1 Introduction

In this contribution we consider the so-called sharp interface limit, i.e., the limit \(\varepsilon \rightarrow 0\), of the convective Allen–Cahn equation

$$\begin{aligned} \partial _t c^\varepsilon + \mathbf {v}\cdot \nabla c^\varepsilon&= m_\varepsilon \left( \varDelta c^\varepsilon -\varepsilon ^{-2} f(c^\varepsilon )\right) \quad \text{ in } \varOmega \times (0,T), \end{aligned}$$
(1)
$$\begin{aligned} c^\varepsilon |_{\partial \varOmega }&= -1 \quad \text{ on } \partial \varOmega \times (0,T), \end{aligned}$$
(2)
$$\begin{aligned} \left. c^\varepsilon \right| _{t=0}&= c^\varepsilon _0 \quad \text{ in } \varOmega . \end{aligned}$$
(3)

Here \(\mathbf {v}:\varOmega \times [0,T) \rightarrow \mathbb {R}^d\) is a given smooth divergence free velocity field with \(\mathbf {n}\cdot \mathbf {v}|_{\partial \varOmega }=0\) and \(c^\varepsilon :\varOmega \times [0,T) \rightarrow \mathbb {R}\) is an order parameter, which will be close to the “pure states” \(\pm 1\) for small \(\varepsilon >0\). Here \(f=F'\), where \(F:\mathbb {R}\rightarrow \mathbb {R}\) is a suitable double well potential with global minima \(\pm 1\), e.g. \(F(c)=(1-c^2)^2\). \(c^\varepsilon \) can describe the concentration difference of two different phases in the case of phase transitions, where the total mass of each phase is not necessarily conserved. Moreover, \(\varOmega \subseteq {\mathbb {R}}^{d}\) is assumed to be a bounded domain with smooth boundary, \(m_\varepsilon \) is a (constant) mobility coefficient and \(\varepsilon >0\) is a parameter that is proportional to the “thickness” of the diffuse interface \(\{x\in \varOmega : |c^\varepsilon (x,t)|<1-\delta \}\) for \(\delta \in (0,1)\).

The convective Allen–Cahn equation (1) is part of the following diffuse interface model for the two-phase flow of two incompressible Newtonian, partly miscible fluids with phase transition

$$\begin{aligned} \partial _t \mathbf {v}^\varepsilon + \mathbf {v}^\varepsilon \cdot \nabla \mathbf {v}^\varepsilon - \mathrm {div}(\nu (c^\varepsilon ) D\mathbf {v}^\varepsilon ) + \nabla p^\varepsilon&= -\varepsilon \mathrm {div}(\nabla c^\varepsilon \otimes \nabla c^\varepsilon ), \end{aligned}$$
(4)
$$\begin{aligned} \mathrm {div}\, \mathbf {v}^\varepsilon&= 0, \end{aligned}$$
(5)
$$\begin{aligned} \partial _t c^\varepsilon + \mathbf {v}^\varepsilon \cdot \nabla c^\varepsilon&= m_\varepsilon \left( \varDelta c^\varepsilon -\varepsilon ^{-2} f(c^\varepsilon )\right) \end{aligned}$$
(6)

in \(\varOmega \times (0,T)\), where \(\mathbf {v}^\varepsilon :\varOmega \times [0,T)\rightarrow {\mathbb {R}}^{d}\) is the velocity of the mixture, \(D \mathbf {v}^\varepsilon = \frac{1}{2} (\nabla \mathbf {v}^\varepsilon + (\nabla \mathbf {v}^\varepsilon )^T)\), \(p^\varepsilon :\varOmega \times [0,T)\rightarrow \mathbb {R}\) is the pressure, and \(\nu (c^\varepsilon )>0\) is the viscosity of the mixture. This model can be considered as a model for a two-phase flow with phase transition or an approximation of a classical sharp interface model for a two-phase flow of incompressible fluids with surface tension. Here the densities of the two separate fluids are assumed to be the same. A derivation of this model in a more general form with variable densities can be found in Jiang et al. [10]. We refer to Gal and Grasselli [6] for the existence of weak solutions and results on the longtime behavior of solutions for this model and to Giorgini et al. [7] for analytic results for a volume preserving variant with different densities. Mathematically, this system arises if one replaces the Cahn–Hilliard equation in the well-known “model H”, cf. e.g. [1, 8], by an Allen–Cahn equation.

With the aid of formally matched asymptotic expansions one can formally show that solutions of this system converge to solutions of the following free boundary value problem

$$\begin{aligned} \partial _t \mathbf {v}+ \mathbf {v}\cdot \nabla \mathbf {v}- \mathrm {div}(\nu ^\pm D\mathbf {v}) + \nabla p&= 0 \quad \hbox {in } \varOmega ^\pm (t), t\in (0,T) , \end{aligned}$$
(7)
$$\begin{aligned} \mathrm {div}\, \mathbf {v}&= 0 \quad \hbox {in } \varOmega ^\pm (t), t\in (0,T) , \end{aligned}$$
(8)
$$\begin{aligned} \left[ \mathbf {v}\right] _{\varGamma _t}&= 0 \quad \text{ on } \varGamma _t, t\in (0,T) , \end{aligned}$$
(9)
$$\begin{aligned} - \left[ \mathbf {n}_{\varGamma _t} \cdot (\nu ^\pm D\mathbf {v}- p \mathrm {Id})\right] _{\varGamma _t}&= \sigma H_{\varGamma _t}\mathbf {n}_{\varGamma _t} \quad \hbox {on } \varGamma _t, t\in (0,T) , \end{aligned}$$
(10)
$$\begin{aligned} V_{\varGamma _t} - \mathbf {n}_{\varGamma _t} \cdot \mathbf {v}&= m_0 H_{\varGamma _t} \quad \text{ on } \varGamma _t, t\in (0,T) , \end{aligned}$$
(11)

when \(m_\varepsilon =m_0>0\) and

$$\begin{aligned} \partial _t \mathbf {v}+ \mathbf {v}\cdot \nabla \mathbf {v}- \mathrm {div}(\nu ^\pm D\mathbf {v}) + \nabla p&= 0 \quad \text{ in } \varOmega ^\pm (t), t\in (0,T) , \end{aligned}$$
(12)
$$\begin{aligned} \mathrm {div}\mathbf {v}&= 0 \quad \text{ in } \varOmega ^\pm (t), t\in (0,T) , \end{aligned}$$
(13)
$$\begin{aligned} \left[ \mathbf {v}\right] _{\varGamma _t}&= 0 \quad \text{ on } \varGamma _t, t\in (0,T) , \end{aligned}$$
(14)
$$\begin{aligned} - \left[ \mathbf {n}_{\varGamma _t} \cdot (\nu ^\pm D\mathbf {v}- p \mathrm {Id})\right] _{\varGamma _t}&= \sigma H_{\varGamma _t} \mathbf {n}_{\varGamma _t} \quad \text{ on } \varGamma _t, t\in (0,T) , \end{aligned}$$
(15)
$$\begin{aligned} V_{\varGamma _t} - \mathbf {n}_{\varGamma _t} \cdot \mathbf {v}&= 0 \quad \text{ on } \varGamma _t, t\in (0,T) , \end{aligned}$$
(16)

when \(m_\varepsilon =m_0\varepsilon \), \(m_0>0\). We will discuss this formal result in the appendix in more detail, cf. Remark 2 below. Here \(\nu ^\pm >0\) are viscosity constants, \(\varOmega ^\pm (t) \subset \varOmega \) are open and disjoint such that \(\partial \varOmega ^-(t) = \varGamma _t = \partial \varOmega ^+(t) \cap \varOmega \), \(\mathbf {n}_{\varGamma _t}\) denotes the outer normal of \(\partial \varOmega ^-(t)\) and the normal velocity and the mean curvature of \(\varGamma _t\) are denoted by \(V_{\varGamma _t}\) and \(H_{\varGamma _t}\), respectively, taken with respect to \(\mathbf {n}_{\varGamma _t}\). Furthermore, \(\left[ \,.\, \right] _{\varGamma _t} \) denotes the jump of a quantity across the interface in the direction of \(\mathbf {n}_{\varGamma _t}\), i.e., \( \left[ f\right] _{\varGamma _t} (x) = \lim _{h \rightarrow 0}(f(x + h \mathbf {n}_{\varGamma _t}) - f(x - h \mathbf {n}_{\varGamma _t}))\) for \(x \in \varGamma _t\).

In the case \(\nu ^+ = \nu ^-\) and that the Navier–Stokes equation is replaced by a (quasi-stationary) Stokes system Liu and the author proved rigorously in [4] that the convergence holds true in the first case \(m_\varepsilon =m_0>0\) for sufficiently small times and for well-prepared initial data. More precisely, it was shown that in a neighborhood of \(\varGamma _t\)

$$\begin{aligned} c_\varepsilon (x,t)= \theta _0\left( \frac{d_{\varGamma _t}(x)-\varepsilon h_\varepsilon (x,t)}{\varepsilon }\right) + \mathcal {O}(\varepsilon ) \end{aligned}$$
(17)

(even with \(\mathcal {O}(\varepsilon ^2)\)), where \(d_{\varGamma _t}\) is the signed distance function to \(\varGamma _t\) and \(h_\varepsilon \) are correction terms, which are uniformly bounded in \(\varepsilon \in (0,1)\), and \(\theta _0:\mathbb {R}\rightarrow \mathbb {R}\) is the so-called optimal profile that is determined by

$$\begin{aligned} - \theta _0'' + f(\theta _0)&= 0 \text{ in } \mathbb {R},&\theta _0(0)&= 0,&\lim _{z \rightarrow \pm \infty }\theta _0(z)&= \pm 1 . \end{aligned}$$
(18)

This form is important in order to obtain in the limit \(\varepsilon \rightarrow 0\) the Young-Laplace law (15), cf. e.g. [2, Section 4].

It is the goal of the present contribution to show that in the case \(m_\varepsilon =m_0 \varepsilon ^\theta \) with \(\theta >2\) the solutions of the convective Allen–Cahn equation (1)–(2) do not have the form (17) in general. Moreover, we will show that the functional

$$\begin{aligned} \left\langle H^\varepsilon , \varvec{\varphi }\right\rangle : = \varepsilon \int _\varOmega {\nabla c^\varepsilon \otimes \nabla c^\varepsilon : \nabla \varvec{\varphi }\, dx} \end{aligned}$$

does not converge to the mean curvature functional

$$\begin{aligned} 2 \sigma \int _{\varGamma _t}{ \mathbf {n}_{\varGamma _t} \otimes \mathbf {n}_{\varGamma _t} : \nabla \varvec{\varphi }\, d \mathcal {H}^{d-1}}&= - 2 \sigma \int _{\varGamma _t}{ ({\text {Id}}- \mathbf {n}_{\varGamma _t} \otimes \mathbf {n}_{\varGamma _t}) : \nabla \varvec{\varphi }\, d \mathcal {H}^{d-1}}\nonumber \\&= - 2 \sigma \int _{\varGamma _t}{H_{\varGamma _t} \mathbf {n}_{\varGamma _t} \cdot \varvec{\varphi }\, d \mathcal {H}^{d-1}} \end{aligned}$$
(19)

for all \(\varvec{\varphi }\in C^\infty _{0,\sigma }(\varOmega )=\left\{ f \in C^\infty _0(\varOmega )^d : \mathrm {div}f = 0 \right\} \), where

$$\begin{aligned} \sigma = \frac{1}{2} \int _\mathbb {R}{\left( \theta '_0(z) \right) ^2 dz}. \end{aligned}$$

We note that \(H^\varepsilon \) is the weak formulation of the right-hand side of (4), which should converge to a weak formulation of the right-hand side of (15). Therefore there is no hope that solutions of the full system (4)–(6) converge to solutions of the corresponding limit system with (15) as \(\varepsilon \rightarrow 0\) in the case that \(m_\varepsilon = m_0 \varepsilon ^\theta \), \(\theta >2\). We note that this effect was first observed for the corresponding Navier–Stokes/Cahn–Hilliard system by Schaubeck and the author in [5] in the case \(\theta >3\). These results are also contained in the PhD-thesis of Schaubeck [11]. It is not difficult to show that \(\left\langle H^\varepsilon , \varvec{\varphi }\right\rangle \) converges to (19) if (17) holds true in a sufficiently strong sense. Moreover, in the case \(\theta >3\) non-convergence of the Navier–Stokes/Cahn–Hilliard system in the case of radial symmetry and an inflow boundary condition was shown by Lengeler and the author in [3, Section 4]. We note that the latter counter example can be adapted to the present case of a Navier–Stokes/Allen–Cahn equation in the case \(\theta >2\).

The structure of this contribution is as follows: in Sect. 2 we summarize some preliminaries and notation. Afterwards we prove the nonconvergence result in Sect. 3. Finally, in Sect. 4 we discuss briefly the sharp interface limit of the convective Allen–Cahn equation in the case \(m_\varepsilon = m_0 \varepsilon ^\theta \) with \(\theta =0,1\).

2 Preliminaries and notation

We denote \(a \otimes b = \left( a_i b_j \right) ^d_{i,j=1}\) for \(a,b \in \mathbb {R}^d\) and \(A:B = \sum ^d_{i,j=1}{A_{ij} B_{ij}}\) for \(A,B \in \mathbb {R}^{d \times d}\). We assume that \(\varOmega \subset \mathbb {R}^d \) is a bounded domain with smooth boundary \(\partial \varOmega \). Furthermore, we define \(\varOmega _T = \varOmega \times (0,T)\) and \(\partial _T \varOmega = \partial \varOmega \times (0,T)\) for \(T>0\). Moreover, \(\mathbf {n}_{\partial \varOmega }\) denotes the exterior unit normal on \(\partial \varOmega \). For a hypersurface \(\varGamma _t \subset \varOmega \), \(t\in [0,T]\), without boundary such that \(\varGamma _t = \partial \varOmega ^-(t)\) for a domain \(\varOmega ^-(t) \subset \subset \varOmega \), the interior domain is denoted by \(\varOmega ^-(t)\) and the exterior domain by \(\varOmega ^+(t) := \varOmega \backslash (\varOmega ^-(t) \cup \varGamma _t)\), i.e., \(\varGamma _t\) separates \(\varOmega \) into an interior and an exterior domain. \(\mathbf {n}_{\varGamma _t}\) is the exterior unit normal on \(\partial \varOmega ^-(t)=\varGamma _t\). The mean curvature of \(\varGamma _t\) with respect to \(\mathbf {n}_{\varGamma _t}\) is denoted by \(H_{\varGamma _t}\). In the following \(d_{\varGamma _t}\) is the signed distance function to \(\varGamma _t\) chosen such that \(d_{\varGamma _t} < 0\) in \(\varOmega ^-(t)\) and \(d_{\varGamma _t}>0\) in \(\varOmega ^+(t) \). By this convention we obtain \(\nabla d_{\varGamma _t} = \mathbf {n}_{\varGamma _t}\) on \(\varGamma _t\). Moreover, we define

$$Q^\pm := \left\{ (x,t) \in \varOmega _T : d(x,t) \gtrless 0 \right\} .$$

The “double-well” potential \(F: \mathbb {R} \rightarrow \mathbb {R}\) is a smooth function taking its global minimum 0 at \(\pm 1\). For its derivative \(f(c)=F'(c)\) we assume

$$\begin{aligned} f(\pm 1)&= 0 ,&f'(\pm 1)&> 0,&\int ^u_{-1}{f(s)\, ds} = \int ^u_1{f(s)\, ds}&> 0 \end{aligned}$$
(20)

for all \(u\in (-1,1)\). In Eq. (1) the given velocity field satisfies \(\mathbf {v}\in C^0_b([0,T]; C^4_b(\overline{\varOmega }))^d\) with \(\mathrm {div}\,\mathbf {v}= 0\) and \(\mathbf {v}\cdot \mathbf {n}_{\partial \varOmega } = 0\) on \(\partial \varOmega \) and the mobility constant \(m_\varepsilon \) has the form \(m_\varepsilon = m_0\varepsilon ^\theta \) for some \(\theta \ge 0\) and \(m_0>0\). In Eq. (3) we choose the special initial value

$$\begin{aligned} \left. c^\varepsilon \right| _{t=0}&= \zeta \!\left( \tfrac{d_{\varGamma _0}}{\delta }\right) \theta _0\!\left( \tfrac{d_{\varGamma _0}}{\varepsilon }\right) + \left( 1- \zeta \!\left( \tfrac{d_{\varGamma _0}}{\delta }\right) \right) \left( 2 \chi _{\left\{ d_{\varGamma _0} \ge 0\right\} } -1 \right)&\text{ in } \varOmega , \end{aligned}$$
(21)

where we determine the constant \(\delta >0 \) later. Here \(\zeta \in C^\infty _0(\mathbb {R})\) is a cut-off function such that

$$\begin{aligned} \zeta (z)&= 1 \text{ if } \left| z \right| < \frac{1}{2} ,&\zeta (z)&= 0 \text{ if } \left| z \right| > 1 ,&z \zeta '(z) \le 0 \text{ in } \mathbb {R} , \end{aligned}$$
(22)

and \(\theta _0\) is the unique solution to (18). This choice of the initial value is natural in view of (17).

3 Nonconvergence result

Our main result is:

Theorem 1

Let \(\theta >2\), \(\varOmega \subset \mathbb {R}^d \) be a bounded domain with smooth boundary \(\partial \varOmega \), \(\varGamma _0\) a smooth hypersurface such that \(\varGamma _0=\partial \varOmega _0^-\) for a domain \(\varOmega _0^-\subset \subset \varOmega \) and let \(c^\varepsilon \) be the solution to the convective Allen–Cahn equation (1), (2) with initial condition (21). Then for every \(T>0\) and for all \(\varvec{\varphi }\in C^\infty ([0,T];\mathcal {D}(\varOmega )^d)\) with \(\mathrm {div}\varvec{\varphi }=0\) we have

$$\begin{aligned} \int ^T_0{\left\langle H^\varepsilon , \varvec{\varphi }\right\rangle dt} \rightarrow _{\varepsilon \rightarrow 0} 2 \sigma \int ^T_0{\int _{\varGamma _t}{ \left| \nabla (d_{\varGamma _0}(X^{-1}_t)) \right| \mathbf {n}_{\varGamma _t} \otimes \mathbf {n}_{\varGamma _t} : \nabla \varvec{\varphi }\, d \mathcal {H}^{d-1} } \, dt} . \end{aligned}$$

Here the evolving hypersurface \(\varGamma _t\), \(t\in [0,T]\), is the solution of the evolution equation

$$\begin{aligned} V_{\varGamma _t}(x) = \mathbf {n}_{\varGamma _t}(x,t) \cdot \mathbf {v}(x,t) \text{ for } x\in \varGamma _t, t \in (0,T], \quad \varGamma (0)=\varGamma _{0}, \end{aligned}$$

where \(V_{\varGamma _t}\) is the normal velocity of \(\varGamma _t\), and \(X_t:\varOmega \rightarrow \varOmega \) is defined by \(X_t(y_0)=y(t;y_0)\) for \(y_0\in \varOmega \), \(t\in [0,T]\), where \(y(\cdot ;y_0)\) is the solution of

$$\begin{aligned} \frac{d}{ds} y(s;y_0)= \mathbf {v}(y(s;y_0),s), s\in [0,T],\quad y(0;y_0)= y_0. \end{aligned}$$

Moreover, it holds

$$\begin{aligned} \left\| c^\varepsilon -(2 \chi _{Q^+} -1 ) \right\| ^2_{L^2(\varOmega _T)} = \mathcal {O}(\varepsilon ) \quad \text {as }\varepsilon \rightarrow 0. \end{aligned}$$

Remark 1

In general \(\left| \nabla (d_{\varGamma _0}(X^{-1}_t)) \right| = \left| D X^{-T}_t \nabla d_{\varGamma _0} \circ X^{-1}_t \right| \ne 1\), we refer to [5, Remark 1] for a proof. This shows that the weak formulation of \(H^\varepsilon \) does not converge to the weak formulation of the right-hand side of the Young-Laplace law (15) in general.

To prove the theorem we follow the same strategy as in [5]: First we construct a family of approximate solutions \(\left\{ c^\varepsilon _A\right\} _{0<\varepsilon \le 1}\). Afterwards we estimate the difference \(\nabla (c^\varepsilon -c^\varepsilon _A)\), which will enable us to prove the assertion of the theorem. We start with the observation that \(\varGamma _t:= X_t(\varGamma _0)\) is the solution to the evolution equation.

Lemma 1

Let \(\varGamma _{0} \subset \varOmega \) be a given smooth hypersurface such that \(\varGamma _0=\partial \varOmega _0^-\) for a domain \(\varOmega _0^-\subset \subset \varOmega \). Then the evolving hypersurface \(\varGamma _t:= X_t\left( \varGamma _{0}\right) \subset \varOmega \), \(t\in [0,T]\), is the solution to the problem

$$\begin{aligned} V_{\varGamma _t}= \mathbf {n}_{\varGamma _t} \cdot \mathbf {v}\quad \text{ on } \varGamma _t, t \in (0,T), \quad \varGamma (0)=\varGamma _{0}. \end{aligned}$$

We refer to [5, Lemma 3] for the proof.

For the following let \(P_{\varGamma _t}(x)\) be the orthogonal projection of x onto \(\varGamma _t\). Then there exists a constant \(\delta >0 \) such that \(\varGamma _t(\delta ):= \left\{ x \in \varOmega : \left| d_{\varGamma _t}(x))\right| < \delta \right\} \subset \varOmega \) and \(\tau _t :\varGamma _t(\delta ) \rightarrow (-\delta ,\delta ) \times \varGamma _t\) defined by \(\tau _t(x) = (d_{\varGamma _t}(x),P_{\varGamma _t}(x))\) is a smooth diffeomorphism, cf. e.g. [9, Chapter 4.6].

We will need the following result:

Lemma 2

For \(e :\bigcup _{t \in [0,T]} X_t(\varGamma _0(\delta )) \times \left\{ t \right\} \rightarrow \mathbb {R}\) defined by \(e(x,t):= d_{\varGamma _0}(X_t^{-1}(x))\) the following properties hold:

  1. 1.

    \( \frac{d}{dt} e(x,t) = - \mathbf {v}(x,t) \cdot \nabla e(x,t) \) for all \((x,t) \in \bigcup _{t \in [0,T]} X_t(\varGamma _0(\delta )) \times \left\{ t \right\} \) .

  2. 2.

    e(x, t) is a level set function for \(\varGamma _t \), i.e., \(e(x,t)=0 \) if and only if \( x \in \varGamma _t\).

We refer to [5, Lemma 4] for the proof.

As mentioned in Sect. 2, let \(\theta _0\) be the solution to (18) and let \(\zeta \) be a cut-off function as in (22). Then we define

$$\begin{aligned} c_A^\varepsilon (x,t):= \left\{ \begin{array}{l@{\;}l} \pm 1 &{} \text{ in } \overline{Q^\pm } \cap \bigcup \limits _{t \in [0,T]} \overline{X_t(\varOmega \backslash \varGamma _0(\delta ))} \times \{ t \} , \\ \zeta \!\left( \tfrac{e}{\delta }\right) \theta _0\!\left( \tfrac{e}{\varepsilon }\right) \pm (1-\zeta \!\left( \tfrac{e}{\delta }\right) ) &{} \text{ in } Q^\pm \cap \bigcup \limits _{t \in [0,T]} X_t(\varGamma _0(\delta ) \backslash \varGamma _0\!\left( \tfrac{\delta }{2}\right) ) \times \{ t \}, \\ \theta _0\!\left( \tfrac{e}{\varepsilon }\right) &{} \text{ in } \bigcup \limits _{t \in [0,T]} X_t(\varGamma _0\!\left( \tfrac{\delta }{2}\right) ) \times \{t\} . \end{array} \right. \end{aligned}$$

Then we have \(c^\varepsilon _A(.,0) = c^\varepsilon (.,0)\) since \(e(.,0) = d_{\varGamma _0}\) and

$$\begin{aligned} \partial _t c^\varepsilon _A + \mathbf {v}\cdot \nabla c^\varepsilon _A = 0\quad \text {in }\varOmega _T \end{aligned}$$

since \(\partial _t e + \mathbf {v}\cdot \nabla e = 0\). Moreover, by the construction

$$\begin{aligned} c^\varepsilon _A=0 \quad \text {on }\partial \varOmega . \end{aligned}$$

Furthermore, we define the approximate mean curvature functional by

$$\begin{aligned} \left\langle H^\varepsilon _A ,\varvec{\varphi }\right\rangle = \varepsilon \int _\varOmega {\nabla c^\varepsilon _A \otimes \nabla c^\varepsilon _A : \nabla \varvec{\varphi }\, dx} . \end{aligned}$$

for all \(\varvec{\varphi }\in \mathcal {D}(\varOmega )^d\) with \(\mathrm {div}\varvec{\varphi }=0\). Then we have:

Lemma 3

Let \(c^\varepsilon _A\) be defined as above. Then there exists some constant \(C>0\) independent of \(\varepsilon \) and \(\varepsilon _0 \in (0,1] \) such that the estimates

$$\begin{aligned} \left\| \varDelta c^\varepsilon _A(.,t) \right\| _{L^2(\varOmega )}\le & {} C \varepsilon ^{- \frac{3}{2}} , \end{aligned}$$
(23)
$$\begin{aligned} \left\| \nabla c^\varepsilon _A(.,t) \right\| _{L^2(\varOmega )}\le & {} C \varepsilon ^{- \frac{1}{2}} , \end{aligned}$$
(24)
$$\begin{aligned} \left\| f(c^\varepsilon _A(.,t)) \right\| _{L^2(\varOmega )}\le & {} C \varepsilon ^{\frac{1}{2}} , \end{aligned}$$
(25)
$$\begin{aligned} \left\| c^\varepsilon _A(.,t) - (2 \chi _{Q^+}(.,t) -1) \right\| _{L^2(\varOmega )}\le & {} C \varepsilon ^{\frac{1}{2}} \end{aligned}$$
(26)

hold for all \(t \in [0,T]\) and \(\varepsilon \in (0,\varepsilon _0)\).

We refer to [5, Lemma 5] for the proof.

Now we are able to prove the central lemma for the proof of Theorem 1.

Lemma 4

Let \(c_A^\varepsilon \) be defined as above and let \(c^\varepsilon \) be the unique solution to (1), (2) with initial condition (21). Then, for \(\theta \ge 2\), there exists some constant \(C >0 \) independent of \(\varepsilon \) and \(\varepsilon _0>0\) such that

$$\begin{aligned} \varepsilon \left\| \nabla (c^\varepsilon - c^\varepsilon _A) \right\| ^2_{L^2(\varOmega _T)}&\le C \varepsilon ^{\theta -2} \quad \text {and} \end{aligned}$$
(27)
$$\begin{aligned} \left\| c^\varepsilon - c^\varepsilon _A \right\| _{L^\infty (0,T;L^2(\varOmega ))}&\le C \varepsilon ^{\theta - \frac{3}{2}} \end{aligned}$$
(28)

for all \(\varepsilon \in (0,\varepsilon _0]\).

Proof

First of all, we note that \(c^\varepsilon (x,t),c^\varepsilon _A(x,t)\in [-1,1]\) for all \(x\in \varOmega \), \(t\in (0,T)\). For \(c^\varepsilon _A\) this follows from the construction and for \(c^\varepsilon \) by the maximum principle.

We denote by \(u = c^\varepsilon - c^\varepsilon _A\) the difference between exact and approximate solution, which solves

$$\begin{aligned} \partial _t c^\varepsilon _A + \mathbf {v}\cdot \nabla c^\varepsilon _A = 0\quad \text {in }\varOmega _T. \end{aligned}$$

We multiply the difference of the differential equations for \( c^\varepsilon \) and \( c^\varepsilon _{A}\) by u and integrate the resulting equation over \(\varOmega \). Then we get for all \(t \in (0,T)\)

$$\begin{aligned} 0&= \int _\varOmega u\left[ \partial _t u + \mathbf {v}\cdot \nabla u - m_0\varepsilon ^{\theta } \varDelta u - m_0\varepsilon ^{\theta } \varDelta c^\varepsilon _A + m_0\varepsilon ^{\theta - 2} f(c^\varepsilon ) \right] dx \\&= \int _\varOmega \left( \partial _t \tfrac{|u|^2}{2} - \mathbf {v}\cdot \nabla \tfrac{|u|^2}{2}+ m_0 \varepsilon ^{\theta } |\nabla u|^2\right) \, dx\\&\quad + \int _\varOmega \left( m_0\varepsilon ^{\theta } \nabla u \cdot \nabla c^\varepsilon _A + m_0\varepsilon ^{\theta -2} f(c^\varepsilon ) u\right) \, dx \\&= \frac{1}{2} \frac{d}{dt} \int _\varOmega | u|^2 dx + m_0 \varepsilon ^\theta \int _\varOmega |\nabla u|^2 dx - \int _\varOmega \left( m_0 \varepsilon ^{\theta } u \varDelta c^\varepsilon _A - m_0\varepsilon ^{\theta -2 } u f(c^\varepsilon )\right) dx , \end{aligned}$$

where we have used \(u = 0\) on \(\partial \varOmega \) as well as \(\mathrm {div}\, \mathbf {v}= 0\) in \(\varOmega \). By Hölder’s and Young’s inequalities we obtain

$$\begin{aligned}&\frac{1}{2} \frac{d}{dt} \int _\varOmega { \left| u \right| ^2 dx} + \frac{m_0}{2}\varepsilon ^{\theta } \int _\varOmega {\left| \nabla u \right| ^2 dx }\nonumber \\&\quad \le \frac{1}{2} \Vert u\Vert _{L^2(\varOmega )}^2 + \frac{m_0^2 \varepsilon ^{2\theta }}{2} \Vert \varDelta c_A^\varepsilon \Vert _{L^2(\varOmega )}^2 + m_0 \varepsilon ^{\theta -2} \left| \int _{\varOmega }{f(c^\varepsilon ) u \, dx} \right| \end{aligned}$$
(29)

for all \(\varepsilon \in (0,\varepsilon _0)\), where

$$\begin{aligned} \left| \int _{\varOmega }{f(c^\varepsilon ) u \, dx}\right|&\le \left| \int _{\varOmega }{f(c_A^\varepsilon ) u \, dx}\right| +C\Vert u\Vert _{L^2(\varOmega )}^2 \nonumber \\&\le \left\| f(c^\varepsilon _A) \right\| _{L^2(\varOmega )}\Vert u\Vert _{L^2(\varOmega )} + C \left\| u\right\| ^2_{L^2(\varOmega )} \nonumber \\&\le C \varepsilon ^{\frac{1}{2}}\Vert u\Vert _{L^2(\varOmega )} + C \left\| u\right\| ^2_{L^2(\varOmega )} \end{aligned}$$
(30)

since \(f'\) is Lipschitz continuous on \([-1,1]\). Hence (29) together with (30) and (23) yield

$$\begin{aligned}&\frac{1}{2} \frac{d}{dt} \int _\varOmega { \left| u \right| ^2 dx} + m_0\varepsilon ^{\theta } \int _\varOmega {\left| \nabla u \right| ^2 dx } \nonumber \\&\quad \le C \left( \left\| u \right\| ^2_{L^2(\varOmega )} + \varepsilon ^{2\theta - 3} +\varepsilon ^{\theta -2}\Vert u\Vert _{L^2(\varOmega )}^2 \right) \le C_1 \left( \left\| u \right\| ^2_{L^2(\varOmega )} + \varepsilon ^{2\theta - 3} \right) \end{aligned}$$

since \(\theta \ge 2\) for some \(C_1>0\) independent of \(\varepsilon \) and \(t \in [0,T]\). Hence the Gronwall inequality implies

$$\begin{aligned} \sup _{0 \le t \le T} \left\| u \right\| ^2_{L^2(\varOmega )} + \varepsilon ^\theta \Vert \nabla u\Vert _{L^2((0,T)\times \varOmega )}^2 \le C \varepsilon ^{2\theta -3} \end{aligned}$$

for some \(C= C(T) >0\) independent of \(\varepsilon \). Therefore the lemma is proved. \(\square \)

Now we can show that \(H^\varepsilon -H^\varepsilon _A\) converges to 0 as \(\varepsilon \) goes to zero.

Lemma 5

Let \(H^\varepsilon \) and \(H^\varepsilon _A\) be defined as above and let \(\theta > 2\). Then it holds

$$\begin{aligned} \left| \int _0^T{\left\langle H^\varepsilon - H^\varepsilon _A , \varvec{\varphi }\right\rangle dt} \right| \rightarrow _{\varepsilon \rightarrow 0} 0 , \end{aligned}$$

for all \(\varvec{\varphi }\in C^\infty ([0,T];\mathcal {D}(\varOmega )^d)\).

Proof

The proof is almost the same as in [5, Lemma 6]. But we include it for the convenience of the reader since the argument is central for our main result. Let \(\varvec{\varphi }\in C^\infty ([0,T];\mathcal {D}(\varOmega )^d)\) and set \(u = c^\varepsilon -c^\varepsilon _A\). Then

$$\begin{aligned}&\varepsilon \left| \int _{\varOmega _T}{\left( \nabla c^\varepsilon \otimes \nabla c^\varepsilon - \nabla c^\varepsilon _A \otimes \nabla c^\varepsilon _A \right) : \nabla \varvec{\varphi }\, dx} \right| \\&\quad \le \varepsilon \left| \int _{\varOmega _T}{\left( \nabla c^\varepsilon \otimes \nabla u \right) : \nabla \varvec{\varphi }\, dx} \right| + \varepsilon \left| \int _{\varOmega _T}{\left( \nabla u \otimes \nabla c^\varepsilon _A \right) : \nabla \varvec{\varphi }\, dx} \right| \\&\quad \le \varepsilon \left\| \nabla \varvec{\varphi }\right\| _{L^\infty (\varOmega _T)} \left\| \nabla u\right\| _{L^2(\varOmega _T)} \left( \left\| \nabla c^\varepsilon \right\| _{L^2(\varOmega _T)} + \left\| \nabla c^\varepsilon _A\right\| _{L^2(\varOmega _T)} \right) . \end{aligned}$$

Because of Lemmas 3 and 4, we have

$$\begin{aligned} \left\| \nabla c^\varepsilon \right\| _{L^2(\varOmega _T)} \le \left\| \nabla c^\varepsilon _A \right\| _{L^2(\varOmega _T)} + \left\| \nabla u \right\| _{L^2(\varOmega _T)}\le C\left( \varepsilon ^{-\frac{1}{2}}+ \varepsilon ^{\frac{\theta -3}{2}}\right) . \end{aligned}$$

Using Lemma 4 we conclude

$$\begin{aligned} \left| \int _0^T{\left\langle H^\varepsilon - H^\varepsilon _A , \varvec{\varphi }\right\rangle dt} \right| \le C \varepsilon ^{\frac{ \theta - 2}{2}} \left( 1 + \varepsilon ^{\frac{\theta - 2}{2}} \right) \end{aligned}$$

for some constant \(C=C(\varvec{\varphi })>0\) and for all \(\varepsilon \) small enough. Since \(\theta >2 \), the assertion follows. \(\square \)

Lemma 6

Let \(H^\varepsilon _A\) and \(c^\varepsilon _A\) be defined as above. Then it holds for all \(\varvec{\varphi }\in \mathcal {D}(\varOmega )^d\) and \(t \in [0,T]\)

$$\begin{aligned} \left\langle H^\varepsilon _A, \varvec{\varphi }\right\rangle&\rightarrow _{\varepsilon \rightarrow 0} 2 \sigma \int _{\varGamma _t}{ \left| \nabla (d_{\varGamma _0}(X^{-1}_t)) \right| \mathbf {n}_{\varGamma _t} \otimes \mathbf {n}_{\varGamma _t} : \nabla \varvec{\varphi }\, d \mathcal {H}^{d-1} }. \end{aligned}$$

We refer to [5, Lemma 8] for the proof.

Proof of Theorem 1

The first assertion of the theorem immediately follows by Lemmas 5 and 6 . The second assertion is a consequence of Lemmas 3 and 4 since \(\theta > 2\). \(\square \)

4 Formal asymptotics

In this section we will use the method of formally matched asymptotic expansions to identify the sharp interface limit of the convective Allen–Cahn equation (1), (2) in the cases \(m_\varepsilon = m_0 \varepsilon ^\theta \) for \(\theta =0,1\) and some \(m_0>0\). We follow similar arguments as in [2, Section 4]. In particular we assume that there are smoothly evolving hypersurfaces \(\varGamma _t\), \(t\in (0,T)\), such that \(\varGamma _t= \partial \varOmega ^-(t)\), and we have the following expansions: Outer expansion: “Away from \(\varGamma _t\)” we assume that \(c_\varepsilon \) has an expansion of the form:

$$\begin{aligned} c_\varepsilon (x,t)= \sum _{k=0}^\infty \varepsilon ^k {c_k^\pm (x,t)} \quad \text {for every }x\in \varOmega ^\pm (t). \end{aligned}$$

Inner expansion: In a neighborhood \(\varGamma _t(\delta )\), \(\delta >0\), of \(\varGamma _t\) \(c_\varepsilon \) has an expansion of the form:

$$\begin{aligned} c_\varepsilon (x,t)= \sum _{k=0}^\infty \varepsilon ^k {c_k(\tfrac{d_{\varGamma _t}}{\varepsilon }, P_{\varGamma _t}(x),t)} \quad \text {for all }x\in \varGamma _t(\delta ). \end{aligned}$$

Matching condition:

$$\begin{aligned} \lim _{z\rightarrow \pm \infty } c_k(z,x,t)&= c_k^\pm (x,t) \quad \text {for all }x\in \varGamma _t, k=0,1,\\ \lim _{z\rightarrow \pm \infty } \partial _z c_0(z,x,t)&= 0 \quad \text {for all }x\in \varGamma _t. \end{aligned}$$

Moreover, all functions in the expansions above are assumed to be sufficiently smooth.

In the following we will use the expansions above and the matching conditions, insert them into the convective Allen–Cahn equation (1) and equate all terms of same order in order to determine the leading parts in the inner and outer expansions formally.

4.1 Outer expansion

First we use a power series expansion of \(c_\varepsilon \) due to the outer expansion. Then

$$\begin{aligned} f'(c_\varepsilon (x,t)) = f' (c_0^\pm (x,t)) c_1^\pm (x,t)+ \varepsilon f'' (c_0^\pm (x,t)) c_1^\pm (x,t) + \mathcal {O} (\varepsilon ^2) \end{aligned}$$

and we obtain from (1)

$$\begin{aligned} \frac{1}{\varepsilon ^{2-k}} f' (c_0^\pm (x,t))+ \frac{1}{\varepsilon ^{1-k}} f''(c_0^\pm (x,t))c_1^\pm (x,t) + \mathcal {O}(1) = 0 \end{aligned}$$

for all \(x\in \varOmega ^\pm (t)\). This yields

  1. (i)

    At order \(\frac{1}{\varepsilon ^{2-k}}\) we obtain \(f'(c_0^\pm (x,t)) =0\). Thus \(c_0^\pm (x,t) \in \bigl \{\pm 1,0\bigr \}\). Here we exclude the case \(c_0^\pm (x,t) =0\) since 0 is unstable and define \(\varOmega ^\pm (t)\) such that

    $$\begin{aligned} c_0^\pm (x,t) =\pm 1 \text { for all } x\in \varOmega ^\pm (t). \end{aligned}$$
  2. (ii)

    If \(k=0\), we obtain at order \(\frac{1}{\varepsilon }\) that \(f''(c_0 (x,t)) c_1^\pm (x,t) =0\). Since \(f''(\pm 1)>0\), we conclude

    $$\begin{aligned} c_1^\pm (x,t) =0 \text { for all } x\in \varOmega ^\pm (t). \end{aligned}$$

    If \(k=1\), the corresponding term is of order \(\mathcal {O}(1)\) and we do not use this information. Moreover, we will not determine \(c^\pm _1\) and \(c_1\) in this case.

4.2 Inner expansion

In \(\varGamma _t(\delta )\) we use the inner expansion in (1) in order to determine the leading coefficients \(c_0(\rho ,s,t)\) and, in the case \(k=0\), \(c_1(\rho ,s,t)\), where \(s:=s(x):= P_{\varGamma _t} (x)\). To this end we use

$$\begin{aligned} \mathbf {v}\cdot \nabla c_j (\rho , s,t)&= \frac{1}{\varepsilon } \mathbf {v}\cdot \nabla d_{\varGamma _t} (\rho ,s,t) +\mathcal {O}(1),\\ \varDelta c_j (\rho ,s,t)&= \frac{1}{\varepsilon ^2} (\partial _\rho ^2 c_j) \left( \rho ,s,t\right) + \frac{1}{\varepsilon } (\partial _\rho c_j) \left( \rho , s,t\right) \varDelta d_{\varGamma _t} (x) + \mathcal {O}(1),\\ \partial _t c_j (\rho ,s,t)&= \frac{1}{\varepsilon } (\partial _\rho c_j) \left( \rho ,s,t\right) \partial _t d_{\varGamma _t} (x) +\mathcal {O}(1) \end{aligned}$$

on \(\varGamma _t\), where \(\rho = \tfrac{d_{\varGamma _t} (x,t)}{\varepsilon }\) and

$$\begin{aligned} \nabla d_{\varGamma _t}= \mathbf {n}_{\varGamma _t},\quad \varDelta d_{\varGamma _t} = -H_{\varGamma _t},\quad \partial _t d_{\varGamma _t} =-V_{\varGamma _t} \quad \text {on }\varGamma _t. \end{aligned}$$

Hence inserting the inner expansion in (1) and equating terms of the same order yields for all \(x \in {\varGamma _t}\):

$$\begin{aligned}&m_0\left[ -\partial _\rho ^2 c_0 (\rho ,s,t) + f'(c_0(\rho ,s,t))\right] \cdot \frac{1}{\varepsilon ^2}\\&\quad + m_0\left[ -\partial _\rho ^2 c_1 (\rho ,s,t) + f''(c_0(\rho ,s,t)) c_1(\rho ,s,t)\right] \cdot \frac{1}{\varepsilon }\\&\quad + \left[ - \partial _\rho c_0 (\rho ,s,t) (V_{\varGamma _t} -\mathbf {n}_{\varGamma _t}\cdot \mathbf {v}- m_0 H_{\varGamma _t} )\right] \cdot \frac{1}{\varepsilon }= O (1) \end{aligned}$$

in the case \(k=0\) and

$$\begin{aligned}&\left[ m_0\left( -\partial _\rho ^2 c_0 (\rho ,s,t) + f'(c_0(\rho ,s,t))\right) - (\partial _\rho c_0) (\rho ,s,t) (V_{\varGamma _t}-\mathbf {n}_{\varGamma _t}\cdot \mathbf {v})\right] \cdot \frac{1}{\varepsilon } = O (1) \end{aligned}$$

in the case \(k=1\). For the following we distinguish the cases \(k=0,1\).

Case \(k=0\): The \(\mathcal {O}(\frac{1}{\varepsilon ^2})\)-terms yield

$$\begin{aligned} - \partial _\rho ^2 c_0 (\rho ,s,t) + f'(c_0(\rho ,s,t)) =0\quad \text {for all }\rho \in \mathbb {R}, s\in \varGamma _t, t\in [0,T]. \end{aligned}$$

Because of the matching condition, we obtain

$$\begin{aligned} \lim _{\rho \rightarrow \pm \infty } c_0(\rho ,s,t) =c_0^\pm (s,t) = \pm 1 \quad \text {for all }s\in \varGamma _t, t\in [0,T]. \end{aligned}$$

In order to obtain that \(\varGamma _t\) approximates the zero-level set of \(c_\varepsilon (x,t)= c_0(\tfrac{d_{\varGamma _t}}{\varepsilon },s(x),t)+\mathcal {O}(\varepsilon )\) sufficiently well, we obtain \(c_0(0,s,t) = 0\). Hence

$$\begin{aligned} c_0(\rho ,x,t)&= \theta _0 (\rho ) \quad \text {for all } x \in \varGamma _t, \rho \in \mathbb {R}. \end{aligned}$$

Furthermore, the \(\mathcal {O}(\frac{1}{\varepsilon })\)-terms yield

$$\begin{aligned} m_0\left( -\partial _\rho ^2 c_1 (\rho ,x,t) + f''(\theta _0 (\rho )) c_1 (\rho ,x,t)\right) = \theta '_0(\rho ) (V_{\varGamma _t} - \mathbf {n}_{\varGamma _t}\cdot \mathbf {v}- m_0H_{\varGamma _t})=:g(\rho ) \end{aligned}$$

Since \(\theta '_0\) is in the kernel of the differential operator \(-\partial _\rho ^2 +f''(\theta _0)\), this ODE has a bounded solution if and only if

$$\begin{aligned} \int _{\mathbb {R}} g(\rho ) \theta '_0 (\rho ) d \rho =0, \end{aligned}$$
(31)

which is equivalent to

$$\begin{aligned} V_{\varGamma _t} - \mathbf {n}_{\varGamma _t}\cdot \mathbf {v}= H_{{\varGamma _t} }\quad \text { on } {\varGamma _t}. \end{aligned}$$

Now the matching condition yields \(c_1 (\rho ,x,t) \rightarrow _{\rho \rightarrow \pm \infty } c_1^\pm \equiv 0\). Hence \(c_1 \equiv 0\) since the solution is unique. Altogether we obtain for the inner expansion

$$\begin{aligned} c_\varepsilon (x,t) = \theta _0 \left( \frac{d_{\varGamma _t} (x)}{\varepsilon }\right) + O (\varepsilon ^2) \end{aligned}$$

close to \(\varGamma _t\).

Case \(k=1\): The \(\mathcal {O}(\frac{1}{\varepsilon })\)-terms yield

$$\begin{aligned}&m_0\left( - \partial _\rho ^2 c_0 (\rho ,s,t) + f'(c_0(\rho ,s,t))\right) \nonumber \\&\quad -\partial _\rho c_0(\rho ,s,t) (V_{\varGamma _t} (s)-\mathbf {n}_{\varGamma _t}(s)\cdot \mathbf {v}(s,t)) =0 \end{aligned}$$
(32)

for all \(s\in \varGamma _t\). Testing with \(\partial _\rho c_0(\rho ,x,t)\) yields

$$\begin{aligned} 0 = \int _{\mathbb {R}}|\partial _\rho c_0(\rho ,s,t)|^2\, d\rho \left( V_{\varGamma _t} (s)-\mathbf {n}_{\varGamma _t}\cdot \mathbf {v}(s,t) \right) \end{aligned}$$

since

$$\begin{aligned} \int _{\mathbb {R}}\partial _\rho \left( \frac{|\partial _\rho c_0(\rho ,s,t)|^2}{2} + f(c_0(\rho ,s,t))\right) d\rho =0 \end{aligned}$$

because of the matching condition for \(\partial _\rho c_0\). Because of \(c_0(\rho ,s,t)\rightarrow _{\rho \rightarrow \pm \infty } \pm 1\), \(\partial _\rho c_0\) does not vanish and we obtain

$$\begin{aligned} V_{\varGamma _t} = \mathbf {n}_{\varGamma _t}\cdot \mathbf {v}\quad \text { on } {\varGamma _t}. \end{aligned}$$

Moreover, we obtain from (32)

$$\begin{aligned} - \partial _\rho ^2 c_0 (\rho ,s,t) + f'(c_0(\rho ,s,t))= 0 \quad \text {for all }s\in \varGamma _t,\rho \in \mathbb {R}. \end{aligned}$$

Hence we can conclude as in the case \(k=0\) that \(c_0(\rho ,s,t)= \theta _0(\rho )\) for all \(\rho \in \mathbb {R}\) and \(s\in \varGamma _t\), \(t\in [0,T]\).

Remark 2

The formal calculations show that \(c_\varepsilon \) should have an expansion of the form (17) in the case \(\theta =0,1\). This is important to obtain (15) in the limit. Actually, using \(c_0(\rho ,s,t)=\theta _0(\rho )\) one can easily modify the results in [2, Section 4] to show formally convergence of the Navier–Stokes/Allen–Cahn system (4)–(6) to (7)–(11) in the case \(\theta =0\) and (12)–(16) in the case \(\theta =1\). A rigorous justification of this convergence under suitable assumptions remains open.