On a Regularized Family of Models for Homogeneous Incompressible Two-Phase Flows

Abstract

We consider a general family of regularized models for incompressible two-phase flows based on the Allen–Cahn formulation in $n$-dimensional compact Riemannian manifolds for $n=2,3$. The system we consider consists of a regularized family of Navier–Stokes equations (including the Navier–Stokes-$\alpha$-like model, the Leray-$\alpha$ model, the modified Leray-$\alpha$ model, the simplified Bardina model, the Navier–Stokes–Voight model, and the Navier–Stokes model) for the fluid velocity $u$ suitably coupled with a convective Allen–Cahn equation for the order (phase) parameter $\phi$. We give a unified analysis of the entire three-parameter family of two-phase models using only abstract mapping properties of the principal dissipation and smoothing operators and then use assumptions about the specific form of the parameterizations, leading to specific models, only when necessary to obtain the sharpest results. We establish existence, stability, and regularity results and some results for singular perturbations, which as special cases include the inviscid limit of viscous models and the $\alpha \rightarrow 0$ limit in $\alpha$ models. Then we show the existence of a global attractor and exponential attractor for our general model and establish precise conditions under which each trajectory $\left( u,\phi \right)$ converges to a single equilibrium by means of a Lojasiewicz–Simon inequality. We also derive new results on the existence of global and exponential attractors for the regularized family of Navier–Stokes equations and magnetohydrodynamics models that improve and complement the results of Holst et al. (J Nonlinear Sci 20(5):523–567, 2010). Finally, our analysis is applied to certain regularized Ericksen–Leslie models for the hydrodynamics of liquid crystals in $n$-dimensional compact Riemannian manifolds.

Appendix

In this section, we include some supporting material on Grönwall-type inequalities, Sobolev inequalities, some definitions, and abstract results. The first lemma is a slight generalization of the usual Grönwall-type inequality (Temam 1988); its proof is quite elementary and thus omitted.

Lemma 8.1

Let $\mathcal {E}:\mathbb {R}_{+}\rightarrow \mathbb {R}_{+}$ be an absolutely continuous function satisfying

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}\mathcal {E}(t)+2\eta \mathcal {E}(t)\le h(t)\mathcal {E}(t)+l\left( t\right) +k, \end{aligned}$$

where $\eta >0$, $k\ge 0$, and $\int \limits _{s}^{t}h\left( \tau \right) \hbox {d}\tau \le \eta (t-s)+m$ for all $t\ge s\ge 0$ and some $m\in \mathbb {R}$, and $\int \limits _{t}^{t+1}l\left( \tau \right) \hbox {d}\tau \le \gamma <\infty$. Then, for all $t\ge 0$,

$$\begin{aligned} \mathcal {E}(t)\le \mathcal {E}(0)e^{m}e^{-\eta t}+\frac{2\gamma e^{m+\eta }}{e^{\eta }-1}+\frac{ke^{m}}{\eta }. \end{aligned}$$

With $s,p\in \mathbb {R}_{+}$, let $W^{s,p}$ be the standard Sobolev space on an $n$-dimensional compact Riemannian manifold, with $n\ge 2$. The following result states the classical Gagliardo–Nirenberg–Sobolev inequality (cf. Bakry et al. 1995; Hebey 1999; Nirenberg 1959; Ceccon and Montenegro 2008, 2013).

Lemma 8.2

Let $0\le k<m$, with $k,m\in \mathbb {N}$ and numbers $p,q,q\in \left[ 1,\infty \right]$, satisfy

$$\begin{aligned} k-\frac{n}{p}=\tau \left( m-\frac{n}{q}\right) -\left( 1-\tau \right) \frac{n}{r}. \end{aligned}$$

Then there exists a positive constant $C$ independently of $u$ such that

$$\begin{aligned} \left\| u\right\| _{W^{k,p}}\le C\left\| u\right\| _{W^{m,q}}^{\tau }\left\| u\right\| _{L^{r}}^{1-\tau }, \end{aligned}$$

with $\tau \in \left[ \frac{k}{m},1\right]$, provided that $m-k-\frac{n}{r}\notin \mathbb {N}_{0},$ and $\tau =\frac{k}{m}$, provided that $m-k-\frac{n}{r}\in \mathbb {N}_{0}.$

We state here a standard result on the pointwise multiplication of functions in Sobolev spaces (see Maxwell 2004; cf. also Holst et al. 2010).

Lemma 8.3

Let $s$, $s_{1}$, and $s_{2}$ be real numbers satisfying

$$\begin{aligned} s_{1}+s_{2}\ge 0,\qquad min(s_{1},s_{2})\ge s,\qquad \text {and}\qquad s_{1}+s_{2}-s>\frac{n}{2}, \end{aligned}$$

where the strictness of the last two inequalities can be interchanged if $s\in \mathbb {N}_{0}$. Then the pointwise multiplication of functions extends uniquely to a continuous bilinear map

$$\begin{aligned} H^{s_{1}}\otimes H^{s_{2}}\rightarrow H^{s}. \end{aligned}$$

We have the following definition of exponential attractor (also known as inertial set).

Definition 8.4

Let $\left( S\left( t\right) , \mathcal {K}\right)$ be a dynamical system on a given Banach space $\mathcal {K}$. A set $\mathcal {M}\subset \mathcal {K}$ is said to be an exponential attractor (also known as an inertial set) for the semigroup $S\left( t\right)$ provided that the following statements hold:

($i$):

The sets $\mathcal {M}$ are positively invariant with respect to the semigroup $S\left( t\right) ,$ that is, $S\left( t\right) \mathcal {M}\subseteq \mathcal {M}$, for all $\,t\ge 0$;

($ii$):

The fractal dimension of the sets $\mathcal {M}$ is finite, that is, $\dim _{F}\left( \mathcal {M},\mathcal {K}\right) \le C<\infty ,$ where $C>0$ can be computed explicitly;

($iii$):

Each $\mathcal {M}$ attracts exponentially any bounded subset of $\mathcal {K},$ that is, there exist a positive constant $\rho$ and a monotone nonnegative function $Q$ such that for every bounded subset $B$ of $\mathcal {K}$ we have

$$\begin{aligned} dist_{\mathcal {K}}\left( S\left( t\right) B,\mathcal {M}\right) \le Q(\left\| B\right\| _{\mathcal {K}})e^{-\rho t}, \end{aligned}$$

where $\displaystyle dist_{\mathcal {K}}\left( X,Y\right) :=\sup _{x\in X}\inf _{y\in Y}\left\| x-y\right\| _{\mathcal {K}}$ is the Hausdorff semidistance.

We report the following basic abstract results (see Gatti et al. 2010, Theorem 4.4; Gal and Grasselli 2010b, a; cf. also Miranville and Zelik 2008), which are needed to prove Theorem 5.8 when $\theta >0$ and Theorem 5.18 in the case $\theta =0$.

Theorem 8.5

Let ${\mathcal {X}}_{1}$ and ${\mathcal {X}}_{2}$ be two Banach spaces such that ${\mathcal {X}_{2}}$ is compactly embedded in ${\mathcal {X}}_{1}.$ Let $X_{0}$ be a bounded subset of $\mathcal {X}_{2}$, and consider a nonlinear map $\Sigma :X_{0}\rightarrow X_{0}$ satisfying the smoothing property

$$\begin{aligned} \left\| \Sigma \left( x_{1}\right) -\Sigma \left( x_{2}\right) \right\| _{{\mathcal {X}}_{2}}\le d\left\| x_{1}-x_{2}\right\| _{{\mathcal {X}_{1}}} \end{aligned}$$

(8.1)

for all $x_{1},x_{2}\in X_{0},$ where $d>0$ depends on $X_{0}.$ Then the discrete dynamical system $(X_{0},\Sigma ^{n})$ possesses a discrete exponential attractor $\mathcal {E}_{M}^{*}\subset {\mathcal {X}_{2}},$ that is, a compact set in ${\mathcal {X}_{1}}$ with finite fractal dimension such that

$$\begin{aligned} \Sigma \left( \mathcal {E}_{M}^{*}\right) \subset \mathcal {E}_{M}^{*}, \end{aligned}$$

(8.2)

$$\begin{aligned} dist_{{\mathcal {X}_{1}}}\left( \Sigma ^{n}\left( X_{0}\right) , \mathcal {E}_{M}^{*}\right) \le d_{X}e^{-\rho _{*}n},\quad n\in {\mathbb {N}}, \end{aligned}$$

(8.3)

where $d_{X}$ and $\rho _{*}$ are positive constants independent of $n,$ with the former depending on $X_{0}.$

Theorem 8.6

Let $\mathcal {K}$ and $\mathcal {K}_{c}$ be two Banach spaces such that $\mathcal {K}_{c}$ is compactly embedded in $\mathcal {K}$, and let $\left( S\left( t\right) , \mathcal {K}\right)$ be a dynamical system. Assume that the following hypotheses hold:

• (H1) There exists a bounded subset $\mathbb {B}\subset \mathcal {K}$ that is positively invariant for $S(t)$ and attracts any bounded set of $\mathcal {K}$ exponentially fast.

• (H2) There exists a positive constant $C$ independently of time such that

  $$\begin{aligned} \left\| S\left( t\right) \varphi _{1}-S\left( t\right) \varphi _{1}\right\| _{\mathcal {K}}\le \rho \left( t\right) \left\| \varphi _{1}-\varphi _{2}\right\| _{\mathcal {K}}, \end{aligned}$$

  for every $t\ge 0$ and every $\varphi _{1},\varphi _{2}\in \mathbb {B}$, where $\rho :\mathbb {R}_{+}\rightarrow \mathbb {R}_{+}$ is some continuous function with $\rho \left( 0\right) >0.$

• (H3) There exist a positive constant $C,$ $\kappa \in \left( 0,1\right)$, and a time $t^{*}>0$ such that

  $$\begin{aligned} \left\| S(t)\varphi _{0}-S(\tilde{t})\varphi _{0}\right\| _{\mathcal {K}}\le C|t-\tilde{t}|^{\kappa }, \end{aligned}$$

  for all $t,\tilde{t}\in \left[ t^{*},2t^{*}\right]$ and any $\varphi _{0}\in \mathbb {B}.$

• (H4) For every $\varphi _{01},\varphi _{02}\in \mathbb {B}$, $S(t)$ can be decomposed as follows:

  $$\begin{aligned} S(t)\varphi _{01}-S(t)\varphi _{02}=D\left( t\right) \left( \varphi _{01},\varphi _{02}\right) +N\left( t\right) \left( \varphi _{01},\varphi _{02}\right) , \end{aligned}$$

  where for all $t\ge 0$ we have

  $$\begin{aligned} \left\{ \begin{array}{l} \left\| D\left( t\right) \left( \varphi _{01},\varphi _{02}\right) \right\| _{\mathcal {K}}\le Ke^{-\varkappa t}\left\| \varphi _{01}-\varphi _{02}\right\| _{\mathcal {K}}, \\ \left\| N\left( t\right) \left( \varphi _{01},\varphi _{02}\right) \right\| _{\mathcal {K}_{c}}\le \rho \left( t\right) \left\| \varphi _{01}-\varphi _{02}\right\| _{\mathcal {K}}, \end{array} \right. \end{aligned}$$

  for some positive constants $\varkappa , K$ independent of time and some positive continuous function $\rho :\mathbb {R}_{+}\rightarrow \mathbb {R}_{+}$, $\rho \left( 0\right) >0.$ Then, if (H1)–(H4) are satisfied, there exists an exponential attractors $\mathcal {M}$ for $\left( S\left( t\right) , \mathcal {K}\right)$ in the sense of Definition 8.4 (i)–(iii).