Abstract
We study an initial-boundary value problem for the incompressible Navier–Stokes–Cahn–Hilliard system with non-constant density proposed by Abels, Garcke and Grün in 2012. This model arises in the diffuse interface theory for binary mixtures of viscous incompressible fluids. This system is a generalization of the well-known model H in the case of fluids with unmatched densities. In three dimensions, we prove that any global weak solution (for which uniqueness is not known) exhibits a propagation of regularity in time and stabilizes towards an equilibrium state as \(t \rightarrow \infty \). More precisely, the concentration function \(\phi \) is a strong solution of the Cahn–Hilliard equation for (arbitrary) positive times, whereas the velocity field \({\varvec{{u}}}\) becomes a strong solution of the momentum equation for large times. Our analysis hinges upon the following key points: a novel global regularity result (with explicit bounds) for the Cahn–Hilliard equation with divergence-free velocity belonging only to \(L^2(0,\infty ; {\textbf{H}}^1_{0,\sigma }(\Omega ))\), the energy dissipation of the system, the separation property for large times, a weak-strong uniqueness type result, and the Lojasiewicz–Simon inequality. Additionally, in two dimensions, we show the existence and uniqueness of global strong solutions for the full system. Finally, we discuss the existence of global weak solutions for the case of the double obstacle potential.
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1 Introduction
We consider the initial-boundary value problem for the diffuse interface model describing a two-phase flow of incompressible viscous Newtonian fluids in the case of general densities, which was introduced by Abels et al. [8] (also called AGG-model nowadays). This model leads to the system
in \(\Omega \times (0,\infty )\), where \(\Omega \) is a bounded domain in \({\mathbb {R}}^d\), with \(d=2\) and \(d=3\). The system (1.1) is completed with the following boundary and initial conditions
Here, \({\varvec{{n}}}\) is the unit outward normal vector on \(\partial \Omega \), and \(\partial _{\varvec{{n}}}\) denotes the outer normal derivative on \(\partial \Omega \). The state variables of the system are the volume averaged velocity \({\varvec{{u}}}:\Omega \times [0,\infty )\rightarrow {\mathbb {R}}^d, (x,t)\mapsto {\varvec{{u}}}(x,t)\), the pressure of the mixture \(P:\Omega \times [0,\infty )\rightarrow {\mathbb {R}}, (x,t)\mapsto P(x,t)\), and the difference of the fluids volume fractions \(\phi :\Omega \times [0,\infty )\rightarrow [-1,1], (x,t)\mapsto \phi (x,t)\). Here \(D {\varvec{{u}}}= \frac{1}{2} (\nabla {\varvec{{u}}}+(\nabla {\varvec{{u}}})^T)\) is the symmetrized gradient of \({\varvec{{u}}}\). The flux term \({\widetilde{{\textbf{J}}}}\), the density \(\rho \) and the viscosity \(\nu \) of the mixture are given by
where \(\rho _1\), \(\rho _2\) and \(\nu _1\), \(\nu _2\) are the positive homogeneous density and viscosity parameters of the two fluids, respectively. Moreover, \(m:[-1,1]\rightarrow [0,\infty )\) is a mobility coefficient, which in general might depend on \(\phi \). The homogeneous free energy density \(\Psi \) is the Flory-Huggins potential
for \(s \in [-1,1]\) where \(\theta \) and \(\theta _0\) are constant positive parameters.
As mentioned before, system (1.1) and (1.2) is a diffuse interface model for the flow of two incompressible and viscous Newtonian fluids, in which the macroscopically immiscible fluids are considered to be (partly) miscible on a small length scale \(\varepsilon >0\). Here the parameter \(\varepsilon \) is for simplicity set to 1. This model is thermodynamically consistent as shown in [8] (see also [9]). The total energy associated to system (1.1) is
and the corresponding energy balance reads as
for sufficiently smooth solutions \(({\varvec{{u}}}, P, \phi )\). Let us note that in the case \(\rho _1=\rho _2\), the flux term \({\widetilde{{\textbf{J}}}}={\textbf{0}}\) and the model simplifies to the well-known “Model H” (cf. [26, 28]). An alternative and thermodynamically consistent model was derived before by Lowengrub and Truskinovski in [31]. From the mathematical viewpoint, the latter has the disadvantage that the mass averaged (barycentric) velocity is not divergence free, which is the case in the present model based on a volume averaged velocity. Moreover, the pressure enters the equation for the chemical potential, which leads to additional difficulties for the construction of weak solutions, cf. [2], and a strong coupling of the linearized systems, cf. [4], where existence of strong solutions locally in time was shown. To the best of our knowledge, there are no results on existence of strong solutions for large times (in two space dimensions), regularity for weak solutions or qualitative behavior for large times for this model. But from a physical point of view it has the advantage that by the choice of the barycentric velocity the linear momentum of the sum of both fluids is conserved, which is not the case in general for the model under consideration (unless \(\rho _1=\rho _2\)). See also [35, Remark 2.2]. We also refer the interested reader to [13, 16, 27, 34] for further (different) diffuse interface systems modeling two-phase flows with unmatched densities. In the recent work [35], the reader can find a unified derivation and comparison of the known diffuse interface models from the physical point of view.
The existence of global weak solutions to the model (1.1) and (1.2) was proved in [6, Theorem 3.4] in the case of a strictly positive mobility m. The corresponding result in the case of a degenerate mobility in the Cahn–Hilliard equation (1.1)\(_{3-4}\) was shown in [7]. The convergence of a fully discrete numerical scheme to weak solutions was shown by Grün et al. [25]. The existence of strong solutions for regular free energy densities \(\Psi \) and small times was proved by Weber [36] (cf. also Abels and Weber [10]). The existence of weak solutions in the case of dynamic boundary conditions was proved by Gal et al. [20]. Results on well-posedness of this system with the free energy density (1.4) in two-space dimensions, locally in time for bounded domains and globally in time in the case of periodic boundary conditions, were achieved in [21]. In three space dimensions, the existence of strong solutions locally in time was obtained in [22]. The existence of weak solutions for a variant of (1.1) with a non-local Cahn–Hilliard equation was shown by Frigeri [18, 19] for non-degenerate and degenerate mobility, respectively, and for non-Newtonian fluids in Abels and Breit [5]. Finally, a model for a two-phase flow with magnetic fluids, where (1.1) is coupled to a gradient flow of the magnetization vector was studied by Kalousek et al. [29], where the existence of weak solutions was proven.
The main goal of the present contribution is to show regularity properties of weak solutions of (1.1) and (1.2) and to study the convergence for large times in the case of constant positive mobility. Let us first recall the result on existence of weak solutions shown in [6] (in the specific case \(a(\cdot )\equiv 1\)). We refer to the end of this introduction for the notation and, in particular, for the function spaces used in the following theorems.
Theorem 1.1
(Global existence of weak solutions) Let \(\Omega \) be a bounded domain in \({\mathbb {R}}^d\), \(d=2,3\), with boundary \(\partial \Omega \) of class \(C^2\) and \(m:[-1,1]\rightarrow (0,\infty )\) be continuous. Assume that \({\varvec{{u}}}_0 \in {\textbf{L}}_\sigma ^2(\Omega )\), \(\phi _0 \in H^1(\Omega )\) with \(\Vert \phi _0\Vert _{L^\infty (\Omega )}\le 1\) and \(\left| \overline{\phi _0} \right| <1\). Then, there exists a global weak solution \(({\varvec{{u}}},\phi )\) to (1.1) and (1.2) defined on \(\Omega \times [0,\infty )\) such that
which satisfies
for all \({\varvec{{w}}}\in C_{0,\sigma }^\infty (\Omega \times (0,\infty ))\), and
for all \(v \in C_0^\infty ((0,\infty ); C^1({\overline{\Omega }}))\), where
as well as \({\varvec{{u}}}(\cdot , 0)= {\varvec{{u}}}_0(\cdot )\) and \(\phi (\cdot , 0)= \phi _0\) in \(\Omega \). In addition, the energy inequality
holds for all \(t \in [s, \infty )\) and almost all \(s \in [0,\infty )\) (including \(s=0\)).
Remark 1.2
The regularity properties (1.7) and the weak formulation (1.9), combined with a density argument, entail that \(\partial _t \phi \in L^2(0, \infty ; H^1(\Omega )')\). As a consequence, (1.9) is equivalent to
Furthermore, arguing as in [1, 24] (cf. also [23]), it follows from (1.10) that \(\phi \in L^2_{{\text {uloc}}} ([0,\infty ); W^{2,p}(\Omega ) ) \cap L^4_{{\text {uloc}}} ([0,\infty ); H^2(\Omega ))\) and \(\Psi '(\phi ) \in L^2_{{\text {uloc}}} ([0,\infty ); L^{p}(\Omega ) )\), for any \(p \in [2,\infty )\) if \(d=2\) and \(p=6\) if \(d=3\). Finally, it holds that \(\partial _{\varvec{{n}}}\phi =0\) almost everywhere on \(\partial \Omega \times (0,\infty )\).
Throughout the manuscript we assume \(m\equiv 1\) for simplicity. Because of the energy dissipation (1.6), it is inherently expected that, as time t goes to infinity, the velocity \({\varvec{{u}}}(t)\) tends to zero and \(\phi (t)\) converges to an equilibrium of the Cahn–Hilliard equation/a critical point of the free energy \(E_{\text {free}}\). Moreover, one predicts the solution to become regular for sufficiently large times. In the case of matched densities, i.e., \(\rho _1=\rho _2\), such a result was shown in [1]. However, in the case of non-matched densities \(\rho _1\ne \rho _2\), this result for (1.1) was unknown so far. It is the purpose of this contribution to provide such a result. Our main result describes the global regularity features and the large time behavior of each weak solution given by Theorem 1.1 as follows.
Theorem 1.3
(Regularity and asymptotic behavior of weak solutions) Let \(\Omega \) be a bounded domain in \({\mathbb {R}}^d\), \(d=2,3\), with boundary \(\partial \Omega \) of class \(C^3\) and \(m\equiv 1\). Consider a global weak solution \(({\varvec{{u}}},\phi )\) given by Theorem 1.1. Then, the following results hold:
-
(i)
Global regularity of the concentration: for any \(\tau >0\), we have
$$\begin{aligned} \begin{aligned}&\phi \in L^\infty ( \tau ,\infty ; W^{2,p}(\Omega )), \quad \partial _t \phi \in L^2( \tau ,\infty ;H^1(\Omega )),\\&\mu \in L^{\infty }(\tau ,\infty ; H^1(\Omega ))\cap L_{{\text {uloc}}}^2([\tau ,\infty );H^3(\Omega )), \quad F'(\phi ) \in L^\infty (\tau ,\infty ; L^p(\Omega )), \end{aligned} \end{aligned}$$(1.12)for any \(2 \le p <\infty \) if \(d=2\) and \(p=6\) if \(d=3\). The Eq. (1.1)\(_{3,4}\) are satisfied almost everywhere in \(\Omega \times (0,\infty )\) and the boundary condition \(\partial _{\varvec{{n}}}\mu =0\) holds almost everywhere on \(\partial \Omega \times (0,\infty )\).
-
(ii)
Separation property: there exist \(T_{SP}>0\) and \(\delta >0\) such that
$$\begin{aligned} |\phi (x,t)|\le 1-\delta , \quad \forall \, (x,t) \in {\overline{\Omega }}\times [T_{SP},\infty ). \end{aligned}$$(1.13) -
(iii)
Large time regularity of the velocity: if \(\Omega \) is a \(C^4\) domain, then there exists \(T_R>0\) (possibly larger than \(T_{SP}\)) such that
$$\begin{aligned} {\varvec{{u}}}\in L^\infty (T_R,\infty ; {\textbf{H}}^1_{0,\sigma }(\Omega )) \cap L^2(T_R,\infty ; {\textbf{H}}^2(\Omega ))\cap H^1( T_R,\infty ; {\textbf{L}}^2_\sigma (\Omega )). \end{aligned}$$(1.14) -
(iv)
Convergence to a stationary solution: \(({\varvec{{u}}}(t),\phi (t)) \rightarrow ({\textbf{0}}, \phi _\infty )\) in \({\textbf{L}}^2_\sigma (\Omega ) \times W^{2-\varepsilon , p}(\Omega )\) as \(t \rightarrow \infty \), for any \(\varepsilon >0\), with any \(2 \le p <\infty \) if \(d=2\) and \(p=6\) if \(d=3\), where \(\phi _\infty \in W^{2,p}(\Omega )\) is a solution to the stationary Cahn–Hilliard equation
$$\begin{aligned} \begin{aligned} -\Delta \phi _\infty + \Psi '(\phi _\infty )&=\mu _\infty \qquad \text {in } \Omega ,\\ \partial _{{\varvec{{n}}}} \phi _\infty&=0 \qquad \quad \text {on } \partial \Omega ,\\ \frac{1}{|\Omega |}\int _\Omega \phi _\infty (x) \textrm{d}x&= \overline{\phi _0}, \end{aligned} \end{aligned}$$(1.15)where \(\mu _\infty \in {\mathbb {R}}\).
The structure of this contribution is as follows: in Sect. 2 we show a result on improved regularity for the solutions to the Cahn–Hilliard equation (1.1)\(_{3-4}\) with given velocity \({\varvec{{u}}}\in L^2(0,\infty ; {\textbf{H}}^1_{0,\sigma }(\Omega ))\), which implies (1.12) and is the basis for the following analysis. Next, we study the \(\omega \)-limit set of a weak solution to (1.1) in Sect. 3. In particular, it is shown that \({\varvec{{u}}}(t)\rightarrow _{t\rightarrow \infty } {\textbf{0}}\) in \({\textbf{L}}^2(\Omega )\) and that the \(\omega \)-limit set of the concentration \(\phi \) consists of stationary solutions of the Cahn–Hilliard equation (cf. (1.15)). Moreover, we prove the strict separation property (1.13). In order to achieve the regularity for large times (1.14), we first prove a result on weak-strong uniqueness in Sect. 4. Combining this with the local existence result of strong solutions from [22], we obtain that every weak solutions becomes a strong solution for sufficiently large times in Sect. 5. The convergence to an equilibrium of the system is shown with the aid of the Lojasiewicz–Simon inequality in the same Sect. 6. Furthermore, we prove the global regularity in two space dimensions in Sect. 7. Finally, we study the limit \(\theta \rightarrow 0\) in (1.4) and show that weak solutions converge (for a suitable subsequence) to weak solutions to (1.1) in the case of a double obstacle potential in Sect. 8.
Notation. In the sequel, we will use the following notation
We denote \(a\otimes b = (a_i b_j)_{i,j=1}^d\) for \(a,b\in {\mathbb {R}}^d\). If X is a Banach space and \(X'\) is its dual, then
denotes the duality product.
For a measurable set \(M\subseteq {\mathbb {R}}^d\) and \(1\le q \le \infty \), \(L^q(M)\) denotes the Lebesgue space and \(\Vert \cdot \Vert _{L^q(M)}\) its norm. We define
If \(q=2\), \((\cdot , \cdot )\) stands for the inner product in \(L^2(M)\). For any \(f \in L^1(M)\) with \(|M|<\infty \), the total mass is defined as \({\overline{f}}=\frac{1}{|M|} \int _{M} f(x) \, \textrm{d}x\). The space \({\textbf{L}}^q(M):= L^q(M; {\mathbb {R}}^d)\) consists of all q-integrable/essentially bounded vector-fields. For simplicity of notation, we denote the norm in \({\textbf{L}}^q(M)\) by \(\Vert \cdot \Vert _{L^q(M)}\) and the inner product in \({\textbf{L}}^2(M)\) by \((\cdot , \cdot )\). Let X be a Banach space, \(L^q(M;X)\) denotes the set of all strongly measurable q-integrable functions/essentially bounded functions with values in X. If \(M=(a,b)\), we write for simplicity \(L^q(a,b;X)\) and \(L^q(a,b)\). Furthermore, \(f\in L^q_{{\text {loc}}}([0,\infty );X)\) if and only if \(f\in L^q(0,T;X)\) for every \(T>0\) and \(L^q_{{\text {uloc}}}([0,\infty ); X)\) denotes the uniformly local variant of \(L^q(0,\infty ;X)\) consisting of all strongly measurable \(f:[0,\infty )\rightarrow X\) such that
For \(T<\infty \), we set \(L^q_{{\text {uloc}}}([0,T); X):= L^q(0,T;X)\).
Let \(\Omega \subset {\mathbb {R}}^d\) be an open bounded and connected domain with Lipschitz boundary. The set \(W^{m,q}(\Omega )\), \(m\in {\mathbb {R}}_+\), \(1\le q\le \infty \), denotes the \(L^q\)-Sobolev space, and \({\textbf{W}}^{m,q}(\Omega )= W^m_q(\Omega ; {\mathbb {R}}^d)\) is the corresponding space for vector-fields. In both cases, the corresponding norms are denoted by \(\Vert \cdot \Vert _{W^{m,q}(\Omega )}\). The space \(W^{m,q}_{0}(\Omega )\) is the closure of \(C^\infty _0(\Omega )\) in \(W^{m,q}(\Omega )\), \(W^{-m,q}(\Omega )= W^{m,q'}_{0}(\Omega )'\) (where \(q'\) is such that \(\frac{1}{q} +\frac{1}{q'}=1\)). As usual, \(H^m(\Omega )= W^{m,2}(\Omega )\) and \({\textbf{H}}^m(\Omega )= {\textbf{W}}^{m,2}(\Omega )\). Next, \({\textbf{L}}^2_\sigma (\Omega )\) and \({\textbf{H}}^1_{0,\sigma }(\Omega )\) are the closure of \(C^\infty _{0,\sigma }(\Omega ; {\mathbb {R}}^d)=\{\varvec{\varphi }\in C_0^\infty (\Omega ; {\mathbb {R}}^d ): {\text {div}} \varvec{\varphi }=0\}\) in \({\textbf{L}}^2(\Omega )\) and \({\textbf{H}}^1(\Omega )\), respectively.
Let \(I=[0,T]\) with \(0<T< \infty \) or let \(I=[0,\infty )\) if \(T=\infty \) and X be a Banach space. The set BC(I; X) is the Banach space of all bounded and continuous \(f:I\rightarrow X\) equipped with the supremum norm, and BUC(I; X) is the subspace of all bounded and uniformly continuous functions. We define \(BC_{\textrm{w}}(I;X)\) as the topological vector space of all bounded and weakly continuous functions \(f:I\rightarrow X\). We denote by \(C^\infty _0(0,T;X)\) the vector space of all smooth functions \(f:(0,T)\rightarrow X\) with \({\text {supp}}f \overset{c}{\subset }\ (0,T)\). Moreover, for \(1\le p <\infty \), \(W^{1,p}(0,T;X)\) is the space of all \(f\in L^p(0,T;X)\) with \(\partial _t f \in L^p(0,T;X)\), where \(\partial _t\) denotes the vector-valued distributional derivative of f. The set \(W^{1,p}_{{\text {uloc}}}([0,\infty );X)\) is defined in the same way, replacing \(L^p(0,T;X)\) by \(L^p_{{\text {uloc}}}([0,\infty );X)\). Lastly, we set \(H^1(0,T;X)= W^{1,2}(0,T;X)\) as well as \(H^1_{{\text {uloc}}}([0,\infty );X):= W^{1,2}_{{\text {uloc}}}([0,\infty );X)\).
2 On the Cahn–Hilliard equation with divergence-free drift
In this section we prove new regularity results for the Cahn–Hilliard equation with divergence-free drift
subject to the boundary and initial conditions
We first report the following well-posedness result proved in [1, Theorem 6].
Theorem 2.1
Let \(\Omega \) be a bounded domain in \({\mathbb {R}}^d\), \(d=2,3\) with \(C^3\) boundary. Assume that \(\phi _0\in H^1(\Omega )\cap L^{\infty }(\Omega )\) with \(\Vert \phi _0\Vert _{L^\infty (\Omega )}\le 1\) and \(\left| \overline{\phi _0}\right| <1\), and \({\varvec{{u}}}\in L^\infty (0,\infty ; {\textbf{L}}^2_\sigma (\Omega ))\cap L^2(0,\infty ;{\textbf{H}}^1_{0,\sigma }(\Omega ))\). Then, there exists a unique global weak solution \(\phi \) to (2.1) and (2.2) such that:
-
(1)
The weak solution satisfies
$$\begin{aligned}&\phi \in BC( [0,\infty ); H^1(\Omega ))\cap L_{{\text {uloc}}}^4([0,\infty );H^2(\Omega ))\cap L_{{\text {uloc}}}^2([0,\infty );W^{2,p}(\Omega )),\\&\phi \in L^{\infty }(\Omega \times (0,\infty )) \text { such that } |\phi (x,t)|<1 \ \text {a.e. in }\Omega \times (0,\infty ),\\&\partial _t \phi \in L^2(0,\infty ; H^1(\Omega )'),\\&\mu \in L_{{\text {uloc}}}^2([0,\infty );H^1(\Omega )), \quad F'(\phi )\in L_{{\text {uloc}}}^2([0,\infty );L^p(\Omega )), \end{aligned}$$for any \(2\le p <\infty \) if \(d=2\) and \(p=6\) if \(d=3\).
-
(2)
The weak solution solves (2.1) in a variational sense as follows:
$$\begin{aligned} \langle \partial _t \phi ,v\rangle + ( {\varvec{{u}}}\cdot \nabla \phi ,v) + (\nabla \mu , \nabla v )=0, \quad \forall \, v \in H^1(\Omega ), \ \text {a.e. in } (0,\infty ), \end{aligned}$$(2.3)where \(\mu \) is given by \( \mu =-\Delta \phi +\Psi '(\phi ). \) Moreover, \(\partial _{\varvec{{n}}}\phi =0\) almost everywhere on \(\partial \Omega \times (0,\infty )\), and \(\phi (\cdot ,0)=\phi _0\) in \(\Omega \).
-
(3)
The weak solution satisfies the energy equality
$$\begin{aligned} E_{\textrm{free}}(\phi (t))+ \int _\tau ^t \Vert \nabla \mu (s)\Vert _{L^2(\Omega )}^2 \, \textrm{d}s = E_{\textrm{free}}(\phi (\tau )) - \int _{\tau }^t ({\varvec{{u}}}\cdot \nabla \phi , \mu ) \, \textrm{d}s, \end{aligned}$$(2.4)for every \(0\le \tau < t \le \infty \).
-
(4)
Let \(Q= \Omega \times (0,\infty )\). The following estimates holds
$$\begin{aligned}&\Vert \phi \Vert _{L^\infty (0,\infty ;H^1(\Omega ))}^2 + \Vert \partial _t \phi \Vert _{L^2(0,\infty ; H^1(\Omega )')}^2 + \Vert \nabla \mu \Vert _{L^2(Q)}^2 \le C \left( 1+ E_{\textrm{free}}(\phi _0) + \Vert {\varvec{{u}}}\Vert _{L^2(Q)}^2 \right) , \end{aligned}$$(2.5)$$\begin{aligned}&\Vert \phi \Vert _{L_{{\text {uloc}}}^2([0,\infty ); W^{2,p}(\Omega )) }^2 + \Vert F'(\phi )\Vert _{L_{{\text {uloc}}}^2([0,\infty ); L^p(\Omega ))}^2 \le C_p \left( 1+ E_{\textrm{free}}(\phi _0) + \Vert {\varvec{{u}}}\Vert _{L^2(Q)}^2 \right) , \end{aligned}$$(2.6)$$\begin{aligned}&\Vert \phi \Vert _{L_{{\text {uloc}}}^4([0,\infty ); H^2(\Omega ))}^4 \le C \left( 1+ E_{\textrm{free}}(\phi _0) + \Vert {\varvec{{u}}}\Vert _{L^2(Q)}^2 \right) ^2, \end{aligned}$$(2.7)for \(2\le p <\infty \) if \(d=2\) and \(p=6\) if \(d=3\). The constants C and \(C_p\) are independent of \(\theta \), \({\varvec{{u}}}\) and \(\phi _0\).
Remark 2.2
The following comments concerning Theorem 2.1 are in order:
-
(i)
A closer look at the proof of [1, Theorem 6] reveals that \({\varvec{{u}}}\in L^2(0,\infty ; {\textbf{H}}^1_{0,\sigma }(\Omega ))\) is sufficient to show the desired claim in Theorem 2.1.
-
(ii)
The regularity \(\phi \in L_{{\text {uloc}}}^4(0,\infty ;H^2(\Omega ))\) is not shown in [1], but the proof can be found in [23].
The propagation of regularity of the weak solutions and the existence of strong solutions have been first shown in [1, Lemma 3]. We report it here below for clarity of presentation.
Theorem 2.3
Let the assumptions of Theorem 2.1 hold. Assume that \(\kappa \equiv 1\) if \(\phi _0 \in H^2(\Omega )\), \(\mu _0=-\Delta \phi _0+\Psi '(\phi _0) \in H^1(\Omega )\) and \(\partial _{\varvec{{n}}}\phi _0=0\) on \(\partial \Omega \), whereas \(\kappa (t)= \left( \frac{t}{1+t}\right) ^\frac{1}{2}\) otherwise.
-
1.
If \(\partial _t {\varvec{{u}}}\in L^1_{{\text {uloc}}}([0,\infty ); {\textbf{L}}^2(\Omega ))\), then the weak solution to (2.1) and (2.2) satisfies
$$\begin{aligned}&\kappa \partial _t \phi \in L^\infty (0,\infty ; H^1(\Omega )')\cap L^2_{{\text {uloc}}}([0,\infty ); H^1(\Omega )),\\&\kappa \phi \in L^\infty (0,\infty ; W^{2,p}(\Omega )), \quad \kappa F'(\phi ) \in L^\infty (0,\infty ; L^p(\Omega )),\\&\kappa \mu \in L^\infty (0,T; H^1(\Omega )), \end{aligned}$$for any \(2 \le p <\infty \) if \(d=2\) and \(p=6\) if \(d=3\).
-
2.
If \({\varvec{{u}}}\in B^\alpha _{\frac{4}{3} \, \infty , {\text {uloc}}}([0,\infty ); {\textbf{H}}^s(\Omega ) )\cap BC_{\textrm{w}}([0,\infty ); {\textbf{L}}_\sigma ^2(\Omega ))\) for some \(-\frac{1}{2} < s \le 0\) and \(\alpha \in (0,1)\), then the weak solution to (2.1) and (2.2) fulfills
$$\begin{aligned} \kappa \phi \in C^\alpha ([0,\infty ); H^1(\Omega )')\cap B^\alpha _{2 \, \infty , {\text {uloc}}}([0,\infty );H^1(\Omega )). \end{aligned}$$
Theorem 2.3 provides two regularity results for the weak solutions to the Cahn–Hilliard equation with divergence-free drift by requiring that either \(\partial _t {\varvec{{u}}}\in L_{{\text {uloc}}}^1([0,\infty ); {\textbf{L}}^2(\Omega ))\) or \({\varvec{{u}}}\in B^\alpha _{\frac{4}{3} \infty , {\text {uloc}}}([0,\infty );{\textbf{H}}^s(\Omega ))\) for some \(-\frac{1}{2} < s \le 0\) and \(\alpha \in (0,1)\), in addition to \({\varvec{{u}}}\in L^\infty (0,\infty ; {\textbf{L}}^2_\sigma (\Omega ))\cap L^2(0,\infty ;{\textbf{H}}^1_{0,\sigma }(\Omega ))\). Although the assumption in the second part of Theorem 2.3 can be proven for weak solutions of (1.1) in the case of matched densities (namely for the Model H) as in [1], it does not seem possible for the case with unmatched densities since the weak formulation only gives some control of \(\partial _t (\rho {\varvec{{u}}})|_{{\textbf{H}}^1_{0,\sigma }(\Omega )}\). To overcome this issue, we will now show that the condition \({\varvec{{u}}}\in L^2(0,\infty ;{\textbf{H}}^1_{0,\sigma }(\Omega ))\) is sufficient to gain full regularity for the solutions to (2.1) and (2.2). We expect that such result will be useful for other diffuse interface models with hydrodynamics.
Theorem 2.4
Let \(\Omega \) be a bounded domain in \({\mathbb {R}}^d\), \(d=2,3\), with \(C^3\) boundary and the initial condition \(\phi _0 \in H^2(\Omega )\) be such that \(\Vert \phi _0\Vert _{L^\infty (\Omega )}\le 1\), \(\left| \overline{\phi _0}\right| <1\), \(\mu _0=-\Delta \phi _0+\Psi '(\phi _0) \in H^1(\Omega )\) and \(\partial _{\varvec{{n}}}\phi _0=0\) on \(\partial \Omega \). Assume that \({\varvec{{u}}}\in L^2(0,\infty ;{\textbf{H}}^1_{0,\sigma }(\Omega ))\). Then, there exists a unique global (strong) solution to (2.1) and (2.2) such that
for any \(2\le p <\infty \) if \(d=2\) and \(p=6\) if \(d=3\). The strong solution satisfies (2.1) almost everywhere in \(\Omega \times (0,\infty )\) and \(\partial _{\varvec{{n}}}\phi =\partial _{\varvec{{n}}}\mu =0\) almost everywhere on \(\partial \Omega \times (0,\infty )\). Moreover, there exists a positive constant C depending only on \(\Omega \), \(\theta \), \(\theta _0\), and \(\overline{\phi _0}\) such that
and
In particular, the constant C is bounded whenever \(\theta \) is restricted to a bounded interval.
In addition, if \({\varvec{{u}}}\in L^\infty (0, \infty ;{\textbf{L}}^2_\sigma )\cap L^2(0,\infty ;{\textbf{H}}^1_{0,\sigma }(\Omega ))\), then \(\partial _t \phi \in L^\infty (0,\infty ; H^1(\Omega )')\).
Proof
Let us assume first that \({\varvec{{u}}}\in {C_0^\infty }(0,T; {\textbf{H}}^1_{0,\sigma }(\Omega ) \cap {\textbf{H}}^2(\Omega ))\) and the initial condition \(\phi _0\) is such that
For any \(\alpha \in (0,1)\), we consider the viscous Cahn–Hilliard system with divergence-free drift
which is equipped with the boundary and initial conditions
Thanks to [22, Theorem A.1], there exists a unique solution such that
The pair \((\phi ,\mu )\) satisfies (2.12) almost everywhere in \(\Omega \times (0,T)\), the boundary conditions \(\partial _{\varvec{{n}}}\phi =\partial _{\varvec{{n}}}\mu =0\) almost everywhere on \(\partial \Omega \times (0,T)\) and \(\phi (\cdot , 0)= \phi _0(\cdot )\) in \(\Omega \).
We now proceed with the conservation of mass and the first energy estimate. Integrating (2.12)\(_1\) over \(\Omega \), exploiting the incompressibility and the no-slip boundary condition of the velocity field \({\varvec{{u}}}\), we infer that
Multiplying (2.12)\(_1\) by \(\mu \), integrating over \(\Omega \) and exploiting the definition of \(\mu \), we find for almost every \(t\in (0,T)\)
Since \({\varvec{{u}}}(t)\) belongs to \({\textbf{H}}^1_{0,\sigma }(\Omega )\), we have that \(\int _{\Omega } {\varvec{{u}}}\cdot \nabla \phi \, \mu \, \textrm{d}x=- \int _{\Omega } {\varvec{{u}}}\cdot \nabla \mu \, \phi \, \textrm{d}x\). Then, thanks to the \(L^\infty \) bound of \(\phi \) in (2.14), we easily reach
An integration in time on [0, t], with \(0<t\le T\), yields
By using (2.15), we obtain
where the constant \(C_0\) depends only on \(E_{\text {free}}(\phi _0)\), \(|\overline{\phi _0}|\), \(\theta _0\), \(\Omega \) and \(\Vert {\varvec{{u}}}\Vert _{L^2(0,T; L^2(\Omega ))}\), but is independent of \(\alpha \) and depends on \(\theta \) only through \(E_{\text {free}}(\phi _0)\).
Next, we derive some preliminary estimates which will play a crucial role for the subsequent part. We recall the well-known inequality (see, for instance, [33])
where the positive constant \(C_1\) only depends only on \(\overline{\phi _{0}}\) and \(C_2\) only depends on \(\theta \) and \(\overline{\phi _{0}}\). We also observe that \(C_2\) can be chosen to depend only on \(\overline{\phi _{0}}\) if we restrict \(\theta \) to lie in a bounded interval. This can be seen if we consider (2.19) for \(\theta =1\) and then multiply the equation by \(\theta \). Multiplying (2.12)\(_2\) by \(\phi - \overline{\phi _{0}}\) (cf. (2.15)), we find
By the generalized Poincaré inequality and the \(L^\infty \) bound of \(\phi \), we reach
where C only depends on the Poincaré constant, \(\theta _0\), \(C_1\), \(C_2\) and \(\Omega \). Then, since \({\overline{\mu }}= \overline{F'(\phi )}- \theta _0 \overline{\phi _{0}}\), we infer from (2.19) and (2.20) that
As a consequence, we deduce that
Besides, multiplying (2.12)\(_2\) by \(|F'(\phi )|^{p-2}F'(\phi )\), with \(p\ge 2\), and integrating over \(\Omega \), we find
Here these computations are justified since \(\phi \) is separated from the pure phases (cf. (2.14)). Then, by the Hölder inequality, it follows that
Combining (2.22) with (2.12)\(_2\), the elliptic regularity of the Neumann problem yields that
In addition, we also find by comparison in (2.12)\(_1\) that
The positive constants C in (2.21)–(2.24) may vary from line to line, but they only depend on \(\overline{\phi _0}\), \(\theta \), \(\theta _0\), \(\Omega \) and \(C_1, C_2\). In particular, they are all independent of \(\alpha \) and stay bounded for \(\theta \) belonging to a bounded interval.
We now carry out the higher-order Sobolev energy estimates. Multiplying (2.12)\(_1\) by \(\partial _t \mu \) and integrating over \(\Omega \), we find
By definition of \(\mu \), we observe that
Similarly, by exploiting the incompressibility and the no-slip boundary condition of the velocity field, we notice that
Then, we arrive at
In order to estimate the terms on the right-hand side of (2.25), combining (2.22) and (2.23) with (2.21) through the Sobolev embedding theorem, we have
where \(2\le p <\infty \) if \(d=2\) and \(p=6\) if \(d=3\). Here the positive constants C and \(C_p\) are independent of \(\alpha \) and remains bounded for \(\theta \) in a bounded interval. Recalling that \(\partial _t \phi \) is mean-free, thanks to (2.24) and the generalized Poincaré inequality, we obtain
By using the \(L^\infty \) bound of \(\phi \) in (2.14), we also get
Exploiting (2.26), the Gagliardo–Nirenberg interpolation inequality and the Sobolev embedding theorem, we infer that
and
as well as
By using (2.14), we find that
The positive constants C in all the above estimates depend on the parameters of the system, such as \(\theta _0\), \(\theta \), \(\Omega \) and \(\overline{\phi _0}\), but are independent of \(\alpha \), \({\varvec{{u}}}\), T and the norms of the initial condition \(\phi _0\). They also are uniformly bounded for \(\theta \) from a bounded interval. Therefore, collecting the above inequalities, we end up with the differential inequality
In light of (2.18), we observe that
Owing to this, we rewrite (2.27) as follows
where
Since \({\varvec{{u}}}\in {C_0^\infty }(0,T; {\textbf{H}}_{0,\sigma }^1(\Omega )\cap {\textbf{H}}^2(\Omega ))\) by assumption, it is easily seen that \({\mathcal {F}}_1\), \({\mathcal {F}}_2 \in L^1(0,T)\). Thanks to the Gronwall lemma, we deduce that
By exploiting (2.28), we then arrive at
We observe that \(\partial _t \phi \in BC([0,T];H^1(\Omega ))\) and \( \mu \in BC([0,T];H^1(\Omega ))\) due to (2.14). By comparison in (2.12)\(_2\), it follows that \(-\Delta \phi +\Psi '(\phi ) \in BC([0,T];H^1(\Omega ))\). Now, multiplying (2.12)\(_2\) by \(\partial _t \phi \) and integrating over \(\Omega \), we have
By using (2.12)\(_1\), we notice that
Integrating by parts and exploiting the boundary conditions of \(\mu \) and \({\varvec{{u}}}\), we deduce that
By continuity of each term in the above equation and recalling that \({\varvec{{u}}}\in {C_0^\infty }(0,T; {\textbf{H}}_{0,\sigma }^1(\Omega ))\), we infer that
which, in turn, entails that
As a consequence, we find
Besides, integrating (2.29) on [0, T], and exploiting (2.28), (2.31) and (2.32), we have
Since \(\alpha \in (0,1]\), \({\varvec{{u}}}\in {C_0^\infty }(0,T; {\textbf{H}}_{0,\sigma }^1(\Omega )\cap {\textbf{H}}^2(\Omega ))\), \(-\Delta \phi _0 + \Psi '(\phi _0) \in H^1(\Omega )\), and (2.18), we thus conclude that
It easily follows from (2.21) and (2.26) that
and
where \(2\le p <\infty \) if \(d=2\) and \(p=6\) if \(d=3\). The constants \(K_0\), \(K_1\) and \(K_p\) (for p as mentioned above) are independent of \(\alpha \) and remain bounded for \(\theta \) from a bounded interval.
Let us now consider a sequence of real numbers \(\alpha _n \in (0,1]\) such that \(\alpha _n \rightarrow 0\) as \(n \rightarrow \infty \). Thanks to the above analysis, there exists a sequence of pairs \((\phi _{\alpha _n}, \mu _{\alpha _n})\) such that
and \(\partial _{\varvec{{n}}}\phi _{\alpha _n}=\partial _{\varvec{{n}}}\mu _{\alpha _n}=0\) almost everywhere on \(\partial \Omega \times (0,T)\), \( \phi _{\alpha _n}(\cdot ,0)=\phi _{0}\) in \(\Omega \). Each pair \((\phi _{\alpha _n}, \mu _{\alpha _n})\) satisfies (2.34)–(2.36) replacing \((\phi , \mu )\) with \((\phi _{\alpha _n}, \mu _{\alpha _n})\). Thus, there exists a subsequence (still denoted in the same way) \((\phi _{\alpha _n}, \mu _{\alpha _n})\) such that
for any \(2\le p <\infty \) if \(d=2\) and \(p=6\) if \(d=3\), and, by the Aubin-Lions theorem,
for all \(2\le q <\infty \) if \(d=2\) and \(2\le q < 6\) if \(d=3\). In order to pass to the limit in the logarithmic function \(F'\), we recall from (2.14) that
Thanks to (2.39), we infer that \(\phi _{\alpha _n} \rightarrow \phi \) almost everywhere in \(\Omega \times (0,T)\). As a consequence, we have that
Then, \(F'(\phi _{\alpha _n}) \rightarrow \widetilde{F'}(\phi )\) almost everywhere in \(\Omega \times (0,T)\), where \(\widetilde{F'}(s)=F'(s)\) if \(s \in (-1,1)\) and \(\widetilde{F'}(\pm 1)=\pm \infty \). By the Fatou lemma and (2.36), \(\int _{\Omega \times (0,T)} |\widetilde{F'}(\phi )|^2 \, \textrm{d}x \textrm{d}s\le K_2^2\), which implies that \(\widetilde{F'}(\phi )\in L^2(\Omega \times (0,T))\). This entails that
and \(\widetilde{F'}(\phi )= F'(\phi )\) almost everywhere in \(\Omega \times (0,T)\). Owing to this, and by (2.36), we conclude that
for any p as above. Thus, letting \(n \rightarrow \infty \) in (2.37), we obtain that \((\phi , \mu )\) solves the Cahn–Hilliard system with divergence-free drift
In addition, \(\partial _{\varvec{{n}}}\phi =\partial _{\varvec{{n}}}\mu =0\) almost everywhere on \(\partial \Omega \times (0,T)\) and \(\phi (\cdot , 0)=\phi _{0}\) in \(\Omega \). Furthermore, by the lower semi-continuity of the norm with respect to the weak convergence, we can pass to the limit in the left-hand side of (2.32) and (2.33). In order to pass to the limit on the right-hand side containing \(\int _{0}^{T}\Vert \nabla \mu _{\alpha _n}(s)\Vert _{L^2(\Omega )}^2 \, \textrm{d}s\), we need to show that, up to a subsequence,
To this end, by the above convergences, we first observe that
and
The latter follows from (2.39) and \(F \in C^0([-1,1])\). Therefore, by integrating the energy identity (2.16) written for \((\phi _{\alpha _n}, \mu _{\alpha _n})\) and taking the limit as \(n \rightarrow \infty \), we obtain
On the other hand, thanks to Theorem 2.1, the solution \((\phi ,\mu )\) to (2.40) satisfies the following energy identity
Computing the difference of the two equations above, we deduce (2.41). Thus, we conclude that
and
where the positive constant C depends on \(\theta _0\), \(\theta \), \(\Omega \) and \(\overline{\phi _0}\), but is independent of \({\varvec{{u}}}\), T and the norms of the initial condition \(\phi _0\). In particular, C is bounded if \(\theta \) belongs to a bounded interval. Then, repeating the argument to obtain (2.20) and (2.21) without the (\(\alpha \)) viscous term, we easily recover that
Since \(\partial _t \phi \) is mean-free, we also have
By recalling the inequality proven in [1, Lemma 2] for (2.40)\(_2\)
where \(2\le p <\infty \) if \(d=2\) and \(p=6\) if \(d=3\), we obtain that
Finally, since
we easily obtain by the elliptic regularity theory applied to (2.40)\(_1\) that
Here the positive constants \(R_1\), \(R_2\), \(R_3\) and \(R_4\) depend on \(\theta _0\), \(\theta \), \(\Omega \), \(\overline{\phi _0}\), \(\big \Vert \nabla \big (-\Delta \phi _{0} +\Psi '(\phi _{0}) \big ) \big \Vert _{L^2(\Omega )}\) and \(\Vert {\varvec{{u}}}\Vert _{L^2(0,T;{\textbf{H}}_{0,\sigma }^1(\Omega ))}\). They all remain bounded for \(\theta \) from a bounded interval. Additionally, \(R_4\) is bounded for T bounded.
We are left to show that the existence of the unique solution \(\phi \) to (2.1) and (2.2) satisfying (2.8) holds for any divergence-free velocity \({\varvec{{u}}}\in L^2(0,\infty ;{\textbf{H}}_{0,\sigma }^1(\Omega ))\) and for any initial condition \(\phi _0 \in H^2(\Omega )\) such that \(\Vert \phi _0\Vert _{L^\infty (\Omega )}\le 1\), \(\left| \overline{\phi _0}\right| <1\), \(\mu _0=-\Delta \phi _0+\Psi '(\phi _0) \in H^1(\Omega )\) and \(\partial _{\varvec{{n}}}\phi _0=0\) on \(\partial \Omega \). For this purpose, since \({C_0^\infty }(0,T; {\textbf{H}}_{0,\sigma }^1(\Omega )\cap {\textbf{H}}^2(\Omega ))\) is dense in \(L^2(0,T; {\textbf{H}}_{0,\sigma }^1(\Omega ))\), there exists a sequence \(\lbrace {\varvec{{u}}}_n \rbrace \subset {C_0^\infty }(0,T; {\textbf{H}}_{0,\sigma }^1(\Omega )\cap {\textbf{H}}^2(\Omega ))\) such that \({\varvec{{u}}}_n \rightarrow {\varvec{{u}}}\) in \(L^2(0,T; {\textbf{H}}^1_{0,\sigma }(\Omega ))\) as \(n\rightarrow \infty \). Besides, it was shown in [24, proof of Theorem 4.1] that there exists a sequence \( \lbrace \phi _{0,n} \rbrace \subset H^3(\Omega )\) such that
-
1.
There exist \(m\in (0,1)\) depending only on \(\overline{\phi _0}\) and \(\delta =\delta (n) \in (0,1)\) such that
$$\begin{aligned} \left| \overline{\phi _{0,n}} \right| \le m, \quad \Vert \phi _{0,n}\Vert _{L^\infty (\Omega )}\le 1-\delta , \quad \forall \, n \in {\mathbb {N}}. \end{aligned}$$ -
2.
\(\phi _{0,n}\rightarrow \phi _0\) in \(H^1(\Omega )\), \(\phi _{0,n}\rightharpoonup \phi _0\) weakly in \(H^2(\Omega )\) and \(-\Delta \phi _{0,n}+ F'(\phi _{0,n}) \rightarrow -\Delta \phi _{0}+ F'(\phi _{0}) \) in \(H^1(\Omega )\) with
$$\begin{aligned} \left\| -\Delta \phi _{0,n}+ F'(\phi _{0,n}) \right\| _{H^1(\Omega )} \le \left\| -\Delta \phi _{0}+ F'(\phi _{0} \right\| _{H^1(\Omega )}. \end{aligned}$$ -
3.
For any \(n \in {\mathbb {N}}\), \(\partial _{\varvec{{n}}}\phi _{0,n}=0\) almost everywhere on \(\partial \Omega \).
Now, owing to the first part of the proof, for any \(n \in {\mathbb {N}}\), there exists a pair \((\phi _n, \mu _n)\) solving the Cahn–Hilliard system with divergence-free drift
and \(\partial _{\varvec{{n}}}\phi _n=\partial _{\varvec{{n}}}\mu _n=0\) almost everywhere on \(\partial \Omega \times (0,T)\), as well as \(\phi _n(\cdot , 0)=\phi _{0,n}\) in \(\Omega \). It follows from (2.42) and (2.43) that
and
In light of the properties of \({\varvec{{u}}}_n\) and \(\phi _{0,n}\), we simply obtain that
where the positive constants \(Q_1\) and \(Q_2\) are independent of n. By reasoning as above to get (2.44)–(2.48), it follows that
where the positive constants \(Q_3\), \(Q_4(T)\), \(Q_5(p)\) and \(Q_6\) are independent of n. Thanks to these bounds, in a similar way as for the vanishing viscosity limit, we obtain by compactness the existence of two limit functions \(\phi \in L^\infty (0,T;W^{2,p}(\Omega ))\cap H^1(0,T;H^1(\Omega ))\), such that \(F'(\phi )\in L^\infty (0,T; L^p(\Omega ))\) with p as above, and \(\mu \in L^\infty (0,T;H^1(\Omega ))\cap L^2(0,T;H^3(\Omega ))\), which solve (2.1) almost everywhere on \(\Omega \times (0,T)\) and (2.2) almost everywhere on \(\partial \Omega \times (0,T)\). The uniqueness of such solution is inferred from [1, Theorem 6] (see Theorem 2.1 and Remark 2.2). By the arbitrariness of T, we can extend the solution on \((0,\infty )\) and obtain the conclusion as stated in Theorem 2.4. The estimates (2.9) and (2.10) simply follows by passing to the limit as \(n \rightarrow \infty \) in (2.50) and (2.51) and letting \(T\rightarrow \infty \).
Finally, if \({\varvec{{u}}}\in L^\infty (0, \infty ;{\textbf{L}}^2_\sigma (\Omega ))\cap L^2(0,\infty ;{\textbf{H}}^1_{0,\sigma }(\Omega ))\), we deduce from
that \(\partial _t \phi \in L^\infty (0,\infty ; H^1(\Omega )')\). The proof is complete. \(\square \)
In order to derive a further global estimate for the gradient of the chemical potential in (2.1), we report the following well-known result for the Neumann problem (see, e.g., [32]).
Lemma 2.5
(Neumann problem) Let \(\Omega \) be a bounded domain in \({\mathbb {R}}^d\), \(d=2,3\), with \(C^3\) boundary. Given \(f \in W^{k,p}(\Omega )\) and \(g \in W^{k+1-1/p,p}(\partial \Omega )\), where \(k=0,1\) and \(p \in (1,\infty )\), such that
there exists a \(u \in W^{k+2,p}(\Omega )\) which is unique up to a constant satisfying
In addition,
where the positive constant C only depends on k, p and \(\Omega \).
Using this, we derive the following conclusion for the gradient of the chemical potential. Beyond its intrinsic interest, it will play a crucial role to control the flux \({\widetilde{{\textbf{J}}}}\) in the momentum equation (1.1).
Corollary 2.6
Let the assumptions of Theorem 2.4. Then, we have the following estimate for \(k=1,2\)
where the positive constant C depends only on k, \(\Omega \), \(\theta \), \(\theta _0\), F, and \(\overline{\phi _0}\).
Proof
Since
and \(\int _\Omega {\partial _t \phi + {\varvec{{u}}}\cdot \nabla \phi }(t) \, \textrm{d}x=0\) for all \(t \in (0,\infty )\), it follows from Lemma 2.5, the generalized Poincaré inequality and (2.46) that
Similarly, we find
Thus, in light of (2.9) and (2.10), for \(k=1,2\), we deduce the desired estimate (2.58). \(\square \)
We are now in the position to prove the first part of our main result
Proof of Theorem 1.3-(i) Global regularity of the concentration
Let \(({\varvec{{u}}},\phi )\) be a global weak solution to (1.1) and (1.2) given by Theorem 1.1. Consider an arbitrary positive time \(\tau \). Thanks to the energy inequality (1.11) and the Korn inequality, we infer that
In addition, we have that \(\mu \in L_{{\text {uloc}}}^2([0,\infty ); H^1(\Omega ))\), \(\phi \in L^4_{{\text {uloc}}}([0,\infty ; H^2(\Omega ))\) and \(\partial _{\varvec{{n}}}\phi =0\) almost everywhere on \(\partial \Omega \times (0,\infty )\) (cf. Remark 1.2). Thus, there exists \(\tau ^\star \in (0,\tau ]\) such that \(\phi (\tau ^\star ) \in H^2(\Omega )\) with \(\Vert \phi (\tau ^\star )\Vert _{L^\infty (\Omega )}\le 1\), \(|\overline{\phi (\tau ^\star )}|<1\), \(\mu (\tau ^\star )=-\Delta \phi (\tau ^\star )+\Psi '(\phi (\tau ^\star )) \in H^1(\Omega )\) and \(\partial _{\varvec{{n}}}\phi (\tau ^\star )=0\) on \(\partial \Omega \). An application of Theorem 2.4 on the interval \([\tau ^\star ,\infty )\), together with the uniqueness of weak solutions (cf. Remark 2.2), gives us the desired conclusion. \(\square \)
3 Large time behaviour and strict separation property
For any \(m \in {\mathbb {R}}\), let us define the function spaces
Notice that \(H^1_{(m)}(\Omega )\) is not a linear space if \(m\ne 0\). Nevertheless, it can be identified with \(H^1_{(0)}(\Omega )\) by simply translation with the constant m. Moreover, the tangent space of \(H^1_{(m)}(\Omega )\) is \(H^1_{(0)}(\Omega )\). Hence, if \(G:H^1_{(m)}(\Omega )\rightarrow {\mathbb {R}}\) is differentiable, then \(DG(f):H^1_{(0)}(\Omega )\rightarrow {\mathbb {R}}\) is linear and bounded, i.e., \(DG(f)\in H^{1}_{(0)}(\Omega )'\) for all \(f\in H^1_{(m)}(\Omega )\).
We now introduce
for \(\varphi \in {\text {dom}}(E_0)= \lbrace f\in H^1_{(m)}(\Omega ): |f(x)|\le 1 \text { a.e. in } \Omega \rbrace \), where F is the “convex part” of \(\Psi \) (cf. (1.4)). We recall that \(E_0\) is convex and lower semi-continuous. Thanks to [11, Theorem 4.3], the subgradient \(\partial E_0 (\varphi )\in L^2_{(0)}(\Omega )\) for all \(\varphi \in {\mathcal {D}}(\partial E_0)\), where
is given as
Let us now consider \(\phi _\infty \in {\mathcal {D}}(\partial E_0)\), which solves the stationary Cahn–Hilliard equation
We recall that solutions to (3.1)–(3.3) are critical points of the functional \(E_{\text {free}}\) on \(H^1_{(m)}(\Omega )\). In addition, they are separated from the pure phases \(\pm 1\) as stated in the next result, whose proof can be found in [11, Proposition 6.1] and in [15, Lemma A.1].
Proposition 3.1
Let \(\phi _\infty \in {\mathcal {D}}(\partial E_0)\) be a solution to (3.1)–(3.3). Then, there exist two constants \(M_j,\ j=1,2\), such that
The first important step towards the longtime stabilization of weak solutions to (1.1) and (1.2) is to show
Lemma 3.2
Let \(({\varvec{{u}}},\phi )\) be a weak solution to (1.1) and (1.2) in the sense of Theorem 1.1. Then \({\varvec{{u}}}(t)\rightarrow \textbf{0}\) in \({\textbf{L}}^2(\Omega )\) as \(t\rightarrow \infty \).
Proof
First of all, we have by (1.11) that the energy inequality for the full system
holds for almost every \(\tau \ge 0\) including \(\tau =0\) and every \(t\ge \tau \). Subtracting the energy equality (2.4) for the Cahn–Hilliard equation with divergence-free drift, we obtain
for almost every \(\tau \ge 0\) including \(\tau =0\) and every \(t\ge \tau \). Now, let \(\varepsilon >0\) be arbitrary. Since \(\nabla \mu \in L^2(\Omega \times (0,\infty ))\) by the energy estimate, there is some \(T'>0\) such that \(\Vert \nabla \mu \Vert _{L^2(\Omega \times (T,\infty ))} \le \varepsilon \) for all \(T\ge T'\). Moreover, since \({\varvec{{u}}}\in L^2(0,\infty ;{\textbf{H}}_{0,\sigma }^1(\Omega ))\), the set of all \(T\ge T'\) such that \(\Vert {\varvec{{u}}}(T)\Vert _{L^2(\Omega )}\le \varepsilon \) has positive measure. Hence there is some \(T\ge T'\) such that \(\Vert \nabla \mu \Vert _{L^2(\Omega \times (T,\infty ))}\le \varepsilon \) and \(\Vert {\varvec{{u}}}(T)\Vert _{L^2(\Omega )}\le \varepsilon \) hold true. Because of (3.6), and exploiting the \(L^\infty \) bound of \(\phi \) (cf. (1.7)), we deduce that
for almost every \(T>0\). Therefore, by the Cauchy-Schwarz, Korn and Young inequalities we obtain
where C is independent of T and \(\varepsilon \). This shows the claim. \(\square \)
We highlight that the proof of Lemma 3.2 consists of a direct argument which only relies on the validity of the total energy inequality for (weak) solutions of (1.1) and the free energy equality for solutions to the Cahn–Hilliard equation with drift. In comparison with [1, Lemma 11], no square-summability of the time derivative of \({\varvec{{u}}}\) is requested.
Next, we proceed with the proof of the second part of our main result.
Proof of Theorem 1.3-(ii) Separation property
Let us recall from the first part of the proof (cf. (1.12)) that \(\phi \in L^\infty (\tau ,\infty ;W^{2,p}(\Omega ))\) for any \(\tau >0\), and for arbitrary \(2\le p<\infty \) if \(d=2\) and \(p=6\) if \(d=3\). Now we define the \(\omega \)-limit set of \(({\varvec{{u}}},\phi )\) as
where \(\varepsilon >0\). Since \(\phi \in BUC([\tau ,\infty ); W^{2-\varepsilon ',p}(\Omega ))\) for any \(\varepsilon '\in (0,\varepsilon )\) and \(\tau >0\) by interpolation, and \({\varvec{{u}}}(t)\rightarrow _{t\rightarrow \infty } {\textbf{0}}\) by Lemma 3.2, we easily obtain that \(\omega ({\varvec{{u}}},\phi )\) is a non-empty, compact, and connected subset of \({\textbf{L}}^2_\sigma (\Omega )\times W^{2-\varepsilon ,p}(\Omega ))\) (cf. [14, Definition 1.4.1 and Theorem 1.4.7]). In addition, owing to the fact that E is a strict Lyapunov functional for (1.1) and following [1, Lemma 11], we are able to prove:
Lemma 3.3
Let \(({\varvec{{u}}},\phi )\) be a weak solution to (1.1) and (1.2) in the sense of Theorem 1.1. Then, we have
Proof
Thanks to (2.4), we recall that
for all \(t\ge 0\). Since \({\varvec{{u}}}\cdot \nabla \mu \, \phi \in L^1(\Omega \times (0,\infty ))\), the limit \(\lim _{t\rightarrow \infty } E_{\text {free}}(\phi (t))\) exists. In addition, owing to \({\varvec{{u}}}(t)\rightarrow _{t\rightarrow \infty } {\textbf{0}}\) in \({\textbf{L}}^2_\sigma (\Omega )\), \(\lim _{t\rightarrow \infty } E_{\text {kin}}({\varvec{{u}}}(t))=0\), and thereby the limit \( E_\infty := \lim _{t\rightarrow \infty } E({\varvec{{u}}}(t), \phi (t))\) exists.
Let us consider a sequence \(\lbrace t_n \rbrace _{n\in {\mathbb {N}}}\subset {\mathbb {R}}\) such that \(0\le t_n\nearrow \infty \). We set \(\lim _{n\rightarrow \infty } ({\varvec{{u}}}(t_n),\phi (t_n))= ({\varvec{{u}}}'_0,\phi '_0)\). By Lemma 3.2, it clearly follows that \({\varvec{{u}}}'_0=\textbf{0}\). Now define \(({\varvec{{u}}}_n(t),\phi _n(t)):=({\varvec{{u}}}(t+t_n),\phi (t+t_n))\) for \(t\in [0,\infty )\). Due to [1, Theorem 6], \(\lbrace \phi _n \rbrace _{n\in {\mathbb {N}}}\) converges weakly in \(L^2(0,T;W^2_p(\Omega ))\cap H^1(0,T;H^{1}(\Omega )')\), for every \(T>0\), to a limit function \(\phi '\), which is a weak solution (as defined in Theorem 2.1) to (2.1) and (2.2) with \({\varvec{{u}}}=\textbf{0}\), chemical potential \(\mu '\) and initial value \(\phi '|_{t=0}= \phi _0'\). In particular, \(\phi _n\rightarrow _{n\rightarrow \infty } \phi '\) in \(L^2(0,T;H^1(\Omega ))\) for every \(T>0\). Hence, there is a subsequence such that
On the other hand, \( \lim _{n\rightarrow \infty }E({\varvec{{u}}}_n(t),\phi _n(t))= \lim _{n\rightarrow \infty }E_{\text {free}}(\phi _n(t))= E_\infty \) since \({\varvec{{u}}}_n(t)\rightarrow _{n\rightarrow \infty } \textbf{0}\) in \({\textbf{L}}^2(\Omega )\). Thus, \(E_{\text {free}}(\phi '(t))=E_\infty \) for a.e. \(t\in [0,\infty )\). As a consequence, by the energy identity for the convective Cahn–Hilliard system (2.4) with \({\varvec{{u}}}\equiv \textbf{0}\), we deduce that \(\nabla \mu '(t)=0\) for almost all \(t\in [0,\infty )\). Therefore, \(\partial _t \phi '(t)=0\), and \(\phi '(t)\equiv \phi _0'\) solves the stationary Cahn–Hilliard equation (3.1)–(3.3) with \(m=\overline{\phi _0}\). \(\square \)
Finally, we are in the position to show the desired separation property (1.13). In fact, thanks to the above characterization of \(\omega ({\varvec{{u}}},\phi )\), Proposition 3.1, the embedding \(W^{2-\varepsilon ,p}(\Omega )\hookrightarrow C({\overline{\Omega }})\) for \(\varepsilon >0\) sufficiently small, and the compactness of \(\omega ({\varvec{{u}}},\phi )\) in \({\textbf{L}}_\sigma ^2(\Omega )\times W^{2-\varepsilon ,p}(\Omega )\), there exists some \(\delta '>0\) such that
Observing that \(\lim _{t\rightarrow \infty }{\text {dist}}(({\varvec{{u}}}(t),\phi (t)),\omega ({\varvec{{u}}},\phi ))=0\) in the norm of \({\textbf{L}}^2_\sigma (\Omega )\times W^{2-\varepsilon ,p}(\Omega )\), we conclude that, for every \(\delta \in (0,\delta ')\), there is some \(T_{SP}>0\) such that
\(\square \)
4 Weak-strong uniqueness result
In this section we demonstrate a weak-strong uniqueness result for the system (1.1) and (1.2) in both two and three dimensions. This will be essential to achieve the large time regularity of the velocity of each weak solution. Due to the presence of the non-constant density, our approach is inspired by [17]. Moreover, the separation property plays a crucial role in our argument.
Theorem 4.1
Let \(\Omega \) be a bounded domain in \({\mathbb {R}}^d\), \(d=2,3\), of class \(C^4\). Assume that \({\varvec{{u}}}_0 \in {\textbf{H}}_{0,\sigma }^1(\Omega )\), \(\phi _0 \in H^2(\Omega )\) be such that \(\Vert \phi _0\Vert _{L^\infty (\Omega )}< 1\), \(|\overline{\phi _0}|<1\), \(\mu _0=-\Delta \phi _0+\Psi '(\phi _0) \in H^1(\Omega )\) and \(\partial _{\varvec{{n}}}\phi _0=0\) on \(\partial \Omega \). In addition, we suppose that:
-
\(({\varvec{{u}}}, \phi )\) is a solution on [0, T] such that
$$\begin{aligned} \begin{aligned}&{\varvec{{u}}}\in BC_{\textrm{w}}([0,T]; {\textbf{L}}_\sigma ^2(\Omega ))\cap L^2(0,T;{\textbf{H}}_{0,\sigma }^1(\Omega )),\\&\phi \in L^\infty (0,T;H^3(\Omega ))\cap L^2(0,T;H^4(\Omega )), \quad \partial _t \phi \in L^2(0,T;H^1(\Omega )),\\&\phi \in C( {\overline{\Omega }}\times [0,T]),\quad \text {with}\quad \max _{t \in [0,T]} \Vert \phi (t)\Vert _{C({\overline{\Omega }})}<1, \\&\mu \in L^{\infty }(0,T; H^1(\Omega ))\cap L^2(0,T;H^3(\Omega )), \end{aligned} \end{aligned}$$(4.1)which satisfies
$$\begin{aligned} \begin{aligned}&\int _{\Omega } \rho (\phi (t)) {\varvec{{u}}}(t)\cdot {\varvec{{w}}}(t) \, \textrm{d}x - \int _{\Omega } \rho (\phi _0) {\varvec{{u}}}_0 \cdot {\varvec{{w}}}(0) \, \textrm{d}x- \int _0^t (\rho (\phi ) {\varvec{{u}}}, \partial _t {\varvec{{w}}}) \, \textrm{d}\tau \\&\quad -\int _0^t (\rho (\phi ) {\varvec{{u}}}\otimes {\varvec{{u}}}, \nabla {\varvec{{w}}}) \, \textrm{d}\tau + \int _0^t (\nu (\phi ) D {\varvec{{u}}}, D {\varvec{{w}}}) \, \textrm{d}\tau \\&\quad - \int _0^t ({\varvec{{u}}}\otimes {\widetilde{{\textbf{J}}}}, \nabla {\varvec{{w}}}) \, \textrm{d}\tau = \int _0^t (\mu \nabla \phi , {\varvec{{w}}}) \, \textrm{d}\tau , \end{aligned} \end{aligned}$$(4.2)where \({\widetilde{{\textbf{J}}}}= -\rho '(\phi ) \nabla \mu \), for any \(t \in (0,T)\), for all \({\varvec{{w}}}\in W^{1,2}(0,T;{\textbf{L}}_\sigma ^2(\Omega ))\cap L^4(0,T; {\textbf{H}}^1_{0,\sigma }(\Omega ))\), and
$$\begin{aligned} \partial _t \phi + {\varvec{{u}}}\cdot \nabla \phi = \Delta \mu , \quad \mu = -\Delta \phi + \Psi '(\phi ) \quad \text {a.e. in } \Omega \times (0,T). \end{aligned}$$(4.3)Moreover, \(({\varvec{{u}}}(0),\phi (0))=({\varvec{{u}}}_0,\phi _0)\) and the energy inequality holds
$$\begin{aligned} E({\varvec{{u}}}(t), \phi (t)) +\int _0^t \int _{\Omega } \nu (\phi ) | D {\varvec{{u}}}|^2 \, \textrm{d}x \textrm{d}\tau + \int _0^t \int _{\Omega } |\nabla \mu |^2 \, \textrm{d}x \textrm{d}\tau \le E({\varvec{{u}}}_0, \phi _0)\nonumber \\ \end{aligned}$$(4.4)for any \(t \in [0,T]\).
-
\(({\varvec{{U}}}, \Pi , \Phi )\) is a strong solution on [0, T] such that
$$\begin{aligned} \begin{aligned}&{\varvec{{U}}}\in C([0,T]; {\textbf{H}}^1_{0,\sigma }(\Omega ))\cap L^2(0,T;{\textbf{H}}^2(\Omega ))\cap W^{1,2}(0,T;{\textbf{L}}^2_\sigma (\Omega )),\\ {}&\Pi \in L^2(0,T;H^1(\Omega )), \\&\Phi \in L^\infty (0,T;H^3(\Omega ))\cap L^2(0,T;H^4(\Omega )), \quad \partial _t \Phi \in L^2(0,T;H^1(\Omega )),\\&\Phi \in C( {\overline{\Omega }}\times [0,T]),\quad \text {with}\quad \max _{t \in [0,T]} \Vert \Phi (t)\Vert _{C({\overline{\Omega }})}<1, \\ {}&M \in L^{\infty }(0,T; H^1(\Omega ))\cap L^2(0,T;H^3(\Omega )), \end{aligned} \end{aligned}$$(4.5)which satisfies
$$\begin{aligned} \begin{aligned}&\rho (\Phi ) \partial _t {\varvec{{U}}}+ \rho (\Phi ) ({\varvec{{U}}}\cdot \nabla ){\varvec{{U}}}+( {\textbf{J}}^\star \cdot \nabla ) {\varvec{{U}}}- \textrm{div}(\nu (\Phi )D{\varvec{{U}}})\\ {}&+ \nabla \Pi = M \nabla \Phi \quad \text {a.e. in } \Omega \times (0,T), \end{aligned} \end{aligned}$$(4.6)where \({\textbf{J}}^\star = -\rho '(\Phi ) \nabla M\), and
$$\begin{aligned} \partial _t \Phi + {\varvec{{U}}}\cdot \nabla \Phi = \Delta M, \quad M= -\Delta \Phi + \Psi '(\Phi ) \quad \text {a.e. in } \Omega \times (0,T), \end{aligned}$$(4.7)as well as \(({\varvec{{U}}}(0),\Phi (0))=({\varvec{{u}}}_0,\phi _0)\).
Then, \({\varvec{{u}}}={\varvec{{U}}}\) and \(\phi =\Phi \) on [0, T].
Remark 4.2
Notice that if \(({\varvec{{u}}},\phi )\) is a weak solution to (1.1) and (1.2) (as in Theorem 1.1) with \(\mu \in L^\infty (0,T; H^1(\Omega ))\) and \({\varvec{{u}}}\in L^2(0,T;{\textbf{H}}^2(\Omega ))\), then one can choose \({\varvec{{u}}}\) as a test function in (1.8). Exploiting (2.4), we then obtain the energy identity
for every \(t \in [0,T]\).
Proof
First of all, thanks to Remark 4.2, we have the energy identity (4.8) for \(({\varvec{{U}}},\Phi ,M)\) (i.e. replacing \(({\varvec{{u}}},\phi ,\mu )\) with \(({\varvec{{U}}},\Phi ,M)\)). In addition, Theorem 2.1 entails the validity of the following free energy equalities
and
for all \(t\in [0,T]\). Thus, we deduce from (4.4) and (4.8) (for \(({\varvec{{U}}}, \Phi , M)\)) that
and
for all \(t \in [0,T]\). Now, taking \({\varvec{{w}}}= - {\varvec{{U}}}\) in (4.2), we obtain
for all \(t \in [0,T]\). We recall the following basic relations
and
Summing (4.9), (4.10) and (4.11), and exploiting (4.12) and (4.13), we find
for all \(t \in [0,T]\). The conservation law of the densities reads as
For any function \(v \in C^1([0,T]; C^1({\overline{\Omega }}))\), we have
for all \(t \in [0,T]\). By a density argument, we take \(v= \frac{1}{2} |{\varvec{{U}}}|^2\) in (4.16) obtaining
for all \(t \in [0,T]\). Substituting (4.17) in (4.14), we deduce that
On the other hand, multiplying (4.6) by \({\varvec{{u}}}\) and integrating over \(\Omega \times (0,t)\), we infer that
for all \(t \in [0,T]\). Summing (4.18) and (4.19), we arrive at
for all \(t \in [0,T]\). We notice that
Thanks to (4.20)–(4.24), we reach
for all \(t \in [0,T]\). The right-hand side can be estimated as in [21, Section 6] and in [22, Section 4]. As a result, we obtain
On the other hand, arguing as in [22, Section 4], we have the differential equation
We observe that
and
By exploiting (4.1) and (4.5), it follows that
As a consequence, integrating (4.27) on [0, t] and using the above estimates, we infer that
Then, summing (4.26) and (4.28), and exploiting the assumption on \(\nu \), we end up with
where
Finally, since \({\mathcal {F}}(t)\in L^1(0,T)\) in light of (4.1) and (4.5), the desired conclusion \({\varvec{{u}}}={\varvec{{U}}}\) and \(\phi =\Phi \) on [0, T] directly follows from the Gronwall lemma. \(\square \)
5 Large time regularity in three dimensions
This section is devoted to the third part of our main result, namely the large time regularity of the velocity \({\varvec{{u}}}\). The proof is based on a continuation argument for some Sobolev norms of \({\varvec{{u}}}\) when the initial data and forcing terms are small, for which we will make use of Corollary 2.6. Notice that, in the case of matched densities, a general result has been formulated in [1, Theorem 8] for the Navier–Stokes equation with non-constant viscosity. Instead, for unmatched densities, we are forced to consider the full system.
Proof of Theorem 1.3-(iii) Large time regularity of the velocity
Let \(\Omega \) be a bounded domain in \({\mathbb {R}}^d\), \(d=2,3\) with \(\partial \Omega \) of class \(C^4\). Consider a global weak solution \(({\varvec{{u}}},\phi )\) to (1.1) and (1.2) given by Theorem 1.1. We first summarize from the previous parts (i)–(ii) of Theorem 1.3 that, for any \(\tau >0\),
where \(2\le p <\infty \) if \(d=2\) and \(p=6\) if \(d=3\). Furthermore, the energy inequality (1.11) entails that
for some positive constant \(C_0\) depending only on \(E({\varvec{{u}}}_0,\phi _0)\) and \(\Omega \), and the strict separation property
holds for some \(\delta \in (0,1)\) and \(T_{SP}>0\). It immediately follows from (5.3) that \(F''(\phi )\) and \(F'''(\phi )\) are globally bounded in \([T_{SP}, \infty )\). As a consequence, it is easily seen from (1.1)\(_4\) and (5.1) that \(\partial _t \mu \in L^2(T_{SP},\infty ; H^1(\Omega )')\), which, in turn, gives \(\mu \in BC([T_{SP}, \infty ); H^1(\Omega ))\). Then, we infer from the regularity theory of the Laplace equation with homogeneous Neumann boundary condition that \(\phi \in L^\infty (T_{SP},\infty ; H^3(\Omega ))\cap L_{{\text {uloc}}}^2([T_{SP};\infty ); H^4(\Omega ))\). Finally, we observe from the achieved regularity that \(\partial _{\varvec{{n}}}\phi (t)=0\) on \(\partial \Omega \) for any \(t\ge T_{SP}\).
Let us now fix \(0<{\widetilde{\varepsilon }}<\varepsilon <1\), whose specific (small) values will be chosen later on. Thanks to (5.2), there exists a positive time \({\mathcal {T}}={\mathcal {T}}({\widetilde{\varepsilon }},\varepsilon )\) (which can be taken, without loss of generality, larger than \(T_{SP}\)) such that
Furthermore, we infer from the above regularity properties that \(\phi ({\mathcal {T}}) \in H^2(\Omega )\), \(\Vert \phi ({\mathcal {T}})\Vert _{L^\infty (\Omega )}< 1\), \(|\overline{\phi ({\mathcal {T}})}|<1\), \(\mu ({\mathcal {T}})=-\Delta \phi ({\mathcal {T}})+\Psi '(\phi ({\mathcal {T}})) \in H^1(\Omega )\) and \(\partial _{\varvec{{n}}}\phi ({\mathcal {T}})=0\) on \(\partial \Omega \). Exploiting once again (5.2), up to a possible redefinition of \({\mathcal {T}}\), we also have
Thanks to (2.10) of Theorem 2.4 and Corollary 2.6, we obtain the following estimates
and
where \(\widetilde{K_1}\) and \(\widetilde{K_2}\) are two positive constants depending only on \(\Omega \), \(\theta \), \(\theta _0\), F, and \(\overline{\phi _0}\).
Next, by [22, Theorem 1.1] there exists a local strong solution \(({\varvec{{U}}}, \Pi , \Phi )\) defined on the maximal interval \([{\mathcal {T}}, {\mathcal {T}}_{\mathrm{{max}}})\) such that
for any \({\mathcal {T}}\le \widetilde{{\mathcal {T}}}<{\mathcal {T}}_{\mathrm{{max}}}\), which satisfies (1.1) almost everywhere in \(({\mathcal {T}},{\mathcal {T}}_{\mathrm{{max}}})\) and the energy identity in \([{\mathcal {T}},\widetilde{{\mathcal {T}}}]\) for any \({\mathcal {T}}\le \widetilde{{\mathcal {T}}}<{\mathcal {T}}_{\mathrm{{max}}}\) (cf. Remark 4.2). On the other hand, in light of Theorem 4.1, we deduce that \(({\varvec{{u}}},\phi )=({\varvec{{U}}},\Phi )\) (and so \(P=\Phi \)) in the interval \([{\mathcal {T}},{\mathcal {T}}_{\mathrm{{max}}})\). This implies that
and (1.1)\(_1\) is solved almost everywhere in \(({\mathcal {T}},{\mathcal {T}}_{\mathrm{{max}}})\).
We are left to show that \({\mathcal {T}}_{\mathrm{{max}}}= \infty \) provided that \(\varepsilon \), \({\widetilde{\varepsilon }}\) and \({\mathcal {T}}\) are suitably chosen. To this end, we notice that the regularity properties in (5.1) and the separation property in (5.3) hold globally in time (from \(T_{SP}\) on), and so they are independent of \({\mathcal {T}}_{\mathrm{{max}}}\). Thus, we only need to prove that the norms of the spaces in (5.9) do not blow up as \(\widetilde{{\mathcal {T}}}\) is replaced by \({\mathcal {T}}_{\mathrm{{max}}}\). We now write the momentum equation (1.1)\(_1\) as follows
where
Multiplying (5.10) by \(\partial _t {\varvec{{u}}}\) and integrating over \(\Omega \), we find
for some positive constant C depending only on \(\rho _1\), \(\rho _2\), \(\nu _1\) and \(\nu _2\). Besides, by the regularity theory of the Stokes operator with variable viscosity (see [1, Lemma 4]), we have
for some positive constant C depending on \(\rho _1\), \(\rho _2\), \(\nu _1\), \(\nu _2\), \(\Omega \) and \(\Vert \phi \Vert _{BC([{\mathcal {T}},\infty ); W^{1,4}(\Omega ))}\). For \(\vartheta >0\), we observe that
and
as well as
for some positive constant \(C_\vartheta \) depending only on \(\vartheta \) and \(\Omega \). Combining the above estimates and choosing \(\vartheta \) sufficiently small, we arrive at the differential inequality
where \(\varpi \) is a (possibly small) positive constant and \(C_R\) is a (possibly large) positive constant. They depends on \(\Vert \phi \Vert _{BC( [{\mathcal {T}},\infty ); W^{1,4}(\Omega ))}\), \(\Vert \nabla \mu \Vert _{L^\infty ({\mathcal {T}}, \infty ; L^2(\Omega ))}\), \(\rho _1\), \(\rho _2\), \(\nu _1\), \(\nu _2\) and \(\Omega \), but are independent of \(\varepsilon \), \({\widetilde{\varepsilon }}\) and \({\mathcal {T}}\). Let us now define
Since \(\sqrt{\frac{4 \nu ^*}{\nu _*}}>1\), it is easily seen that \({\mathcal {T}}_*> {\mathcal {T}}\). We choose \(\varepsilon \in (0,1) \) such that
Recalling that
we infer from (5.16) and (5.18) that, for almost all \(t\in [{\mathcal {T}},{\mathcal {T}}_*]\),
Then, integrating in time (5.20), multiplying by \(\frac{4}{\nu _*}\) and using (5.5), (5.6), (5.7) and (5.19), we obtain that
for any \(t \in [{\mathcal {T}}, {\mathcal {T}}_*]\). Setting now \({\widetilde{\varepsilon }}\in (0,\varepsilon )\) such that
we eventually infer that
In particular, \(\Vert {\varvec{{u}}}({\mathcal {T}}_*)\Vert _{{\textbf{H}}^1_{0,\sigma }(\Omega )} < \sqrt{\frac{ 4 \nu ^*}{\nu _*}} \varepsilon \). However, by continuity, there exists \({\widetilde{\tau }}>0\) such that \(\Vert {\varvec{{u}}}(t)\Vert _{{\textbf{H}}^1_{0,\sigma }(\Omega )} \le \sqrt{\frac{ 4 \nu ^*}{\nu _*}} \varepsilon \) in \([{\mathcal {T}}_*, {\mathcal {T}}_*+{\widetilde{\tau }}]\). Thus, we found a contradiction with the definition of \({\mathcal {T}}_*\) in (5.17). We conclude that
In addition, exploiting (5.21) and (5.22), we are led to
In light of (5.24) and (5.25), we have that
As a result, by a classical argument, it is possible to define \({\varvec{{u}}}({\mathcal {T}}_{\mathrm{{max}}}) \in {\textbf{H}}^1_{0,\sigma }\). Since \(\phi ({\mathcal {T}}_{\mathrm{{max}}}) \in H^2(\Omega )\), \(\Vert \phi ({\mathcal {T}}_{\mathrm{{max}}})\Vert _{L^\infty (\Omega )}< 1\), \(|\overline{\phi ({\mathcal {T}}_{\mathrm{{max}}})}|<1\), \(\mu ({\mathcal {T}}_{\mathrm{{max}}}) \in H^1(\Omega )\) and \(\partial _{\varvec{{n}}}\phi ({\mathcal {T}}_{\mathrm{{max}}})=0\) on \(\partial \Omega \), a further application of [22, Theorem 1.1] and Theorem 4.1 ensure the existence of a strong solution beyond the maximal time \({\mathcal {T}}_{\mathrm{{max}}}\), which contradicts with the definition of \({\mathcal {T}}_{\mathrm{{max}}}\). Hence, \(({\varvec{{u}}}, P, \phi )\) is a strong solution defined on the whole interval \([{\mathcal {T}},\infty )\) such that
Therefore, the desired claim in Theorem 1.3-(iii) follows with \(T_R= {\mathcal {T}}(\varepsilon )\) where \(\varepsilon \in (0,1)\) is given by (5.18). \(\square \)
6 Convergence to equilibrium
In this section, we complete the proof of Theorem 1.3 by showing that each weak solution converges to an equilibrium (stationary state) of the system (1.1) and (1.2).
Proof of Theorem 1.3-(iv) Convergence to a stationary solution
Let us first recall (cf. Theorem 1.3-(ii)) that
Now let \({\widetilde{\Psi }}\) be the smooth and bounded function such that \({\widetilde{\Psi }}|_{[1+\delta ,1-\delta ]}=\Psi |_{[-1+\delta ,1-\delta ]}\), and
for all \(\varphi \in H^1_{(m)}(\Omega )\) with \(|\varphi (x)|\le 1\) for almost every \(x\in \Omega \). Then, we report the following result whose proof can be found in [11, Proposition 6.1].
Proposition 6.1
(Lojasiewicz–Simon inequality) Let \(\phi '\in {\mathcal {D}}(\partial E_0)\) be a solution to (3.1)–(3.3). Then, there exist three constants \(\theta \in \left( 0,\frac{1}{2}\right] , C, \kappa >0\) such that
where \(D{\widetilde{E}}_{\textrm{free}}:H^1_{(m)}(\Omega )\rightarrow {H^1_{(0)}(\Omega )'}\) denotes the Frechét derivative of \({\widetilde{E}}_{\textrm{free}}:H^1_{(m)}(\Omega )\rightarrow {\mathbb {R}}\).
Next, we note that the only critical points of the energy
are \(({\varvec{{v}}},\varphi )\) with \({\varvec{{v}}}={\textbf{0}}\) and \(\varphi \) arbitrary. Moreover, obviously
Hence, by Proposition 6.1, for any \(\phi '\in {\mathcal {D}}(\partial E_0)\) solution to (3.1)–(3.3) and any \(R>0\), there exist three constants \(\theta \in \left( 0,\frac{1}{2}\right] , C, \kappa >0\) such that \({\widetilde{E}}({\varvec{{v}}},\varphi )=E_{\text {kin}}({\varvec{{v}}},\varphi )+ {\widetilde{E}}_{\text {free}}(\varphi )\) satisfies
for all \(\Vert \varphi -\phi '\Vert _{H^1_{(0)}}\le \kappa \), and \({\varvec{{v}}}\in {\textbf{L}}^2_\sigma (\Omega )\) with \(\Vert {\varvec{{v}}}\Vert _{L^2(\Omega )}\le R\).
Since \(\omega ({\varvec{{u}}},\phi )\) is a compact subset of \({\textbf{L}}^2_\sigma (\Omega )\times W^{2-\varepsilon ,p}(\Omega )\) for every \(\varepsilon >0\), and because of Lemma 3.3, for every \(R>0\) there exist three universal constants \(\theta \in \left( 0,\frac{1}{2}\right] , C, \kappa >0\) such that (6.2) holds for all \(({\textbf{0}},\phi ') \in \omega ({\varvec{{u}}},\phi )\), \(\varphi \in H_{(m)}^1 (\Omega )\) such that \(\Vert \varphi -\phi '\Vert _{H^1_{(0)}}\le \kappa \), and \({\varvec{{v}}}\in {\textbf{L}}^2_\sigma (\Omega )\) with \(\Vert {\varvec{{v}}}\Vert _{L^2(\Omega )}\le R\). Furthermore, by choosing \(T_C\ge \max \lbrace T_{SP}, T_R \rbrace \) sufficiently large, we obtain \({\text {dist}}(({\varvec{{u}}}(t),\phi (t)), \omega ({\varvec{{u}}},\phi ))\le \kappa \) with respect to the norm of \({\textbf{L}}^2_\sigma (\Omega )\times H^1(\Omega )\) for all \(t\ge T_C\). Thus, we infer that
On the other hand, since \(\omega ({\varvec{{u}}},\phi )\) is connected, \({\widetilde{E}}({\textbf{0}},\phi ')\) is independent of \(\phi '\) with \(({\textbf{0}},\phi ')\in \omega ({\varvec{{u}}},\phi ))\). Thus, we set such a value as \(E_\infty \), and we conclude that
Lastly, in order to prove the convergence as \(t\rightarrow \infty \), we consider
where \(\theta \) is as in (6.3). By the energy identity (4.8), which holds due to Remark 4.2 and Theorem 1.3-(iii), H(t) is non-increasing and
for all \(t\ge T_C\). Now we use that
since \(|\phi (t)|\le 1-\delta \). In light of \(\Vert P_0\mu (t)\Vert _{H^1_{(0)}(\Omega )'}\le C\Vert \nabla \mu (t)\Vert _{L^2(\Omega )}\) and \(\Vert {\varvec{{u}}}(t)\Vert _{L^2(\Omega )}\le C \Vert D{\varvec{{u}}}(t)\Vert _{L^2(\Omega )}\) by Korn’s inequality, \( - \frac{\textrm{d}}{\textrm{d}t}H(t) \ge C \left( \Vert \nabla \mu (t)\Vert _{L^2(\Omega )}+\Vert D{\varvec{{u}}}(t)\Vert _{L^2(\Omega )}\right) , \) for some positive constant C. In turn, this implies that
Therefore, by (2.54), we deduce that
which entails that \(\partial _t \phi \in L^1(T_C,\infty ;H^1(\Omega )')\). Thus, we infer that
In particular, \(\omega ({\varvec{{u}}},\phi )= \{(0,\phi _\infty )\}\) and \(\phi _\infty \) solves the stationary Cahn–Hilliard equation (3.1)–(3.3) thanks to Lemma 3.3. Since \(({\varvec{{u}}},\phi )\in BC( [T_R, \infty ); {\textbf{H}}_{0,\sigma }^1(\Omega ) \times W^{2,p}(\Omega )\), for any \(2\le p<\infty \) if \(d=2\) and \(p=6\) if \(d=3\), we conclude that \(({\varvec{{u}}}(t),\phi (t))\) converges weakly to \((0,\phi _\infty )\) in \({\textbf{H}}_{0,\sigma }^{1}(\Omega ) \times W^{2,p}(\Omega )\). This finally proves Theorem 1.3-(iv). \(\square \)
7 Global regularity for the AGG model in two dimensions
In this section we prove the global well-posedness of the AGG model in any generic two dimensional bounded domain. We recall that the local well-posedness has been proven in [21, Theorem 3.1]. We show herein that the local strong solutions to system (1.1) and (1.2) are in fact globally defined.
Theorem 7.1
Let \(\Omega \) be a bounded domain of class \(C^3\) in \({\mathbb {R}}^2\). Assume that \({\varvec{{u}}}_0 \in {\textbf{H}}^1_{0,\sigma }(\Omega )\) and \(\phi _0 \in H^2(\Omega )\) such that \(\Vert \phi _0\Vert _{L^\infty (\Omega )}\le 1\), \(|\overline{\phi _0}|<1\), \(\mu _0= -\Delta \phi _0+ \Psi '(\phi _0) \in H^1(\Omega )\), and \(\partial _{\varvec{{n}}}\phi _0=0\) on \(\partial \Omega \). Then, there exists a unique global strong solution \(({\varvec{{u}}}, P, \phi )\) to system (1.1) and (1.2) defined on \(\Omega \times [0,\infty )\) in the following sense:
-
(i)
The solution \(({\varvec{{u}}}, P, \phi )\) satisfies
$$\begin{aligned}&{\varvec{{u}}}\in BC([0,\infty ); {\textbf{H}}^1_{0,\sigma }(\Omega )) \cap L^2(0,\infty ;{\textbf{H}}^2(\Omega )) \cap W^{1,2}(0,\infty ;{\textbf{L}}^2_\sigma (\Omega )), \nonumber \\&P \in L_{{\text {uloc}}}^2([0,\infty );H^1(\Omega )),\nonumber \\&\phi \in L^\infty (0,\infty ;H^3(\Omega )), \ \partial _t \phi \in L^\infty (0,\infty ; H^1(\Omega )') \cap L^2(0,\infty ;H^1(\Omega )),\nonumber \\&\phi \in L^\infty (\Omega \times (0,\infty )) \text { with } |\phi (x,t)|<1 \ \text {a.e. in } \ \Omega \times (0,\infty ),\nonumber \\&\mu \in BC([0,\infty );H^1(\Omega ))\cap L_{{\text {uloc}}}^2([0,\infty );H^3(\Omega ))\cap W_{{\text {uloc}}}^{1,2}([0,\infty ); H^1(\Omega )'), \nonumber \\&F'(\phi ), F''(\phi ), F'''(\phi ) \in L^\infty (0,\infty ;L^p(\Omega )), \ \forall \, p \in [1,\infty ). \end{aligned}$$(7.1) -
(ii)
The solution \(({\varvec{{u}}}, P, \phi )\) fulfills the system (1.1) almost everywhere in \(\Omega \times (0,\infty )\) and the boundary conditions \(\partial _{\varvec{{n}}}\phi =\partial _{\varvec{{n}}}\mu =0\) almost everywhere in \(\partial \Omega \times (0,\infty )\).
-
(iii)
The solution \(({\varvec{{u}}}, P, \phi )\) is such that \({\varvec{{u}}}(\cdot , 0)={\varvec{{u}}}_0\) and \(\phi (\cdot , 0)=\phi _0\) in \(\Omega \).
Proof
Given an initial condition \(({\varvec{{u}}}_0, \phi _0)\) satisfying the required assumptions, the result in [21, Theorem 3.1] guarantees the existence and uniqueness of a local strong solution \(({\varvec{{u}}}, P, \phi )\) to system (1.1) and (1.2) originating from \(({\varvec{{u}}}_0, \phi _0)\). We consider the maximal interval of existence \([0,T_{\mathrm{{max}}})\) of such solution. That is, the solution \(({\varvec{{u}}}, P, \phi )\) satisfies (7.1) in the interval [0, T] for any \(T<T_{\mathrm{{max}}}\), the system (1.1) almost everywhere in \(\Omega \times (0,T_{\mathrm{{max}}})\) and the boundary conditions \(\partial _{\varvec{{n}}}\phi =\partial _{\varvec{{n}}}\mu =0\) almost everywhere in \(\partial \Omega \times (0,T_{\mathrm{{max}}})\). Furthermore, \({\varvec{{u}}}(0)={\varvec{{u}}}_0\) and \(\phi (0)=\phi _0\) in \(\Omega \). Our goal is to show that \(T_{\mathrm{{max}}}=\infty .\)
We assume by contradiction that \(T_{\mathrm{{max}}}<\infty \). First, integrating (1.1)\(_3\) over \(\Omega \times (0,t)\) with \(t<T_{\mathrm{{max}}}\), we obtain
Multiplying (1.1)\(_1\) and (1.1)\(_3\) by \({\varvec{{u}}}\) and \(\mu \), respectively, integrating over \(\Omega \) and summing the resulting equation, we find the energy identity
Since \(E({\varvec{{u}}}_0,\phi _0) <\infty \), we infer that, for all \(0<t<T_{\mathrm{{max}}}\),
Here the positive constant \(\widetilde{C_0}\) depends on \(E({\varvec{{u}}}_0,\phi _0)\), but it is independent of t and \(T_{\mathrm{{max}}}\). In light of (7.3), owing to Theorem (2.4) and Corollary 2.6, it follows that
for all \(0<t<T_{\mathrm{{max}}}\) and \(2\le p<\infty \), where the positive constants \(\widetilde{C_1}\), \(\widetilde{C_1}(p)\) are independent of \(T_{\mathrm{{\max }}}\). Furthermore, by [15, Lemma A.6] (see also [33]) and (7.5)–(7.7), we get
for p as above and some \(\widetilde{C_2}(p)\) independent of \(T_{\mathrm{{max}}}\).
Next, exploiting (1.1)\(_3\), we rewrite (1.1)\(_1\) in non-conservative form (cf. (5.10)) as follows
Multiplying (7.9) by \(\partial _t {\varvec{{u}}}\) and integrating over \(\Omega \), we obtain
Using the Ladyzhenskaya inequality, the Korn inequality and (7.3), we find
where \({\widetilde{C}}\) stands for a generic positive constant, whose value may change from line to line, which depends on the norms of the initial data, the parameters of the system and \(\varpi >0\). The (small) value of \(\varpi \) will be chosen subsequently. Similarly, we have
Besides, by the Sobolev embedding and (7.7)
and
On the other hand, exploiting the regularity theory of the Stokes equation with concentration-depending viscosity in [1, Lemma 4] and owing to (7.7), we infer that
Arguing in a similar way as above, we deduce that
Thus, using the Young inequality, we arrive at
Now, combining (7.10) with (7.11)–(7.14), summing the resulting equation by \(3 \varpi \times \)(7.17), and setting \(\varpi =\frac{\rho _*}{12 {\widetilde{C}}}\), we eventually reach the differential inequality
where
The Gronwall lemma yields
In light of (7.3), (7.5) and (7.6), \({\widetilde{G}} \in L^\infty (0,T_{\mathrm{{max}}})\). On the other hand, by the assumption \(T_{\mathrm{{max}}}< \infty \), it is easily seen that
Otherwise, the solution could be extended beyond the time \(T_{\mathrm{{max}}}\) thanks to [21, Theorem 3.1]. This contradicts (7.5) and (7.18). Thus, we conclude that \(T_{\mathrm{{max}}}=\infty \), and the solution \(({\varvec{{u}}}, P, \phi )\) exists on \([0,\infty )\). In particular, since the estimates (7.3)–(7.8) holds on \([0,\infty )\), it follows that \(\Vert {\widetilde{G}}\Vert _{L^\infty (0,\infty )}< \infty \) and thereby
In addition, since \(F''(\phi ) \in L^\infty (0,\infty ; L^p(\Omega ))\) for any \(2\le p < \infty \), by comparison with terms in (1.1)\(_4\), we also deduce that \(\partial \mu \in L_{{\text {uloc}}}^2([0,\infty ); H^1(\Omega )')\), which, in turn, implies that \(\mu \in BC([0,\infty ); H^1(\Omega ))\). \(\square \)
8 Double obstacle potential: the limit \(\theta \searrow 0\)
In this final section we study the double obstacle version of the system (1.1) and (1.2) which is obtained by passing to the limit \(\theta \searrow 0\) in the Flory–Huggins potential \(\Psi \), cf. (1.4). The limiting free energy is then given by
Here \(I_{[-1,1]}\) is the indicator function of the interval \([-1,1]\) given by
In this case equation (1.1)\(_4\) has to be replaced by
almost everywhere in \(\Omega \times (0,\infty )\) where \(\partial I_{[-1,1]}\) is the subdifferential of \(I_{[-1,1]}\). The inclusion (8.1) is equivalent to the variational inequality
which has to hold for almost all t and all \(\zeta \in H^1(\Omega )\) which fulfill \(|\zeta (x)|\le 1\) almost everywhere in \(\Omega \), see [12].
We first state a result on the double obstacle limit for the convective Cahn–Hilliard equation which has been shown in [3, Theorem 3.10].
Theorem 8.1
(Double obstacle limit of the convective Cahn–Hilliard equation) Let \(\Omega \) be a bounded domain in \({\mathbb {R}}^d\), \(d=2,3\), with \(C^2\) boundary and \(0<\theta _k\le 1\), \( k\in {\mathbb {N}}\), be such that \(\lim _{k\rightarrow \infty }\theta _k = 0\). Moreover, assume \( \phi _0, \phi _{0,k} \in H^1(\Omega )\) with \(\Vert \phi _{0,k}\Vert _{L^\infty (\Omega )}\le 1\) and \( \sup _{k\in {\mathbb {N}}} \left| \overline{\phi _{0,k}} \right| <1\), and \({\varvec{{u}}}, {\varvec{{u}}}_k \in L^2(0,\infty ;{\textbf{H}}^1_{0,\sigma }(\Omega ))\) be such that as k tends to infinity
for all \(T\in (0,\infty )\). Furthermore, let \((\phi _k, \mu _k)\) be the sequence of weak solutions to (2.1) and (2.2) with \(({\varvec{{u}}},\phi _0, F)\) replaced by \(({\varvec{{u}}}_k,\phi _{0,k}, F_k)\) where F is defined with \(\theta _k\) instead of \(\theta \). Then, it holds in the limit k tending to \(\infty \)
for all \(T\in (0,\infty )\), where \(F^* = \mu +\Delta \phi +\theta _c \phi \in \partial I_{[-1,1]} (\phi )\) almost everywhere in \(\Omega \times (0,\infty )\), \(p=6\) if \(d=3\), \(p\in [2,\infty )\) is arbitrary if \(d=2\) and \((\phi ,\mu )\in C( [0,\infty ); H^1(\Omega ))\cap L_{{\text {uloc}}}^4([0,\infty );H^2(\Omega ))\cap L_{{\text {uloc}}}^2([0,\infty );W^{2,p}(\Omega ))\times L_{{\text {uloc}}}^2([0,\infty );H^1(\Omega ))\) is the unique weak solution of
completed with the following boundary and initial conditions
The weak solution satisfies the free energy equality
for every \(0\le \tau < t \le \infty \).
The formulation in [3, Theorem 3.10] is slightly different. However, e.g., the variable mean values \(\overline{\phi _{0,k}}\) do not effect the arguments substantially.
We now formulate our result stating the higher regularity for the Cahn–Hilliard equation with divergence-free drift in the double obstacle case.
Theorem 8.2
Let \(\Omega \) be a bounded domain in \({\mathbb {R}}^d\), \(d=2,3\), with \(C^3\) boundary and the initial condition \(\phi _0 \in H^2(\Omega )\) be such that \(\Vert \phi _0\Vert _{L^\infty (\Omega )}\le 1\), \(\left| \overline{\phi _0} \right| <1\). Furthermore, we assume that a function \(\mu _0\in H^1(\Omega )\) exists such that
and \(\partial _{\varvec{{n}}}\phi _0=0\) on \(\partial \Omega \). Assume that \({\varvec{{u}}}\in L^2(0,\infty ;{\textbf{H}}^1_{0,\sigma }(\Omega ))\). Then, there exists a unique global solution to
such that
for any \(2\le p <\infty \) if \(d=2\) and \(p=6\) if \(d=3\). The solution satisfies (8.7) almost everywhere in \(\Omega \times (0,\infty )\) and \(\partial _{\varvec{{n}}}\mu =0\) almost everywhere on \(\partial \Omega \times (0,\infty )\). Moreover, there exists a positive constant C depending only on \(\Omega \), \(\theta _0\),
and \(\overline{\phi _0}\) such that
and
In addition, if \({\varvec{{u}}}\in L^\infty (0, \infty ;{\textbf{L}}^2_\sigma (\Omega ))\cap L^2(0,\infty ;{\textbf{H}}^1_{0,\sigma }(\Omega ))\), then \(\partial _t \phi \in L^\infty (0,\infty ; H^1(\Omega )')\).
Proof
The argument relies on the limit \(\theta =1/k \searrow 0\), with \(k\in {\mathbb {N}}\), in the problem (2.1)-(2.2). To this end, we will use Theorem 2.4 with appropriate initial data \(\phi _0^k\). To fulfill the assumptions on the initial data in Theorem 2.4, we solve the elliptic boundary value problem
where we choose \(F_0(s)= \frac{1}{2} [ (1+s)\log (1+s)+(1-s)\log (1-s)]\). Some of the following estimates are formal but can be justified by approximating \(F_0\) by smooth functions with quadratic growth. Testing (8.11) with \(\frac{1}{k} F_0^\prime (\phi _0^k)\) and using the fact that \(F_0\) is monotone gives that \(\frac{1}{k} F_0^\prime (\phi _0^k)\) is uniformly in k bounded in \(L^2(\Omega )\). Now elliptic regularity theory gives that \(\phi _0^k\) is uniformly in k bounded in \(H^2(\Omega )\). As \(F_0^\prime (\phi _0^k) \in L^2(\Omega )\), we obtain that \(|\phi _0^k(x)|<1\) almost everywhere in \(\Omega \). For a subsequence, we obtain that \(\phi _{0}^k\) converges to \(\phi ^*\) weakly in \(H^2(\Omega )\) and strongly in \(H^1(\Omega )\). We now choose \({\widetilde{\eta }}\in H^1(\Omega )\) with \(|{\widetilde{\eta }}(x)|\le 1\) almost everywhere in \(\Omega \) and use \(\eta ={\widetilde{\eta }}-\phi _0^k\) as test function in the weak formulation of (8.11) obtaining
As \(F_0\) is convex, we have
Then, the right hand side in (8.12) is greater or equal than
which converges to zero for k tending to infinity. The above convergence properties now give the inequality
which gives that
together with zero Neumann boundary conditions. As the operator \(\phi \mapsto -\Delta \phi +\phi + \partial I_{[-1,1]}(\phi ) \) is strictly monotone we obtain from (8.6) that \(\phi _0=\phi ^*\).
We now consider the sequence of strong solutions \(\lbrace (\phi ^k,\mu ^k) \rbrace \) given by Theorem 2.4 with \(\theta =\frac{1}{k}\) and \(\phi _0^k\) instead of \(\phi _0\). In order to derive estimates, which are uniform in k, from (2.9) and(2.10), we need to control \( -\Delta \phi _0^k+\frac{1}{k} F_0^\prime (\phi _0^k)-\theta _0 \phi _0^k \) in \(H^1(\Omega )\). In fact, by construction in (8.11), we notice that
and we observe that \(\mu _0^k\) is uniformly bounded in \(H^1(\Omega )\) in k. Setting \(\Psi _{\frac{1}{k}} (s) =\frac{1}{k} \frac{1}{2}\big [ (1+s)\log (1+s)+(1-s)\log (1-s)\big ] -\frac{\theta _0}{2}s^2 =\frac{1}{k} F_0(s) - \frac{\theta _0}{2}s^2 \), we infer that
and
Here, the convergence of \(\Vert \nabla \mu ^k \Vert _{L^2(0,\infty ;L^2(\Omega ))}\) to \(\Vert \nabla \mu \Vert _{L^2(0,\infty ;L^2(\Omega ))}\) as \(k\rightarrow \infty \) can be deduced as in the proof of Theorem 2.4. In particular, the constant C depends on \(\Omega \), \(\theta _0\), and \(\overline{\phi _0} \), but is independent of k. In the limit as k tends to \(\infty \), we obtain that \(\nabla \mu \in L^\infty (0,\infty ;L^2(\Omega ))\) and \(\nabla \partial _t \phi \in L^2(0,\infty ;L^2(\Omega ))\). In addition, thanks to [3, Proposition 3.3], we deduce that \(\phi \in L^\infty (0, \infty ; W^{2,p}(\Omega ))\) for \(2\le p < \infty \) if \(d=2\) and \(p=6\) if \(d=3\). Also, the estimates (8.9) and (8.10) hold due to the fact that \(\phi _0^k \rightarrow \phi _0\) in \(H^1(\Omega )\) and the lower semicontinuity of the norm. The rest of the proof follows by arguing similarly as in Theorem 2.4 (see also [3]). \(\square \)
We now study the double obstacle limit of the full Navier–Stokes/Cahn–Hilliard system (1.1) with the boundary conditions (1.2). The limiting system is given as
in \(\Omega \times (0,\infty )\), subject to the initial and boundary conditions
Combining the arguments of [3, 6] and the methods of this paper we obtain the following result.
Theorem 8.3
(Weak solutions in the double obstacle case) Let \(\Omega \) be a bounded domain in \({\mathbb {R}}^d\), \(d=2,3\), with boundary \(\partial \Omega \) of class \(C^2\). Moreover, let \(0<\theta _k\le 1\), \( k\in {\mathbb {N}}\), be such that \(\lim _{k\rightarrow \infty }\theta _k = 0\). Assume that \(\phi _0, \phi _{0,k} \in H^1(\Omega )\) with \(\Vert \phi _{0,k}\Vert _{L^\infty (\Omega )}\le 1\) and \( \sup _{k\in {\mathbb {N}}}|\overline{\phi _{0,k}}|<1\), and \({\varvec{{u}}}_0, {\varvec{{u}}}_{0,k} \in {\textbf{H}}^1_{0,\sigma }(\Omega )\) be such that
In addition, let \(({\varvec{{u}}}_k, \phi _k,\mu _k)\) be weak solutions to (1.1) and (1.2) with initial values \(({\varvec{{u}}}_{0,k}, \phi _{0,k})\). Then, there exists a subsequence \(k_j\), \(j\in {\mathbb {N}}\), with \(k_j\rightarrow \infty \) as \(j\rightarrow \infty \), such that as j tends to infinity
for all \(T\in (0,\infty )\), with \(p\in [2,\infty )\) arbitrary if \(d=2\) and \(p=6\) if \(d=3\), and \(({\varvec{{u}}}, \phi ,\mu )\) is a weak solution to (8.15)–(8.16). Furthermore, the energy inequality
holds for all \(t \in [s, \infty )\) and almost all \(s \in [0,\infty )\) (including \(s=0\)). Here, the total energy \(E^{\textrm{do}}({\varvec{{u}}},\phi )\) is given by
The following result now states additional regularity for the concentration of any weak solution obtained in the previous Theorem 8.3. The proof of which can be performed exactly in the same way as in the proof of Theorem 1.3 (i), see the end of Sect. 2.
Theorem 8.4
(Regularity of weak solutions in the double obstacle case) Let \(\Omega \) be a bounded domain in \({\mathbb {R}}^d\), \(d=2,3\), with boundary \(\partial \Omega \) of class \(C^3\). Consider a global weak solution \(({\varvec{{u}}},\phi )\) given by Theorem 8.3. Then, for any \(\tau >0\), we have
where \(2\le p <\infty \) if \(d=2\) and \(p=6\) if \(d=3\). Moreover, the equations (8.15)\(_{3-4}\) are satisfied almost everywhere in \(\Omega \times (0,\infty )\) and the boundary conditions \(\partial _{\varvec{{n}}}\phi =\partial _{\varvec{{n}}}\mu =0\) holds almost everywhere on \(\partial \Omega \times (0,\infty )\).
Let us finally comment about the longtime behavior. In the double obstacle case, we cannot obtain enough large time regularity to show the same results as in the case of a logarithmic potential. This is mainly due to the fact that a separation property does not hold in this case. In addition, as typical for obstacle problems, the variable \(\phi \) in space will not lie in \(H^3(\Omega )\). This is already true for simple stationary solutions, see, e.g., [12]. In addition, no Lojasiewicz–Simon inequality is known so far in the double obstacle case and hence it is not possible to show that the solution converges for large time to a single stationary solution. However, we can still characterize the \(\omega \)-limit set of a weak solution. As in Lemma 3.2 one can show that the velocity for large times converges to 0. Concerning the concentration \(\phi \), its long time limits consist of critical points of the energy \(E^{\text {do}}_{\text {free}}\). Defining the convex part of \(E^{\text {do}}_{\text {free}}\) as
we now recall the characterization of the subgradient \(\partial E_0^{\text {do}}\) given in [3, 30].
Theorem 8.5
The domain of definition of the subgradient \(\partial E_0^{{\text {do}}}\) with respect to \(L^2_{(0)}(\Omega )\) of \( E_0^{{\text {do}}}\) is given as
For any \(\phi _0\in {\mathcal {D}}(\partial E_{0}^{{\text {do}}})\), we have \(\mu _0\in \partial E_{0}^{{\text {do}}}(\phi )\) if and only if
for some constant \({{\overline{\mu }}}\).
The stationary solutions \(\phi _\infty \in {\mathcal {D}}(\partial E_{0}^{{\text {do}}})\) of the double obstacle version of the Cahn–Hilliard equation fulfill (with a suitable constant \(\overline{\mu }\))
We notice that the solutions to (8.20)–(8.22) are critical points of the functional \(E^{{\text {do}}}_{{\text {free}}}\) on \(H^1_{(m)}(\Omega )\). For the long time behavior, we find with the help of the energy inequality (8.18) and the free energy identity (8.5) similar as in Lemmas 3.2 and 3.3 the following result
Theorem 8.6
Let \(({\varvec{{u}}},\phi )\) be a weak solution to (8.15) and (8.16) in the sense of Theorem 8.3. Then, we have
and
where \(m = \overline{\phi _0}\) and \(\omega ({\varvec{{u}}},\phi )\) is the \(\omega \)-limit set with respect to the norm of \({\textbf{L}}^2_\sigma \times W^{2-\varepsilon ,p}(\Omega )\) for an arbitrary \(\varepsilon >0\) and p as before.
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Acknowledgements
Part of this work was done while AG was visiting HA and HG at the Department of Mathematics of the University of Regensburg whose hospitality is gratefully acknowledged. AG is a member of Gruppo Nazionale per l’Analisi Matematica, la Probabilitá e le loro Applicazioni (GNAMPA) of Istituto Nazionale per l’Alta Matematica (INdAM). AG is supported by the MUR grant Dipartimento di Eccellenza 2023-2027. Finally, we are grateful to the anonymous referees for their helpful comments, which improved the manuscript, and to A. Poiatti for noticing a missing norm in the proof of Theorem 2.4.
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Abels, H., Garcke, H. & Giorgini, A. Global regularity and asymptotic stabilization for the incompressible Navier–Stokes-Cahn–Hilliard model with unmatched densities. Math. Ann. (2023). https://doi.org/10.1007/s00208-023-02670-2
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DOI: https://doi.org/10.1007/s00208-023-02670-2