Abstract
We study the Abels–Garcke–Grün (AGG) model for a mixture of two viscous incompressible fluids with different densities. The AGG model consists of a Navier–Stokes–Cahn–Hilliard system characterized by a (non-constant) concentration-dependent density and an additional flux term due to interface diffusion. In this paper we address the well-posedness problem in the two-dimensional case. We first prove the existence of local strong solutions in general bounded domains. In the space periodic setting we show that the strong solutions exist globally in time. In both cases we prove the uniqueness and the continuous dependence on the initial data of the strong solutions. Lastly, we show a stability result for the strong solutions to the AGG model and the model H in terms of the density values.
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1 Introduction
In this article we consider the Abels–Garcke–Grün (AGG) model
The AGG system is studied in \(\Omega \times (0,T)\), where \(\Omega \) is either a bounded domain in \({\mathbb {R}}^2\) or the two-dimensional torus \({\mathbb {T}}^2\). The state variables are the volume averaged velocity \({{\varvec{u}}}={{\varvec{u}}}(x,t)\), the pressure of the mixture \(P=P(x,t)\), and the difference of the fluids concentrations \(\phi =\phi (x,t)\). The symmetric gradient is \(D=\frac{1}{2} (\nabla +\nabla ^T)\). The flux term \({\widetilde{{\mathbf {J}}}}\), the mean density \(\rho \) and the mean viscosity \(\nu \) of the mixture are given by
where \(\rho _1\), \(\rho _2\) and \(\nu _1\), \(\nu _2\) are the homogeneous densities and viscosities of the two fluids. The nonlinear function \(\Psi \) is the Flory-Huggins potential
where the constant parameters \(\theta \) and \(\theta _0\) fulfill the conditions \(0<\theta <\theta _0\). Notice that (1.1)\(_1\) can be rewritten in the non-conservative form as
In a bounded domain \(\Omega \), the system is subject to the classical boundary conditions
where \({{\varvec{n}}}\) is the unit outward normal vector on \(\partial \Omega \), and \(\partial _{{\varvec{n}}}\) denotes the outer normal derivative on \(\partial \Omega \). In the case \(\Omega ={\mathbb {T}}^2\), the state variables satisfy periodic boundary conditions. In both cases, the system (1.1) is supplemented with the initial conditions
The total energy associated to system (1.1) is defined as
and the corresponding energy equation reads as
The Abels–Garcke–Grün system is a fundamental diffuse interface model which describes the motion of two viscous incompressible Newtonian fluids with unmatched densities (i.e. \(\rho _1\ne \rho _2\)). The model was derived in the seminal paper [7]. The AGG model is a thermodynamically consistent generalization of the well-known model H (see [25] for the derivation and [1, 23] for the mathematical analysis). In fact, the classical Navier–Stokes–Cahn–Hilliard system is recovered in the matched density case (i.e. \(\rho _1=\rho _2\)) since the flux \({\widetilde{{\mathbf {J}}}}={\mathbf {0}}\) and the density \(\rho (\phi )\) is constant. As for the model H, the fluid mixture in the AGG system is driven by the capillarity forces \(-\mathrm {div}\,(\nabla \phi \otimes \nabla \phi )\) due to the surface tension effect. In addition, a partial diffusive mixing is assumed in the interfacial region, which is modeled by \(\Delta \mu \), being the chemical potential \(\mu = \frac{\delta E_{\text {free}}(\phi )}{\delta \phi }\). The specificity of the AGG model lies in the presence of the flux term \({\widetilde{{\mathbf {J}}}}\). In contrast to the one-phase flow, the (average) density \(\rho (\phi )\) in (1.1) does not satisfy the continuity equation with respect to the flux associated with the velocity \({{\varvec{u}}}\). Instead, the density \(\rho (\phi )\) satisfies the continuity equation with a flux given by the sum of the transport term \(\rho (\phi ){{\varvec{u}}}\) and the term \({\widetilde{{\mathbf {J}}}}\). The latter is due to the diffusion of the concentration in the unmatched densities case. For the connection of the AGG model with the classical sharp interface two-phase problem and the so-called sharp interface limit, we refer the reader to the review in [8]. It is important to mention that the theory of diffuse interface models for mixtures of fluids has been widely developed in the past decades. Several systems have been proposed to model binary mixtures with non-constant density in view of their applications in engineering and physics. We mention the models derived in [12, 15, 20, 26, 29, 31] and the theoretical analysis achieved in [2, 3, 11, 24, 27].
The mathematical analysis of the AGG system has been focused so far on the existence of weak solutions in two and three dimensional bounded domains. More precisely, global solutions with finite energy for the system (1.1) with boundary and initial conditions (1.4)–(1.5) were proven in [5] and [6]. In the former the mobility coefficient \(m(\phi )\) is non-constant and strictly positive, whereas in the latter \(m(\phi )\) is degenerate.Footnote 1 Later on, the existence of global weak solutions have been generalized in [4] for viscous non-Newtonian binary fluids and in [19] for the case of dynamic boundary conditions describing moving contact lines. Furthermore, non-local variants of the AGG system have been investigated in [9] and in [16, 17], where the gradient term \(\frac{1}{2} | \nabla \phi |^2\) in the local free energy \( E_{\text {free}}(\phi )\) has been replaced by different non-local operators. Lately, in the recent work [10] (see also [34]) the local well-posedness of strong solutions is proven in three dimensions for regular potentials \(\Psi \) provided that \(\phi _0 \in (L^p(\Omega ),W^{4}_{p,N}(\Omega ))_{1-\frac{1}{p},p}\) for \(4<p<6\) such that \(\Vert \phi _0\Vert _{L^\infty (\Omega )}\le 1\). Notice that, in this range of p, \(\phi _0 \in W_p^{4(1-\frac{1}{p})}(\Omega ) \subset H^3(\Omega )\) (cf. Remark 3.4). In addition, the solution in [10] may not satisfy \(| \phi (x,t)|\le 1\) for positive times, thereby the system may degenerate.Footnote 2
The aim of this contribution is to present the first well-posedness result for the AGG model with logarithmic Flory-Huggins potential. In our analysis we show existence, uniqueness and continuous dependence on the initial data of the strong solutions in the two-dimensional case. In comparison with the notion of weak solutions studied in the previous works on the AGG model, such strong solutions are more regular and solve the system (1.1) pointwise almost everywhere. These solutions depart from initial data \({{\varvec{u}}}_0 \in {\mathbf {H}}^1(\Omega )\) with \(\mathrm {div}\,{{\varvec{u}}}_0=0\), and \(\phi _0 \in H^2(\Omega )\) such that \(-1\le \phi _0(x)\le 1\) in \(\Omega \) and \(-\Delta \phi _0+\Psi '(\phi _0)\in H^1(\Omega )\), which satisfy suitable boundary or periodic conditions. We first prove the existence of local-in-time strong solutions in a general bounded domain (see Theorem 3.1). Our proof relies on the existence of suitable (global) approximate solutions to system (1.1) constructed through a semi-Galerkin formulation. In this framework the modified Navier–Stokes equations (1.1)\(_1\)-(1.1)\(_2\) are solved in finite-dimensional (spacial) spaces, whereas the convective Cahn–Hilliard system (1.1)\(_3\)-(1.1)\(_4\) is fully solved (i.e. not approximated). The advantage of this approach is that the approximate velocity fields \({{\varvec{u}}}_m\) is regular in the space variable, and the approximate concentrations \(\phi _m\) take values in the physical interval \([-1,1]\) which, in turn, ensures that \(\rho '(\phi )=\frac{\rho _1-\rho _2}{2}\).Footnote 3 It is worth pointing out that our strategy entirely exploit the regularity properties of the Cahn–Hilliard equation with logarithmic potential in two dimensions. More precisely, the control of \(\Psi ''(\phi )\) in \(L^p\) spaces (available in the two dimensional setting) allows us to recover the time continuity of the chemical potential \(\mu \), which is needed to solve the approximated problem. Once the existence of the approximate solutions is shown, we employ the energy method to deduce uniform estimates and the necessary compactness to obtain the existence of a local solution to (1.1). Next, in the periodic boundary setting we demonstrate that the strong solutions exist globally in time (see Theorem 3.3). The key observation to obtain the propagation of regularity for all times is that global-in-time higher-order estimates for the full system as in [23, 24] are out of reach due to the presence of the nonlinear term \((\nabla \mu \cdot \nabla ) {{\varvec{u}}}\) (cf. the term \(I_3\) in (4.46)). Notice that, since \(\nabla \mu \) belongs to \(L^2(0,T;L^2({\mathbb {T}}^2))\) [cf. (1.6)], \(\nabla \mu \) has a lower regularity than \({{\varvec{u}}}\). Therefore, the idea is to split the argument by first improving the regularity of the concentration \(\phi \) relying on the energy estimates obtained from (1.6), and then showing more regularity properties for the velocity field. A similar idea was used in [1] for the model H. However, the argument in [1, Lemma 3] is based on the integrability properties of \(\partial _t {{\varvec{u}}}\) or the fractional in time regularity of \({{\varvec{u}}}\), which are not known for the weak solutions to (1.1). Nevertheless, it is possible to overcome this issue by exploiting the fine structure of the incompressible Navier–Stokes equations in the periodic setting. The crucial term involving the time derivative of the velocity is rewritten in (5.18) in such a way that the highest space derivative acting on the velocity is of order one, and boundary terms do not appear when integrating by parts. Such technique requires an estimate of the pressure P in \(L^2\), which is deduced from the incompressibility condition (1.1)\(_2\) and the crucial estimate (5.7) for the Cahn–Hilliard equation. In both cases (bounded domains and periodic setting) we show the uniqueness of the strong solutions and their continuous dependence on the initial data. Lastly, we rigorously justify the model H as the matched densities approximation of the AGG model through a stability result. Specifically, we study the difference in the energy norm between the strong solutions to the AGG model and the model H (departing from the same initial datum), and we prove that the error is proportional to the difference of the density values.
Plan of the paper. We report in Sect. 2 the function spaces and the notation used in this paper. In Sect. 3 we state the main results. Section 4 is devoted to the local existence of strong solutions in bounded domains. In Sect. 5 we prove the global existence of strong solutions in the space periodic setting. In Sect. 6 we address the uniqueness and the continuous dependence on the initial data of the strong solutions. The last Sect. 7 is devoted to a stability result of the solutions to the AGG model and the model H with respect to the density parameters.
2 Preliminaries
For a real Banach space X, its norm is denoted by \(\Vert \cdot \Vert _{X}\). The symbol \(\langle \cdot , \cdot \rangle _{X',X}\) stands for the duality pairing between X and its dual space \(X'\). The boldface letter \(\varvec{X}\) denotes the vectorial space endowed with the product structure. We assume that \(\Omega \) is a bounded domain in \({\mathbb {R}}^2\) with boundary \(\partial \Omega \) of class \(C^3\) or the flat torus \({\mathbb {T}}^2=({\mathbb {R}}/ 2\pi {\mathbb {Z}})^2\). We denote the Lebesgue spaces by \(L^p(\Omega )\) \((p\ge 1)\) with norms \(\Vert \cdot \Vert _{L^p(\Omega )}\). The inner product in the Hilbert space \(L^2(\Omega )\) is denoted by \((\cdot , \cdot )\). For \(s \in {\mathbb {N}}\), \(p\ge 1\), \(W^{s,p}(\Omega )\) is the Sobolev space with norm \(\Vert \cdot \Vert _{W^{s,p}(\Omega )}\). If \(p=2\), we use the notation \(W^{s,p}(\Omega )=H^s(\Omega )\). For \(s=1\) we denote the duality between \(H^1(\Omega )\) and the dual space \((H^1(\Omega ))'\) by \(\langle \cdot , \cdot \rangle \). In the case \(\Omega ={\mathbb {T}}^2\), we recall that the functions are characterized by their Fourier expansion
where \({\overline{z}}^c\) is the complex conjugate of \(z \in {\mathbb {C}}\). We report that \(\big ( \sum _{k \in {\mathbb {Z}}^2} (1+|k|^{2s}) |{\widehat{f}}_k|^2 \big )^\frac{1}{2}\) is a norm on \(H^s({\mathbb {T}}^2)\), \(s \in {\mathbb {N}}\), which is equivalent to the standard norm. For every \(f\in (H^1(\Omega ))'\), we denote by \({\overline{f}}\) the generalized mean value over \(\Omega \) defined by \({\overline{f}}=|\Omega |^{-1}\langle f,1\rangle \). If \(f\in L^1(\Omega )\), then \({\overline{f}}=|\Omega |^{-1}\int _\Omega f \, \mathrm{d}x\). By the generalized Poincaré inequality, there exists a positive constant C such that
We recall the Ladyzhenskaya, Agmon and Gagliardo-Nirenberg interpolation inequalities in two dimensions
Next, we introduce the Hilbert spaces of solenoidal vector-valued functions. In the case of a bounded domain \(\Omega \subset {\mathbb {R}}^2\), we define
We also use \(( \cdot ,\cdot )\) and \(\Vert \cdot \Vert _{L^2(\Omega )}\) for the inner product and the norm in \({\mathbf {H}}_\sigma \). The space \({{\mathbf {V}}}_\sigma \) is endowed with the inner product and norm \(( {{\varvec{u}}},{{\varvec{v}}})_{{{\mathbf {V}}}_\sigma }= ( \nabla {{\varvec{u}}},\nabla {{\varvec{v}}})\) and \(\Vert {{\varvec{u}}}\Vert _{{{\mathbf {V}}}_\sigma }=\Vert \nabla {{\varvec{u}}}\Vert _{L^2(\Omega )}\), respectively. We report the Korn inequality
which implies that \(\Vert D {{\varvec{u}}}\Vert _{L^2(\Omega )}\) is a norm on \({{\mathbf {V}}}_\sigma \) equivalent to \(\Vert {{\varvec{u}}}\Vert _{{{\mathbf {V}}}_\sigma }\). We introduce the space \({\mathbf {W}}_\sigma = {\mathbf {H}}^2(\Omega )\cap {{\mathbf {V}}}_\sigma \) with inner product and norm \( ( {{\varvec{u}}},{{\varvec{v}}})_{{\mathbf {W}}_\sigma }=( {\mathbf {A}}{{\varvec{u}}}, {\mathbf {A}}{{\varvec{v}}})\) and \(\Vert {{\varvec{u}}}\Vert _{{\mathbf {W}}_\sigma }=\Vert {\mathbf {A}}{{\varvec{u}}}\Vert \), where \({\mathbf {A}}={\mathbb {P}}(-\Delta )\) is the Stokes operator and \({\mathbb {P}}\) is the Leray projection from \({\mathbf {L}}^2(\Omega )\) onto \({\mathbf {H}}_\sigma \). We recall that there exists a positive constant \(C>0\) such that
In the space periodic case \(\Omega ={\mathbb {T}}^2\), we defineFootnote 4
which are endowed with the norms \(\Vert {{\varvec{u}}}\Vert _{{\mathbb {H}}_\sigma }= \Vert {{\varvec{u}}}\Vert _{L^2({\mathbb {T}}^2)}\), \(\Vert {{\varvec{u}}}\Vert _{{\mathbb {V}}_\sigma }= \Vert {{\varvec{u}}}\Vert _{H^1({\mathbb {T}}^2)}\), and \(\Vert {{\varvec{u}}}\Vert _{{\mathbb {W}}_\sigma }= \Vert {{\varvec{u}}}\Vert _{H^2({\mathbb {T}}^2)}\). Since
it follows that \((\Vert {{\varvec{u}}}\Vert _{L^2({\mathbb {T}}^2)}^2+ \Vert D {{\varvec{u}}}\Vert _{L^2({\mathbb {T}}^2)}^2 )^\frac{1}{2}\) is a norm on \({\mathbb {V}}_\sigma \), which is equivalent to \(\Vert {{\varvec{u}}}\Vert _{{\mathbb {V}}_\sigma }\). We recall that
Throughout this paper we make use of the following notation:
-
We define the positive constants
$$\begin{aligned} \rho _*=\min \lbrace \rho _1,\rho _2\rbrace , \quad \rho ^*=\max \lbrace \rho _1,\rho _2\rbrace , \quad \nu _*=\min \lbrace \nu _1,\nu _2 \rbrace , \quad \nu ^*=\max \lbrace \nu _1,\nu _2 \rbrace . \end{aligned}$$ -
We denote the convex part of the Flory-Huggins potential by F, namely
$$\begin{aligned} F(s)=\frac{\theta }{2}\big [ (1+s)\log (1+s)+(1-s)\log (1-s)\big ], \quad s \in [-1,1]. \end{aligned}$$ -
The symbol C denotes a generic positive constant whose value may change from line to line. The specific value depends on the domain \(\Omega \) and the parameters of the system, such as \(\rho _*\), \(\rho ^*\), \(\nu _*\), \(\nu ^*\), \(\theta \) and \(\theta _0\). Further dependencies will be specified when necessary.
3 Main results
In this section we formulate the main results of this paper. We start with the local well-posedness of the strong solutions to system (1.1) in a bounded domain \(\Omega \subset {\mathbb {R}}^2\) subject to the boundary conditions (1.4).
Theorem 3.1
Let \(\Omega \) be a bounded domain of class \(C^3\) in \({\mathbb {R}}^2\). Assume that \({{\varvec{u}}}_0 \in {{\mathbf {V}}}_\sigma \) 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 exist \(T_0>0\), depending on the norms of the initial data, and a unique strong solution \(({{\varvec{u}}}, P, \phi )\) to system (1.1) subject to (1.4)–(1.5) on \((0,T_0)\) in the following sense:
-
(i)
The solution \(({{\varvec{u}}}, P, \phi )\) satisfies the properties
$$\begin{aligned} \begin{aligned}&{{\varvec{u}}}\in C([0,T_0]; {{\mathbf {V}}}_\sigma ) \cap L^2(0,T_0;{\mathbf {W}}_\sigma )\cap W^{1,2}(0,T_0;{\mathbf {H}}_\sigma ),\\&P \in L^2(0,T_0;H^1(\Omega )),\\&\phi \in L^\infty (0,T_0;H^3(\Omega )), \ \partial _t \phi \in L^\infty (0,T_0;(H^1(\Omega ))')\cap L^2(0,T_0;H^1(\Omega )),\\&\phi \in L^\infty (\Omega \times (0,T_0)) : |\phi (x,t)|<1 \ \text {a.e. in } \ \Omega \times (0,T_0),\\&\mu \in C([0,T_0];H^1(\Omega ))\cap L^2(0,T_0;H^3(\Omega ))\cap W^{1,2}(0,T_0;(H^1(\Omega ))'), \\&F'(\phi ), F''(\phi ), F'''(\phi ) \in L^\infty (0,T_0;L^p(\Omega )), \ \forall \, p \in [1,\infty ). \end{aligned} \end{aligned}$$(3.1) -
(ii)
The solution \(({{\varvec{u}}}, P, \phi )\) fulfills the system (1.1) almost everywhere in \(\Omega \times (0,T_0)\) and the boundary conditions \(\partial _{{\varvec{n}}}\phi =\partial _{{\varvec{n}}}\mu =0\) almost everywhere in \(\partial \Omega \times (0,T_0)\).
-
(iii)
The solution \(({{\varvec{u}}}, P, \phi )\) is such that \({{\varvec{u}}}(\cdot , 0)={{\varvec{u}}}_0\) and \(\phi (\cdot , 0)=\phi _0\) in \(\Omega \). Moreover, \(({{\varvec{u}}},\phi )\) depends continuously on the initial data in \({\mathbf {H}}_\sigma \times H^1(\Omega )\) on \([0,T_0]\).
Remark 3.2
The AGG system (1.1) corresponds to the model H in the case of matched densities (i.e. \(\rho =\rho _1=\rho _2\)). In this case, under the same assumptions of Theorem 3.1 regarding the domain \(\Omega \) and the initial data \(({{\varvec{u}}}_0,\phi _0)\), it is proven in [23, Theorem 4.1] that the (unique) strong solution exists globally in time.
In the space periodic setting we establish the global well-posedness of the strong solutions.
Theorem 3.3
Let \(\Omega = {\mathbb {T}}^2\). Assume that \({{\varvec{u}}}_0 \in {\mathbb {V}}_{\sigma }\) and \(\phi _0 \in H^2({\mathbb {T}}^2)\) such that \(\Vert \phi _0\Vert _{L^\infty ({\mathbb {T}}^2)}\le 1\), \(|\overline{\phi _0}|<1\), \(\mu _0=-\Delta \phi _0+ \Psi '(\phi _0) \in H^1({\mathbb {T}}^2)\). Then, there exists a unique global strong solution \(({{\varvec{u}}}, P, \phi )\) to system (1.1) with periodic boundary conditions and initial conditions (1.5) in the following sense:
-
(i)
For all \(T>0\), the solution \(({{\varvec{u}}}, P, \phi )\) is such that
$$\begin{aligned} \begin{aligned}&{{\varvec{u}}}\in C([0,T]; {\mathbb {V}}_\sigma ) \cap L^2(0,T;{\mathbb {W}}_\sigma )\cap W^{1,2}(0,T;{\mathbb {H}}_\sigma ),\\&P \in L^2(0,T;H^1({\mathbb {T}}^2)),\\&\phi \in L^\infty (0,T;H^3({\mathbb {T}}^2)), \ \partial _t \phi \in L^\infty (0,T;(H^1({\mathbb {T}}^2))')\cap L^2(0,T;H^1({\mathbb {T}}^2)),\\&\phi \in L^\infty ({\mathbb {T}}^2\times (0,T)) : |\phi (x,t)|<1 \ \text {a.e. in } \ {\mathbb {T}}^2\times (0,T),\\&\mu \in C([0,T];H^1({\mathbb {T}}^2))\cap L^2(0,T;H^3({\mathbb {T}}^2))\cap W^{1,2}(0,T;(H^1({\mathbb {T}}^2))'), \\&F'(\phi ), F''(\phi ), F'''(\phi ) \in L^\infty (0,T;L^p({\mathbb {T}}^2)),\ \forall \, p \in [1,\infty ). \end{aligned} \end{aligned}$$(3.2) -
(ii)
The solution \(({{\varvec{u}}}, P, \phi )\) satisfies the system (1.1) almost everywhere in \({\mathbb {T}}^2 \times (0,T)\).
-
(iii)
The solution \(({{\varvec{u}}}, P, \phi )\) fulfills \({{\varvec{u}}}(\cdot , 0)={{\varvec{u}}}_0\) and \(\phi (\cdot , 0)=\phi _0\) in \({\mathbb {T}}^2\). In addition, for all \(T>0\), \(({{\varvec{u}}},\phi )\) depends continuously on the initial data in \({\mathbb {H}}_\sigma \times H^1({\mathbb {T}}^2)\) on [0, T].
Remark 3.4
The assumption on the initial chemical potential \(\mu _0\) required in both Theorems 3.1 and 3.3 is satisfied if \(\phi _0 \in H^3(\Omega )\) such that \(F''(\phi _0) \in L^2(\Omega )\).
Finally, we prove a stability result in terms of the density values between the strong solutions to the AGG model and the model H departing from the same initial datum.
Theorem 3.5
Let \(\Omega \) be a bounded domain of class \(C^3\) in \({\mathbb {R}}^2\). Given an initial datum \(({{\varvec{u}}}_0, \phi _0)\) as in Theorem 3.1, we consider the strong solution \(({{\varvec{u}}}, P, \phi )\) to the AGG model with density (1.2) defined on \([0,T_0]\) and the strong solution \(({{\varvec{u}}}_H, P_H, \phi _H)\) to the model H with density \({\overline{\rho }}\). Then, there exists a constant C, which depends on the norm of the initial data, the time \(T_0\) and the parameters of the systems, such that
Remark 3.6
Assuming that \(\rho _1={\overline{\rho }}\) and \(\rho _2={\overline{\rho }}+\varepsilon \) for (small) \(\varepsilon >0\), the stability estimate (3.3) reads as
Remark 3.7
The statement of Theorem 3.5 is also valid in the space periodic setting. In particular, thanks to the global well-posedness, the stability estimate (3.3) holds on [0, T] for any \(T>0\).
4 Proof of Theorem 3.1: local existence in bounded domains
In this section, we prove the existence of local strong solutions to system (1.1) with boundary and initial conditions (1.4)–(1.5) in a bounded domain \(\Omega \) in \({\mathbb {R}}^2\). We first present the semi-Galerkin approximation scheme, then prove the solvability of the approximated system through a fixed point argument, and finally carry out the uniform estimates of the approximate solutions which allow the passage to the limit in the approximate formulation.
4.1 Definition of the approximate problem
We consider the family of eigenfunctions \(\lbrace {{\varvec{w}}}_j\rbrace _{j=1}^\infty \) and eigenvalues \(\lbrace \lambda _j\rbrace _{j=1}^\infty \) of the Stokes operator \({\mathbf {A}}\). For any integer \(m\ge 1\), we define the finite-dimensional subspaces of \({{\mathbf {V}}}_\sigma \) by \({{\mathbf {V}}}_m= \text {span}\lbrace {{\varvec{w}}}_1,\ldots ,{{\varvec{w}}}_m\rbrace \). We denote by \({\mathbb {P}}_m\) the orthogonal projection on \({{\mathbf {V}}}_m\) with respect to the inner product in \({\mathbf {H}}_\sigma \). Since \(\Omega \) is of class \(C^3\), it follows that \({{\varvec{w}}}_j \in {\mathbf {H}}^3(\Omega )\cap {{\mathbf {V}}}_\sigma \) for all \(j\in {\mathbb {N}}\). Moreover, we report the inverse Sobolev embedding inequalities in \({{\mathbf {V}}}_m\)
Let us fix \(T>0\). For any \(m \in {\mathbb {N}}\), we determine the approximate solution \(({{\varvec{u}}}_m, \phi _m)\) to the system (1.1) with boundary and initial conditions (1.4)–(1.5) as follows:
for all \(p \in [2,\infty )\), such that
for all \({{\varvec{w}}}\in {{\mathbf {V}}}_m\) and \(t \in [0,T]\), and
The approximate solution \(({{\varvec{u}}}_m,\phi _m)\) satisfies the boundary and initial conditions
4.2 Existence of approximate solutions
We perform a fixed point argument to show the existence of the approximate solutions satisfying (4.2)–(4.5). To this aim, we take \({{\varvec{v}}}\in W^{1,2}(0,T;{{\mathbf {V}}}_m)\). We consider the convective Cahn–Hilliard system
which is equipped with the boundary and initial conditions
It is proven in [1, Theorem 6 and Lemma 3] that there exists a unique solution to (4.6)–(4.7) such that
for any \(p\in [2,\infty )\). Thanks to [14, Lemma A.6], it follows that \(F''(\phi )\in L^\infty (0,T;L^p(\Omega ))\) for any \(p \in [2, \infty )\). In addition, by comparison in (4.6)\(_1\) and (4.6)\(_2\), we infer that \(\mu \in L^2(0,T;H^3(\Omega ))\) and \(\partial _t \mu _m\in L^2(0,T;(H^1(\Omega ))')\) (see, e.g., [21, Proof of Theorem 5.1]). Therefore, we have
We report the following estimates for the system (4.6)–(4.7) (see [1], cf. also [14, 22]):
-
1.
\(L^2\) estimate:
$$\begin{aligned} \sup _{t \in [0,T]}\Vert \phi _m(t)\Vert _{L^2(\Omega )}^2 + \int _0^T \Vert \Delta \phi _m(\tau )\Vert _{L^2(\Omega )}^2 \, \mathrm{d}\tau \le \Vert \phi _0\Vert _{L^2(\Omega )}^2 + \frac{\theta _0^2}{2}T. \end{aligned}$$(4.10) -
2.
Energy estimate:
$$\begin{aligned} \begin{aligned}&\sup _{t\in [0,T]} \int _{\Omega } \frac{1}{2} | \nabla \phi _m(t)|^2 + F(\phi _m(t)) \, \mathrm{d}x + \frac{1}{2} \int _0^T \Vert \nabla \mu (\tau )\Vert _{L^2(\Omega )}^2 \, \mathrm{d}\tau \\&\quad \le E_{\text {free}}(\phi _0)+ \frac{1}{2} \int _0^T \Vert {{\varvec{v}}}(\tau )\Vert _{L^2(\Omega )}^2 \, \mathrm{d}\tau + \frac{\theta _0}{2} \Vert \phi _0\Vert _{L^2(\Omega )}^2 + \frac{\theta _0^3}{4}T. \end{aligned} \end{aligned}$$(4.11) -
3.
Time derivative estimateFootnote 5
$$\begin{aligned} \begin{aligned}&\Vert \partial _t \phi _m \Vert _{L^\infty (0,T;(H^1(\Omega ))')}^2 +\int _0^T \Vert \nabla \partial _t \phi _m(\tau )\Vert _{L^2(\Omega )}^2 \, \mathrm{d}\tau \\&\quad \le C\Big (1+\Vert \nabla \mu _0\Vert _{L^2(\Omega )}^2+ \Vert {{\varvec{v}}}\Vert _{L^\infty (0,T;L^2(\Omega ))}^2\\&\qquad + \int _0^T \Vert \partial _t {{\varvec{v}}}(\tau )\Vert _{L^2(\Omega )}^2 \, \mathrm{d}\tau \Big ) \mathrm {e}^{C\int _0^T 1+\Vert {{\varvec{v}}}(\tau )\Vert _{L^2(\Omega )}^2\, \mathrm{d}\tau }, \end{aligned} \end{aligned}$$(4.12)where the constant C only depends on \(\Omega \) and \(\theta _0\).
Next, we look for the approximated velocity field
that solves the Galerkin approximation of (1.1)\(_1\) as follows
which is completed with the initial condition \({{\varvec{u}}}_m(\cdot , 0)= {\mathbb {P}}_m{{\varvec{u}}}_0\). Setting \({\mathbf {A}}^m(t)=(a_1^m(t), \ldots , a_m^m(t))^T\), (4.13) is equivalent to the system of differential equations
where the matrices \({\mathbf {M}}^m(t)\), \({\mathbf {L}}^m(t)\) and the vector \({\mathbf {G}}^m(t)\) are given by
and the initial condition is
Thanks to (4.8), it follows that \(\phi _m \in C([0,T]; W^{1,4}(\Omega ))\). This, in turn, implies that \(\rho (\phi _m), \nu (\phi ) \in C(\overline{\Omega \times [0,T]})\). In addition, we recall that \({{\varvec{v}}}\in C([0,T]; {\mathbf {H}}_\sigma )\) and \(\nabla \mu _m\in C([0,T];L^2(\Omega ))\). As a consequence, it follows that \({\mathbf {M}}^m\) and \({\mathbf {L}}^m\) belong to \(C([0,T];{\mathbb {R}}^{m \times m})\), and \({\mathbf {G}}^m \in C([0,T];{\mathbb {R}}^m)\). Furthermore, the matrix \({\mathbf {M}}^m(\cdot )\) is definite positive on [0, T], and so the inverse \(({\mathbf {M}}^m)^{-1} \in C([0,T]; {\mathbb {R}}^{m\times m})\). Therefore, the classical existence and uniqueness theorem for system of linear ODEs entails that there exists a unique vector \({\mathbf {A}}^m \in C^1([0,T];{\mathbb {R}}^m)\) that solves (4.14) on [0, T]. This implies that the problem (4.13) has a unique solution \({{\varvec{u}}}_m \in C^1([0,T];{{\mathbf {V}}}_m)\).
Next, multiplying (4.13) by \(a_l^m\) and summing over l, we find
By integration by parts, we have
Since \(\rho '(\phi _m)= \frac{\rho _1-\rho _2}{2}\) and \(\mathrm {div}\,{{\varvec{v}}}=0\), by using (4.6)\(_1\), we observe that
Thus, we deduce that
By using (2.4), (2.6) and (4.8), we have
for some constant C depending only on \(\Omega \) and \(\nu _*\). Since \(\Vert \phi _m\Vert _{H^2(\Omega )}\le C(1+ \Vert \Delta \phi _m\Vert _{L^2(\Omega )})\), we arrive at
In light of (4.10), we infer that
This, in turn, implies that
where the constant \(R_0\) depends on \(\rho _*\), \(\rho ^*\), \(\nu _*\), \(\theta _0\), \(\Vert {{\varvec{u}}}_0 \Vert _{L^2(\Omega )}\), T, \(\Omega \). As an immediate consequence, we deduce that
Next, we proceed in estimating the time derivative of \({{\varvec{u}}}_m\). To this aim, multiplying (4.13) by \(\frac{\mathrm{d}}{\mathrm{d}t}a_l^m\) and summing over l, we obtain
By exploiting (2.4) and (4.1), we find
Then, by (4.10), (4.11), (4.15) we eventually infer that
Thus, there exist two positive constants \(R_2\) and \(R_3\), depending only on \(\rho _*\), \(\rho ^*\), \(\nu _*\), \(\theta _0\), \(\Vert {{\varvec{u}}}_0 \Vert _{L^2(\Omega )}\), \(E_{\text {free}}(\phi _0)\), T, \(\Omega \), m, such that
We are now in a position to state the setting of the fixed point argument. Let us define \(R_4= \sqrt{R_2 R_1^2+R_3}\). We introduce the set
and the map
where \({{\varvec{u}}}_m\) is the solution to the system (4.13). Thanks to (4.15) and (4.17), we deduce that \( \Lambda : S \rightarrow S. \) We notice that S is a convex set. In addition, by [32, Theorem 1, Section 3], S is compact set in \(L^2(0,T;{{\mathbf {V}}}_m)\).
We are left to prove that the map \(\Lambda \) is continuous. Let us consider a sequence \(\lbrace {{\varvec{v}}}_n\rbrace \subset S\) such that \({{\varvec{v}}}_n \rightarrow {\widetilde{{{\varvec{v}}}}}\) in \(L^2(0,T;{{\mathbf {V}}}_m)\). By arguing as above, there exists a sequence \(\lbrace (\psi _n, \mu _n) \rbrace \) and \(({\widetilde{\psi }}, {\widetilde{\mu }})\) that solve the convective Cahn–Hilliard equation (4.6)–(4.7), where \({{\varvec{v}}}\) is replaced by \({{\varvec{v}}}_n\) and \({\widetilde{{{\varvec{v}}}}}\), respectively. Since \(\lbrace {{\varvec{v}}}_n \rbrace \) and \({\widetilde{{{\varvec{v}}}}}\) belong to S, and \( E_{\text {free}}(\phi _0)<\infty \), we infer from [1, Theorem 6] that
On the other hand, using again that \(\lbrace {{\varvec{v}}}_n \rbrace \) and \({\widetilde{{{\varvec{v}}}}}\) belong to S, together with the continuous embedding \(W^{1,2}(0,T;{{\mathbf {V}}}_m) \hookrightarrow C([0,T];{{\mathbf {V}}}_m)\), it follows from (4.12) that
for some constant C which depends on \(\phi _0\), T, \(R_1\), \(R_4\), \(\theta _0\), \(\Omega \), but is independent of n. By comparison in (4.6)\(_1\), it is easily seen that
By exploiting [14, Lemma A.4 and Lemma A.6], we obtain
for all \(p\in [2,\infty )\), where the constant \(C_p\) depends on p, \(\phi _0\), T, \(R_1\), \(R_4\), \(\theta _0\), \(\Omega \), but is independent of n. Thanks to the above estimates, we infer that
which, in turn, gives us
for some constant C independent of n. By standard interpolation, we deduce from (4.18) and (4.20) that
As a consequence, by using the definition of \(\mu _n-{\widetilde{\mu }}\) and the above estimates, we eventually obtain
Next, we introduce \({{\varvec{u}}}_n = \Lambda ( {{\varvec{v}}}_n) \in S\), for any \(n \in {\mathbb {N}}\), and \({\widetilde{{{\varvec{u}}}}}=\Lambda ({\widetilde{{{\varvec{v}}}}})\in S\). We define \( {{\varvec{u}}}= {{\varvec{u}}}_n-{\widetilde{{{\varvec{u}}}}}\), \( \psi = \psi _n-{\widetilde{\psi }}\), \( {{\varvec{v}}}={{\varvec{v}}}_n - {\widetilde{{{\varvec{v}}}}}\), and \( \mu = \mu _n-{\widetilde{\mu }}\). We have the system
for all \({{\varvec{w}}}\in {{\mathbf {V}}}_m\), for all \(t \in [0,T]\). Taking \({{\varvec{w}}}={{\varvec{u}}}\), we obtain
By using (2.6) and the Sobolev embedding, we have
and
Since \({{\varvec{v}}}_n\), \({\widetilde{{{\varvec{v}}}}}\) and \({{\varvec{u}}}_n \) belong to S, by (2.6) and (4.1) we get
In a similar way, we find
and
By Sobolev embedding and (4.20), we have
Combining the above inequalities, we arrive at the differential inequality
where
Therefore, an application of the Gronwall lemma yields
Owing to (4.19), (4.21), (4.22), and the convergence \({{\varvec{v}}}_n \rightarrow {\widetilde{{{\varvec{v}}}}}\) in \(L^2(0,T;{{\mathbf {V}}}_m)\), we deduce that \({{\varvec{u}}}_n \rightarrow {\widetilde{{{\varvec{u}}}}}\) in \(L^\infty (0,T;{{\mathbf {V}}}_m)\), which entails that the map \(\Lambda \) is continuous. Finally, we conclude from the Schauder fixed point theorem that the map \(\Lambda \) has a fixed point in S. This implies the existence of the approximate solution \(({{\varvec{u}}}_m, \phi _m)\) on [0, T] satisfying (4.2)–(4.5) for any \(m \in {\mathbb {N}}\).
4.3 A priori estimates for the approximate solutions
First, we observe that
Taking \({{\varvec{w}}}={{\varvec{u}}}_m\) in (4.3) and integrating by parts, we obtain
Thanks to (4.4)\(_1\), the first term of the right-hand side in the above equality is zero. We recall that
Then, we have
Multiplying (4.6) by \(\mu _m\), integrating over \(\Omega \) and using the definition of \(\mu _m\), we get
By summing (4.25) and (4.26), we obtain
Integrating in time, we find
Since
and recalling that \(\phi _m \in L^\infty (\Omega \times (0,T))\) such that \(|\phi _m(x,t)|<1\) almost everywhere in \(\Omega \times (0,T)\), we deduce that
where the positive constant C depends on \(\Vert {{\varvec{u}}}_0\Vert _{L^2(\Omega )}\) and \( E_{\text {free}}(\phi _0)\), but is independent of m. Multiplying (4.6) by \(-\Delta \phi _m\) and integrating over \(\Omega \), we have
Thanks to the regularity of \(\phi _m\), \(F'(\phi _m)\) and \(F''(\phi _m)\) [cf. (4.8) and its consequences], it follows that \(\nabla F'(\phi _m)= F''(\phi _m) \nabla \phi _m\),Footnote 6 for almost every \(t\in (0,T)\), and \(F'(\phi _m)\in L^\infty (0,T; H^1(\Omega ))\). This allows us to integrate by parts in the second term of the left-hand side, thus we obtain that \(-\int _{\Omega } F'(\phi _m) \Delta \phi _m \, \mathrm{d}x= \int _{\Omega } F''(\phi _m) |\nabla \phi _m|^2 \,\mathrm{d}x\). Since \(F''(s)>0\) for \(s\in (-1,1)\), by using (4.29), we get
for some C independent of m. Then, it follows from (4.30) that
By the form of \(F'\), thanks to [30, Eq. (A4), Prop. A1], we have
where the constants \(c_1, c_2\) depends on \(\overline{\phi _0}\). Then, multiplying (4.6)\(_2\) by \(\phi _m - \overline{\phi _0}\) [cf. (4.24)], we obtain
By the Poincaré inequality and (4.29), we find
Since \(\overline{\mu _m}= \overline{F'(\phi _m)}- \theta _0 \overline{\phi _0}\), by combining (4.33) and (4.34), we have
Thanks to (2.1), we are led to
which, in turn, implies
for some constant C independent of m. In addition, using the boundary conditions (4.5) and (4.28), we deduce that
which entails that
Furthermore, by using [1, Lemma 2] or [14, Lemma A.4], we infer that, for all \(p\in (2,\infty )\),
As a consequence, it holds
Next, taking \({{\varvec{w}}}=\partial _t {{\varvec{u}}}_m\) in (4.3) we get
Computing the duality between \(\partial _t \mu _m\) and (4.6), we find
Notice that
and
Then, we obtain
By summing (4.40) and (4.41), we have
where
By (2.2), (2.6), (4.28), (4.29), and (4.35),
for some \(C_0\) independent of m. Then, we infer that
We now proceed in estimating the terms \(I_i\), \(i=1, \ldots ,7\). Let \(\varpi _1\) and \(\varpi _2\) be two positive constant whose values will be determined later. Exploiting (2.2), (2.6) and (4.43), we have
By interpolation of Sobolev spaces and (2.1), (2.2), (4.37), we obtain
By using (2.3) and (4.35), we get
Exploiting (4.31), (4.37) and (4.38), we find
and
Thanks to (4.31) and (4.35), we deduce that
and
Combining (4.42) with (4.43) and the above estimates of \(I_i\), we arrive at
where the positive constant C depends on the values of \(\varpi _1\) and \(\varpi _2\) but is independent of m. We are left to control the norms \(\Vert {\mathbf {A}}{{\varvec{u}}}_m\Vert _{L^2(\Omega )}\) and \(\Vert \mu _m\Vert _{H^3(\Omega )}\). To this end, taking \({{\varvec{w}}}={\mathbf {A}}{{\varvec{u}}}_m\) in (4.13), we have
By the theory of the Stokes problem (see, e.g., [18] and [23, Appendix B, Lemma B.2]), there exists \(\pi _m \in C([0,T];H^1(\Omega ))\) such that \(-\Delta {{\varvec{u}}}_m + \nabla \pi _m= {\mathbf {A}}{{\varvec{u}}}_m\) almost everywhere in \(\Omega \times (0,T)\) and
where C is independent of m. Thus, we deduce that
By Young’s inequality, we have
By using (2.2), (2.3), (2.6) and (4.28), we find
and
In light of (4.31) and (4.35), we have
and
Owing to (4.38) and (4.53), we obtain
Thus, we are led to
Next, taking the gradient of (4.4)\(_1\), and using (4.38), we find
Since
for some positive constant \(C_S\) independent of m, we infer from (4.55) that
Let us now set
Multiplying (4.54) and (4.56) by \(\varepsilon _1\) and \(\varepsilon _2\), respectively, and summing the resulting inequalities to (4.51), we deduce the differential inequality
where
Hence, whenever \({\widetilde{T}}>0\) satisfies
we have
We observe that
for a positive constant \(C_2\) independent of m. Therefore, setting
it yields that
Notice that \(T_0\) is independent of m. Thanks to (4.35) and (4.43), we infer that
where \(K_1\) is a positive constant that depends on \(E({{\varvec{u}}}_0,\phi _0)\), \(\Vert {{\varvec{u}}}_0\Vert _{{{\mathbf {V}}}_\sigma }\), \(\Vert \mu _0\Vert _{H^1(\Omega )}\) and the parameters of the system, but is independent of m. Recalling (4.38), and using [14, Lemma A.6], we immediately obtain for any \(p \in [2,\infty )\)
As a consequence, we have
Integrating (4.57) we deduce that
Finally, it follows from (4.60) and (4.62) that
Here, the constants \(K_2\), ..., \(K_5\) depend on the same factors as \(K_1\).
4.4 Passage to the limit and existence of strong solutions
We are in a position to pass to the limit as \(m\rightarrow \infty \). More precisely, thanks to the above estimates (4.59)–(4.63), we deduce the following convergences (up to a subsequence)
The strong convergences of \({{\varvec{u}}}_m\), \(\phi _m\) and \(\mu _m\) are recovered through the Aubin-Lions lemma which yields
As a consequence, since \(\rho (\cdot )\) and \(\nu (\cdot )\) are linear functions, we infer that
Furthermore, it follows from \(\phi _m \rightarrow \phi \) almost everywhere in \(\Omega \times (0,T_0)\) and the continuity of \(F'\) in \((-1,1)\) that \(F'(\phi _m)\rightarrow F'(\phi )\) almost everywhere in \(\Omega \times (0,T_0)\). At the same time, by exploiting (4.4) and (4.65), we observe that \(F'(\phi _m)= \mu _m +\Delta \phi _m + \theta _0 \phi _m \rightarrow \mu +\Delta \phi + \theta _0 \phi \) in \(C([0,T_0]; L^2(\Omega ))\), which implies that \(F'(\phi _m)\rightarrow w\) in \(C([0,T_0]; L^2(\Omega ))\). Thus, by the uniqueness of the almost everywhere convergence, we conclude that \(w= F'(\phi )\). In particular, we also have
The above properties entail the convergence of the nonlinear terms in (4.3), which allows us to pass to the limit as \(m\rightarrow \infty \) in (4.3)–(4.4) (see, e.g., [33] for the limit in the Galerkin formulation). By the weak lower semicontinuity of the norm and the time continuity properties of the solution, there exists a constant \({\overline{K}}\) depending only on the norm of the initial data, the time \(T_0\) and the parameters of the system, such that
and
In addition, since \(-\Delta \phi +F'(\phi )= \mu + \theta _0 \phi \) in \(\Omega \times (0,T)\) and \(\partial _{{\varvec{n}}}\phi =0\) on \(\Omega \times (0,T)\), by using [14, Lemma A.6] for any \(p \in [1,\infty )\), there exists \({\overline{K}}(p)\) such that
Here \({\overline{K}}(p)\) also depends on the norm of the initial data and the time \(T_0\). Lastly, since
for all \({{\varvec{w}}}\in {\mathbf {H}}_\sigma \), there exists \(P \in L^2(0,T_0;H^1(\Omega ))\), \({\overline{P}}(t)=0\) (see, e.g., [18]) such that
Remark 4.1
The proof of Theorem 3.1 holds true in the boundary periodic setting. In particular, the orthogonal dense set in \({\mathbb {H}}_\sigma \) can be chosen as the eigenfunctions of the Stokes operator (see [33]) augmented by the constant function. Moreover, in order to recover the norm of \({{\varvec{u}}}_m\) in \(H^2({\mathbb {T}}^2)\) [cf. (4.52)], it is sufficient to take \(-\Delta {{\varvec{u}}}_m\) in (4.13) (instead of \({\mathbf {A}}{{\varvec{u}}}_m\)). In turn, the term \(I_{13}\) involving the pressure \(\pi _m\) does not appear. The rest of the proof remains valid with few minor changes.
5 Proof of Theorem 3.3: global existence in the space periodic setting
In this section we address the global existence of the strong solutions to the AGG system (1.1) in \({\mathbb {T}}^2\). We consider a strong solution \(({{\varvec{u}}},P, \phi )\) to system (1.1) defined on the maximal interval of existence \((0,T_*)\). This satisfies for all \(0<T<T_*\)
for all \(p\in [2,\infty )\), and
almost everywhere in \({\mathbb {T}}^2\times (0,T^*)\).
The aim is to show that \(T_*=\infty \). We assume by contradiction that \(T_*<\infty \). In the rest of this section, we prove that the norms related to the functional spaces in (5.1) are uniformly bounded on \((0,T_*)\). In turn, this entails that \({{\varvec{u}}}(T_*) \in {\mathbb {V}}_\sigma \), \(\phi (T_*) \in H^2({\mathbb {T}}^2)\) such that \(\Vert \phi (T_*)\Vert _{L^\infty ({\mathbb {T}}^2)}\le 1\), \(|{\overline{\phi }}(T_*)|<1\) and \(\mu (T_*)=-\Delta \phi (T_*)+\Psi '(\phi (T_*)) \in H^1({\mathbb {T}}^2)\). Thus, by the local existence result in Theorem 3.1, it is possible to extend the solution beyond the time \(T_*\). As a consequence, the solution exists globally in time.
5.1 Energy estimates
We report some basic energy estimates similar to those obtained in Sect. 4 [cf. (4.24)–(4.39)]. First, combining (5.2)\(_1\) and (5.2)\(_3\), the solution satisfies (1.1)\(_1\) almost everywhere in \({\mathbb {T}}^2\times (0,T^*)\). Integrating over \({\mathbb {T}}^2 \times (0,t)\) with \(t<T_*\), we obtain
Similarly, integrating (5.2)\(_3\) over \({\mathbb {T}}^2 \times (0,t)\) with \(t<T_*\), we get
Thanks to the energy identity (1.6), we have
Since \(E({{\varvec{u}}}_0,\phi _0) <\infty \), we find for all \(0<T<T_*\)
Here the constant C depends on \(E_({{\varvec{u}}}_0,\phi _0)\), but it is independent of \(T_*\). Arguing as in Sect. 4, we have
and
The latter implies
In addition, we recall that
and
As a consequence, it follows that, for all \(T<T_*\),
5.2 High-order estimates for the concentration
Taking the duality between \(\partial _t \mu \) and (5.2)\(_3\), we obtain [cf. (4.41)]
Since
arguing as in (4.55), we infer that
Let us set \(\varepsilon =\frac{1}{4C_S}\). Multiplying (5.15) by \(\varepsilon \) and adding the resulting inequality to (5.14), we get
By interpolation of Sobolev spaces and (5.10)
By using (5.9), we have
Thus, we preliminary obtain
We observe that
Here the periodic boundary conditions played a crucial role to avoid any boundary term. We now proceed in estimating the terms \(W_i\), \(i=1, \ldots ,5\). By using (2.2), (5.5) and (5.9), we have
and
We observe that \(\rho (\phi )-\phi \rho '(\phi )= \frac{\rho _1+\rho _2}{2}\). By interpolation of Sobolev spaces and (5.9), we find
By (5.7) we obtain
Similarly, by (2.2) and (5.9), we find
We are now left to find an estimate of the pressure P. We introduce the function q as the solution to
Since \(P\in L^2(0,T; H^1({\mathbb {T}}^2))\), for all \(0<T<T_*\), such that \({\overline{P}}(t)=0\) for all \(t\in (0,T_*)\), and \(\rho (\phi )\ge \rho _*\), the existence of q follows from the Lax-Milgram theorem. In particular, we have \( q \in L^2(0,T_*;H^1({\mathbb {T}}^2))\), and \({\overline{q}}(t)=0\) for all \(t\in (0,T_*)\). In addition, by elliptic regularity, we have the following estimates [cf. [21, Theorem 2.1]]
The latter, together with (5.12), entails that \(q\in L^1(0,T_*;H^2({\mathbb {T}}^2))\). Multiplying (5.2) by \(\frac{\nabla q}{\rho (\phi )}\), we find
Integrating by parts and using the periodic boundary conditions, and then exploiting (5.24), we deduce that
Exploiting (2.2), (2.3), (5.5) and (5.9), we find
Thus, we are led to
Inserting (5.27) in (5.23), we obtain
where
By (5.7), (5.11) and the Young inequality, we have
Combining (5.28) with (5.29)–(5.34), we infer that
Collecting (5.19)–(5.21) and (5.35) together, we find
Hence, it follows from (5.17) and the above inequality that
We now set
Thanks to (5.5), we observe that
Then, there exists a positive constant \({\overline{C}}\) depending on \(E({{\varvec{u}}}_0,\phi _0)\) such that
Therefore, we deduce the differential inequality
where
In light of (5.1) and (5.12), we infer from the Gagliardo-Nirenberg inequality (2.5) with \(s=3\) that \(\Vert \phi \Vert _{L^\frac{8}{3}(0,T_*;W^{1,\infty }({\mathbb {T}}^2))}\le C(1+T_*)\). In turn, it gives \(\Vert Y\Vert _{L^1(0,T_*)}\le C(1+T_*)\) [cf. (5.5) and (5.12)]. Thus, the Gronwall lemma yields
which entails that
where \(K_T\) stands for a generic constant depending on the parameters of the system, the initial energy \(E({{\varvec{u}}}_0,\phi _0)\), the norms of the initial data \(\Vert {{\varvec{u}}}_0\Vert _{H^1({\mathbb {T}}^2)}\) and \(\Vert \mu _0\Vert _{H^1({\mathbb {T}}^2)}\), and the time T. In particular, \(K_T\) is finite for any \(T<\infty \). Integrating in time (5.37), we infer that
As a consequence, we obtain from (5.11) that for all \(p\in [2, \infty )\)
Finally, as in Section 4, by exploiting [14, Lemma A.6], we immediately deduce that for all \(p \in [2,\infty )\)
which implies that
5.3 High-order estimates for the velocity field
Multiplying (5.2) by \(\partial _t {{\varvec{u}}}\) and integrating over \({\mathbb {T}}^2\), we obtain [cf. (4.40)]
On the other hand, multiplying (5.2) by \(-\Delta {{\varvec{u}}}\), we find
By the Young inequality, we simply have
Multiplying (5.46) by \(\frac{\nu _*}{2\rho ^*}\) and adding the resulting inequality to (5.44), we reach
Notice that
due to (2.8), (2.9) and (5.5). By (2.2), (5.5), (5.41), we can estimate the terms \(L_i\) as follows
Hence, it follows that on [0, T], for all \(T<T_*\),
where
In light of (5.5) and (5.40), an application of the Gronwall lemma yields
for all \(T <T_*\), where
for some positive constant C depending on \(\nu _*\), \(\rho _*\) and \(\rho ^*\).
5.4 Global existence of strong solutions
The uniform-in-time estimates (5.39)–(5.41) and (5.56) entails that the solution does not blowup as T approaches \(T_*\). More precisely, since \(K_{T_*}\) and \({\widetilde{K}}_{T_*}\) are finite, we infer from (5.40), (5.43) and (5.56) that \(\mu \in L^2(0,T_*;H^3({\mathbb {T}}^2))\cap W^{1,2}(0,T_*; (H^1({\mathbb {T}}^2))')\) and \({{\varvec{u}}}\in L^2(0,T_*; {\mathbb {W}}_\sigma )\cap W^{1,2}(0,T_*; {\mathbb {H}}_\sigma )\). Then, it follows from [28, Theorem 3.1 and Theorem 12.5] that \(\mu \in C([0,T_*]; H^1({\mathbb {T}}^2))\) and \({{\varvec{u}}}\in C([0,T_*];{\mathbb {V}}_\sigma )\). This implies that \(\mu (T_*)\) and \({{\varvec{u}}}(T_*)\) are well-defined. Then, the solution can be continued beyond \(T_*\) into a solution which satisfies (5.1) and (5.2) on an interval \((0,{\overline{T}})\) for some \({\overline{T}}>T_*\). This contradicts the maximality of \(T_*\). Hence, \(T_*=\infty \).
6 Uniqueness
In this section we show the uniqueness and the continuous dependence on the initial data for the strong solutions proved in Theorem 3.1 and Theorem 3.3. We demonstrate hereafter the case of a general bounded domain \(\Omega \subset {\mathbb {R}}^2\). The proof in the case \(\Omega ={\mathbb {T}}^2\) can be adapted with minor changes.
Let \(({{\varvec{u}}}_1,P_1,\phi _1)\) and \(({{\varvec{u}}}_2,P_2,\phi _2)\) be two strong solutions to system (1.1) with boundary conditions (1.4) defined on a common interval \([0,T_0]\) given by Theorem 3.1. We consider \({{\varvec{u}}}={{\varvec{u}}}_1-{{\varvec{u}}}_2\), \(P=P_1-P_2\) and \(\phi =\phi _1-\phi _2\). It is clear that
almost everywhere in \(\Omega \times (0,T_0)\). Multiplying (6.1) by \({{\varvec{u}}}\) and integrating over \(\Omega \), we find
Here we have used that
Taking the gradient of (6.2), multiplying the resulting equation by \(\nabla \phi \) and integrating over \(\Omega \), then using the boundary conditions (1.4), we obtain
Since \( \frac{\mathrm{d}}{\mathrm{d}t}{\overline{\phi }}=0, \) by (6.3) and (6.4) we reach
We recall that \(\Vert \phi \Vert _{H^3(\Omega )} \le C\big (\Vert \phi \Vert _{H^1(\Omega )} + \Vert \nabla \Delta \phi \Vert _{L^2(\Omega )})\). By exploiting (2.2), (2.6) and the regularity of the strong solutions, we infer that
Therefore, by (6.5)–(6.13), we find the differential inequality
where the constant \({\overline{C}}\) depends on the norm of the initial data and \(T_0\). Thanks to the Gronwall lemma, together with (2.1), we deduce for all \(t \in [0,T_0]\) that
The above inequality proves the continuous dependence of the solutions on the initial data. In particular, when \({{\varvec{u}}}(0)={\mathbf {0}}\) and \(\phi (0)=0\), it follows that \({{\varvec{u}}}(t)={\mathbf {0}}\) and \(\phi (t)=0\) for all \(t\in [0,T_0]\). Thus, the strong solution is unique.
7 Stability
In this section we prove Theorem 3.5, which states a stability result for the strong solutions to the AGG model and the model H. We denote by \(({{\varvec{u}}},P,\phi )\) and \(({{\varvec{u}}}_H,P_H,\phi _H)\) the strong solutions to the AGG model with density \(\rho (\phi )\) and the model H with constant density \({\overline{\rho }}\) (see [7, Eqs. (1.1)-(1.4)]), respectively, defined on a common interval \([0,T_0]\). For simplicity, we assume that the viscosity function is given by \(\nu (s)= \nu _1 \frac{1+s}{2}+\nu _2 \frac{1-s}{2}\) [cf. (1.2)] for both systems. We define \({{\varvec{v}}}={{\varvec{u}}}-{{\varvec{u}}}_H\), \(p=P-P_H\), \(\varphi =\phi -\phi _H\), and the difference of the chemical potentials \(w= \mu - \mu _H\). They solve the system
almost everywhere in \(\Omega \times (0,T_0)\). In addition, we have the boundary and initial conditions
We recall that \(({{\varvec{u}}}_H,P_H,\phi _H)\) fulfills the same regularity properties of \(({{\varvec{u}}},P,\phi )\) as stated in Theorem () [cf. [23, Theorem 4.1]]. In particular, there exists a constant \(K_H\), which depends on the norm of the initial condition, the time \(T_0\) and the parameters of the system (\({\overline{\rho }}\), \(\nu _1\), \(\nu _2\), \(\theta \), \(\theta _0\)), such that [cf. (4.68)–(4.70)]
and
In addition, for any \(p\in [1,\infty )\), there exists a constant \(K_{H}(p)\), which depends on the same factors as \(K_H\), such that
Arguing as in the proof of uniqueness (cf. Section 6), we multiply (7.1) by \({{\varvec{v}}}\) and (7.2) by \(-\Delta \varphi \), and we sum the resulting equations. We observe that the following equalities hold
and
Then, we eventually end up with the differential equality
Before proceeding with the estimate of \(V_i\), \(i=1, \ldots ,8\), we notice that \({\overline{\varphi }}(t)=0\) for all \(t\in [0,T_0]\) and
Also, we recall that \(\Vert \nabla \varphi \Vert _{L^2(\Omega )}\), \(\Vert \Delta \varphi \Vert _{L^2(\Omega )}\) and \(\Vert \nabla \Delta \varphi \Vert _{L^2(\Omega )}\) are norms in \(H^1(\Omega )\), \(H^2(\Omega )\) and \(H^3(\Omega )\), respectively, which are equivalent to the usual ones due to \({\overline{\varphi }}=0\). Thanks to (4.68), (4.70), (7.4), (7.5), (7.6) and (7.8), we deduce that
Therefore, we find the differential inequality
where
with the positive constant C depending on the norm of the initial data and the time \(T_0\). Using the Gronwall lemma, together with the initial conditions (7.3), we infer that
Thus, in light of (7.4), the above inequality implies that
where the positive constant \(K^*\) depends on the norm of the initial data, the time \(T_0\) and the parameters of the systems.
Notes
A different approximation leading to a concentration \(\phi _m\) with values outside the interval \([-1,1]\) may need a suitable extension of \(\rho (\cdot )\) outside the interval \([-1,1]\), and, in general, it may happen that \(\rho '(\phi )\ne \frac{\rho _1-\rho _2}{2}\).
In contrast to the classical periodic setting for the incompressible Navier–Stokes [cf. [33]], we do not require that \({\widehat{{{\varvec{u}}}}}_0=0\) in the definition of \({\mathbb {H}}_\sigma \) and \({\mathbb {V}}_\sigma \). This is due to the fact that \({\overline{{{\varvec{u}}}}}=\frac{1}{|\Omega |}\int _{\Omega } {{\varvec{u}}}\, \mathrm{d}x\) is not conserved by the flow of (1.1).
The estimate (4.12) is derived from [1, Lemma 3] and [22, Lemma 5.1]. More precisely, [22, Eq. (5.6)] yields
$$\begin{aligned}&\frac{1}{2} \frac{\mathrm{d}}{\mathrm{d}t}\Vert \nabla A^{-1} \partial _t^h \phi _m\Vert _{L^2(\Omega )}^2 + \frac{1}{2} \Vert \nabla \partial _t^h \phi _m\Vert _{L^2(\Omega )}^2 \le C\Vert \nabla A^{-1} \partial _t^h \phi _m|_{L^2(\Omega )}^2 \\&\qquad + ({{\varvec{v}}}(t+h)\partial _t^h \phi _m, \nabla A^{-1}\partial _t^h \phi _m) +(\partial _t^h {{\varvec{v}}}\phi _m, \nabla A^{-1}\partial _t^h \phi _m), \end{aligned}$$for \(t\in (0,T-h\), where \(\partial _t^h f (\cdot )= \frac{1}{h}(f(\cdot +h) -f(\cdot ))\) and \(A^{-1}\) is the inverse of the Laplace operator with Neumann boundary conditions. Since \(\phi _m\) is bounded [cf. (4.8)] and following [22, Lemma 5.1], we obtain (cf. also [1, Eq. (3.19)])
$$\begin{aligned}&\frac{1}{2} \frac{\mathrm{d}}{\mathrm{d}t}\Vert \nabla A^{-1} \partial _t^h \phi _m\Vert _{L^2(\Omega )}^2 + \frac{1}{4} \Vert \nabla \partial _t^h \phi _m\Vert _{L^2(\Omega )}^2 \\&\quad \le C(1+ \Vert {{\varvec{v}}}(t+h)\Vert _{L^2(\Omega )}^2) \Vert \nabla A^{-1} \partial _t^h \phi _m|_{L^2(\Omega )}^2+ C \Vert \partial _t^h {{\varvec{v}}}\Vert _{L^2(\Omega )}^2, \end{aligned}$$where C is independent of h. Thanks to \(\Vert \nabla A^{-1} \partial _t^h \phi (0)\Vert _L^2(\Omega )\le C(1+ \Vert \nabla \mu _0\Vert _{L^2(\Omega )}+ \Vert {{\varvec{v}}}\Vert _{L^\infty (0,T; L^2(\Omega ))})\) [cf. [1, Eq. (3.18)] and [14, Eq. 5.8]], we conclude from the Gronwall lemma that (4.12) holds replacing \(\partial _t \phi _m\) with \(\partial _t^h \phi _m\). Then, passing to the limit \(h\rightarrow 0\) as in [1, Lemma 3], we arrive at (4.12).
Let us consider the Lipschitz truncation of \(\phi _m\) defined by \(\phi _m^k=h_k(\phi _m)\), where \(h_k(s)=s\) for \(s\in (-1+\frac{1}{k},1-\frac{1}{k})\), \(h_k(s)= 1-\frac{1}{k}\) for \(s\in [1-\frac{1}{k},1)\), and \(h_k(s)=-1+\frac{1}{k}\) for \(s \in (-1, -1 +\frac{1}{k}]\). Thanks to (4.8), for almost every \(t\in (0,T)\), \(\phi _m^k \rightarrow \phi _m\) almost everywhere in \(\Omega \), thereby \(F'(\phi _m^k) \rightarrow F'(\phi _m)\), \(F''(\phi _m^k) \rightarrow F''(\phi _m)\) almost everywhere in \(\Omega \). Since \(|F'(\phi _m^k)|\le |F'(\phi _m)|\) and \(F'(\phi _m)\in L^\infty (0,T;L^2(\Omega ))\), we infer that \(F'(\phi _m^k) \rightarrow F'(\phi _m)\) in \(L^2(\Omega )\) for almost every \(t \in (0,T)\). Then, for any test function \(\varphi \in C_c^\infty (\Omega )\)
$$\begin{aligned} \int _{\Omega } F'(\phi _m) \partial _{x_i} \varphi \, \mathrm{d}x&= -\lim _{k\rightarrow \infty } \int _{\Omega }F'(\phi ^k_m) \partial _{x_i} \varphi \, \mathrm{d}x = \lim _{k \rightarrow \infty } \int _{\Omega } F''(\phi ^k_m)\partial _{x_i} \phi ^k_m \, \varphi \, \mathrm{d}x\\&= \lim _{k \rightarrow \infty } \int _{\Omega } F''(\phi ^k_m) \chi _{\lbrace \phi _m \in (-1+\frac{1}{k}, 1-\frac{1}{k})\rbrace } (\phi _m) \partial _{x_i} \phi _m \, \varphi \, \mathrm{d}x = \int _{\Omega } F''(\phi _m) \partial _{x_i} \phi _m \, \varphi \, \mathrm{d}x, \end{aligned}$$where the last limit follows from Lebesgue’s dominated convergence theorem since \(|F''(\phi ^k_m) \chi _{\lbrace \phi _m \in (-1+\frac{1}{k}, 1-\frac{1}{k})\rbrace } (\phi _m)|\le |F''(\phi _m)|\) and \(F''(\phi _m) \in L^\infty (0,T;L^2(\Omega ))\).
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Giorgini, A. Well-posedness of the two-dimensional Abels–Garcke–Grün model for two-phase flows with unmatched densities. Calc. Var. 60, 100 (2021). https://doi.org/10.1007/s00526-021-01962-2
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DOI: https://doi.org/10.1007/s00526-021-01962-2