \times (0,t)\cap \varGamma (2\delta ))}^{2}\) on the left-hand side of the latter estimate. Hence choosing T, ɛ 0 sufficiently small, one obtains that (9.48) remains true for some T ɛ ≥ T and all ɛ ∈ (0, ɛ 0], which shows (9.45). Afterwards the rest of the statements of Theorem 2 follow from the construction of c A and v A and estimates for v ɛ − v A in dependence of (9.45). □

5 Works in Progress and Perspectives

The convergence result in the last section should be considered as a first step since only a simplified model in two dimensions is treated. More research needs to be done in order to extend this result to more realistic models and situations as e.g. to the three dimensional case, to a Navier-Stokes/Cahn-Hilliard system, to the case of different viscosities and to the case of different densities. Moreover, we expect that the convergence result holds true as long as the limit system possesses a sufficiently smooth solution, as it is the case for the Cahn-Hilliard equation. But even in this simplified situation the proof is already quite long and involved. Extending it in one of the above mentioned directions will probably result in the need of approximative solutions (v A , c A ) of higher order than in the last section. Moreover, additional perturbation terms might require new ideas and refinements.

The original goal of this research project was to obtain a convergence result for the model H (9.1)– (9.4) for short times and for a suitable scaling of m ɛ . In the case m ɛ ≡ m 0 > 0 this is work in progress and will be treated in the third author’s PhD-thesis. So far (9.1)– (9.4) is considered in a two-dimensional torus with constant viscosity ν and neglected material time derivative of the velocity. In this case convergence has been proved provided a suitable approximative solution is constructed. The construction of the approximative solution and the inclusion of the convection term is work in progress. But even when this work is finished the same kind of extensions as for the Stokes/Allen-Cahn system remain open. Furthermore, other scalings of the mobility as e.g. m ɛ = m 0 ɛ remain to be discussed.

Since the rigorous analysis of such sharp interface limits is rather involved and consists of tedious and technical constructions of approximative solutions, it is very desirable to develop a more systematic calculus for such constructions, which could be applied to general classes of diffuse interface models. Finally, even if the approximative solutions are constructed, suitable estimates of the coupling terms and results for the linearized operators are needed. There might still be a great potential for improvements concerning these results and estimates.