Diffuse Interface models for incompressible binary fluids and the mass-conserving Allen-Cahn approximation
aa r X i v : . [ m a t h . A P ] M a y DIFFUSE INTERFACE MODELS FOR INCOMPRESSIBLE BINARY FLUIDSAND THE MASS-CONSERVING ALLEN-CAHN APPROXIMATION
ANDREA GIORGINI † , MAURIZIO GRASSELLI ‡ & HAO WU ∗† Department of Mathematics & Institute for Scientific Computing and Applied MathematicsIndiana UniversityBloomington, IN 47405, USA ‡ Dipartimento di MatematicaPolitecnico di MilanoMilano 20133, Italy ∗ School of Mathematical SciencesKey Laboratory of Mathematics for Nonlinear Sciences (Fudan University), Ministry of EducationShanghai Key Laboratory for Contemporary Applied MathematicsFudan UniversityShanghai 200433, China A BSTRACT . This paper is devoted to the mathematical analysis of some Diffuse Interface systems whichmodel the motion of a two-phase incompressible fluid mixture in presence of capillarity effects in abounded smooth domain. First, we consider a two-fluids parabolic-hyperbolic model that accounts forunmatched densities and viscosities without diffusive dynamics at the interface. We prove the existenceand uniqueness of local solutions. Next, we introduce dissipative mixing effects by means of the mass-conserving Allen-Cahn approximation. In particular, we consider the resulting nonhomogeneous Navier-Stokes-Allen-Cahn and Euler-Allen-Cahn systems with the physically relevant Flory-Huggins potential.We study the existence and uniqueness of global weak and strong solutions and their separation property.In our analysis we combine energy and entropy estimates, a novel end-point estimate of the product oftwo functions, and a logarithmic type Gronwall argument.
1. I
NTRODUCTION
The flow of a two-phase or multicomponent incompressible mixture is nowadays one of the mostattractive theoretical and numerical problems in Fluid Mechanics (see, for instance, [8, 32, 36, 50, 64]and the references therein). This is mainly due to the interplay between the motion of the interface
E-mail address : [email protected], [email protected], [email protected] . Date : May 18, 2020.2010
Mathematics Subject Classification.
Key words and phrases.
Diffuse Interface models, Navier-Stokes equations, Korteweg force, conserved Allen-Cahnequation, non-constant density, non-constant viscosity, logarithmic potential, inviscid flow. separating the two fluids (or phases) and the surrounding fluids. A natural description of this phenom-enon is based on a free-boundary formulation. Let Ω be a bounded domain in R d with d = 2 , , and T > . We assume that Ω is filled by two incompressible fluids (e.g. two liquids or a liquid and a gas),and we denote by Ω = Ω ( t ) and Ω = Ω ( t ) the subsets of Ω containing, respectively, the first andthe second fluid portions for any time t ≥ . The equations of motion are ( ρ (cid:0) ∂ t u + u · ∇ u (cid:1) − ν div D u + ∇ p = 0 , div u = 0 , in Ω × (0 , T ) ,ρ (cid:0) ∂ t u + u · ∇ u (cid:1) − ν div D u + ∇ p = 0 , div u = 0 , in Ω × (0 , T ) . (1.1)Here, u , u and p and p are, respectively, the velocities and pressures of the two fluids, while ρ , ρ and ν , ν are the (constant) densities and viscosities of the two fluids, respectively. The symmetricgradient is D = ( ∇ + ∇ t ) . The effect of the gravity are neglected for simplicity. Denoting by Γ = Γ( t ) the (moving) interface between Ω and Ω , system (1.1) can be equipped with the classicalfree boundary conditions u = u , (cid:0) ν D u − ν D u (cid:1) · n Γ = ( p − p + σH ) n Γ on Γ × (0 , T ) , (1.2)together with the no-slip boundary condition u = , u = on ∂ Ω × (0 , T ) . (1.3)The vector n Γ in (1.2) is the unit normal vector of the interface from ∂ Ω ( t ) , H is the mean curvatureof the interface ( H = − div n Γ ). In this setting, Γ( t ) is assumed to move with the velocity given by V Γ( t ) = u · n Γ( t ) . (1.4)The coefficient σ > is the surface tension, which introduces a discontinuity in the normal stressproportional to the mean curvature of the surface. Since dd t H d − (Γ( t )) = − R Γ( t ) HV Γ d H d − , where H d − is the d − -dimensional Hausdorff measure, the (formal) energy identity for system (1.1)-(1.2)is dd t n X i =1 , Z Ω i ( t ) ρ i | u i | d x + σ H d − (Γ( t )) o + X i =1 , Z Ω i ( t ) ν i | D u i | d x = 0 . (1.5)We refer the reader to [1, 19, 62–64, 71, 72] for the analysis of classical and varifold solutions to thesystem (1.1)-(1.3).The twofold Lagrangian and Eulerian nature of system (1.1)-(1.3) has led to the breakthrough idea(mainly from numerical analysts, see the review [67]) to reformulate the above system in the Euleriandescription by interpreting the effect of the surface tension as a singular force term localized at theinterface. Let us introduce the so-called level set function φ : Ω × (0 , T ) → R such that φ > in Ω × (0 , T ) , φ < in Ω × (0 , T ) , φ = 0 on Γ × (0 , T ) , namely the interface is the zero level set of φ . We consider the Heaviside type function K ( φ ) = φ > , φ = 0 , − φ < , (1.6) IFFUSE INTERFACE MODELS FOR INCOMPRESSIBLE BINARY FLUIDS 3 and we denote by u the velocity such that u = u in Ω × (0 , T ) and u = u in Ω × (0 , T ) . It wasshown in [17, Section 2] that the system (1.1)-(1.3) is formally equivalent to ρ ( φ ) (cid:0) ∂ t u + u · ∇ u (cid:1) − div ( ν ( φ ) D u ) + ∇ P = σH ( φ ) ∇ φδ ( φ ) , div u = 0 ,∂ t φ + u · ∇ φ = 0 , in Ω × (0 , T ) , (1.7)together with the boundary condition (1.3). Here ρ ( φ ) = ρ K ( φ )2 + ρ − K ( φ )2 , ν ( φ ) = ν K ( φ )2 + ν − K ( φ )2 , H ( φ ) = div (cid:18) ∇ φ |∇ φ | (cid:19) . Here, δ is the Dirac distribution, and ∇ φ is oriented as n Γ . The equation (1.7) represents the motionof the interface Γ that is simply transported by the flow. This follows from the immiscibility condition,which translates into ( u , ∈ Tan { ( x, t ) ∈ Ω × (0 , T ) : x ∈ Γ( t ) } . Although (1.7) seems to be moreamenable than (1.1)-(1.2), the presence of the Dirac mass still makes the analysis challenging. In theliterature, two different approaches have been used to overcome the singular nature of the right-handside of (1.7) , which both rely on the idea of continuous transition at the interface. The first approachis the Level Set method developed in the seminal works [17, 60, 61, 70] (see also the review [67]). Thisapproach consists in approximating the Heaviside function K ( φ ) by a smoothing regularization K ε ( φ ) .More precisely, for a given ε > , we introduce the function K ε ( φ ) = φ > ε, h φε + π sin (cid:0) πφε (cid:1)i | φ | ≤ ε, − φ < − ε. (1.8)The resulting approximating system reads as follows ρ ε ( φ ) (cid:0) ∂ t u + u · ∇ u (cid:1) − div ( ν ε ( φ ) D u ) + ∇ P = σH ( φ ) ∇ φδ ε ( φ ) , div u = 0 ,∂ t φ + u · ∇ φ = 0 , in Ω × (0 , T ) , (1.9)where ρ ε ( φ ) = ρ K ε ( φ )2 + ρ − K ε ( φ )2 , ν ε ( φ ) = ν K ε ( φ )2 + ν − K ε ( φ )2 , δ ε = d K ε ( φ )d φ . As a consequence of the approximation (1.8), the thickness of the interface is approximately ε |∇ φ | .This necessarily requires that |∇ φ | = 1 when | φ | ≤ ε , namely φ is a signed-distance function nearthe interface. However, even though the initial condition is suitably chosen, the evolution under thetransport equation (1.9) does not guarantee that this property remains true for all time. This fact hadled to different numerical algorithms aiming to avoid the expansion of the interface (see [67] and thereferences therein). In addition, as pointed out in [54], another drawback of this approach is that thedynamics is sensitive to the particular choice of the approximation for the surface stress tensor.The second approach is the so-called Diffuse Interface method (see [8, 24, 32]). This is based onthe postulate that the interface is a layer with positive volume, whose thickness is determined by theinteractions of particles occurring at small scales. In this context, the auxiliary function φ representsthe difference between the fluids concentrations (or rescaled density/volume fraction). This function GIORGINI-GRASSELLI-WU may exhibit a smooth transition at the interface, which is identified as intermediate level sets betweenthe two values and − . The evolution equations for the state variables (density, velocity, concen-tration) are derived by combining the theory of binary mixtures and the energy-based formalism fromthermodynamics and statistical mechanics. In this framework, the surface stress tensor is replaced bya diffuse stress tensor whose action is essentially localized in the regions of high gradients, namely, − σ div ( ∇ φ ⊗ ∇ φ ) . This tensor is known as (Korteweg) capillary tensor (cf., e.g., [8]). The resultingDiffuse Interface system, also called “complex fluid” model (see, e.g., [50, Sec.5]), is the following ρ ( φ ) (cid:0) ∂ t u + u · ∇ u (cid:1) − div ( ν ( φ ) D u ) + ∇ P = − σ div ( ∇ φ ⊗ ∇ φ ) , div u = 0 ,∂ t φ + u · ∇ φ = 0 , in Ω × (0 , T ) , (1.10)equipped with the no-slip boundary condition u = on ∂ Ω × (0 , T ) . (1.11)Here ρ ( φ ) = ρ φ ρ − φ , ν ( φ ) = ν φ ν − φ . (1.12)The energy associated to system (1.10) is defined as E ( u , φ ) = Z Ω ρ ( φ ) | u | + σ (cid:16) |∇ φ | + Ψ( φ ) (cid:17) d x, where Ψ is a double-well potential from [ − , → R , and the corresponding energy identity is dd t E ( u , φ ) + Z Ω ν ( φ ) | D u | d x = 0 . (1.13)This model dissipates energy due to viscosity, but there are no regularization effects for φ . It is worthnoting that (1.10) is also related to the models for viscoelastic fluids (see, for instance, [51]) or tothe two-dimensional incompressible MHD system without magnetic diffusion (cf., e.g., [65] and thereferences therein). Notice that, after rescaling the capillary tensor and the free energy by a parameter ε , it is possible to recognize the connection between (1.1)-(1.2) and (1.10). Indeed, we have formallythe convergences of the stress tensor (see, for instance, [5, 50, 64] for further details on the sharpinterface limit) − Z Ω σε div ( ∇ φ ⊗ ∇ φ ) · v d x ε → −−→ Z Γ σH n Γ · v d H d − , where the limit integral corresponds to the weak formulation of (1.2), and of the (Helmholtz) freeenergy R Ω (cid:0) ε |∇ φ | + ε Ψ( φ ) (cid:1) d x to the area functional H d − (Γ) (see [57]). Before proceeding withthe introduction of diffusive relaxations of the transport equation and their physical motivations, it isimportant to point out two main properties of (1.10) -(1.10) :1. Conservation of mass: Z Ω φ ( t ) d x = Z Ω φ d x, ∀ t ∈ [0 , T ] . (1.14)2. Conservation of L ∞ (Ω) -norm: k φ ( t ) k L ∞ (Ω) = k φ k L ∞ (Ω) , ∀ t ∈ [0 , T ] , (1.15) IFFUSE INTERFACE MODELS FOR INCOMPRESSIBLE BINARY FLUIDS 5 which implies that − ≤ φ ( x ) ≤ a.e. in Ω ⇒ − ≤ φ ( x, t ) ≤ a.e. in Ω × (0 , T ) . (1.16)The theory of binary mixtures takes into accounts dissipative mechanisms occurring at the inter-faces. The molecules of two fluids interact at a miscoscopic scale, and their disposition is the resultof a competition between the diffusion of molecules and the attraction of molecules of the same fluid(mixing vs demixing or “philic” vs “phobic” effects). This liquid-liquid phase separation phenome-non, though already well-known in Materials Science, has recently become a sort of paradigm in CellBiology (see, for instance, [6, 11, 42, 68]). This competition is described in the Helmholtz free energyof the system E ( φ ) defined by E ( φ ) = Z Ω |∇ φ | + Ψ( φ ) d x. The first term describes weakly non-local interactions (see [14], cf. also [21]). The potential Ψ is theFlory-Huggins free energy density Ψ( s ) = θ s ) log(1 + s ) + (1 − s ) log(1 − s )] − θ s , s ∈ [ − , . (1.19)We consider hereafter the case < θ < θ , which implies, in particular, that Ψ is a non-convexpotential . It is worth mentioning that the Landau theory that leads to the well-known Ginzburg-Landau free energy is just an approximation of the above E ( φ ) obtained through a Taylor expansion ofthe logarithmic potential Ψ . This choice is very common in the related literature (see, for instance, [13] For a system of finite number of molecules A and B occupying a lattice with M sites, the thermodynamic propertiesof the system of molecules are derived from the partition function Z = X Ω e (cid:16) H ( σ ,...,σM ) kBT (cid:17) (1.17)where the Hamiltonian H ( σ , . . . , σ M ) denotes the energy of the arrangement σ , . . . , σ M ( σ n = 1 if the lattice is occupiedby molecule A , σ n = 0 otherwise), and Ω is the set of all possible arrangements. Here k B is the Boltzmann constant and T is the temperature. It is common to describe only nearest neighbor interactions between particles, which lead to theparticular Hamiltonian H ( { σ } ) = 12 X m,n (cid:16) e AA σ m σ n + e BB (1 − σ m )(1 − σ n ) + e AB (cid:0) σ m (1 − σ n ) + σ n (1 − σ m ) (cid:1)(cid:17) , (1.18)where e AA , e BB , and e AB are coefficients. In the Mean Field approximation the arrangements σ n and − σ n are approx-imated by the probability (average) that a site is occupied by a molecule A and B , namely φ A = N A M and φ B = N B M ( N A and N B are the number of molecules of type A and B , and M = N A + N B ). Then, the partition function is given by Z = M ! N A ! N B ! e − H( φ A ,φ B)kBT , H( φ A , φ B ) = zM2 (cid:0) e AA φ + 2e AB φ A φ B + e BB φ (cid:1) , where z is the number of neighbors in a lattice. By using the Stirling approximation, the free energy density reads as f ( φ A , φ B ) = − k B T ln ZM ≈ k B Tν h φ A ln φ A + φ B ln φ B i + z ν h e AA φ A + 2 e AB φ A φ B + e BB φ B i , where ν is the volume of molecules. By defining φ = φ A − φ B (with range [ − , ), and setting appropriately the constants θ and θ , the Flory-Huggins potential (1.19) immediately follows. As usual, the function Ψ is meant as the continuousextension at the values s = ± . Roughly speaking, the logarithmic term accounts for the entropy after mixing and thequadratic perturbation represents the internal energy after mixing. For more details, we refer the reader to [46]. In the case, θ ≥ θ , mixing prevails over demixing, and no separation takes place. GIORGINI-GRASSELLI-WU and [22]). However, it has the main drawback that the solution does not belong in general to thephysical interval [ − , (cf. (1.16)).In order to include dissipative mechanisms in the dynamics of the concentration, we define the firstvariation of the Helmholtz free energy. This is called chemical potential and it is given by µ = δ E ( φ ) δφ = − ∆ φ + Ψ ′ ( φ ) . Two fundamental relaxation models proposed in the Diffuse Interface theory for binary mixture are thefollowing modifications of the transport equation (1.10) : . Mass-conserving Allen-Cahn dynamics ( [66, 79]) ∂ t φ + u · ∇ φ + γ (cid:0) µ − µ (cid:1) = 0 in Ω × (0 , T ) , ∂ n φ = 0 on ∂ Ω × (0 , T );2 . Cahn-Hilliard dynamics ( [14, 15]) ∂ t φ + u · ∇ φ − γ ∆ µ = 0 in Ω × (0 , T ) , ∂ n φ = ∂ n µ = 0 on ∂ Ω × (0 , T ) . Here µ is the spatial average defined by µ = 1 | Ω | Z Ω µ d x, and γ is the elastic relaxation time. We point out that from the thermodynamic viewpoint the relax-ation terms describe dissipative diffusional flux at the interface (cf. [38, 54]). As for the transportequation, both the mass-conserving Allen-Cahn and Cahn-Hilliard equations satisfy the conservationproperties (1.14) and (1.16). In addition, their dynamics maintain the integrity of the interface: themixing-demixing mechanism (which also translates into µ ) allows a balance which avoids uncontrolledexpansion or shrinkage of the interface layer (cf. [24]).In this work, we study a Diffuse Interface model that has been recently derived in [32, Part I, Chap.2,4.2.1]. It accounts for unmatched densities and viscosities of the fluids, as well as dissipation due tointerface mixing. The dynamics of φ is described through the following modification of the transportequation ∂ t φ + u · ∇ φ + γ (cid:16) µ + ρ ′ ( φ ) | u | − ξ (cid:17) = 0 , in Ω × (0 , T ) , where u denotes the volume averaged fluid velocity and ξ ( t ) = 1 | Ω | Z Ω µ + ρ ′ ( φ ) | u | x, in (0 , T ) . Here the dissipation mechanism is similar to that of the mass-conserving Allen-Cahn dynamics, butit also includes an extra term due to the difference of densities. We thus have the nonhomogeneousNavier-Stokes-Allen-Cahn system ρ ( φ ) (cid:0) ∂ t u + u · ∇ u (cid:1) − div ( ν ( φ ) D u ) + ∇ P = − σ div ( ∇ φ ⊗ ∇ φ ) , div u = 0 ,∂ t φ + u · ∇ φ + γ (cid:0) µ + ρ ′ ( φ ) | u | − ξ (cid:1) = 0 , in Ω × (0 , T ) . (1.20) This equation differs from the classical Allen-Cahn equation due to the presence of term µ (see [66, 79], cf. also[7, 12, 18, 35, 58, 74]). IFFUSE INTERFACE MODELS FOR INCOMPRESSIBLE BINARY FLUIDS 7
This system is usually subject to a no-slip boundary condition for u and a homogeneous Neumannboundary condition for φ , namely u = , ∂ n φ = 0 on ∂ Ω × (0 , T ) . (1.21)In the last part of this work, we will consider the mass-conserving Euler-Allen-Cahn system ∂ t u + u · ∇ u + ∇ P = − σ div ( ∇ φ ⊗ ∇ φ ) , div u = 0 ,∂ t φ + u · ∇ φ + γ (cid:0) µ − µ (cid:1) = 0 , in Ω × (0 , T ) , (1.22)endowed with the boundary conditions u · n = 0 , ∂ n φ = 0 on ∂ Ω × (0 , T ) . (1.23)The above model is obtained from (1.20) in the case of inviscid flow and matched densities. We observethat other boundary conditions can be considered, for instance, periodic (cf. [32, Part I, Chap.2, 4.2.3])and also [55] for moving contact lines.The mathematical literature concerning systems similar to the Navier-Stokes-Allen-Cahn system(1.20)-(1.21) has been widely developed in last years, in terms of both physical modeling and well-posedness analysis. First, we report that there are different ways of accounting for the unmatcheddensities for incompressible binary mixtures. Among the existing literature, we mention [4, 10, 27,38, 39, 54]. The model herein studied is derived via an energetic variational approach in [37] (seealso [32] and, for the Navier-Stokes-Cahn-Hilliard system [53]). The system (1.20)-(1.21) has beeninvestigated in [37] in the case of constant viscosity and standard Allen-Cahn equation with regularLandau potential Ψ ( s ) = ( s − and no mass conservation. The authors prove the existence of aglobal weak solution in three dimensions and the existence as well as uniqueness of the global strongsolution in two dimensions. In the latter case, they also show the convergence of a weak solution to asingle stationary state and they establish the existence of a global attractor. Thanks to their choice ofpotential and the absence of mass constraint, the authors can easily ensure that φ takes values in thephysical range [ − , . This fact is crucial for their proofs. However, the mass constraint would notallow to establish a comparison principle even if the double-well potential is smooth. We also mentionthe previous contributions [29, 30, 40, 76–78] for the case with constant density, and [5] for the sharpinterface limit in the Stokes case. Additionally, there are works devoted to Navier-Stokes-Allen-Cahnmodels in which density is regarded as an independent variable (see, for instance, [23, 47, 48]). Inthese works the potential is the classical Landau double-well and there is no mass conservation. The(non-conserved) compressible case (see [9, 27] for modeling issues) has been analyzed, for instance,in [20, 25, 44, 80] (see also [75] for sharp interface limits). On the other hand, in comparison with theviscous case above-mentioned, only few works have been addressed with the Euler-Allen-Cahn system(1.22)-(1.23). In this respect, we mention [81] (see also [28] for a nonlocal model), where the authorsprove local existence of smooth solutions for the Euler-Allen-Cahn in the case of no-mass conservationand Landau potential.The aim of this paper is to address the existence, uniqueness and (possibly) regularity of the solutionsto the aforementioned Diffuse Interface systems : the complex fluid model (1.10)-(1.11), the Navier-Stokes-Allen-Cahn system (1.20)-(1.21), and the Euler-Allen-Cahn system (1.22)-(1.23). On one hand,the purpose of our analysis is to stay as close as possible to a thermodynamically grounded framework Without loss of generality, we consider the values of the parameters σ = γ = 1 in our analysis. GIORGINI-GRASSELLI-WU by keeping densities and viscosities to be dependent on φ , and the physically relevant Flory-Hugginspotential (1.19). Although this choice requires some technical efforts, it provides results which arephysically more reasonable. On the other hand, by working in this general setting, we demonstratethat the dynamics originating from a general initial condition become global (in time) when the mass-conserving Allen-Cahn relaxation is taken into account. The latter is achieved in three dimensions forfinite energy (weak) solutions, and in two dimensions even for more regular solutions in the case ofnon-constant density and viscosity and of constant density and zero viscosity.Before concluding this introduction, we make some more precise comments on the analysis and onthe main novelties of our techniques. First, we recall that the existence and uniqueness of local (in time)regular solutions to the complex fluid system (1.10)-(1.11) has been proven in [51, 52] for constantdensity and viscosity. Here we generalize this result by allowing ρ and ν to depend on φ and taking amore general initial datum ( u , φ ) ∈ ( V σ ∩ H (Ω)) × W ,p (Ω) , with p > in two dimensions and p > in three dimensions (see Theorem 3.1). Next, we study the Navier-Stokes-Allen-Cahn system(1.20)-(1.21). We prove the existence of a global weak solution with ( u , φ ) ∈ H σ × H (Ω) (seeTheorems 4.1 and 4.2), and the existence of a global strong solution with ( u , φ ) ∈ V σ × H (Ω) suchthat Ψ ′ ( φ ) ∈ L (Ω) (see Theorem 5.1). For the latter, we combine a classical energy approach, anew end-point estimate of the product of two functions in L (Ω) (see Lemma 2.1 below), and a newestimate for the Stokes system with non-constant viscosity. The proof is concluded with a logarithmicGronwall argument that leads to double-exponential control. However, in light of the singularity of theFlory-Huggins potential, the uniqueness of these strong solutions seem to be a hard task. To overcomethis issue, we then establish global estimates on the derivatives of the entropy (entropy estimates)provided that k ρ ′ k L ∞ ( − , is small enough and F ′′ ( φ ) ∈ L (Ω) . These estimates allow us to provethat F ′′ ( φ ) log(1 + F ′′ ( φ )) ∈ L (Ω × (0 , T )) , and, in turn, the uniqueness of such strong solutions.As a consequence of these entropy estimates, we achieve the so-called uniform separation property.The latter says that φ stays uniformly away from the pure states in finite time . This fact, besidesbeing physically relevant, entails further regularity properties of the solution. Note that in the case ofa smooth potential and no mass conservation (cf. [37]) this issue is trivial since the potential is smoothand a comparison principle holds. Finally, we consider the inviscid case, namely the Euler-Allen-Cahn system (1.22)-(1.23). Although this system turns out to be similar to the MHD equations withmagnetic diffusion and without viscosity, the classical argument in the literature (see, e.g., [16]) doesnot apply because of the logarithmic potential. In our proof, it is crucial to make use of the structure ofthe incompressible Euler equations (1.22) -(1.22) , and the end-point estimate of the product (Lemma2.1). This gives the existence of global solutions with ( u , φ ) ∈ ( H σ ∩ H (Ω)) × H (Ω) in twodimensions. Next, in light of the entropy estimates, we also prove the existence of smoother globalsolutions originating from ( u , φ ) ∈ ( H σ ∩ W ,p (Ω)) × H (Ω) provided that p > and ∇ µ := ∇ ( − ∆ φ + Ψ ′ ( φ )) ∈ L (Ω) . Plan of the paper.
In Section 2 we introduce the notation, some functional inequalities and thenprove an estimate for the product of two functions. In Section 3 we show the local well-posedness of We define the (mixing) entropy as F ( s ) = θ [(1 + s ) log(1 + s ) + (1 − s ) log(1 − s )] , for s ∈ [ − , . This corre-sponds to the convex part of (1.19). It is worth pointing out that the initial concentration φ for strong solutions is not separated from the pure phases.Indeed, the imposed conditions F ′ ( φ ) ∈ L (Ω) or F ′′ ( φ ) ∈ L (Ω) allow φ being arbitrarily close to +1 and − . IFFUSE INTERFACE MODELS FOR INCOMPRESSIBLE BINARY FLUIDS 9 system (1.10)-(1.11). Section 4 is devoted to the existence of global weak solutions for the Navier-Stokes-Allen-Cahn system (1.20)-(1.21). In Section 5 we study the existence and uniqueness of strongsolutions to the Navier-Stokes-Allen-Cahn system (1.20)-(1.21). Section 6 is devoted to the globalexistence of solutions to the Euler-Allen-Cahn system (1.22)-(1.23). In Appendix A we prove a resulton the Stokes problem with variable viscosity, and in Appendix B we recall the Osgood lemma andtwo logarithmic versions of the Gronwall lemma.2. P
RELIMINARIES
Notation.
For a real Banach space X , its norm is denoted by k · k X . The symbol h· , ·i X ′ ,X stands for the duality pairing between X and its dual space X ′ . The boldface letter X denotes thevectorial space endowed with the product structure. We assume that Ω ⊂ R d , d = 2 , , is a boundeddomain with smooth boundary ∂ Ω , n is the unit outward normal vector on ∂ Ω , and ∂ n denotes the outernormal derivative on ∂ Ω . We denote the Lebesgue spaces by L p (Ω) ( p ≥ with norms k · k L p (Ω) .When p = 2 , the inner product in the Hilbert space L (Ω) is denoted by ( · , · ) . For s ∈ R , p ≥ , W s,p (Ω) is the Sobolev space with corresponding norm k · k W s,p (Ω) . If p = 2 , we use the notation W s,p (Ω) = H s (Ω) . For every f ∈ ( H (Ω)) ′ , we denote by f the generalized mean value over Ω defined by f = | Ω | − h f, i ( H (Ω)) ′ ,H (Ω) . If f ∈ L (Ω) , then f = | Ω | − R Ω f d x . Thanks to thegeneralized Poincar´e inequality, there exists a positive constant C = C (Ω) such that k f k H (Ω) ≤ C (cid:16) k∇ f k L (Ω) + (cid:12)(cid:12)(cid:12) Z Ω f d x (cid:12)(cid:12)(cid:12) (cid:17) , ∀ f ∈ H (Ω) . (2.1)We introduce the Hilbert space of solenoidal vector-valued functions H σ = { u ∈ L (Ω) : div u = 0 , u · n = 0 on ∂ Ω } = C ∞ ,σ (Ω) L (Ω) , V σ = { u ∈ H (Ω) : div u = 0 , u = on ∂ Ω } = C ∞ ,σ (Ω) H (Ω) , where C ∞ ,σ (Ω) is the space of divergence free vector fields in C ∞ (Ω) . We also use ( · , · ) and k · k L (Ω) for the inner product and the norm in H σ . The space V σ is endowed with the inner product and norm ( u , v ) V σ = ( ∇ u , ∇ v ) and k u k V σ = k∇ u k L (Ω) , respectively. We denote by V ′ σ its dual space. Werecall the Korn inequality k∇ u k L (Ω) ≤ √ k D u k L (Ω) ≤ √ k∇ u k L (Ω) , ∀ u ∈ V σ , (2.2)where D u = (cid:0) ∇ u + ( ∇ u ) t (cid:1) . We define the Hilbert space W σ = H (Ω) ∩ V σ with inner product andnorm ( u , v ) W σ = ( A u , A v ) and k u k W σ = k A u k , where A is the Stokes operator. We recall that thereexists C > such that k u k H (Ω) ≤ C k u k W σ , ∀ u ∈ W σ . (2.3)2.2. Analytic tools.
We recall the Ladyzhenskaya, Agmon, Gagliardo-Nirenberg, Brezis-Gallouet-Wainger and trace interpolation inequalities: k f k L (Ω) ≤ C k f k L (Ω) k f k H (Ω) , ∀ f ∈ H (Ω) , d = 2 , (2.4) k f k L p (Ω) ≤ Cp k f k p L (Ω) k f k − p H (Ω) , ∀ f ∈ H (Ω) , ≤ p < ∞ , d = 2 , (2.5) k f k L p (Ω) ≤ C ( p ) k f k − p p L (Ω) k f k p − p H (Ω) , ∀ f ∈ H (Ω) , ≤ p ≤ , d = 3 , (2.6) k f k L ∞ (Ω) ≤ C k f k L (Ω) k f k H (Ω) , ∀ f ∈ H (Ω) , d = 2 , (2.7) k∇ f k W , (Ω) ≤ C k f k H (Ω) k f k L ∞ (Ω) , ∀ f ∈ H (Ω) , d = 3 , (2.8) k f k L ∞ (Ω) ≤ C k f k H (Ω) log (cid:16) e k f k H (Ω) k f k H (Ω) (cid:17) , ∀ f ∈ H (Ω) , d = 2 , (2.9) k f k L ∞ (Ω) ≤ C ( p ) k f k H (Ω) log (cid:16) C ( p ) k f k W ,p (Ω) k f k H (Ω) (cid:17) , ∀ f ∈ W ,p (Ω) , p > , d = 2 , (2.10) k f k L ( ∂ Ω) ≤ C k f k L (Ω) k f k H (Ω) , ∀ f ∈ H (Ω) , d = 2 . (2.11)Here, the constant C depends only on Ω , whereas the constant C ( p ) depends on Ω and p .We now prove the following end-point estimate for the product of two functions, which will play animportant role in the subsequent analysis. This is a generalization of [34, Proposition C.1]. Lemma 2.1.
Let Ω be a bounded domain in R with smooth boundary. Assume that f ∈ H (Ω) and g ∈ L p (Ω) for some p > , g is not identical to . Then, we have k f g k L (Ω) ≤ C (cid:16) pp − (cid:17) k f k H (Ω) k g k L (Ω) log (cid:16) e | Ω | p − p k g k L p (Ω) k g k L (Ω) (cid:17) , (2.12) for some positive constant C depending only on Ω .Proof. Let us consider the Neumann operator A = − ∆ + I on L (Ω) with domain D ( A ) = { u ∈ H (Ω) : ∂ n u = 0 on ∂ Ω } . By the classical spectral theory, there exists a sequence of positive eigenval-ues λ k ( k ∈ N ) associated with A such that λ = 1 , λ k ≤ λ k +1 and λ k → + ∞ as k goes to + ∞ . Thesequence of eigenfunctions w k ∈ D ( A ) satisfying Aw k = λ k w k forms an orthonormal basis in L (Ω) and an orthogonal basis in H (Ω) . Let us fix N ∈ N whose value will be chosen later. We write f asfollows f = N X n =0 f n + f ⊥ N , (2.13)where f n = X k : e n ≤√ λ k 2) log e | Ω | p − p k g k L p (Ω) k g k L (Ω) ! ≤ N + 1 < p p − 2) log e | Ω | p − p k g k L p (Ω) k g k L (Ω) ! . We observe that the logarithm term in the above relations is greater than for any function g ∈ L p (Ω) with p > , g = 0 . Then by using the choice of N in (2.14), we infer that k f g k L (Ω) ≤ C k f k H (Ω) k g k L (Ω) e " p p − 2) log (cid:16) e | Ω | p − p k g k L p (Ω) k g k L (Ω) (cid:17) + 2 pe ( p − | Ω | p − p ! ≤ C k f k H (Ω) k g k L (Ω) e p ( p − 2) log (cid:16) e | Ω | p − p k g k L p (Ω) k g k L (Ω) (cid:17) + 2 pe ( p − | Ω | p − p ! , which implies the desired conclusion. (cid:3) Remark 2.2. The conclusion of Lemma 2.1 holds as well in T . Remark 2.3. It is well-known that H (Ω) is not an algebra in two dimensions. An interesting appli-cation of Lemma 2.1 together with the Brezis-Gallouet-Wainger inequality (2.10) is that k f g k H (Ω) ≤ C k f k H (Ω) k g k H (Ω) log (cid:16) C k g k W ,p (Ω) k g k H (Ω) (cid:17) , for any f ∈ H (Ω) , g ∈ W ,p (Ω) for some p > , where C and C are two positive constantsdepending only on Ω and p . Remark 2.4. Lemma 2.1 can be regarded as a generalization of H¨older and Young inequalities. Thisinequality is sharp since the product between f and g is not defined in L (Ω) if f ∈ H (Ω) and g ∈ L (Ω) . Indeed, we have the following counterexample in R : g ( x ) = 1 | x | log (cid:0) | x | (cid:1) , f ( x ) = log − (cid:16) | x | (cid:17) , < x ≤ . We notice that g ∈ L ( B R (0 , since Z B R (0 , | g ( x ) | d x = 2 π Z r log ( r (cid:1) d r = 2 π Z + ∞ s log ( s ) d s < + ∞ . However, g / ∈ L p ( B R (0 , for any p > because Z B R (0 , | g ( x ) | p d x = 2 π Z r p − log p ( r (cid:1) d r = 2 π Z + ∞ s − p log p ( s ) d s = + ∞ . We easily observe that f ∈ L ( B R (0 , , but f / ∈ L ∞ ( B R (0 , since lim | x |→ f ( x ) = + ∞ . This,in turn, implies that f / ∈ W ,p ( B R (0 , for any p > , due to the Sobolev embedding theorem.Nonetheless, we have ∂ x i f ( x ) = (cid:16) − (cid:17) x i | x | + (cid:0) | x | (cid:1) , i = 1 , , such that Z B R (0 , | ∂ x i f ( x ) | d x ≤ π (cid:16) − (cid:17) Z r log + ) (cid:0) r (cid:1) d r ≤ C Z r log (cid:0) r (cid:1) < + ∞ . Thus, we have f ∈ W , ( B R (0 , . Finally, we observe that Z B R (0 , | g ( x ) f ( x ) | d x = Z B R (0 , log − (cid:0) | x | (cid:1) | x | log (cid:0) | x | (cid:1) d x = 2 π Z r log + (cid:0) r (cid:1) d r = + ∞ , IFFUSE INTERFACE MODELS FOR INCOMPRESSIBLE BINARY FLUIDS 13 namely, the product f g / ∈ L ( B R (0 , . The above counterexample can be generalized to any pair offunctions g ( x ) = 1 | x | log α (cid:0) | x | (cid:1) , f ( x ) = log β (cid:0) | x | (cid:1) , x ∈ B R (0 , , where < α < and β < such that α − β < . 3. C OMPLEX F LUIDS M ODEL : L OCAL W ELL - POSEDNESS In this section we consider the complex fluids system ρ ( φ )( ∂ t u + u · ∇ u ) − div ( ν ( φ ) D u ) + ∇ P = − div ( ∇ φ ⊗ ∇ φ ) , div u = 0 ,∂ t φ + u · ∇ φ = 0 , in Ω × (0 , T ) , (3.1)subject to the boundary condition u = on ∂ Ω × (0 , T ) , (3.2)and to the initial conditions u ( · , 0) = u , φ ( · , 0) = φ in Ω . (3.3)We recall that u is the (volume) averaged velocity of the binary mixture, P is the pressure, and φ denotes the difference of the concentrations (volume fraction) of the two fluids. The coefficients ρ ( · ) and ν ( · ) represent the density and the viscosity of the mixture depending on φ . Throughout this work,motivated by the linear interpolation density and viscosity functions in (1.12), we assume that ρ, ν ∈ C ([ − , ρ ( s ) ∈ [ ρ , ρ ] , ν ( s ) ∈ [ ν , ν ] for s ∈ [ − , , (3.4)where ρ , ρ and ν , ν are, respectively, the (positive) densities and viscosities of two homogeneous(different) fluids. In addition, we will use the notation ρ ∗ = min { ρ , ρ } > , ν ∗ = min { ν , ν } > . The aim of this section is to prove the local existence and uniqueness of regular solutions to problem(3.1)-(3.3) with general initial data. This generalizes [52, Theorem 1.1] to the case with unmatcheddensities and viscosities depending on the concentration, and to initial data φ belonging to W ,p (Ω) ,instead of φ ∈ H (Ω) (see also [51, Theorem 2.2] for the Cauchy problem in R ). Theorem 3.1 (Local well-posedness in 2D) . Let Ω be a smooth bounded domain in R . For any initialdatum ( u , φ ) such that u ∈ V σ ∩ H (Ω) , φ ∈ W ,p (Ω) for some p > , with | φ ( x ) | ≤ , forall x ∈ Ω , there exists a positive time T , which depends only on the norms of the initial data, and aunique solution ( u , φ ) to problem (3.1) - (3.3) on [0 , T ] such that u ∈ L ∞ (0 , T ; V σ ∩ H (Ω)) ∩ L pp − (0 , T ; W ,p (Ω)) ∩ W , (0 , T ; V σ ) ∩ W , ∞ (0 , T ; H σ (Ω)) ,φ ∈ L ∞ (0 , T ; W ,p (Ω)) ∩ W , ∞ (0 , T ; H (Ω) ∩ L ∞ (Ω)) , | φ ( x, t ) | ≤ × [0 , T ] . Proof. We perform some a priori estimates for the solutions to problem (3.1)-(3.3), and then we provethe uniqueness. With these arguments, the existence of local solutions to (3.1)-(3.3) follows from themethod of successive approximations (Picard’s method). This relies on the definition of a suitablesequence ( u k , φ k ) via an iteration scheme, a priori bounds on ( u k , φ k ) in terms of ( u k − , φ k − ) , anduniform estimates of ( u k − u k − , φ k − φ k − ) (by arguing as in the uniqueness proof reported below). Werefer to [45] for the details of this type of argument in the case of the nonhomogeneous Navier-Stokesequations (see also, e.g., [48] for the Navier-Stokes-Allen-Cahn system). First estimate. Multiplying (3.1) by u and integrating over Ω , we find Z Ω ρ ( φ ) ∂ t | u | d x + 12 Z Ω ρ ( φ ) u · ∇ (cid:0) | u | (cid:1) d x + Z Ω ν ( φ ) | D u | d x = − Z Ω ∆ φ ∇ φ · u d x. Taking the gradient of (3.1) , we have ∇ ∂ t φ + ∇ ( u · ∇ φ ) = 0 . Multiplying the above identity by ∇ φ , integrating over Ω and using the no-slip boundary condition of u , we obtain 12 dd t Z Ω |∇ φ | d x − Z Ω ( u · ∇ φ )∆ φ d x = 0 . By adding the two obtained equations, and using the identity ∂ t ρ ( φ ) + div ( ρ ( φ ) u ) = ρ ′ ( φ ) (cid:0) ∂ t φ + u · ∇ φ (cid:1) = 0 , we find the basic energy law 12 dd t Z Ω (cid:16) ρ ( φ ) | u | + |∇ φ | (cid:17) d x + Z Ω ν ( φ ) | D u | d x = 0 . Integrating over [0 , t ] , we obtain E ( u ( t ) , φ ( t )) + Z t Z Ω ν ( φ ) | D u | d x = E ( u , φ ) , ∀ t ≥ . where E ( u , φ ) = 12 Z Ω ρ ( φ ) | u | + |∇ φ | d x. In addition, the transport equation yields that, for all p ∈ [2 , ∞ ] , it holds k φ ( t ) k L p (Ω) = k φ k L p (Ω) , ∀ t ≥ . (3.5)Thus, we infer that u ∈ L ∞ (0 , T ; H σ ) ∩ L (0 , T ; V σ ) , φ ∈ L ∞ (0 , T ; H (Ω) ∩ L ∞ (Ω)) . (3.6) Second estimate. We multiply (3.1) by ∂ t u and integrate over Ω . After integrating by parts andusing the fact that ∂ t u = 0 on ∂ Ω , we obtain 12 dd t Z Ω ν ( φ ) | D u | d x + Z Ω ρ ( φ ) | ∂ t u | d x = 12 Z Ω ν ′ ( φ ) ∂ t φ | D u | d x − Z Ω ρ ( φ )( u · ∇ ) u · ∂ t u d x + Z Ω ( ∇ φ ⊗ ∇ φ ) : ∇ ∂ t u d x. IFFUSE INTERFACE MODELS FOR INCOMPRESSIBLE BINARY FLUIDS 15 Combining (2.4), (2.7) and (3.6), together with the relation ∂ t φ = − u · ∇ φ , we have 12 dd t Z Ω ν ( φ ) | D u | d x + Z Ω ρ ( φ ) | ∂ t u | d x ≤ C k ∂ t φ k L (Ω) k D u k L (Ω) + C k u k L ∞ (Ω) k∇ u k L (Ω) k ∂ t u k L (Ω) + k∇ φ k L (Ω) k∇ ∂ t u k L (Ω) ≤ C k u · ∇ φ k L (Ω) k D u k L (Ω) k u k H (Ω) + C k u k L (Ω) k u k H (Ω) k∇ u k L (Ω) k ∂ t u k L (Ω) + C k∇ φ k L (Ω) k φ k H (Ω) k∇ ∂ t u k L (Ω) ≤ ρ ∗ k ∂ t u k L (Ω) + ν ∗ k D∂ t u k L (Ω) + C k u k L ∞ (Ω) k∇ φ k L (Ω) k D u k L (Ω) k u k H (Ω) + C k∇ u k L (Ω) k u k H (Ω) + C k φ k H (Ω) ≤ ρ ∗ k ∂ t u k L (Ω) + ν ∗ k D∂ t u k L (Ω) + C k D u k L (Ω) k u k H (Ω) + C k D u k L (Ω) k u k H (Ω) + C k φ k H (Ω) . (3.7)Here we have also used that k∇ u k L (Ω) is equivalent to k D u k L (Ω) thanks to (2.2). Next, we rewrite(3.1) -(3.1) as a Stokes problem with non-constant viscosity − div ( ν ( φ ) D u ) + ∇ P = f , in Ω × (0 , T ) , div u = 0 , in Ω × (0 , T ) , u = , on ∂ Ω × (0 , T ) , where f = − ρ ( φ )( ∂ t u + u · ∇ u ) − div( ∇ φ ⊗ ∇ φ ) . By exploiting Theorem A.1 with p = 2 , s = 2 , r = ∞ , we infer that k u k H (Ω) ≤ C k ρ ( φ ) ∂ t u k L (Ω) + C k ρ ( φ )( u · ∇ u ) k L (Ω) + C k div( ∇ φ ⊗ ∇ φ ) k L (Ω) + C k∇ φ k L ∞ (Ω) k D u k L (Ω) ≤ C k ∂ t u k L (Ω) + C k u k L ∞ (Ω) k∇ u k L (Ω) + C k∇ φ k L ∞ (Ω) (cid:0) k φ k H (Ω) + k D u k L (Ω) (cid:1) ≤ C k ∂ t u k L (Ω) + C k u k H (Ω) k D u k L (Ω) + C k∇ φ k L ∞ (Ω) (cid:0) k φ k H (Ω) + k D u k L (Ω) (cid:1) . Here we have used (2.7) and (3.6). Thus, by Young’s inequality we find k u k H (Ω) ≤ C k ∂ t u k L (Ω) + C k D u k L (Ω) + C k∇ φ k L ∞ (Ω) (cid:0) k φ k H (Ω) + k D u k L (Ω) (cid:1) . (3.8)Inserting (3.8) into (3.7), and using again Young’s inequality, we get 12 dd t Z Ω ν ( φ ) | D u | d x + ρ ∗ Z Ω | ∂ t u | d x ≤ ν ∗ k D∂ t u k L (Ω) + C k D u k L (Ω) + C k∇ φ k L ∞ (Ω) ( k φ k H (Ω) + k D u k L (Ω) ) + C k φ k H (Ω) . (3.9) Third estimate. We differentiate (3.1) with respect to the time to obtain ρ ( φ ) ∂ tt u + ρ ( φ ) (cid:0) ∂ t u · ∇ u + u · ∇ ∂ t u (cid:1) + ρ ′ ( φ ) ∂ t φ ( ∂ t u + u · ∇ u ) − div ( ν ( φ ) D∂ t u ) − div ( ν ′ ( φ ) ∂ t φD u ) + ∇ ∂ t P = − div ( ∇ φ ⊗ ∇ ∂ t φ + ∇ ∂ t φ ⊗ ∇ φ ) . Multiplying the above equation by ∂ t u and integrating over Ω , we are led to Z Ω ρ ( φ ) ∂ t | ∂ t u | d x + Z Ω ρ ( φ ) (cid:0) ∂ t u · ∇ u + u · ∇ ∂ t u (cid:1) · ∂ t u d x + Z Ω ρ ′ ( φ ) ∂ t φ ( ∂ t u + u · ∇ u ) · ∂ t u d x + Z Ω ν ( φ ) | D∂ t u | d x + Z Ω ν ′ ( φ ) ∂ t φD u : D∂ t u d x = Z Ω (cid:0) ∇ φ ⊗ ∇ ∂ t φ + ∇ ∂ t φ ⊗ ∇ φ (cid:1) : ∇ ∂ t u d x. Since Z Ω ρ ( φ ) ∂ t | ∂ t u | d x + 12 Z Ω ρ ( φ ) u · ∇| ∂ t u | d x = 12 dd t Z Ω ρ ( φ ) | ∂ t u | d x − Z Ω (cid:0) ∂ t ρ ( φ ) + div ( ρ ( φ ) u ) (cid:1)| {z } =0 | ∂ t u | d x = 12 dd t Z Ω ρ ( φ ) | ∂ t u | d x, we have 12 dd t Z Ω ρ ( φ ) | ∂ t u | d x + Z Ω ν ( φ ) | D∂ t u | d x = − Z Ω ρ ( φ )( ∂ t u · ∇ u ) · ∂ t u d x − Z Ω ρ ′ ( φ ) ∂ t φ ( ∂ t u + u · ∇ u ) · ∂ t u d x − Z Ω ν ′ ( φ ) ∂ t φD u : D∂ t u d x + Z Ω (cid:0) ∇ φ ⊗ ∇ ∂ t φ + ∇ ∂ t φ ⊗ ∇ φ (cid:1) : ∇ ∂ t u d x. (3.10)We now estimate the terms on the right-hand side of the above equality. By using (2.2), (2.4), and theequation (3.1) , we find − Z Ω ρ ( φ )( ∂ t u · ∇ u ) · ∂ t u d x ≤ C k ∂ t u k L (Ω) k∇ u k L (Ω) ≤ ν ∗ k D∂ t u k L (Ω) + C k ∂ t u k L (Ω) k D u k L (Ω) , (3.11)and − Z Ω ρ ′ ( φ ) ∂ t φ ( ∂ t u + u · ∇ u ) · ∂ t u d x ≤ C k ∂ t φ k L (Ω) k ∂ t u k L (Ω) + C k ∂ t φ k L (Ω) k u · ∇ u k L (Ω) k ∂ t u k L (Ω) ≤ C k u · ∇ φ k L (Ω) k ∂ t u k L (Ω) k∇ ∂ t u k L (Ω) + C k u · ∇ φ k L (Ω) k u k L ∞ (Ω) k∇ u k L (Ω) k u k H (Ω) k ∂ t u k L (Ω) k∇ ∂ t u k L (Ω) ≤ ν ∗ k D∂ t u k L (Ω) + C k u k L ∞ (Ω) k ∂ t u k L (Ω) + C k u k L ∞ (Ω) k D u k L (Ω) k u k H (Ω) k ∂ t u k L (Ω) ≤ ν ∗ k D∂ t u k L (Ω) + C k u k H (Ω) k ∂ t u k L (Ω) + C k D u k L (Ω) k u k H (Ω) k ∂ t u k L (Ω) ≤ ν ∗ k D∂ t u k L (Ω) + C k ∂ t u k L (Ω) + C k D u k L (Ω) k ∂ t u k L (Ω) + C k∇ φ k L ∞ (Ω) ( k φ k H (Ω) + k D u k L (Ω) ) k ∂ t u k L (Ω) + C k D u k L (Ω) k ∂ t u k L (Ω) + C k D u k L (Ω) (cid:0) k D u k L (Ω) + k∇ φ k L ∞ (Ω) ( k φ k H (Ω) + k D u k L (Ω) ) (cid:1) k ∂ t u k L (Ω) , (3.12) IFFUSE INTERFACE MODELS FOR INCOMPRESSIBLE BINARY FLUIDS 17 where we have also used (3.6) and (3.8). Moreover, we obtain − Z Ω ν ′ ( φ ) ∂ t φD u : D∂ t u d x ≤ C k ∂ t φ k L (Ω) k D u k L (Ω) k D∂ t u k L (Ω) ≤ ν ∗ k D∂ t u k L (Ω) + C k u · ∇ φ k L (Ω) k∇ ∂ t φ k L (Ω) k D u k L (Ω) k u k H (Ω) ≤ ν ∗ k D∂ t u k L (Ω) + C k∇ ∂ t φ k L (Ω) k D u k L (Ω) k u k H (Ω) , (3.13)and Z Ω (cid:0) ∇ φ ⊗ ∇ ∂ t φ + ∇ ∂ t φ ⊗ ∇ φ (cid:1) : ∇ ∂ t u d x ≤ C k∇ φ k L ∞ (Ω) k∇ ∂ t φ k L (Ω) k D∂ t u k L (Ω) ≤ ν ∗ k D∂ t u k L (Ω) + C k∇ φ k L ∞ (Ω) k∇ ∂ t φ k L (Ω) . (3.14)It is clear that an estimate of ∇ ∂ t φ is needed in order to control of the last two terms in (3.13) and(3.14). For this purpose, we have ∇ ∂ t φ = ( ∇ u ) t ∇ φ + ∇ φ u . Then, we easily deduce that k∇ ∂ t φ k L (Ω) ≤ k∇ u k L (Ω) k∇ φ k L ∞ (Ω) + k φ k W ,p (Ω) k u k L pp − (Ω) ≤ C k D u k L (Ω) (cid:0) k∇ φ k L ∞ (Ω) + k φ k W ,p (Ω) (cid:1) , (3.15)for p > . Combining (3.11)-(3.14) and (3.15) with (3.8)-(3.10), we arrive at 12 dd t Z Ω ρ ( φ ) | ∂ t u | d x + 3 ν ∗ Z Ω | D∂ t u | d x ≤ C k ∂ t u k L (Ω) k D u k L (Ω) + k ∂ t u k L (Ω) + C k∇ φ k L ∞ (Ω) ( k φ k H (Ω) + k D u k L (Ω) ) k ∂ t u k L (Ω) + C k D u k L (Ω) k ∂ t u k L (Ω) + C k D u k L (Ω) (cid:0) k D u k L (Ω) + k∇ φ k L ∞ (Ω) ( k φ k H (Ω) + k D u k L (Ω) ) (cid:1) k ∂ t u k L (Ω) + C k D u k L (Ω) ( k∇ φ k L ∞ (Ω) + k φ k W ,p (Ω) )( k ∂ t u k L (Ω) + k D u k L (Ω) )+ C k D u k L (Ω) ( k∇ φ k L ∞ (Ω) + k φ k W ,p (Ω) ) k∇ φ k L ∞ (cid:0) k φ k H (Ω) + k D u k L (Ω) (cid:1) + C k D u k L (Ω) (cid:0) k∇ φ k L ∞ (Ω) + k φ k W ,p (Ω) (cid:1) k∇ φ k L ∞ (Ω) . (3.16) Fourth estimate. In light of (3.9) and (3.16), we are left to control the W ,p (Ω) -norm of φ . To thisaim, we make use of the following equivalent norm k f k W ,p (Ω) = (cid:16) k f k pL p (Ω) + X | α | =2 k ∂ α f k pL p (Ω) (cid:17) p , where α is a multi-index. Next, we apply ∂ α to the transport equation (3.1) ∂ t ∂ α φ + ∂ α (cid:0) u · ∇ φ (cid:1) = 0 . Multiplying the above equation by | ∂ α φ | p − ∂ α φ and integrating over Ω , we get p dd t Z Ω | ∂ α φ | p d x + Z Ω ∂ α (cid:0) u · ∇ φ (cid:1) | ∂ α φ | p − ∂ α φ d x = 0 . (3.17)Since u is divergence free, the above can be rewritten as p dd t Z Ω | ∂ α φ | p d x + Z Ω (cid:16) ∂ α (cid:0) u · ∇ φ (cid:1) − u · ∇ ∂ α φ (cid:17) | ∂ α φ | p − ∂ α φ d x = 0 . (3.18)By summing over all multi-inder of order , and using (3.5), we find p dd t k φ k pW ,p (Ω) = − X | α | =2 Z Ω (cid:16) ∂ α (cid:0) u · ∇ φ (cid:1) − u · ∇ ∂ α φ (cid:17) | ∂ α φ | p − ∂ α φ d x. (3.19)It is easily seen that the right-hand side can be written as X | α | =2 Z Ω (cid:16) ∂ α (cid:0) u · ∇ φ (cid:1) − u · ∇ ∂ α φ (cid:17) | ∂ α φ | p − ∂ α φ d x = X | α | =2 Z Ω (cid:0) ∂ α u · ∇ φ (cid:1) | ∂ α φ | p − ∂ α φ d x + X | β | =1 , | γ | =1 ,β + γ = α Z Ω (cid:0) ∂ β u · ∇ ∂ γ φ ) | ∂ α φ | p − ∂ α φ d x. (3.20)Observe that X | α | =2 Z Ω (cid:0) ∂ α u · ∇ φ (cid:1) | ∂ α φ | p − ∂ α φ d x ≤ C k u k W ,p (Ω) k∇ φ k L ∞ (Ω) k φ k p − W ,p (Ω) , (3.21)and X | β | =1 , | γ | =1 ,β + γ = α Z Ω (cid:0) ∂ β u · ∇ ∂ γ φ ) | ∂ α φ | p − ∂ α φ d x ≤ C k u k W , ∞ (Ω) k φ k pW ,p (Ω) . (3.22)Collecting (3.19)-(3.22) together, and using the Sobolev embedding W ,p (Ω) ֒ → W , ∞ (Ω) (with p > ), we obtain p dd t k φ k pW ,p (Ω) ≤ C k u k W ,p (Ω) k φ k pW ,p (Ω) . Notice that the above inequality is equivalent to 12 dd t k φ k W ,p (Ω) ≤ C k u k W ,p (Ω) k φ k W ,p (Ω) . (3.23)Next, by exploiting Theorem A.1 with s = p > and r = ∞ , we deduce that k u k W ,p (Ω) ≤ C k ρ ( φ ) ∂ t u + ρ ( φ )( u · ∇ u ) + div( ∇ φ ⊗ ∇ φ ) k L p (Ω) + C k∇ φ k L ∞ (Ω) k D u k L p (Ω) ≤ C (cid:0) k ∂ t u k L p (Ω) + k u k L ∞ (Ω) k D u k L p (Ω) + k φ k W ,p (Ω) k∇ φ k L ∞ (Ω) + k∇ φ k L ∞ (Ω) k D u k L p (Ω) (cid:1) ≤ C k ∂ t u k p L (Ω) k D∂ t u k p − p L (Ω) + C k u k H (Ω) k u k W ,p (Ω) k u k L p (Ω) IFFUSE INTERFACE MODELS FOR INCOMPRESSIBLE BINARY FLUIDS 19 + C k∇ φ k L ∞ (Ω) (cid:0) k φ k W ,p (Ω) + k u k W ,p (Ω) k u k L p (Ω) (cid:1) . Thus, by Young’s inequality we find k u k W ,p (Ω) ≤ C k ∂ t u k p L (Ω) k D∂ t u k p − p L (Ω) + C k φ k W ,p (Ω) + C ( k u k H (Ω) + k∇ φ k L ∞ (Ω) ) k D u k L (Ω) . (3.24)Inserting (3.8) and (3.24) into (3.23), we are led to 12 dd t k φ k W ,p (Ω) ≤ C (cid:0) k ∂ t u k p L (Ω) k D∂ t u k p − p L (Ω) + k φ k W ,p (Ω) (cid:1) k φ k W ,p (Ω) + C ( k u k H (Ω) + k∇ φ k L ∞ (Ω) ) k D u k L (Ω) k φ k W ,p (Ω) ≤ ν ∗ k D∂ t u k L (Ω) + C k ∂ t u k p +2 L (Ω) k φ k pp +2 W ,p (Ω) + C k φ k W ,p (Ω) + C (cid:0) k ∂ t u k L (Ω) + k φ k H (Ω) k∇ φ k L ∞ (Ω) + k∇ φ k L ∞ (Ω) (cid:1) k D u k L (Ω) k φ k W ,p (Ω) + C ( k D u k L (Ω) + k∇ φ k L ∞ (Ω) k D u k L (Ω) ) k φ k W ,p (Ω) . (3.25) Final estimate. By adding (3.9), (3.16) and (3.25) together, and using the embeddings W ,p (Ω) ֒ → W , ∞ (Ω) , W ,p (Ω) ֒ → H (Ω) for p > , we deduce that dd t Y ( t ) + ρ ∗ Z Ω | ∂ t u | d x + ν ∗ Z Ω | D∂ t u | d x ≤ C (1 + Y ( t )) , (3.26)where Y ( t ) = Z Ω ν ( φ ( t )) | D u ( t ) | d x + Z Ω ρ ( φ ( t )) | ∂ t u ( t ) | d x + k φ ( t ) k W ,p (Ω) . Concerning the initial data, we observe from (3.1) that Z Ω ρ ( φ (0)) | ∂ t u (0) | d x ≤ C (cid:0) k u k L ∞ (Ω) + k φ k W ,p (Ω) (cid:1) k∇ u k L (Ω) + C k u k H (Ω) + C k φ k W ,p (Ω) , which, in turn, implies Y (0) ≤ Q ( k u k H (Ω) , k φ k W ,p (Ω) ) , where Q is a positive continuous and increasing function of its arguments.Finally, we deduce from (3.26) that there exists a positive time T < C (1+ Y (0)) , which depends onthe parameters of the system and on the norms of the initial data k u k H (Ω) and k φ k W ,p (Ω) , such that Z Ω | D u ( t ) | d x + Z Ω | ∂ t u ( t ) | d x + k φ ( t ) k W ,p (Ω) + Z t k ∂ t u ( τ ) k H (Ω) d τ ≤ C , (3.27)for all t ∈ [0 , T ] , where C is a positive constant depending on T , k u k H (Ω) , k φ k W ,p (Ω) . In addition,we learn from (3.24) and (3.27) that Z t k u ( τ ) k pp − W ,p (Ω) d τ ≤ C, ∀ t ∈ [0 , T ] . Similarly, we also deduce from (3.8), (3.15), and (3.27) that k u ( t ) k H (Ω) + k ∂ t φ ( t ) k H (Ω) + k ∂ t φ ( t ) k L ∞ (Ω) ≤ C, ∀ t ∈ [0 , T ] . We have obtained all the necessary a priori estimates. Then the existence result follows as outlinedat the beginning of the proof. Uniqueness. Let ( u , φ ) and ( u , φ ) be two solutions to problem (3.1)-(3.3) originating from thesame initial datum. The difference of solutions ( u , φ, P ) := ( u − u , φ − φ , P − P ) solves thesystem ρ ( φ ) (cid:0) ∂ t u + u · ∇ u + u · ∇ u (cid:1) − div (cid:0) ν ( φ ) D u (cid:1) + ∇ P = − div ( ∇ φ ⊗ ∇ φ ) − div ( ∇ φ ⊗ ∇ φ ) − ( ρ ( φ ) − ρ ( φ ))( ∂ t u + u · ∇ u )+ div (cid:0) ( ν ( φ ) − ν ( φ )) D u (cid:1) , (3.28) ∂ t φ + u · ∇ φ + u · ∇ φ = 0 , (3.29)for almost every ( x, t ) ∈ Ω × (0 , T ) , together with the incompressibility constraint div u = 0 . Multi-plying (3.28) by u and integrating over Ω , we find 12 dd t Z Ω ρ ( φ ) | u | d x + Z Ω ρ ( φ )( u · ∇ ) u · u d x + Z Ω ρ ( φ )( u · ∇ ) u · u d x + Z Ω ν ( φ ) | D u | d x = Z Ω ( ∇ φ ⊗ ∇ φ + ∇ φ ⊗ ∇ φ ) : ∇ u d x − Z Ω ( ρ ( φ ) − ρ ( φ ))( ∂ t u + u · ∇ u ) · u d x − Z Ω ( ν ( φ ) − ν ( φ )) D u : D u d x + 12 Z Ω | u | ρ ′ ( φ ) ∂ t φ d x. (3.30)Noticing the identity Z Ω ρ ( φ )( u · ∇ ) u · u d x = Z Ω ρ ( φ ) u · ∇ (cid:16) | u | (cid:17) d x = − Z Ω ρ ′ ( φ )( ∇ φ · u ) | u | d x, since φ solves the transport equation (3.1) , we can rewrite (3.30) as follows 12 dd t Z Ω ρ ( φ ) | u | d x + Z Ω ν ( φ ) | D u | d x = Z Ω ( ∇ φ ⊗ ∇ φ + ∇ φ ⊗ ∇ φ ) : ∇ u d x − Z Ω ( ρ ( φ ) − ρ ( φ ))( ∂ t u + u · ∇ u ) · u d x − Z Ω ( ν ( φ ) − ν ( φ )) D u : D u d x − Z Ω ρ ( φ )( u · ∇ ) u · u d x. (3.31)By using the embedding W ,p (Ω) ֒ → W , ∞ (Ω) for p > , we find that Z Ω ( ∇ φ ⊗ ∇ φ + ∇ φ ⊗ ∇ φ ) : ∇ u d x ≤ (cid:0) k∇ φ k L ∞ (Ω) + k∇ φ k L ∞ (Ω) (cid:1) k∇ φ k L (Ω) k∇ u k L (Ω) ≤ ν ∗ k D u k L (Ω) + C k∇ φ k L (Ω) . Next, by H¨older’s inequality, we have − Z Ω ( ρ ( φ ) − ρ ( φ ))( ∂ t u + u · ∇ u ) · u d x ≤ C k φ k L (Ω) k ∂ t u + u · ∇ u k L (Ω) k u k L (Ω) ≤ C (cid:0) k ∂ t u k L (Ω) + k u k L ∞ (Ω) k∇ u k L (Ω) (cid:1) k φ k H (Ω) k u k L (Ω) , IFFUSE INTERFACE MODELS FOR INCOMPRESSIBLE BINARY FLUIDS 21 − Z Ω ( ν ( φ ) − ν ( φ )) D u : D u d x ≤ C k φ k L (Ω) k D u k L (Ω) k D u k L (Ω) ≤ ν ∗ k D u k L (Ω) + C k D u k L (Ω) k φ k H (Ω) , and − Z Ω ρ ( φ )( u · ∇ ) u · u d x ≤ C k∇ u k L ∞ (Ω) k u k L (Ω) . Collecting the above estimates together, we deduce from (3.31) that dd t Z Ω ρ ( φ ) | u | d x + 3 ν ∗ Z Ω | D u | d x ≤ C (1 + k ∂ t u k L (Ω) + k u k L ∞ (Ω) k∇ u k L (Ω) + k D u k L (Ω) + k∇ u k L ∞ (Ω) ) × (cid:0) k u k L (Ω) + k φ k H (Ω) (cid:1) . (3.32)Next, we multiply (3.29) by φ and get 12 dd t k φ k L (Ω) + Z Ω ( u · ∇ φ ) φ d x = 0 . Then taking the gradient of (3.29) and multiplying the resulting equation by ∇ φ , we find 12 dd t k∇ φ k L (Ω) + Z Ω ∇ (cid:0) u · ∇ φ (cid:1) · ∇ φ d x + Z Ω ∇ (cid:0) u · ∇ φ (cid:1) · ∇ φ d x = 0 . By adding the last two equations, we obtain 12 dd t k φ k H (Ω) + Z Ω ( ∇ u ∇ φ ) · ∇ φ d x + Z Ω ( ∇ u ∇ φ ) · ∇ φ d x + Z Ω ( ∇ φ u ) · ∇ φ d x + Z Ω ( u · ∇ φ ) φ d x = 0 . We have − Z Ω ( ∇ u ∇ φ ) · ∇ φ d x ≤ k∇ u k L ∞ (Ω) k∇ φ k L (Ω) , and by W ,p (Ω) ֒ → W , ∞ (Ω) , p > , we get − Z Ω ( ∇ u ∇ φ ) · ∇ φ d x ≤ k∇ u k L (Ω) k∇ φ k L ∞ (Ω) k∇ φ k L (Ω) ≤ ν ∗ k D u k L (Ω) + C k∇ φ k L (Ω) , − Z Ω ( u · ∇ φ ) φ d x ≤ k u k L (Ω) k∇ φ k L ∞ (Ω) k φ k L (Ω) ≤ C k u k L (Ω) + C k φ k L (Ω) . Using (2.5), we obtain − Z Ω ( ∇ φ u ) · ∇ φ d x ≤ k∇ φ k L p (Ω) k u k L pp − (Ω) k∇ φ k L (Ω) ≤ C k φ k W ,p (Ω) k u k p − p L (Ω) k∇ u k p L (Ω) k∇ φ k L (Ω) ≤ C k u k p − p L (Ω) k∇ u k p L (Ω) k∇ φ k L (Ω) ≤ ν ∗ k D u k L (Ω) + C k u k L (Ω) + C k∇ φ k L (Ω) . Collecting the above estimates, we are led to dd t k φ k H (Ω) ≤ ν ∗ k D u k + C (1 + k∇ u k L ∞ (Ω) ) (cid:0) k u k L (Ω) + k φ k H (Ω) (cid:1) . (3.33)By adding (3.32) and (3.33), we end up with the differential inequality dd t (cid:0) k u k L (Ω) + k φ k H (Ω) (cid:1) + ν ∗ k D u k ≤ CR ( t ) (cid:0) k u k L (Ω) + k φ k H (Ω) (cid:1) , (3.34)where R = 1 + k ∂ t u k L (Ω) + k u k L ∞ (Ω) k∇ u k L (Ω) + k D u k L (Ω) + k∇ u k L ∞ (Ω) + k∇ u k L ∞ (Ω) . Since R ∈ L (0 , T ) , then the uniqueness of strong solutions follows from Gronwall’s lemma. (cid:3) Remark 3.2. The local well-posedness result stated in Theorem 3.1 is also valid in three dimensionalcase, provided that the initial condition φ ∈ W ,p (Ω) for some p > . The strategy used in theabove proof can be adapted to the this case by using the corresponding Sobolev inequalities in threedimensions. 4. M ASS -C ONSERVING N AVIER -S TOKES -A LLEN -C AHN S YSTEM : W EAK S OLUTIONS In this section, we consider the Navier-Stokes-Allen-Cahn system for a binary mixture of two incom-pressible fluids with different densities. This model was proposed in [32, Section 4.2.2] and derivedthrough an energetic variational approach (see also [37] for the case with no mass constraint). Thesystem reads as follows ρ ( φ ) (cid:0) ∂ t u + u · ∇ u (cid:1) − div (cid:0) ν ( φ ) D u (cid:1) + ∇ P = − div ( ∇ φ ⊗ ∇ φ ) , div u = 0 ,∂ t φ + u · ∇ φ + µ + ρ ′ ( φ ) | u | ξ,µ = − ∆ φ + Ψ ′ ( φ ) , ξ = µ + ρ ′ ( φ ) | u | , in Ω × (0 , T ) , (4.1)subject to the boundary conditions u = , ∂ n φ = 0 on ∂ Ω × (0 , T ) , (4.2)and to the initial conditions u ( · , 0) = u , φ ( · , 0) = φ in Ω . (4.3)Here, ρ ( φ ) and ν ( φ ) are, respectively, the density and the viscosity of the mixture, which satisfy theassumptions (3.4). The nonlinear function Ψ is the Flory-Huggins potential (1.19). The total energy ofsystem (4.1)-(4.2) is given by E ( u , φ ) = Z Ω ρ ( φ ) | u | + 12 |∇ φ | + Ψ( φ ) d x. (4.4)The main results of this section concern with the existence of global weak solutions. IFFUSE INTERFACE MODELS FOR INCOMPRESSIBLE BINARY FLUIDS 23 Theorem 4.1 (Global weak solution) . Let Ω be a bounded domain in R d with smooth boundary, d =2 , . Assume that the initial datum ( u , φ ) satisfies u ∈ H σ , φ ∈ H (Ω) ∩ L ∞ (Ω) with k φ k L ∞ (Ω) ≤ and | φ | < . Then, there exists a global weak solution ( u , φ ) to system (4.1) - (4.3) in the followingsense: (i) For all T > , the pair ( u , φ ) satisfies u ∈ L ∞ (0 , T ; H σ ) ∩ L (0 , T ; V σ ) ,φ ∈ L ∞ (0 , T ; H (Ω)) ∩ L q (0 , T ; H (Ω)) , ∂ t φ ∈ L q (0 , T ; L (Ω)) ,φ ∈ L ∞ (Ω × (0 , T )) : | φ ( x, t ) | < a.e. in Ω × (0 , T ) ,µ ∈ L q (0 , T ; L (Ω)) , with q = 2 if d = 2 , q = if d = 3 . (ii) For all T > , the system (4.1) is solved as follows − Z T Z Ω ( ρ ′ ( φ ) ∂ t φη ( t ) + ρ ( φ ) η ′ ( t )) u · v d x d t + Z T Z Ω (cid:0) ρ ( φ ) u · ∇ u (cid:1) · v η ( t ) d x d t + Z T Z Ω ν ( φ )( D u : D v ) η ( t ) d x d t = Z Ω ρ ( φ ) u v η (0) d x + Z T Z Ω (cid:0) ( ∇ φ ⊗ ∇ φ ) : ∇ v (cid:1) η ( t ) d x d t, for v ∈ V σ , η ∈ C ([0 , T ]) with η ( T ) = 0 , and ∂ t φ + u · ∇ φ − ∆ φ + Ψ ′ ( φ ) + ρ ′ ( φ ) | u | ′ ( φ ) + ρ ′ ( φ ) | u | , a.e. in Ω × (0 , T ) . (iii) The pair ( u , φ ) fulfills the regularity u ∈ C ([0 , T ]; ( H σ ) w ) and φ ∈ C ([0 , T ]; ( H (Ω)) w ) , forall T > , and u | t =0 = u , φ | t =0 = φ in Ω . In addition, ∂ n φ = 0 on ∂ Ω × (0 , T ) for all T > . (iv) The energy inequality E ( u ( t ) , φ ( t )) + Z t Z Ω ν ( φ ) | D u | d x d τ + Z t k ∂ t φ + u · ∇ φ k L (Ω) d τ ≤ E ( u , φ ) holds for all t ≥ . Next, we investigate the special case with matched densities (i.e. ρ = ρ , so that ρ ≡ ). Theresulting model is the homogeneous mass-conserving Navier-Stokes-Allen-Cahn system ∂ t u + u · ∇ u − div ( ν ( φ ) D u ) + ∇ p = − div ( ∇ φ ⊗ ∇ φ ) , div u = 0 ,∂ t φ + u · ∇ φ + µ = µ,µ = − ∆ φ + Ψ ′ ( φ ) , in Ω × (0 , T ) . (4.5)This system is associated with the boundary and the initial conditions u = , ∂ n φ = 0 on ∂ Ω × (0 , T ) , u ( · , 0) = u , φ ( · , 0) = φ in Ω . (4.6)We first state the existence of global weak solutions, whose proof follows from similar a priori esti-mates as the ones obtained for the nonhomogeneous case in the proof of Theorem 4.1 below. Theorem 4.2 (Global weak solution) . Let Ω be a bounded domain in R d , d = 2 , , with smoothboundary. Assume that the initial datum ( u , φ ) satisfies u ∈ H σ , φ ∈ H (Ω) ∩ L ∞ (Ω) with k φ k L ∞ (Ω) ≤ and | φ | < . Then there exists a global weak solution ( u , φ ) to problem (4.5) - (4.6) .This is, the solution ( u , φ ) satisfies, for all T > , u ∈ L ∞ (0 , T ; H σ ) ∩ L (0 , T ; V σ ) ,∂ t u ∈ L (0 , T ; V ′ σ ) if d = 2 , ∂ t u ∈ L (0 , T ; V ′ σ ) if d = 3 ,φ ∈ L ∞ (0 , T ; H (Ω)) ∩ L (0 , T ; H (Ω)) ,φ ∈ L ∞ (Ω × (0 , T )) : | φ ( x, t ) | < a.e. in Ω × (0 , T ) ,∂ t φ ∈ L (0 , T ; L (Ω)) if d = 2 , ∂ t φ ∈ L (0 , T ; L (Ω)) if d = 3 , and h ∂ t u , v i + ( u · ∇ u , v ) + ( ν ( φ ) D u , ∇ v ) = ( ∇ φ ⊗ ∇ φ, ∇ v ) , ∀ v ∈ V σ , a.e. t ∈ (0 , T ) ,∂ t φ + u · ∇ φ − ∆ φ + Ψ ′ ( φ ) = Ψ ′ ( φ ) , a.e. ( x, t ) ∈ Ω × (0 , T ) . Moreover, the initial and boundary conditions and the energy inequality hold as in Theorem 4.1. Furthermore, due to the particular form of the density function, we are able to prove a uniquenessresult in dimension two. Theorem 4.3 (Uniqueness of weak solutions in 2D) . Assume d = 2 . Let ( u , φ ) and ( u , φ ) be twoweak solutions to problem (4.5) - (4.6) on [0 , T ] subject to the same initial condition ( u , φ ) which sat-isfies the assumptions of Theorem 4.2. Moreover, we assume that φ satisfies the additional regularity L γ (0 , T ; H (Ω)) with γ > . Then ( u , φ ) = ( u , φ ) on [0 , T ] . Remark 4.4. The existence of strong solutions obtained in Section 5 (cf. Remark 5.4), which yieldsthe regularity φ ∈ L γ (0 , T ; H (Ω)) , where γ > , entails that the Theorem 4.3 can be regarded asa weak-strong uniqueness result for problem (4.5) - (4.6) in two dimensions. That is, the weak solutionoriginating from an initial condition ( u , φ ) such that u ∈ V σ and φ ∈ H (Ω) with Ψ ′ ( φ ) ∈ L (Ω) coincides with the (unique) strong solution departing from the same initial datum. Proof of Theorem 4.1. First, we derive a priori estimates of problem (4.1)-(4.3) that will becrucial to prove the existence of global weak solutions. Mass conservation and energy dissipation. First, integrating the equation (4.1) over Ω and usingthe definition of ξ , we observe that | Ω | Z Ω φ ( t ) d x = 1 | Ω | Z Ω φ d x, ∀ t ≥ . Next, we derive the energy equation associated with (4.1). Multiplying (4.1) by u and integrating over Ω , we obtain Z Ω ρ ( φ ) ∂ t | u | d x + Z Ω ρ ( φ )( u · ∇ ) u · u d x + Z Ω ν ( φ ) | D u | d x = − Z Ω ∆ φ ∇ φ · u d x. (4.7) IFFUSE INTERFACE MODELS FOR INCOMPRESSIBLE BINARY FLUIDS 25 Here we have used the relation − ∆ φ ∇ φ = ∇|∇ φ | − div( ∇ φ ⊗ ∇ φ ) and the incompressibilitycondition (4.1) . Thanks to the no-slip boundary condition for u , we observe that Z Ω ρ ( φ )( u · ∇ ) u · u d x = Z Ω ρ ( φ ) u · ∇ (cid:16) | u | (cid:17) d x = − Z Ω div ( ρ ( φ ) u ) | u | d x = − Z Ω ρ ′ ( φ ) ∇ φ · u | u | x. Next, we multiply (4.1) by ∂ t φ + u · ∇ φ and integrate over Ω . Noticing that ∂ t φ + u · ∇ φ = 0 , we get k ∂ t φ + u · ∇ φ k L (Ω) + Z Ω µ (cid:0) ∂ t φ + u · ∇ φ (cid:1) d x + Z Ω ρ ′ ( φ ) | u | (cid:0) ∂ t φ + u · ∇ φ (cid:1) d x = 0 . (4.8)On the other hand, the following equalities hold Z Ω µ ∂ t φ d x = dd t Z Ω |∇ φ | + Ψ( φ ) d x, Z Ω µ u · ∇ φ d x = Z Ω − ∆ φ ∇ φ · u d x + Z Ω u · ∇ Ψ( φ ) d x = Z Ω − ∆ φ ∇ φ · u d x, Z Ω ρ ′ ( φ ) | u | ∂ t φ d x = Z Ω ∂ t ( ρ ( φ )) | u | x. Thus, by adding (4.7) and (4.8), and using the above identities, we obtain the energy equation dd t E ( u , φ ) + Z Ω ν ( φ ) | D u | d x + k ∂ t φ + u · ∇ φ k L (Ω) = 0 . (4.9) Lower-order estimates. We assume that φ ∈ L ∞ (Ω × (0 , T )) such that | φ ( x, t ) | < almosteverywhere in Ω × (0 , T ) (cf. Existence of weak solutions below). Since ρ is strictly positive, it isimmediately seen from (4.9) that E ( u ( t ) , φ ( t )) + Z t Z Ω ν ( φ ) | D u | d x d τ + Z t k ∂ t φ + u · ∇ φ k L (Ω) d τ ≤ E ( u , φ ) , ∀ t ≥ . (4.10)Therefore, we deduce u ∈ L ∞ (0 , T ; H σ ) ∩ L (0 , T ; V σ ) , φ ∈ L ∞ (0 , T ; H (Ω)) (4.11)and ∂ t φ + u · ∇ φ ∈ L (0 , T ; L (Ω)) . (4.12)In light of (4.11) and (4.12), when d = 2 , we have (cid:13)(cid:13)(cid:13) − ρ ′ ( φ ) | u | (cid:13)(cid:13)(cid:13) L (Ω) ≤ C k u k L (Ω) ≤ C k∇ u k L (Ω) , which entails that ρ ′ ( φ ) | u | ∈ L (0 , T ; L (Ω)) . Instead, when d = 3 , we have (cid:13)(cid:13)(cid:13) − ρ ′ ( φ ) | u | (cid:13)(cid:13)(cid:13) L (Ω) ≤ C k u k L (Ω) ≤ C k∇ u k L (Ω) , thus ρ ′ ( φ ) | u | ∈ L (0 , T ; L (Ω)) . Since ρ ′ ( φ ) | u | ∈ L ∞ (0 , T ) , we also learn that µ − µ ∈ L q (0 , T ; L (Ω)) , (4.13) for q = 2 if d = 2 , q = if d = 3 . Thanks to the boundary condition for φ , we see that ∆ φ = 0 . Then,multiplying (4.1) by − ∆ φ and integrating by parts, we have Z Ω | ∆ φ | + F ′′ ( φ ) |∇ φ | d x = θ k∇ φ k L (Ω) − Z Ω ( µ − µ )∆ φ d x, where F is the convex part of the potential Ψ , i.e. F ( s ) = θ [(1 + s ) log(1 + s ) + (1 − s ) log(1 − s )] . By (4.11) and (4.13), we obtain k ∆ φ k L (Ω) ≤ C (1 + k µ − µ k L (Ω) ) . (4.14)Then, from the regularity theory of the Neumann problem, we infer that φ ∈ L q (0 , T ; H (Ω)) . (4.15)From (2.4), (2.6) and the above bounds, we have k u · ∇ φ k L (Ω) ≤ C k u k L (Ω) k∇ φ k L (Ω) ≤ C k u k L (Ω) k∇ u k L (Ω) k∇ φ k L (Ω) k φ k H (Ω) ≤ C k∇ u k L (Ω) k φ k H (Ω) , if d = 2 , and k u · ∇ φ k L (Ω) ≤ C k u k L (Ω) k∇ φ k L (Ω) ≤ C k u k L (Ω) k∇ u k L (Ω) k φ k L ∞ (Ω) k φ k H (Ω) ≤ C k∇ u k L (Ω) k φ k H (Ω) , if d = 3 , which implies u · ∇ φ ∈ L q (0 , T ; L (Ω)) . Thus ∂ t φ ∈ L q (0 , T ; L (Ω)) . (4.16)Moreover, we observe that k µ − µ k L (Ω) ≤ k ∂ t φ k L (Ω) + k u · ∇ φ k L (Ω) + (cid:13)(cid:13)(cid:13) − ρ ′ ( φ ) | u | (cid:13)(cid:13)(cid:13) L (Ω) + | Ω | − (cid:13)(cid:13)(cid:13) − ρ ′ ( φ ) | u | (cid:13)(cid:13)(cid:13) L (Ω) ≤ k ∂ t φ k L (Ω) + C k u k L (Ω) k∇ φ k L (Ω) + C k u k L (Ω) + C k u k L (Ω) ≤ k ∂ t φ k L (Ω) + C k∇ u k L (Ω) k φ k H (Ω) + C k u k L (Ω) k∇ u k L (Ω) + C k u k L (Ω) ≤ k ∂ t φ k L (Ω) + C k∇ u k L (Ω) k φ k H (Ω) + C k∇ u k L (Ω) + C, if d = 2 , (4.17)and k µ − µ k L (Ω) ≤ k ∂ t φ k L (Ω) + C k∇ u k L (Ω) k φ k H (Ω) + C k∇ u k L (Ω) + C, if d = 3 . (4.18)Recalling (4.11) and (4.14), and using Young’s inequality, we find that k φ k H (Ω) ≤ C (1 + k ∂ t φ k L (Ω) + k∇ u k L (Ω) ) , if d = 2 , (4.19)and k φ k H (Ω) ≤ C (1 + k ∂ t φ k L (Ω) + k∇ u k L (Ω) ) , if d = 3 . (4.20) IFFUSE INTERFACE MODELS FOR INCOMPRESSIBLE BINARY FLUIDS 27 In order to recover the full L -norm of µ , we observe that µ = F ′ ( φ ) − θ φ. Since | φ ( t ) | = | φ | < , it is well-known that Z Ω | F ′ ( φ ) | d x ≤ C Z Ω F ′ ( φ )( φ − φ ) d x + C for some positive constant C depending on F and φ . Multiplying (4.1) by φ − φ and using theboundary condition on φ , we obtain k∇ φ k L (Ω) + Z Ω F ′ ( φ )( φ − φ ) = Z Ω µ ( φ − φ ) d x + Z Ω θ φ ( φ − φ ) d x. Combining the above two relations and exploiting the energy bounds (4.11), we reach k F ′ ( φ ) k L (Ω) ≤ C (1 + k µ − µ k L (Ω) ) . (4.21)This actually implies that µ ∈ L q (0 , T ; L (Ω)) (4.22)and, in view of (4.15), we also have F ′ ( φ ) ∈ L q (0 , T ; L (Ω)) , (4.23)where q = 2 if d = 2 , q = if d = 3 .Besides, we have the following estimate for the time translations of u : Lemma 4.5. For any δ ∈ (0 , T ) , the following bound holds Z T − δ k u ( t + δ ) − u ( t ) k L (Ω) d t ≤ Cδ . (4.24) Proof. We only present the proof for the case d = 3 . The case d = 2 follows along the same lines. Itfollows from (4.11) and the interpolation (2.6) with p = 3 that u ∈ L (0 , T ; L (Ω)) . Similar to [49](see also [37, Lemma 3.5]), we have k p ρ ( φ ( t + δ ))( u ( t + δ ) − u ( t )) k L (Ω) ≤ − Z Ω ( ρ ( φ ( t + δ )) − ρ ( φ ( t ))) u ( t ) · ( u ( t + δ ) − u ( t )) d x − Z t + δt Z Ω ρ ( φ ( τ ))( u ( τ ) · ∇ ) u ( τ ) · ( u ( t + δ ) − u ( t )) d x d τ − Z t + δt Z Ω ν ( φ ( τ )) D u ( τ ) : D ( u ( t + δ ) − u ( t )) d x d τ + Z t + δt Z Ω ( ∇ φ ( τ ) ⊗ ∇ φ ( τ )) : ∇ ( u ( t + δ ) − u ( t )) d x d τ + Z t + δt Z Ω ρ ′ ( φ ) ∂ τ φ ( τ ) u ( τ ) · ( u ( t + δ ) − u ( t )) d x d τ := X i =1 J i . Observe now Z T − δ J ( t ) d t ≤ Z T − δ Z t + δt Z Ω | ρ ′ ( φ ) || ∂ τ φ ( τ ) || u ( t ) | ( | u ( t + δ ) | + | u ( t ) | ) d x d τ d t ≤ Z T − δ ( k u ( t + δ ) k L (Ω) + k u ( t ) k L (Ω) ) k u ( t ) k L (Ω) Z t + δt k ∂ τ φ ( τ ) k L (Ω) d τ d t ≤ Cδ (cid:18)Z T k∇ u ( t ) k L (Ω) d t (cid:19) (cid:18)Z T k ∂ t φ ( t ) k L (Ω) d t (cid:19) ≤ Cδ , and, in a similar manner, Z T − δ J ( t ) d t ≤ Z T − δ ( k u ( t + δ ) k L (Ω) + k u ( t ) k L (Ω) ) Z t + δt k u ( τ ) k L (Ω) k ∂ τ φ ( τ ) k L (Ω) d τ d t ≤ Cδ (cid:18)Z T k∇ u ( t ) k L (Ω) d t (cid:19) (cid:18)Z T k ∂ t φ ( t ) k L (Ω) d t (cid:19) ≤ Cδ . Next, we have Z T − δ J ( t ) d t ≤ Z T − δ Z t + δt k ρ ( φ ( τ )) k L ∞ (Ω) k u ( τ ) k L (Ω) k∇ u ( τ ) k L (Ω) d τ ( k u ( t + δ ) k L (Ω) + k u ( t ) k L (Ω) ) d t ≤ Cδ Z T − δ (cid:18)Z t + δt k∇ u ( τ ) k L (Ω) d τ (cid:19) ( k u ( t + δ ) k L (Ω) + k u ( t ) k L (Ω) ) d t ≤ Cδ (cid:18)Z T k∇ u ( t ) k L (Ω) d t (cid:19) Z T k u ( t ) k L (Ω) d t ≤ Cδ , and Z T − δ J ( t ) d t ≤ Z T − δ Z t + δt k ν ( φ ( τ )) k L ∞ (Ω) k D u ( τ ) k L (Ω) d τ ( k D u ( t + δ ) k L (Ω) + k D u ( t ) k L (Ω) ) d t ≤ Cδ Z T − δ (cid:18)Z t + δt k∇ u ( τ ) k L (Ω) d τ (cid:19) ( k D u ( t + δ ) k L (Ω) + k D u ( t ) k L (Ω) ) d t ≤ Cδ (cid:18)Z T k∇ u ( t ) k L (Ω) d t (cid:19) Z T k∇ u ( t ) k L (Ω) d t ≤ Cδ , Finally, by using (2.8) we get Z T − δ J ( t ) d t ≤ Z T − δ Z t + δt k∇ φ ( τ ) k L (Ω) d τ ( k∇ u ( t + δ ) k L (Ω) + k∇ u ( t ) k L (Ω) ) d t IFFUSE INTERFACE MODELS FOR INCOMPRESSIBLE BINARY FLUIDS 29 ≤ Cδ Z T − δ (cid:18)Z t + δt k φ ( τ ) k H (Ω) d τ (cid:19) ( k∇ u ( t + δ ) k L (Ω) + k∇ u ( t ) k L (Ω) ) d t ≤ Cδ (cid:18)Z T k φ ( t ) k H (Ω) d t (cid:19) Z T k∇ u ( t ) k L (Ω) d t ≤ Cδ . From the above estimate and the fact that ρ is strictly bounded from below, we obtain the conclusion(4.24). The proof is complete. (cid:3) Existence of weak solutions. With the above a priori estimates, we are able to prove the existence ofa global weak solution by using a semi-Galerkin scheme similar to [37]. More precisely, for any n ∈ N ,we find a local-in-time approximating solution ( u n , φ n ) where u n solves (4.1) as in the classicalGalerkin approximation and φ n is the (non-discrete) solution to the Allen-Cahn equations (4.1) -(4.1) with the velocity u n , the singular potential and the nonlocal term. This is achieved via a Schauder fixedpoint argument. For this approach, it is needed to solve separately a convective nonlocal Allen-Cahnequation. This can be done by introducing a family of regular potentials { Ψ ε } that approximates theoriginal singular potential Ψ by setting (see, e.g., [33]) Ψ ε ( s ) = F ε ( s ) − θ s , ∀ s ∈ R , where F ε ( s ) = X j =0 j ! F ( j ) (1 − ε ) [ s − (1 − ε )] j , ∀ s ≥ − ε,F ( s ) , ∀ s ∈ [ − ε, − ε ] , X j =0 j ! F ( j ) ( − ε ) [ s − ( − ε )] j , ∀ s ≤ − ε. Substituting the regular potential Ψ ε into the original Allen-Cahn equation, we are able to prove theexistence of an approximating solution φ ε to the resulting regularized equation using the semigroupapproach like in [37, Lemma 3.2] or simply by the Galerkin method. For the approximating solution φ ε ,we can derive estimates that are uniform in ε and then pass to the limit as ε → to recover the case withsingular potential. Here, we would like to remark that, thanks to the singular potential, we can showthat the phase field takes values in [ − , (using a similar argument like in [33]), without the additionalassumption sρ ′ ( s ) ≥ for | s | > that was required in [37]. Next, thanks to the a priori estimatesshowed above, it follows that the existence time interval of any solution ( u n , φ n ) is independent to n .From the same argument, we deduce uniform estimates that allows compactness for the phase field φ .Then, the key issue is to obtain uniform estimates of translations R T − δ k u ( t + δ ) − u ( t ) k L (Ω) d t (seeLemma 4.5) that yields compactness of the velocity field in the case of unmatched densities (cf. [49]).The above two-level approximating procedure is standard and we omit the details here. Time continuity and initial condition. We first observe that the regularity properties (4.11) and(4.16), together with the global bound k φ k L ∞ (0 ,T ; L ∞ (Ω)) ≤ , entail that φ ∈ C ([0 , T ]; L p (Ω)) , for any ≤ p < ∞ if d = 2 , . In addition, since φ ∈ L ∞ (0 , T ; H (Ω)) , we also infer from [69, Theorem 2.1] that φ ∈ C ([0 , T ]; ( H (Ω)) w ) . If d = 2 , since φ ∈ L (0 , T ; H (Ω)) ∩ W , (0 , T ; L (Ω)) , we deducethat φ ∈ C ([0 , T ]; H (Ω)) . Next, the weak formulation of (4.1) -(4.1) can be written as dd t h P ( ρ ( φ ) u ) , v i = h e f , v i , for all v ∈ V σ , in the sense of distribution on (0 , T ) , where P is the Leray projection onto H σ and h e f , v i = ( ρ ′ ( φ ) ∂ t φ u , v ) − ( ρ ( φ )( u · ∇ ) u , v ) − ( ν ( φ ) D u , ∇ v ) + ( ∇ φ ⊗ ∇ φ, ∇ v ) . Arguing similarly to the proof of Lemma 4.5, we observe that k e f k V ′ σ ≤ C k ∂ t φ k L (Ω) k u k L (Ω) + C k u k L (Ω) k D u k L (Ω) + C k D u k L (Ω) + C k∇ φ k L (Ω) ≤ C k ∂ t φ k L (Ω) k D u k L (Ω) + C k D u k L (Ω) + k D u k L (Ω) + C k φ k H (Ω) ≤ C (cid:0) k ∂ t φ k L (Ω) + k D u k L (Ω) + C k φ k H (Ω) (cid:1) . In light of the regularity of the weak solution, we infer that e f ∈ L (0 , T ; V ′ σ ) . By definition ofthe weak time derivative, this implies that ∂ t P ( ρ ( φ ) u ) ∈ L (0 , T ; V ′ σ ) . Observing that P ( ρ ( φ ) u ) ∈ L ∞ (0 , T ; H σ ) , we have P ( ρ ( φ ) u ) ∈ C ([0 , T ]; V ′ σ ) . As a consequence, we deduce from [69, Theorem2.1] that P ( ρ ( φ ) u ) ∈ C ([0 , T ]; ( H σ ) w ) . It easily follows from the properties of the Leray operator P that P ( ρ ( φ ) u ) ∈ C ([0 , T ]; ( L (Ω)) w ) . Now, repeating the argument in [3, Section 5.2], we deducethat ρ ( φ ) u ∈ C ([0 , T ]; ( L (Ω)) w ) . Therefore, since ρ ( φ ) ∈ C ([0 , T ]; L (Ω)) and ρ ( φ ) ≥ ρ ∗ > , weconclude that u ∈ C ([0 , T ]; ( L (Ω)) w ) . Finally, thanks to the time continuity of u and φ , a standardargument ensures that u | t =0 = u , φ | t =0 = φ in Ω . (cid:3) Proof of Theorem 4.3. Let us consider two global weak solutions ( u , φ ) and ( u , φ ) to prob-lem (4.5)-(4.6) given by Theorem 4.2. Denote the differences of solutions by u = u − u , φ = φ − φ .Then we have h ∂ t u , v i + ( u · ∇ u , v ) + ( u · ∇ u , v ) + ( ν ( φ ) D u , ∇ v ) + (( ν ( φ ) − ν ( φ )) D u , ∇ v )= ( ∇ φ ⊗ ∇ φ, ∇ v ) + ( ∇ φ ⊗ ∇ φ , ∇ v ) (4.25)for all v ∈ V σ , almost every t ∈ (0 , T ) , and ∂ t φ + u · ∇ φ + u · ∇ φ − ∆ φ + Ψ ′ ( φ ) − Ψ ′ ( φ ) = Ψ ′ ( φ ) − Ψ ′ ( φ ) (4.26)almost every ( x, t ) ∈ Ω × (0 , T ) . Following the same strategy as in [34], we take v = A − u , where A is the Stokes operator, and we find 12 dd t k u k ∗ + ( ν ( φ ) D u , ∇ A − u ) = ( u ⊗ u , ∇ A − u ) + ( u ⊗ u , ∇ A − u ) − (( ν ( φ ) − ν ( φ )) D u , ∇ A − u ) + ( ∇ φ ⊗ ∇ φ, ∇ A − u ) + ( ∇ φ ⊗ ∇ φ , ∇ A − u ) , where k u k ∗ = k∇ A − u k L (Ω) , which is a norm on V ′ σ equivalent to the usual dual norm. Here,we have used the equality u i · ∇ u = div ( u ⊗ u i ) , i = 1 , , due to the incompressibility condition.Multiplying (4.26) by φ , integrating over Ω and observing that Z Ω ( u · ∇ φ ) φ d x = Z Ω u · ∇ φ d x = 0 , Z Ω (Ψ ′ ( φ ) − Ψ ′ ( φ )) φ d x = (Ψ ′ ( φ ) − Ψ ′ ( φ )) φ = 0 , IFFUSE INTERFACE MODELS FOR INCOMPRESSIBLE BINARY FLUIDS 31 we obtain 12 dd t k φ k L (Ω) + k∇ φ k L (Ω) + Z Ω ( u · ∇ φ ) φ d x + Z Ω ( F ′ ( φ ) − F ′ ( φ )) φ d x = θ k φ k L (Ω) . By adding the above two equations and using the convexity of F , we deduce that dd t G ( t ) + ( ν ( φ ) D u , ∇ A − u ) + k∇ φ k L (Ω) ≤ ( u ⊗ u , ∇ A − u ) + ( u ⊗ u , ∇ A − u ) − (( ν ( φ ) − ν ( φ )) D u , ∇ A − u )+ ( ∇ φ ⊗ ∇ φ, ∇ A − u ) + ( ∇ φ ⊗ ∇ φ , ∇ A − u ) − ( u · ∇ φ , φ ) + θ k φ k L (Ω) , (4.27)where G ( t ) = 12 k u ( t ) k ∗ + 12 k φ ( t ) k L (Ω) . In order to recover a L (Ω) -norm of u , which is a key term to control the nonlinear terms on theright-hand side, we obtain by integration by parts (see [34, (3.9)]) ( ν ( φ ) D u , ∇ A − u ) = ( ∇ u , ν ( φ ) D A − u )= − ( u , div ( ν ( φ ) D A − u )= − ( u , ν ′ ( φ ) D A − u ∇ φ ) − 12 ( u , ν ( φ )∆ A − u ) . Here we have used that div ∇ t v = ∇ div v . Notice that, by definition of Stokes operator, there exists ascalar function p ∈ L ∞ (0 , T ; H (Ω)) ∩ L (0 , T ; H (Ω)) (unique up to a constant) such that − ∆ A − u + ∇ p = u for almost any ( x, t ) ∈ Ω × (0 , T ) . Moreover, we have the following estimates from [31]and [34, Appendix B] k p k L (Ω) ≤ C k∇ A − u k L (Ω) k u k L (Ω) , k p k H (Ω) ≤ C k u k L (Ω) , k p k H (Ω) ≤ C k∇ u k L (Ω) . (4.28)Then, we can write − 12 ( u , ν ( φ )∆ A − u ) = 12 ( u , ν ( φ ) u ) − 12 ( u , ν ( φ ) ∇ p )= 12 ( u , ν ( φ ) u ) + 12 (div ( ν ( φ ) u ) , p )= 12 ( u , ν ( φ ) u ) + 12 ( ν ′ ( φ ) ∇ φ · u , p ) . Hence, recalling that ν ( · ) ≥ ν ∗ > , we find ( ν ( φ ) D u , ∇ A − u ) ≥ ν ∗ k u k L (Ω) + 12 ( ν ′ ( φ ) ∇ φ · u , p ) − ( u , ν ′ ( φ ) D A − u ∇ φ ) . Owing to the above estimate, we rewrite (4.27) as follows dd t G ( t ) + ν ∗ k u k L (Ω) + k∇ φ k L (Ω) = ( u ⊗ u , ∇ A − u ) + ( u ⊗ u , ∇ A − u ) + (( ν ( φ ) − ν ( φ )) D u , ∇ A − u )+ ( ∇ φ ⊗ ∇ φ, ∇ A − u ) + ( ∇ φ ⊗ ∇ φ , ∇ A − u ) − ( u · ∇ φ , φ ) + θ k φ k L (Ω) + ( u , ν ′ ( φ ) D A − u ∇ φ ) − 12 ( ν ′ ( φ ) ∇ φ · u , p ) . (4.29)By the Ladyzhenskaya inequality (2.4), together with (2.3) and the bounds for weak solutions, we have ( u ⊗ u , ∇ A − u ) + ( u ⊗ u , ∇ A − u ) ≤ k u k L (Ω) (cid:0) k u k L (Ω) + k u k L (Ω) (cid:1) k∇ A − u k L (Ω) ≤ C (cid:0) k u k H (Ω) + k u k H (Ω) ) k u k L (Ω) k u k ∗ ≤ ν ∗ k u k L (Ω) + C (cid:0) k u k H (Ω) + k u k H (Ω) ) k u k ∗ . Similarly, we obtain ( ∇ φ ⊗ ∇ φ, ∇ A − u ) + ( ∇ φ ⊗ ∇ φ , ∇ A − u ) ≤ (cid:0) k∇ φ k L (Ω) + k∇ φ k L (Ω) (cid:1) k∇ φ k L (Ω) k∇ A − u k L (Ω) ≤ C (cid:0) k φ k H (Ω) + k φ k H (Ω) ) k∇ φ k L (Ω) k u k L (Ω) k u k ∗ ≤ ν ∗ k u k L (Ω) + 112 k∇ φ k L (Ω) + C (cid:0) k φ k H (Ω) + k φ k H (Ω) ) k u k ∗ , and ( u · ∇ φ , φ ) ≤ k u k L (Ω) k∇ φ k L (Ω) k φ k L (Ω) ≤ C k u k L (Ω) k∇ φ k H (Ω) k φ k L (Ω) k∇ φ k L (Ω) ≤ ν ∗ k u k L (Ω) + 112 k∇ φ k L (Ω) + C k φ k H (Ω) k φ k L (Ω) , where we have also used the inequality (2.1) and the conservation of mass φ = 0 . Next, since ν ′ isbounded, by exploiting (2.4) we have ( u , ν ′ ( φ ) D A − u ∇ φ ) ≤ C k u k L (Ω) k D A − u k L (Ω) k∇ φ k L (Ω) ≤ C k u k L (Ω) k u k ∗ k∇ φ k H (Ω) ≤ ν ∗ k u k L (Ω) + C k φ k H (Ω) k u k ∗ . By using the Stokes operator (i.e. A = P ( − ∆) ) and the integration by parts, we infer that − 12 ( ν ′ ( φ ) ∇ φ · u , p ) = 12 (cid:0) ∆ A − u , P ( ν ′ ( φ ) ∇ φ p ) (cid:1) = − Z Ω ( ∇ A − u ) t : ∇ P ( ν ′ ( φ ) ∇ φ p ) d x + 12 Z ∂ Ω (cid:0) ( ∇ A − u ) t P ( ν ′ ( φ ) ∇ φ p ) (cid:1) · n d σ. Thanks to (2.3), (2.11), and the properties of the Leray projection, we find − 12 ( ν ′ ( φ ) ∇ φ · u , p ) ≤ C k∇ A − u k L (Ω) k∇ P ( ν ′ ( φ ) ∇ φ p ) k L (Ω) + C k∇ A − u k L ( ∂ Ω) k P ( ν ′ ( φ ) ∇ φ p ) k L ( ∂ Ω) IFFUSE INTERFACE MODELS FOR INCOMPRESSIBLE BINARY FLUIDS 33 ≤ C k u k ∗ k ν ′ ( φ ) ∇ φ p k H (Ω) + C k u k ∗ k u k L (Ω) k P ( ν ′ ( φ ) ∇ φ p ) k L (Ω) k P ( ν ′ ( φ ) ∇ φ p ) k H (Ω) ≤ C k u k ∗ k ν ′ ( φ ) ∇ φ p k H (Ω) + C k u k ∗ k u k L (Ω) k ν ′ ( φ ) ∇ φ p k L (Ω) k ν ′ ( φ ) ∇ φ p k H (Ω) . (4.30)Owing to (2.4), (2.9), Lemma 2.1 and (4.28), we observe that k ν ′ ( φ ) ∇ φ p k L (Ω) ≤ C k∇ φ k L (Ω) k p k L (Ω) ≤ C k φ k H (Ω) k p k L (Ω) k p k H (Ω) ≤ C k φ k H (Ω) k∇ A − u k L (Ω) k u k L (Ω) , and k ν ′ ( φ ) ∇ φ p k H (Ω) ≤ k ν ′ ( φ ) ∇ φ p k L (Ω) + k ν ′′ ( φ ) ∇ φ ⊗ ∇ φ p k L (Ω) + k ν ′ ( φ ) ∇ φ p k L (Ω) + k ν ′ ( φ ) ∇ φ ⊗ ∇ p k L (Ω) ≤ C k u k L (Ω) log (cid:16) C k∇ u k L (Ω) k u k L (Ω) (cid:17) + C k∇ φ k L (Ω) k p k L ∞ (Ω) + C k φ k H (Ω) k p k L ∞ (Ω) + C k φ k H (Ω) k p k H (Ω) log (cid:16) C k p k H (Ω) k p k H (Ω) (cid:17) ≤ C (cid:0) k φ k H (Ω) (cid:1) k u k L (Ω) log (cid:16) C k∇ u k L (Ω) k u k L (Ω) (cid:17) . Combining the above estimates with (4.30), we are led to − 12 ( ν ′ ( φ ) ∇ φ · u , p ) ≤ C (cid:0) k φ k H (Ω) (cid:1) k u k ∗ k u k L (Ω) log (cid:16) C k∇ u k L (Ω) k u k L (Ω) (cid:17) + C (cid:0) k φ k H (Ω) (cid:1) k u k ∗ k u k L (Ω) log (cid:16) C k∇ u k L (Ω) k u k L (Ω) (cid:17) ≤ ν ∗ k u k L (Ω) + C (cid:0) k φ k H (Ω) (cid:1) k u k ∗ log (cid:16) C k∇ u k L (Ω) k u k L (Ω) (cid:17) + C (cid:0) k φ k H (Ω) (cid:1) k u k ∗ log (cid:16) C k∇ u k L (Ω) k u k L (Ω) (cid:17) . In order to handle the logarithmic terms, we recall that C k∇ u k L k u k L > . Since C ′ k u k L k u k ∗ > , for some C ′ > depending on Ω , we have log (cid:16) C k∇ u k L (Ω) k u k L (Ω) (cid:17) ≤ (cid:16) C k∇ u k L (Ω) k u k L (Ω) (cid:17) ≤ (cid:16) C C ′ k∇ u k L (Ω) k u k ∗ (cid:17) ≤ C + log (cid:0) k∇ u k L (Ω) (cid:1) + log (cid:16) e C k u k ∗ (cid:17) , where e C > is a sufficiently large constant such that log (cid:16) e C k u k ∗ (cid:17) > , which holds true in light of(4.11). Thus, we obtain − 12 ( ν ′ ( φ ) ∇ φ · u , p ) ≤ ν ∗ k u k L (Ω) + C (cid:0) k φ k H (Ω) (cid:1) log (cid:0) k∇ u k L (Ω) (cid:1) k u k ∗ + C (cid:0) k φ k H (Ω) (cid:1) k u k ∗ log (cid:16) e C k u k ∗ (cid:17) . Netx, by using Lemma 2.1, we infer that − (( ν ( φ ) − ν ( φ )) D u , ∇ A − u )= − Z Ω Z ν ′ ( τ φ + (1 − τ ) φ ) d τ φD u : ∇ A − u d x ≤ C k D u k L (Ω) k φ ∇ A − u k L (Ω) ≤ C k u k H (Ω) k∇ φ k L (Ω) k u k ∗ log (cid:16) C k u k L (Ω) k u k ∗ (cid:17) ≤ k∇ φ k L (Ω) + C k u k H (Ω) k u k ∗ log (cid:16) e C k u k ∗ (cid:17) , where e C is chosen sufficiently large as above.Summing up, we arrive at the differential inequality dd t G ( t ) + ν ∗ k u k L (Ω) + 12 k∇ φ k L (Ω) ≤ CS ( t ) G ( t ) log (cid:16) e CG ( t ) (cid:17) , (4.31)where S ( t ) = (cid:16) k u k H (Ω) + k u k H (Ω) + k φ k H (Ω) + k φ k H (Ω) + k φ k H (Ω) (cid:0) (cid:0) k∇ u k L (Ω) (cid:1)(cid:1)(cid:17) . Here we have used that the function s log (cid:16) e Cs (cid:17) is increasing on (0 , e Ce ) . We observe that S ∈ L (0 , T ) provided that φ ∈ L γ (0 , T ; H (Ω)) with γ > . Indeed, we recall that log(1 + s ) ≤ C ( κ )(1 + s ) κ ,for any κ > . Taking κ = γ − γ , we have Z T k φ ( τ ) k H (Ω) log (cid:0) k∇ u ( τ ) k L (Ω) (cid:1) d τ ≤ C Z T k φ ( τ ) k H (Ω) (cid:0) k∇ u ( τ ) k L (Ω) (cid:1) γ − γ d τ ≤ C Z T k φ ( τ ) k γH (Ω) + k∇ u ( τ ) k L (Ω) + k∇ u ( τ ) k L (Ω) d τ. Throughout the rest of the proof, we will assume that S ∈ L (0 , T ) .Integrating (4.31) on the time interval [0 , t ] , we find G ( t ) ≤ G (0) + C Z t S ( s ) G ( s ) log (cid:16) e CG ( s ) (cid:17) d s, IFFUSE INTERFACE MODELS FOR INCOMPRESSIBLE BINARY FLUIDS 35 for almost every t ∈ [0 , T ] . We observe that R 10 1 s log( Cs ) d s = ∞ . Thus, if G (0) = 0 , applying theOsgood lemma B.1, we deduce that G ( t ) = 0 for all t ∈ [0 , T ] , namely u ( t ) = u ( t ) and φ ( t ) = φ ( t ) . This demonstrates the uniqueness of solutions in the class of weak solutions satisfying theadditional regularity φ ∈ L γ (0 , T ; H (Ω)) with γ > . Indeed, we are able to deduce a continuousdependence estimate with respect to the initial datum. To this end, we define M ( s ) = log(log( e Cs )) . bythe Osgood lemma, for G (0) > , we are led to − log (cid:16) log (cid:16) e CG ( t ) (cid:17)(cid:17) + log (cid:16) log (cid:16) e CG (0) (cid:17)(cid:17) ≤ C Z t S ( s ) d s (4.32)for almost every t ∈ [0 , T ] . Taking the double exponential of (4.32), we eventually infer the control G ( t ) ≤ e C (cid:16) G (0) e C (cid:17) exp( − C R t S ( s ) d s ) ∀ t ∈ [0 , T ] , (4.33)where T > is defined by log (cid:16) log (cid:16) e CG (0) (cid:17)(cid:17) ≥ C Z T S ( s ) d s. The proof is complete. (cid:3) Remark 4.6. We note that the same existence result as in Theorem 4.2 holds for Ω = T d , d = 2 , . Inthe particular case Ω = T , the uniqueness of weak solutions can be achieved, without the additionalregularity φ ∈ L γ (0 , T ; H (Ω)) as in Theorem 4.3. Indeed, in this case the solutions of the Stokesoperator A − u and p are given by (see [73, Chapter 2.2]) A − u = X k ∈ Z g k e iπk · xL , p = X k ∈ Z p k e iπk · xL , where g k = − L π | k | (cid:16) u k − ( k · u k ) k | k | (cid:17) , p k = Lk · u k iπ | k | , k ∈ Z , k = 0 ,L > is the cell size. Here u k is the k -mode of u . We observe that we only consider the case k = 0 since u is conserved for (4.5) on T , and so we can choose u = 0 . Moreover, since u ∈ H σ , we havethat k · u k = 0 for any k ∈ Z , which implies that p k = 0 for any k ∈ Z . Thus, following the aboveproof, we are led to the differential inequality (4.29) without the last term on the right-hand side, i.e. − ( ν ′ ( φ ) ∇ φ · u , p ) . Hence, we eventually end up with dd t G ( t ) + ν ∗ k u k L (Ω) + 12 k∇ φ k L (Ω) ≤ C e S ( t ) G ( t ) log (cid:16) e CG ( t ) (cid:17) , where e S ( t ) = (cid:16) k u k H (Ω) + k u k H (Ω) + k φ k H (Ω) + k φ k H (Ω) (cid:17) . Since e S ( t ) ∈ L (0 , T ) for any couple of weak solutions, an application of the Osgood lemma as aboveentails the uniqueness of weak solutions (without additional regularity) and a continuous dependenceestimate with respect to the initial data, i.e. (4.33) . 5. M ASS - CONSERVING N AVIER -S TOKES -A LLEN -C AHN S YSTEM : S TRONG S OLUTIONS This section is devoted to the analysis of global strong solutions to the nonhomogeneous Navier-Stokes-Allen-Cahn system (4.1)-(4.3) in two dimensions. The main results are as follows. Theorem 5.1 (Global strong solution in 2D) . Let Ω be a bounded smooth domain in R . Assume that u ∈ V σ , φ ∈ H (Ω) such that ∂ n φ = 0 on ∂ Ω , F ′ ( φ ) ∈ L (Ω) , k φ k L ∞ (Ω) ≤ and | φ | < . (1) There exists a global strong solution ( u , φ ) to problem (4.1) - (4.3) satisfying, for all T > , u ∈ L ∞ (0 , T ; V σ ) ∩ L (0 , T ; H (Ω)) ∩ H (0 , T ; H σ ) ,φ ∈ L ∞ (0 , T ; H (Ω)) ∩ L (0 , T ; W ,p (Ω)) ,∂ t φ ∈ L ∞ (0 , T ; L (Ω)) ∩ L (0 , T ; H (Ω)) ,F ′ ( φ ) ∈ L ∞ (0 , T ; L (Ω)) ∩ L (0 , T ; L p (Ω)) where p ∈ (2 , ∞ ) . The strong solution satisfies the system (4.1) almost everywhere in Ω × (0 , ∞ ) . Besides, | φ ( x, t ) | < for almost any ( x, t ) ∈ Ω × (0 , ∞ ) and ∂ n φ = 0 on ∂ Ω × (0 , ∞ ) . (2) There exists η > depending only on the norms of the initial data and on the parameters ofthe system: η = η ( E ( u , φ ) , k u k V σ , k φ k H (Ω) , k F ′ ( φ ) k L (Ω) , θ, θ ) . If, in addition, k ρ ′ k L ∞ ( − , ≤ η and F ′′ ( φ ) ∈ L (Ω) , then, for any T > , we have F ′′ ( φ ) ∈ L ∞ (0 , T ; L (Ω)) , F ′′ ∈ L q (0 , T ; L p (Ω)) , (5.1) where p + q = 1 , p ∈ (1 , ∞ ) , and ( F ′′ ( φ )) log(1 + F ′′ ( φ )) ∈ L (Ω × (0 , T )) . (5.2) In particular, the strong solution satisfying (5.2) is unique. Theorem 5.2 (Propagation of regularity in 2D) . Let the assumptions in Theorem 5.1-(1) be satisfied.Assume that k ρ ′ k L ∞ ( − , ≤ η . Given a strong solution from Theorem 5.1-(1), for any σ > , thereholds ( F ′′ ( φ )) log(1 + F ′′ ( φ )) ∈ L (Ω × ( σ, T )) , and ∂ t u ∈ L ∞ ( σ, T ; H σ ) ∩ L ( σ, T ; V σ ) , ∂ t φ ∈ L ∞ ( σ, T ; H (Ω)) ∩ L ( σ, T ; H (Ω)) . In addition, for any σ > , there exists δ = δ ( σ ) > such that − δ ≤ φ ( x, t ) ≤ − δ, ∀ x ∈ Ω , t ≥ σ. Remark 5.3. The smallness assumption on ρ ′ (see (5.38) below for the explicit form) can be reformu-lated in terms of the difference of the (constant) densities of the two fluids when ρ is a linear interpola-tion function. In this case, we have ρ ( s ) = ρ s ρ − s , ρ ′ ( s ) = ρ − ρ ∀ s ∈ [ − , . IFFUSE INTERFACE MODELS FOR INCOMPRESSIBLE BINARY FLUIDS 37 Roughly speaking, the results given by Theorem 5.1 and Theorem 5.2 imply that uniqueness and furtherregularity of strong solutions to the nonhomogeneous system hold provided that the two fluids havesimilar densities ( ρ ≈ ρ ). Remark 5.4 (Matched densities) . It is worth noticing that Theorem 5.1 and Theorem 5.2 hold true inthe case of constant density ρ ≡ (i.e. ρ = ρ ) without any smallness assumption. Proof of Theorem 5.1. We perform higher-order a priori estimates that are necessary for theexistence of global strong solutions. Higher-order estimates. Multiplying (4.1) by ∂ t u , integrating over Ω , and observing that Z Ω ν ( φ ) D u · D∂ t u d x = 12 dd t Z Ω ν ( φ ) | D u | d x − Z Ω ν ′ ( φ ) ∂ t φ | D u | d x, we obtain 12 dd t Z Ω ν ( φ ) | D u | d x + Z Ω ρ ( φ ) | ∂ t u | d x = − ( ρ ( φ ) u · ∇ u , ∂ t u ) − Z Ω ∆ φ ∇ φ · ∂ t u d x + 12 Z Ω ν ′ ( φ ) ∂ t φ | D u | d x. (5.3)Next, differentiating (4.1) in time, multiplying the resultant by ∂ t φ and integrating over Ω , we obtain 12 dd t k ∂ t φ k L (Ω) + Z Ω ∂ t u · ∇ φ ∂ t φ d x + k∇ ∂ t φ k L (Ω) + Z Ω F ′′ ( φ ) | ∂ t φ | d x = θ k ∂ t φ k L (Ω) − Z Ω ρ ′′ ( φ ) | ∂ t φ | | u | x − Z Ω ρ ′ ( φ ) u · ∂ t u ∂ t φ d x + ∂ t ξ Z Ω ∂ t φ d x. (5.4)Since ∂ t φ = 0 , by adding the equations (5.3) and (5.4), we find that dd t H ( t ) + ρ ∗ k ∂ t u k L (Ω) + k∇ ∂ t φ k L (Ω) + Z Ω F ′′ ( φ ) | ∂ t φ | d x ≤ − ( ρ ( φ ) u · ∇ u , ∂ t u ) − Z Ω ∆ φ ∇ φ · ∂ t u d x + 12 Z Ω ν ′ ( φ ) ∂ t φ | D u | d x + θ k ∂ t φ k L (Ω) − Z Ω ∂ t u · ∇ φ ∂ t φ d x − Z Ω ρ ′′ ( φ ) | ∂ t φ | | u | x − Z Ω ρ ′ ( φ ) u · ∂ t u ∂ t φ d x, (5.5)where H ( t ) = 12 Z Ω ν ( φ ) | D u | d x + 12 k ∂ t φ k L (Ω) . (5.6)In (5.5), we have used that ρ is strictly positive ( ρ ( s ) ≥ ρ ∗ > ). In addition, we simply infer from(4.1) that k ∂ t φ k L (Ω) ≤ C (cid:0) k u k H (Ω) (cid:1) k φ k H (Ω) + C k F ′ ( φ ) k L (Ω) + C k u k H (Ω) . Therefore, it follows from the assumptions on the initial data that H (0) < + ∞ .We proceed to estimate the right-hand side of (5.5). By using (2.2) and (2.10), we have − ( ρ ( φ ) u · ∇ u , ∂ t u ) ≤ k ρ ( φ ) k L ∞ (Ω) k u k L ∞ (Ω) k∇ u k L (Ω) k ∂ t u k L (Ω) ≤ C k D u k L (Ω) log (cid:16) C k u k W ,p (Ω) k D u k L (Ω) (cid:17) k ∂ t u k L (Ω) ≤ ρ ∗ k ∂ t u k L (Ω) + C k D u k L (Ω) log (cid:16) C k u k W ,p (Ω) k D u k L (Ω) (cid:17) , for some p > . Moreover, it holds − Z Ω ∆ φ ∇ φ · ∂ t u d x ≤ k ∆ φ k L (Ω) k∇ φ k L ∞ (Ω) k ∂ t u k L (Ω) ≤ C k ∆ φ k L (Ω) k∇ φ k H (Ω) log (cid:16) C k∇ φ k W ,p (Ω) k∇ φ k H (Ω) (cid:17) k ∂ t u k L (Ω) ≤ ρ ∗ k ∂ t u k L (Ω) + C k ∆ φ k L (Ω) k∇ φ k H (Ω) log (cid:16) C k∇ φ k W ,p (Ω) k∇ φ k H (Ω) (cid:17) . Next, by exploiting Lemma 2.1, together with (2.1) and ∂ t φ = 0 , we obtain Z Ω ν ′ ( φ ) ∂ t φ | D u | d x ≤ k ν ′ ( φ ) k L ∞ (Ω) k ∂ t φ | D u |k L (Ω) k D u k L (Ω) ≤ C k∇ ∂ t φ k L (Ω) k D u k L (Ω) log (cid:16) C k D u k L p (Ω) k D u k L (Ω) (cid:17) ≤ k∇ ∂ t φ k L (Ω) + C k D u k L (Ω) log (cid:16) C k D u k L p (Ω) k D u k L (Ω) (cid:17) . It remains to control the last three terms on the right-hand side of (5.5). By using (2.4) and (4.11), weobtain − Z Ω ∂ t u · ∇ φ ∂ t φ d x ≤ k ∂ t u k L (Ω) k∇ φ k L (Ω) k ∂ t φ k L (Ω) ≤ k ∂ t u k L (Ω) k∇ φ k L (Ω) k φ k H (Ω) k ∂ t φ k L (Ω) k∇ ∂ t φ k L (Ω) ≤ ρ ∗ k ∂ t u k L (Ω) + 18 k∇ ∂ t φ k L (Ω) + C k φ k H (Ω) k ∂ t φ k L (Ω) , − Z Ω ρ ′′ ( φ ) | ∂ t φ | | u | x ≤ C k ρ ′′ ( φ ) k L ∞ (Ω) k ∂ t φ k L (Ω) k u k L (Ω) ≤ C k ∂ t φ k L (Ω) k∇ ∂ t φ k L (Ω) k u k L (Ω) k∇ u k L (Ω) ≤ k∇ ∂ t φ k L (Ω) + C k ∂ t φ k L (Ω) k D u k L (Ω) , and − Z Ω ρ ′ ( φ ) u · ∂ t u ∂ t φ d x ≤ C k ρ ′ ( φ ) k L ∞ (Ω) k u k L (Ω) k ∂ t u k L (Ω) k ∂ t φ k L (Ω) ≤ C k u k L (Ω) k∇ u k L (Ω) k ∂ t u k L (Ω) k ∂ t φ k L (Ω) k∇ ∂ t φ k L (Ω) ≤ ρ ∗ k ∂ t u k L (Ω) + 18 k∇ ∂ t φ k L (Ω) + C k ∂ t φ k L (Ω) k D u k L (Ω) . Combining (5.5) and the above inequalities, we deduce that dd t H ( t ) + ρ ∗ k ∂ t u k L (Ω) + 12 k∇ ∂ t φ k L (Ω) IFFUSE INTERFACE MODELS FOR INCOMPRESSIBLE BINARY FLUIDS 39 ≤ C k ∂ t φ k L (Ω) + C (cid:0) k D u k L (Ω) + k φ k H (Ω) (cid:1) k ∂ t φ k L (Ω) + C k D u k L (Ω) log (cid:16) C k u k W ,p (Ω) k D u k L (Ω) (cid:17) + C k ∆ φ k L (Ω) k∇ φ k H (Ω) log (cid:16) C k∇ φ k W ,p (Ω) k∇ φ k H (Ω) (cid:17) . From the inequalities x log (cid:16) Cyx (cid:17) ≤ x log( Cy ) + 1 , ∀ x, y > , (5.7) ν ∗ k D u k L (Ω) + 12 k ∂ t φ k L (Ω) ≤ H ( t ) ≤ C (cid:16) k D u k L (Ω) + k ∂ t φ k L (Ω) (cid:17) , (5.8)and the estimate (4.19), we can rewrite the above differential inequality as follows dd t H ( t ) + ρ ∗ k ∂ t u k L (Ω) + 12 k∇ ∂ t φ k L (Ω) ≤ C (cid:0) H ( t ) + H ( t ) (cid:1) + CH ( t ) log (cid:0) C k u k W ,p (Ω) (cid:1) + C (cid:0) H ( t ) (cid:1) log (cid:0) C k φ k W ,p (Ω) (cid:1) . (5.9)Let us now estimate the argument of the logarithmic terms on the right-hand side of (5.9). First, werewrite (4.1) as a Stokes problem with non-constant viscosity − div ( ν ( φ ) D u ) + ∇ P = f , in Ω × (0 , T ) , div u = 0 , in Ω × (0 , T ) , u = , on ∂ Ω × (0 , T ) , where f = − ρ ( φ ) (cid:0) ∂ t u + u · ∇ u (cid:1) − ∆ φ ∇ φ . We now apply Theorem A.1 with the following choice ofparameters p = 1 + ε , ε ∈ (0 , , and r ∈ (2 , ∞ ) such that r = ε − . We infer that k u k W , ε (Ω) ≤ C (cid:0) k ∂ t u k L ε (Ω) + k u · ∇ u k L ε (Ω) + k ∆ φ ∇ φ k L ε (Ω) (cid:1) + C k D u k L (Ω) k∇ φ k L r (Ω) ≤ C (cid:0) k ∂ t u k L (Ω) + k u k L ε )1 − ε (Ω) k∇ u k L (Ω) + k∇ φ k L ε )1 − ε (Ω) k ∆ φ k L (Ω) (cid:1) + C k D u k L (Ω) k φ k H (Ω) ≤ C k ∂ t u k L (Ω) + C k D u k L (Ω) + k φ k H (Ω) ≤ C k ∂ t u k L (Ω) + C (1 + H ( t )) , where the constant C depends on ε . We recall the Sobolev embedding W , ε (Ω) ֒ → W ,p (Ω) where p = ε − . Therefore, for any p ∈ (2 , ∞ ) there exists a constant C depending on p such that k u k W ,p (Ω) ≤ C k ∂ t u k L (Ω) + C (1 + H ( t )) . (5.10)Next, by reformulating the equation (4.1) as the elliptic problem ( − ∆ φ + F ′ ( φ ) = µ + θ φ, in Ω × (0 , T ) ,∂ n φ = 0 , on ∂ Ω × (0 , T ) . (5.11)We infer from the elliptic regularity theory (see, e.g., [2, Lemma 2] and [34]) that k φ k W ,p (Ω) + k F ′ ( φ ) k L p (Ω) ≤ C (1 + k φ k L (Ω) + k µ + θ φ k L p (Ω) ) ≤ C (1 + k φ k L p (Ω) + k µ k L p (Ω) ) , (5.12)for p ∈ (2 , ∞ ) . On the other hand, from the equation (4.1) , we see that µ = − ∂ t φ − u · ∇ φ − ρ ′ ( φ ) | u | µ + ρ ′ ( φ ) | u | . Observe now that (cid:13)(cid:13)(cid:13) − ρ ′ ( φ ) | u | (cid:13)(cid:13)(cid:13) L p (Ω) ≤ C k u k L p (Ω) ≤ C k∇ u k L (Ω) . Then, owing to Sobolev embedding and (2.1), we have k µ − µ k L p (Ω) ≤ k ∂ t φ k L p (Ω) + k u · ∇ φ k L p (Ω) + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ρ ′ ( φ ) | u | − ρ ′ ( φ ) | u | (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p (Ω) ≤ C k∇ ∂ t φ k L (Ω) + C k u k H (Ω) k φ k H (Ω) + C k∇ u k L (Ω) . In light of (4.17) and (4.21), the above inequality yields k µ k L p (Ω) ≤ C k µ − µ k L p (Ω) + C | µ |≤ C k µ − µ k L p (Ω) + C (1 + k µ − µ k L (Ω) ) ≤ C (1 + k∇ ∂ t φ k L (Ω) + H ( t )) . (5.13)Thus, for any p > , we deduce from the above estimate and (5.12) that k φ k W ,p (Ω) ≤ C (1 + k∇ ∂ t φ k L (Ω) + H ( t )) , (5.14)for some positive constant C depending on p .We recall the generalized Young inequality xy ≤ Φ( x ) + Υ( y ) , ∀ x, y > , (5.15)where Φ( s ) = s log s − s + 1 , Υ( s ) = e s − . Then we have H ( t ) log(1 + k ∂ t u k L (Ω) ) ≤ H ( t ) log H ( t ) + 1 + k ∂ t u k L (Ω) . Thus, using the above estimate and the elementary inequality log( x + y ) < log(1 + x ) + log(1 + y ) , x, y > , we can estimate the second term on the right-hand side of (5.9) as follows CH ( t ) log( C k u k W ,p (Ω) ) ≤ CH ( t ) log (cid:0) C k ∂ t u k L (Ω) + C (1 + H ( t )) (cid:1) ≤ CH ( t ) (cid:0) k ∂ t u k L (Ω) ) + log(1 + H ( t )) (cid:1) ≤ CH ( t ) + CH ( t ) (cid:0) H ( t ) log H ( t ) + 1 (cid:1) + CH ( t ) k ∂ t u k L (Ω) + CH ( t ) log(1 + H ( t )) ≤ ρ ∗ k ∂ t u k L (Ω) + C (cid:0) H ( t ) (cid:1) + CH ( t ) log( e + H ( t )) . (5.16)In a similar manner, we have H ( t ) log(1 + k∇ ∂ t φ k L (Ω) ) ≤ H ( t ) log H ( t ) + 1 + k∇ ∂ t φ k L (Ω) . IFFUSE INTERFACE MODELS FOR INCOMPRESSIBLE BINARY FLUIDS 41 Then, using (5.14), the third term on the right-hand side of (5.9) can be estimated as follows C (cid:0) H ( t ) (cid:1) log (cid:0) C k φ k W ,p (Ω) (cid:1) ≤ C (cid:0) H ( t ) (cid:1) log (cid:0) C (1 + k∇ ∂ t φ k L (Ω) + H ( t )) (cid:1) ≤ C (cid:0) H ( t ) (cid:1) + C log (cid:0) k∇ ∂ t φ k L (Ω) + H ( t ) (cid:1) + H ( t ) log (cid:0) k∇ ∂ t φ k L (Ω) + H ( t ) (cid:1) ≤ C (cid:0) H ( t ) (cid:1) + C (cid:0) k∇ ∂ t φ k L (Ω) + H ( t ) (cid:1) + C k∇ ∂ t φ k L (Ω) H ( t )+ H ( t ) log(1 + H ( t )) ≤ k∇ ∂ t φ k L (Ω) + C (cid:0) H ( t ) (cid:1) + CH ( t ) (cid:0) e + H ( t ) (cid:1) log( e + H ( t )) . (5.17)Hence, by (5.16) and (5.17), we easily deduce from (5.9) that dd t ( e + H ( t )) + ρ ∗ k ∂ t u k L (Ω) + 14 k∇ ∂ t φ k L (Ω) ≤ C + CH ( t )( e + H ( t )) log( e + H ( t )) . (5.18)Thanks to (4.10), (4.19), and (5.8), we obtain Z t +1 t H ( τ ) d τ ≤ Q ( E ) , ∀ t ≥ , (5.19)where Q is independent of t , and E = E ( u , φ ) . We now apply the generalized Gronwall lemma B.2to (5.18) and find the estimate sup t ∈ [0 , H ( t ) ≤ C (cid:0) e + H (0) (cid:1) e Q ( E . Moreover, by using the generalized uniform Gronwall lemma B.3 together with (5.19), we infer that sup t ≥ H ( t ) ≤ Ce ( e + Q ( E )) e (1+ Q ( E . By combining the above inequalities, we get sup t ≥ H ( t ) ≤ Q ( E , k u k V σ , k φ k H (Ω) , k F ′ ( φ ) k L (Ω) ) . (5.20)In addition, integrating (5.18) on the time interval [ t, t + 1] , we have, for all t ≥ , Z t +1 t k ∂ t u ( τ ) k L (Ω) + k∇ ∂ t φ ( τ ) k L (Ω) d τ ≤ Q ( E , k u k V σ , k φ k H (Ω) , k F ′ ( φ ) k L (Ω) ) . (5.21)Then we can deduce that u ∈ L ∞ (0 , T ; V σ ) ∩ H (0 , T ; H σ ) ∂ t φ ∈ L ∞ (0 , T ; L (Ω)) ∩ L (0 , T ; H (Ω)) . (5.22)Thanks to (4.19) and (5.14), we also get, sup t ≥ k φ ( t ) k H (Ω)) ≤ Q ( E , k u k V σ , k φ k H (Ω) , k F ′ ( φ ) k L (Ω) ) , (5.23)and, for all t ≥ , Z t +1 t k φ ( τ ) k W ,p (Ω)) d τ ≤ Q ( E , k u k V σ , k φ k H (Ω) , k F ′ ( φ ) k L (Ω) ) , (5.24) for any p ∈ (2 , ∞ ) . This entails that φ ∈ L ∞ (0 , T ; H (Ω)) ∩ L (0 , T ; W ,p (Ω)) . According to (4.17),(4.21) and (5.13), it follows that µ ∈ L ∞ (0 , T ; L (Ω)) ∩ L (0 , T ; L p (Ω)) and, as a consequence, F ′ ( φ ) ∈ L ∞ (0 , T ; L (Ω)) ∩ L (0 , T ; L p (Ω)) . Finally, by exploiting Theorem A.1 with p = 2 and r = ∞ , together with the regularity of φ obtainedabove, we have, for all t ≥ , Z t +1 t k u ( τ ) k H (Ω) d τ ≤ Q ( E , k u k V σ , k φ k H (Ω) , k F ′ ( φ ) k L (Ω) ) , (5.25)which yields that u ∈ L (0 , T ; H (Ω)) . Entropy bound in L ∞ (0 , T ; L (Ω)) . First of all, we observe that, for all s ∈ ( − , , F ′ ( s ) = θ (cid:16) s − s (cid:17) , F ′′ ( s ) = θ − s , F ′′′ ( s ) = 2 θs (1 − s ) (1 + s ) (5.26)and F (4) ( s ) = 2 θ (1 + 3 s )(1 − s ) (1 + s ) > . (5.27)Next, we compute dd t Z Ω F ′′ ( φ ) d x = Z Ω F ′′′ ( φ ) ∂ t φ d x = Z Ω F ′′′ ( φ ) (cid:16) ∆ φ − u · ∇ φ − F ′ ( φ ) + θ φ − ρ ′ ( φ ) | u | ξ (cid:17) d x. Since Z Ω F ′′′ ( φ ) u · ∇ φ d x = Z Ω u · ∇ ( F ′′ ( φ )) d x = 0 , and exploiting the integration by parts, we rewrite the above equality as follows dd t Z Ω F ′′ ( φ ) d x + Z Ω F (4) ( φ ) |∇ φ | d x + Z Ω F ′′′ ( φ ) F ′ ( φ ) d x = Z Ω F ′′′ ( φ ) (cid:16) θ φ − ρ ′ ( φ ) | u | ξ (cid:17) d x. (5.28)In particular, by using (5.27), we have dd t Z Ω F ′′ ( φ ) d x + Z Ω F ′′′ ( φ ) F ′ ( φ ) d x ≤ Z Ω F ′′′ ( φ ) (cid:16) θ φ − ρ ′ ( φ ) | u | ξ (cid:17) d x. (5.29)It follows from (5.15) that xy ≤ εx log x + e yε , ∀ x > , y > , ε ∈ (0 , . (5.30)which implies Z Ω − F ′′′ ( φ ) ρ ′ ( φ ) | u | x ≤ Z Ω | F ′′′ ( φ ) || ρ ′ ( φ ) | | u | x ≤ ε Z Ω | F ′′′ ( φ ) | log( | F ′′′ ( φ ) | ) d x + Z Ω e | ρ ′ ( φ ) | ε | u | d x. (5.31) IFFUSE INTERFACE MODELS FOR INCOMPRESSIBLE BINARY FLUIDS 43 We observe that, for all s ∈ [0 , , it holds log( | F ′′′ ( s ) | ) = log( F ′′′ ( s )) = log (cid:16) θs (1 − s ) (1 + s ) (cid:17) = 2 log (cid:16) s − s √ θs (1 + s ) (cid:17) ≤ (cid:16) √ θ s − s (cid:17) = log(2 θ ) + 4 θ F ′ ( s ) . Since both F ′ ( s ) and F ′′′ ( s ) are odd, we easily deduce that log( | F ′′′ ( s ) | ) ≤ C + 4 θ | F ′ ( s ) | , ∀ s ∈ ( − , , where C = log(2 θ ) (without loss of generality, we assume in the sequel that C > ). Then, using thefact that F ′′′ ( s ) F ′ ( s ) ≥ for all s ∈ ( − , , we obtain | F ′′′ ( s ) | log( | F ′′′ ( s ) | ) ≤ C | F ′′′ ( s ) | + 4 θ F ′′′ ( s ) F ′ ( s ) , ∀ s ∈ ( − , . Fix the constant α ∈ (0 , such that F ′ ( α ) = 1 . We infer that | F ′′′ ( s ) | log( | F ′′′ ( s ) | ) ≤ C + C F ′′′ ( s ) F ′ ( s ) , ∀ s ∈ ( − , . (5.32)where C = C F ′′′ ( α ) , C = 4 θ + C . Taking ε = C in (5.31), we arrive at Z Ω − F ′′′ ( φ ) ρ ′ ( φ ) | u | x ≤ C | Ω | C + 12 Z Ω F ′′′ ( φ ) F ′ ( φ ) d x + Z Ω e C | ρ ′ ( φ ) || u | d x. (5.33)Arguing in a similar way ( ε = C ), we obtain Z Ω F ′′′ ( φ ) ( θ φ + ξ ) d x ≤ C | Ω | C + 14 Z Ω F ′′′ ( φ ) F ′ ( φ ) d x + Z Ω e C | θ φ + ξ | d x. Since φ is globally bounded ( k φ k L ∞ (Ω × (0 ,T )) ≤ ) and k ξ k L ∞ (0 ,T ) ≤ C ∗ , we get Z Ω F ′′′ ( φ ) ( θ + ξ ) φ d x ≤ Z Ω F ′′′ ( φ ) F ′ ( φ ) d x + C | Ω | C + e C ( θ + C ∗ ) | Ω | . (5.34)Combining (5.29) with (5.33) and (5.34), we deduce that dd t Z Ω F ′′ ( φ ) d x + 14 Z Ω F ′′′ ( φ ) F ′ ( φ ) d x ≤ C | Ω | C + e C ( θ + C ∗ ) | Ω | + Z Ω e C | ρ ′ ( φ ) |k∇ u k L | u | k∇ u k L ! d x. (5.35)In order to control the last term on the right-hand side of (5.35), we shall use the Trudinger-Moserinequality (see, e.g., [59]). Namely, let f ∈ H (Ω) ( d = 2 ) such that R Ω |∇ f | d x ≤ . Then, thereexists a constant C T M = C T M (Ω) (which depends only on the domain Ω ) such that Z Ω e π | f | d x ≤ C T M (Ω) . (5.36) Next, as a consequence of (5.20), we have the following uniform estimate sup t ≥ k∇ u ( t ) k L (Ω) ≤ Q ( E ( u , φ ) , H (0)) =: R , (5.37)where R is independent of time. The exact value of R can be estimated in terms of the norm of theinitial conditions. Now we make the following assumptions: | ρ ′ ( s ) | L ∞ ( − , ≤ πC R . (5.38)Thanks to (5.38), we conclude that dd t Z Ω F ′′ ( φ ) d x + 14 Z Ω F ′′′ ( φ ) F ′ ( φ ) d x ≤ C | Ω | C + e C ( θ + C ∗ ) | Ω | + C T M (Ω) . (5.39)Observe now that, for s ∈ (cid:2) , , F ′′ ( s ) = θ − s = (1 − s )(1 + s )2 s F ′′′ ( s ) ≤ F ′ ( ) F ′′′ ( s ) F ′ ( s ) . This gives F ′′ ( s ) ≤ C + C F ′′′ ( s ) F ′ ( s ) , ∀ s ∈ ( − , , (5.40)where C = F ′′ (cid:16) (cid:17) , C = 34 F ′ ( ) . Hence, we are led to dd t Z Ω F ′′ ( φ ) d x + 14 C Z Ω F ′′ ( φ ) d x ≤ C , where C = 3 C | Ω | C + e C ( θ + C ∗ ) | Ω | + C T M (Ω) + C | Ω | C . We recall that F ′′ ( φ ) ∈ L (Ω) . Then, an application of the Gronwall lemma entails that Z Ω F ′′ ( φ ( t )) d x ≤ k F ′′ ( φ ) k L (Ω) e − t C + 4 C C , ∀ t ≥ . (5.41)In addition, integrating (5.39) on the time interval [ t, t + 1] , we find Z t +1 t Z Ω F ′′′ ( φ ) F ′ ( φ ) d x d τ ≤ k F ′′ ( φ ) k L (Ω) + C , ∀ t ≥ , (5.42)where C = 4 C − C | Ω | C . This allows us to improve the integrability of F ′′ ( φ ) . Indeed, arguing similarly to (5.40), we have for s ∈ (cid:2) , 1) ( F ′′ ( s )) log(1 + F ′′ ( s )) = θ (1 − s ) (1 + s ) log (cid:16) θ − s (cid:17) ≤ θF ′′′ ( s ) log (cid:16) s − s − s + θ (1 + s ) (cid:17) IFFUSE INTERFACE MODELS FOR INCOMPRESSIBLE BINARY FLUIDS 45 ≤ F ′′′ ( s ) F ′ ( s ) + θF ′′′ ( s ) log (cid:16) 12 + 2 θ (cid:17) ≤ C F ′′′ ( s ) F ′ ( s ) . Hence, we infer that ( F ′′ ( s )) log(1 + F ′′ ( s )) ≤ C F ′′′ ( s ) F ′ ( s ) + C , ∀ s ∈ ( − , . In light of (5.42), we deduce (5.2). Indeed, we have Z t +1 t Z Ω ( F ′′ ( φ )) log(1 + F ′′ ( φ )) d x d τ ≤ C k F ′′ ( φ ) k L (Ω) + C C + C , ∀ t ≥ . (5.43)We notice that, by keeping the (non-negative) term F (4) ( φ ) |∇ φ | (cf. (5.28)) on the left-hand sideof (5.39) in the above argument, we can also deduce that Z t +1 t Z Ω F (4) ( φ ) |∇ φ | d x d τ ≤ C , ∀ t ≥ , where C depends on k F ′′ ( φ ) k L (Ω) , R , θ , θ and Ω . Since (cid:16) s √ − s (cid:17) ′ = (1 − s ) − , we infer that Z t +1 t Z Ω (cid:12)(cid:12)(cid:12) ∇ (cid:16) φ p − φ (cid:17)(cid:12)(cid:12)(cid:12) d x d τ ≤ C θ , ∀ t ≥ . Setting ψ = φ √ − φ , and observing that F ′′ ( s ) = θ h(cid:0) s √ − s (cid:1) + 1 i , we have (cf. (5.39)) k ψ ( t ) k L (Ω) + Z t +1 t k∇ ψ ( τ ) k L (Ω) d τ ≤ C , ∀ t ≥ . This implies that ψ ∈ L ∞ (0 , T ; L (Ω)) ∩ L (0 , T ; H (Ω)) . By Sobolev embedding, we also have that ψ ∈ L q (0 , T ; L p (Ω)) where = p + q , p ∈ (2 , ∞ ) . As a consequence, we conclude that Z t +1 t k F ′′ ( φ ( τ )) k qL p (Ω) d τ ≤ C , ∀ t ≥ , (5.44)where p + q , p ∈ (1 , ∞ ) . Uniqueness of strong solutions. Let us consider two strong solutions ( u , φ , P ) and ( u , φ .P ) to system (4.1)-(4.3) satisfying the entropy bound (5.2) and originating from the same initial datum.The solutions difference ( u , φ, P ) := ( u − u , φ − φ , P − P ) solves ρ ( φ ) (cid:0) ∂ t u + u · ∇ u + u · ∇ u (cid:1) − div (cid:0) ν ( φ ) D u (cid:1) + ∇ P = − ∆ φ ∇ φ − ∆ φ ∇ φ − ( ρ ( φ ) − ρ ( φ ))( ∂ t u + u · ∇ u ) + div (cid:0) ( ν ( φ ) − ν ( φ )) D u (cid:1) (5.45)and ∂ t φ + u · ∇ φ + u · ∇ φ − ∆ φ + Ψ ′ ( φ ) − Ψ ′ ( φ )= − ρ ′ ( φ ) | u | ρ ′ ( φ ) | u | ξ − ξ , (5.46)for almost every ( x, t ) ∈ Ω × (0 , T ) , together with the incompressibility constraint div u = 0 . It follows that φ ( t ) = 0 . Multiplying (5.45) by u and integrating over Ω , we obtain dd t Z Ω ρ ( φ )2 | u | d x + Z Ω ρ ( φ )( u · ∇ ) u · u d x + Z Ω ρ ( φ )( u · ∇ ) u · u d x + Z Ω ν ( φ ) | D u | d x = − Z Ω ∆ φ ∇ φ · u d x − Z Ω ∆ φ ∇ φ · u d x − Z Ω ( ρ ( φ ) − ρ ( φ ))( ∂ t u + u · ∇ u ) · u d x − Z Ω ( ν ( φ ) − ν ( φ )) D u : D u d x + Z Ω | u | ρ ′ ( φ ) ∂ t φ d x. (5.47)Next, multiplying (5.46) by − ∆ φ and integrating over Ω , we find dd t Z Ω |∇ φ | d x + k ∆ φ k L (Ω) = Z Ω ( u · ∇ φ ) ∆ φ d x + Z Ω ( u · ∇ φ ) ∆ φ d x + Z Ω ( F ′ ( φ ) − F ′ ( φ ))∆ φ d x + θ k∇ φ k L (Ω) + Z Ω (cid:16) ρ ′ ( φ ) | u | − ρ ′ ( φ ) | u | (cid:17) ∆ φ d x. (5.48)Here we have used the fact that ∆ φ = 0 which implies that R Ω ( ξ − ξ )∆ φ d x = 0 . Adding (5.47) and(5.48), together with the bound from below of the viscosity, we have dd t (cid:16) Z Ω ρ ( φ )2 | u | d x + Z Ω |∇ φ | d x (cid:17) + ν ∗ k D u k L (Ω) + k ∆ φ k L (Ω) ≤ − Z Ω ρ ( φ )( u · ∇ ) u · u d x − Z Ω ρ ( φ )( u · ∇ ) u · u d x − Z Ω ∆ φ ∇ φ · u d x − Z Ω ( ρ ( φ ) − ρ ( φ ))( ∂ t u + u · ∇ u ) · u d x − Z Ω ( ν ( φ ) − ν ( φ )) D u : D u d x + Z Ω | u | ρ ′ ( φ ) ∂ t φ d x + Z Ω ( u · ∇ φ ) ∆ φ d x + Z Ω ( F ′ ( φ ) − F ′ ( φ ))∆ φ d x + θ k∇ φ k L (Ω) + Z Ω (cid:16) ρ ′ ( φ ) | u | − ρ ′ ( φ ) | u | (cid:17) ∆ φ d x. (5.49)We now proceed by estimating the terms on the right hand side of the above differential equality.We would like to mention that most of the bounds obtained below are easy applications of the Sobolevembedding theorem and interpolation inequalities in view of the estimates for global strong solutionsthat have been obtained before. Nevertheless, less standard is the estimate of the term involving thedifference of the nonlinear terms ( F ′ ( φ ) − F ′ ( φ ) ) which makes use of the entropy bound (5.43).By using the regularity of strong solutions, (2.2) and (2.4), we have − Z Ω ρ ( φ )( u · ∇ ) u · u d x ≤ C k u k L ∞ (Ω) k∇ u k L (Ω) k u k L (Ω) ≤ ν ∗ k D u k L (Ω) + C k u k L ∞ (Ω) k u k L (Ω) , − Z Ω ρ ( φ )( u · ∇ ) u · u d x ≤ C k∇ u k L (Ω) k u k L (Ω) ≤ ν ∗ k D u k L (Ω) + C k u k L (Ω) , IFFUSE INTERFACE MODELS FOR INCOMPRESSIBLE BINARY FLUIDS 47 − Z Ω ∆ φ ∇ φ · u d x ≤ C k ∆ φ k L (Ω) k∇ φ k L (Ω) k u k L (Ω) ≤ ν ∗ k D u k L (Ω) + C k ∆ φ k L (Ω) k∇ φ k L (Ω) , − Z Ω ( ρ ( φ ) − ρ ( φ ))( ∂ t u + u · ∇ u ) · u d x ≤ C k φ k L (Ω) k ∂ t u + u · ∇ u k L (Ω) k u k L (Ω) ≤ C k∇ φ k L (Ω) k ∂ t u + u · ∇ u k L (Ω) k∇ u k L (Ω) ≤ ν ∗ k D u k L (Ω) + C k ∂ t u + u · ∇ u k L (Ω) k∇ φ k L (Ω) , − Z Ω ( ν ( φ ) − ν ( φ )) D u : D u d x ≤ C k φ k L (Ω) k D u k L (Ω) k D u k L (Ω) ≤ ν ∗ k D u k L (Ω) + C k D u k L (Ω) k∇ φ k L (Ω) , Z Ω | u | ρ ′ ( φ ) ∂ t φ d x ≤ C k u k L (Ω) k ∂ t φ k L (Ω) ≤ ν ∗ k D u k L (Ω) + C k u k L (Ω) , Z Ω ( u · ∇ φ ) ∆ φ d x ≤ k u k L ∞ (Ω) k∇ φ k L (Ω) k ∆ φ k L (Ω) ≤ k ∆ φ k L (Ω) + C k u k L ∞ (Ω) k∇ φ k L (Ω) , Z Ω (cid:16) ρ ′ ( φ ) | u | − ρ ′ ( φ ) | u | (cid:17) ∆ φ d x = Z Ω (cid:16) ρ ′ ( φ ) − ρ ′ ( φ ) (cid:17) | u | φ d x + Z Ω ρ ′ ( φ )2 (cid:16) u · u + u · u (cid:17) ∆ φ d x ≤ C k φ k L (Ω) k u k L (Ω) k ∆ φ k L (Ω) + C k u k L (Ω) ( k u k L ∞ (Ω) + k u k L ∞ (Ω) ) k ∆ φ k L (Ω) ≤ k ∆ φ k L (Ω) + C k∇ φ k L (Ω) + C ( k u k L ∞ (Ω) + k u k L ∞ (Ω) ) k u k L (Ω) . Using the generalized Young inequality (5.15) and the standard Young inequality, for x > , y > , z > with Cz > y , we obtain x y log (cid:16) Czy (cid:17) ≤ xy (cid:18) x log x + Czy (cid:19) ≤ εz + x y log x + C ε − x y , ∀ ε > . (5.50)By making use of (2.9) and (5.50), we obtain that Z Ω ( F ′ ( φ ) − F ′ ( φ ))∆ φ d x = Z Ω Z F ′′ ( τ φ + (1 − τ ) φ ) d τ φ ∆ φ d x ≤ C (cid:0) k F ′′ ( φ ) k L (Ω) + k F ′′ ( φ ) k L (Ω) (cid:1) k φ k L ∞ (Ω) k ∆ φ k L (Ω) ≤ C (cid:0) k F ′′ ( φ ) k L (Ω) + k F ′′ ( φ ) k L (Ω) (cid:1) k∇ φ k L (Ω) log (cid:16) C k ∆ φ k L (Ω) k∇ φ k L (Ω) (cid:17) k ∆ φ k L (Ω) ≤ k ∆ φ k L (Ω) + C (cid:0) k F ′′ ( φ ) k L (Ω) + k F ′′ ( φ ) k L (Ω) (cid:1) k∇ φ k L (Ω) log (cid:16) C k ∆ φ k L (Ω) k∇ φ k L (Ω) (cid:17) ≤ k ∆ φ k L (Ω) + C k F ′′ ( φ ) k L (Ω) (cid:0) (cid:0) k F ′′ ( φ ) k L (Ω) (cid:1)(cid:1) k∇ φ k L (Ω) + C k F ′′ ( φ ) k L (Ω) (cid:0) (cid:0) k F ′′ ( φ ) k L (Ω) (cid:1)(cid:1) k∇ φ k L (Ω) . Collecting the above bounds, we find the differential inequality dd t (cid:16) Z Ω ρ ( φ )2 | u | d x + Z Ω |∇ φ | d x (cid:17) + ν ∗ Z Ω | D u | d x + 12 k ∆ φ k L (Ω) ≤ W ( t ) Z Ω ρ ( φ )2 | u | d x + W ( t ) k∇ φ k L (Ω) , (5.51)where W ( t ) = C (cid:0) k u k L ∞ (Ω) + k u k L ∞ (Ω) + k ∂ t u + u · ∇ u k L (Ω) (cid:1) , and W ( t ) = C (cid:16) k ∆ φ k L (Ω) + k ∂ t u + u · ∇ u k L (Ω) + k D u k L (Ω) + k u k L ∞ (Ω) (cid:17) + C k F ′′ ( φ ) k L (Ω) log (cid:0) k F ′′ ( φ ) k L (Ω) (cid:1) + C k F ′′ ( φ ) k L (Ω) log (cid:0) k F ′′ ( φ ) k L (Ω) (cid:1) . Here we have used that ρ ( s ) ≥ ρ ∗ for all s ∈ ( − , . In order to apply the Gronwall lemma, we areleft to show that Z T k F ′′ ( φ i ) k L (Ω) log (cid:0) k F ′′ ( φ i ) k L (Ω) (cid:1) d τ ≤ C ( T ) , i = 1 , . (5.52)To this aim, we introduce the function g ( s ) = s log( C ∗ s ) , ∀ s ∈ (0 , ∞ ) , where C ∗ is a positive constant. It is easily seen that g is continuous and convex ( g ′′ ( s ) = s > ). Byapplying Jensen’s inequality, we have g (cid:16) | Ω | Z Ω | F ′′ ( φ ) | d x (cid:17) ≤ | Ω | Z Ω g ( | F ′′ ( φ ) | ) d x. Using the explicit form of g , this is equivalent to | Ω | k F ′′ ( φ ) k L (Ω) log (cid:16) C ∗ | Ω | k F ′′ ( φ ) k L (Ω) (cid:17) ≤ | Ω | Z Ω | F ′′ ( φ ) | log( C ∗ | F ′′ ( φ ) | ) d x. IFFUSE INTERFACE MODELS FOR INCOMPRESSIBLE BINARY FLUIDS 49 Taking C ∗ = | Ω | and integrating the above inequality over [0 , T ] , we find Z T k F ′′ ( φ ) k L (Ω) log (cid:0) k F ′′ ( φ ) k L (Ω) (cid:1) d τ ≤ Z T Z Ω | F ′′ ( φ ) | log( | Ω || F ′′ ( φ ) | ) d x d τ. (5.53)Then, (5.52) immediately follows from the entropy bounds (5.43) and (5.53). As a consequence, both W and W belong to L (0 , T ) . Finally, an application of the Gronwall lemma entails the uniquenessof strong solutions. (cid:3) Remark 5.5 (Entropy Estimates in L p , p > ) . Notice that the entropy estimate in L (Ω) proved inTheorem 5.1-(2) can be generalized to the L p (Ω) case with p > . More precisely, for any p ∈ N , thereexists η p > with the latter depending on the norms of the initial data and on the parameters of thesystem η p = η p ( E ( u , φ ) , k u k V σ , k φ k H (Ω) , k F ′ ( φ ) k L (Ω) , θ, θ ) such that, if k ρ ′ k L ∞ ( − , ≤ η p and F ′′ ( φ ) ∈ L p (Ω) , then, for any T > σ , we have F ′′ ( φ ) ∈ L ∞ (0 , T ; L p (Ω)) , | F ′′ ( φ ) | p − F ′′′ ( φ ) F ′ ( φ ) ∈ L (Ω × (0 , T )) . Such result follows from the above proof by replacing dd t R Ω F ′′ ( φ ) d x by dd t R Ω ( F ′′ ( φ )) p d x , and theobservation that, for any p > , there exist two positive constants C p and C p such that | ( F ′′ ( s )) p − F ′′′ ( s ) | log (cid:0) | ( F ′′ ( s )) p − F ′′′ ( s ) | (cid:1) ≤ C p + C p ( F ′′ ( s )) p − F ′′′ ( s ) F ′ ( s ) , ∀ s ∈ ( − , . Proof of Theorem 5.2. We now prove the propagation of entropy bound as stated in Theorem5.2. For every strong solution given by Theorem 5.1-(1), we have k u · ∇ φ k H (Ω) ≤ k u k L (Ω) k∇ φ k L + k∇ u k L (Ω) k∇ φ k L (Ω) + k u k L ∞ (Ω) k φ k H (Ω) ≤ C + C k u k H (Ω) k φ k H (Ω) + C k φ k H (Ω) , and k ρ ′ ( φ ) | u | k H (Ω) ≤ C k u k L (Ω) + C k∇ φ k L ∞ (Ω) k u k L (Ω) + C k∇ u k L (Ω) k u k L (Ω) ≤ C + C k φ k W , (Ω) + C k u k H (Ω) , which imply that Z t +1 t k u ( τ ) · ∇ φ ( τ ) k H (Ω) + k ρ ′ ( φ ( τ )) | u ( τ ) | k H (Ω) d τ ≤ C, ∀ t ≥ , for some C independent of t . In light of (5.21), it follows that Z t +1 t k − ∆ φ ( τ ) + F ′ ( φ ( τ )) k H (Ω) d τ ≤ C, ∀ t ≥ . By using [33, Lemma 7.4], we infer that, for any p ≥ , there exists C = C ( p ) such that k F ′′ ( φ ) k L p (Ω) ≤ C (cid:16) e C k− ∆ φ + F ′ ( φ ) k H (cid:17) a.e. in (0 , T ) . (5.54) Notice that we are not able to conclude that the right hand side of (5.54) is L (0 , T ) . Nevertheless,since integrable function are finite almost everywhere, the above inequality entails that there existssome σ ∈ (0 , (actually σ can be taken arbitrarily small but positive) such that F ′′ ( φ ( σ )) ∈ L p (Ω) with k F ′′ ( φ ( σ )) k L p (Ω) ≤ C ( p, σ ) , ∀ p ∈ [1 , ∞ ) . (5.55)Then, under the condition (5.38) but without the additional assumption F ′′ ( φ ) ∈ L (Ω) on the initialdatum, we are able to deduce that the previous estimates (5.41)-(5.43) hold for t ≥ σ > . Moreprecisely, we have Z t +1 t Z Ω ( F ′′ ( φ )) log(1 + F ′′ ( φ )) d x d τ ≤ C ( σ ) , ∀ t ≥ . (5.56)Differentiating (4.1) with respect to time and testing the resultant by ∂ t u , integrating over Ω , wehave Z Ω ρ ( φ ) ∂ t | ∂ t u | d x + Z Ω ρ ( φ ) (cid:0) ∂ t u · ∇ u + u · ∇ ∂ t u (cid:1) · ∂ t u d x + Z Ω ρ ′ ( φ ) ∂ t φ ( ∂ t u + u · ∇ u ) · ∂ t u d x + Z Ω ν ( φ ) | D∂ t u | d x + Z Ω ν ′ ( φ ) ∂ t φD u : D∂ t u d x = Z Ω ∂ t ( ∇ φ ⊗ ∇ φ ) : ∇ ∂ t u d x. Since Z Ω ρ ( φ ) ∂ t | ∂ t u | d x = 12 dd t Z Ω ρ ( φ ) | ∂ t u | d x − Z Ω ρ ′ ( φ ) ∂ t φ | ∂ t u | d x, we find 12 dd t Z Ω ρ ( φ ) | ∂ t u | d x + Z Ω ν ( φ ) | D∂ t u | d x = − Z Ω ρ ( φ )( ∂ t u · ∇ u + u · ∇ ∂ t u ) · ∂ t u d x − Z Ω ρ ′ ( φ ) ∂ t φ | ∂ t u | d x − Z Ω ρ ′ ( φ ) ∂ t φ ( u · ∇ u ) · ∂ t u d x − Z Ω ν ′ ( φ ) ∂ t φD u : ∇ ∂ t u + Z Ω ∂ t ( ∇ φ ⊗ ∇ φ ) : ∇ ∂ t u d x. In view of (5.22), by using (2.4), we have − Z Ω ρ ( φ )( ∂ t u · ∇ u ) · ∂ t u d x ≤ C k ∂ t u k L (Ω) k∇ u k L (Ω) ≤ ν ∗ k D∂ t u k L (Ω) + C k ∂ t u k L (Ω) , and − Z Ω ρ ( φ )( u · ∇ ∂ t u ) · ∂ t u d x ≤ C k u k L (Ω) k∇ ∂ t u k L (Ω) k ∂ t u k L (Ω) ≤ ν ∗ k D∂ t u k L (Ω) + C k ∂ t u k L (Ω) . Similarly, we obtain − Z Ω ρ ′ ( φ ) ∂ t φ | ∂ t u | d x ≤ C k ∂ t φ k L (Ω) k ∂ t u k L (Ω) IFFUSE INTERFACE MODELS FOR INCOMPRESSIBLE BINARY FLUIDS 51 ≤ ν ∗ k D∂ t u k L (Ω) + C k ∂ t u k L (Ω) , and − Z Ω ρ ′ ( φ ) ∂ t φ ( u · ∇ u ) · ∂ t u d x ≤ C k ∂ t φ k L (Ω) k u k L (Ω) k∇ u k L (Ω) k ∂ t u k L (Ω) ≤ C k∇ ∂ t φ k L (Ω) k u k H (Ω) k ∂ t u k L (Ω) k D∂ t u k L (Ω) ≤ ν ∗ k D∂ t u k L (Ω) + C k ∂ t u k L (Ω) + C k∇ ∂ t φ k L (Ω) + C k u k H (Ω) . Besides, by means of (2.7), we deduce that − Z Ω ν ′ ( φ ) ∂ t φD u : D∂ t u d x ≤ C k ∂ t φ k L ∞ (Ω) k D u k L (Ω) k D∂ t u k L (Ω) ≤ ν ∗ k D∂ t u k L (Ω) + C k ∂ t φ k L (Ω) k ∂ t φ k H (Ω) ≤ ν ∗ k D∂ t u k L (Ω) + 114 k ∆ ∂ t φ k L (Ω) + C k∇ ∂ t φ k L (Ω) , and Z Ω ∂ t ( ∇ φ ⊗ ∇ φ ) : ∇ ∂ t u d x ≤ k∇ φ k L (Ω) k∇ ∂ t φ k L (Ω) k D∂ t u k L (Ω) ≤ ν ∗ k D∂ t u k L (Ω) + C k∇ ∂ t φ k L (Ω) k∇ ∂ t φ k H (Ω) ≤ ν ∗ k D∂ t u k L (Ω) + 114 k ∆ ∂ t φ k L (Ω) + C k∇ ∂ t φ k L (Ω) . Next, we differentiate (4.1) with respect to time, multiply the resultant by − ∆ ∂ t φ , and integrate over Ω to obtain 12 dd t k∇ ∂ t φ k L (Ω) + k ∆ ∂ t φ k L (Ω) = θ k∇ ∂ t φ k L (Ω) + Z Ω F ′′ ( φ ) ∂ t φ ∆ ∂ t φ d x + Z Ω ( ∂ t u · ∇ φ )∆ ∂ t φ d x + Z Ω ( u · ∇ ∂ t φ )∆ ∂ t φ d x + 12 Z Ω ρ ′′ ( φ ) ∂ t φ | u | ∆ ∂ t φ d x + Z Ω ρ ′ ( φ )( u · ∂ t u )∆ ∂ t φ d x. Here we have used that ∆ ∂ t φ = 0 since ∂ n ∂ t φ = 0 on the boundary ∂ Ω . Exploiting (2.9), we get Z Ω F ′′ ( φ ) ∂ t φ ∆ ∂ t φ d x ≤ k F ′′ ( φ ) k L (Ω) k ∂ t φ k L ∞ (Ω) k ∆ ∂ t φ k L (Ω) ≤ k F ′′ ( φ ) k L (Ω) k∇ ∂ t φ k L (Ω) log (cid:16) C k ∆ ∂ t φ k L (Ω) k∇ ∂ t φ k L (Ω) (cid:17) k ∆ ∂ t φ k L (Ω) ≤ k ∆ ∂ t φ k L (Ω) + C k F ′′ ( φ ) k L (Ω) k∇ ∂ t φ k L (Ω) log (cid:16) C k ∆ ∂ t φ k L (Ω) k∇ ∂ t φ k L (Ω) (cid:17) . Recalling (5.50), we obtain Z Ω F ′′ ( φ ) ∂ t φ ∆ ∂ t φ d x ≤ k ∆ ∂ t φ k L (Ω) + C k F ′′ ( φ ) k L (Ω) log (cid:0) C k F ′′ ( φ ) k L (Ω) (cid:1) k∇ ∂ t φ k L (Ω) . (5.57)Next, using (2.4) and (5.22), we see that Z Ω ( ∂ t u · ∇ φ )∆ ∂ t φ d x ≤ k ∂ t u k L (Ω) k∇ φ k L (Ω) k ∆ ∂ t φ k L (Ω) ≤ ν ∗ k D∂ t u k L (Ω) + 114 k ∆ ∂ t φ k L (Ω) + C k ∂ t u k L (Ω) , and Z Ω ( u · ∇ ∂ t φ )∆ ∂ t φ d x ≤ k u k L (Ω) k∇ ∂ t φ k L (Ω) k ∆ ∂ t φ k L (Ω) ≤ k ∆ ∂ t φ k L (Ω) + C k∇ ∂ t φ k L (Ω) . (5.58)Finally, in a similar manner we find that Z Ω ρ ′′ ( φ ) ∂ t φ | u | ∆ ∂ t φ d x ≤ C k ∂ t φ k L (Ω) k u k L (Ω) k ∆ ∂ t φ k L (Ω) ≤ k ∆ ∂ t φ k L (Ω) + C k∇ ∂ t φ k L (Ω) , and Z Ω ρ ′ ( φ )( u · ∂ t u )∆ ∂ t φ d x ≤ C k u k L (Ω) k ∂ t u k L (Ω) k ∆ ∂ t φ k L (Ω) ≤ ν ∗ k D∂ t u k L (Ω) + 114 k ∆ ∂ t φ k L (Ω) + C k ∂ t u k L (Ω) . From the above estimates, we deduce that dd t L ( t ) + ν ∗ k D∂ t u k L (Ω) + 12 k ∆ ∂ t φ k L (Ω) ≤ CK ( t ) L ( t ) + C k u k H (Ω) , (5.59)where L ( t ) = 12 Z Ω ρ ( φ ) | ∂ t u ( t ) | d x + 12 k∇ ∂ t φ ( t ) k L (Ω) ,K ( t ) = 1 + k F ′′ ( φ ) k L (Ω) log (cid:0) C k F ′′ ( φ ) k L (Ω) (cid:1) . Recalling estimates (5.21) and (5.25), we have Z t +1 t L ( τ ) + k u ( τ ) k H (Ω) d τ ≤ C, ∀ t ≥ , where C is independent of t . As a consequence, there exists σ ∈ (0 , ( σ can be chosen arbitrarysmall but positive) such that L ( σ ) ≤ C ( σ ) . (5.60) IFFUSE INTERFACE MODELS FOR INCOMPRESSIBLE BINARY FLUIDS 53 Notice that, without loss of generality, this value of σ can be chosen equal to the one in (5.55). Then,by exploiting (5.56) and the Jensen inequality (cf. (5.53)), we obtain Z t +1 t K ( τ ) d τ ≤ C, ∀ t ≥ σ, where C depends on σ , but is independent of t . Thus, by using the Gronwall lemma on the time interval [ σ, and the uniform Gronwall lemma for t ≥ , we deduce that L ( t ) + Z t +1 t k D∂ t u k L (Ω) + k ∆ ∂ t φ k L (Ω) d τ ≤ C ( σ ) , ∀ t ≥ σ. Hence we have ∂ t u ∈ L ∞ ( σ, T ; H σ ) ∩ L ( σ, T ; V σ ) , ∂ t φ ∈ L ∞ ( σ, T ; H (Ω)) ∩ L ( σ, T ; H (Ω)) . In light of (5.10) and (5.20), we infer that u ∈ L ∞ ( σ, T ; W ,p (Ω)) , ∀ p ∈ (2 , ∞ ) . An immediate consequence of the above regularity results is that e µ = − ∆ φ + F ′ ( φ ) ∈ L ( σ, T ; L ∞ (Ω)) . Thanks to [33, Lemma 7.2], we deduce that F ′ ( φ ) ∈ L ( σ, T ; L ∞ (Ω)) . This property entails that thereexists σ ′ ∈ ( σ, σ + 1) such that k F ′ ( φ ( σ ′ )) k L ∞ (Ω) ≤ C ( σ ) . (5.61)Note that σ ′ can also be chosen arbitrarily close to σ .Now, we rewrite (4.1) as follows ∂ t φ + u · ∇ φ − ∆ φ + F ′ ( φ ) = U ( x, t ) , where U = θ φ − ρ ′ ( φ ) | u | + ξ . Thanks to the above regularity, it easily seen that U ∈ L ∞ (0 , T ; L ∞ (Ω)) .In particular, sup t ≥ σ k U ( t ) k L ∞ (Ω) ≤ C ( σ ) . For any p ≥ , we compute p dd t Z Ω | F ′ ( φ ) | p d x = Z Ω | F ′ ( φ ) | p − F ′ ( φ ) F ′′ ( φ ) ∂ t φ d x = Z Ω | F ′ ( φ ) | p − F ′ ( φ ) F ′′ ( φ ) (cid:16) − u · ∇ φ + ∆ φ − F ′ ( φ ) + U (cid:17) d x. Since Z Ω | F ′ ( φ ) | p − F ′ ( φ ) F ′′ ( φ ) u · ∇ φ d x = Z Ω u · ∇ (cid:16) p | F ′ ( φ ) | p (cid:17) d x = 0 , we deduce that p dd t Z Ω | F ′ ( φ ) | p d x + Z Ω (cid:16) ( p − | F ′ ( φ ) | p − F ′′ ( φ ) + | F ′ ( φ ) | p − F ′ ( φ ) F ′′′ ( φ ) (cid:17) |∇ φ | d x + Z Ω | F ′ ( φ ) | p F ′′ ( φ ) d x = Z Ω | F ′ ( φ ) | p − F ′ ( φ ) F ′′ ( φ ) U d x. We notice that the second term on the left-hand side is non-negative. Next, we observe that F ′′ ( s ) ≤ θ e θ | F ′ ( s ) | , ∀ s ∈ ( − , . Owing to the above inequality, and using the fact that s ≤ e s for s ≥ , we deduce that log (cid:16) | F ′ ( s ) | p − F ′′ ( s ) (cid:17) ≤ log( θ ) + (cid:16) θ (cid:17) ( p − | F ′ ( s ) | , ∀ s ∈ ( − , . Thus, we get | F ′ ( s ) | p − F ′′ ( s ) log (cid:16) | F ′ ( s ) | p − F ′′ ( s ) (cid:17) ≤ C p | F ′ ( s ) | p F ′′ ( s ) + C , ∀ s ∈ ( − , , (5.62)for some C , C > independent of p . Recalling xy ≤ εx log x + e yε , ∀ x > , y > , ε ∈ (0 , , and taking ε = C p , we arrive at p dd t Z Ω | F ′ ( φ ) | p d x + 12 Z Ω | F ′ ( φ ) | p F ′′ ( φ ) d x ≤ C | Ω | C + Z Ω e C p | U | d x. Since U is globally bounded, we obtain C | Ω | C + Z Ω e C p | U | d x ≤ C | Ω | C + | Ω | e C p ≤ C e C p , for some C , C > independent of p and t . Observing that F ′′ ( s ) ≥ θ for all s ∈ ( − , , we rewritethe above differential inequality for p ≥ as follows dd t Z Ω | F ′ ( φ ) | p d x + θ Z Ω | F ′ ( φ ) | p d x ≤ C pe C p . By applying the Gronwall lemma on the time interval [ σ ′ , ∞ ) , we infer that k F ′ ( φ ( t )) k pL p (Ω) ≤ k F ′ ( φ ( σ ′ )) k pL p (Ω) e − θ ( t − σ ′ ) + C pe C p θ , ∀ t ≥ σ ′ . (5.63)We recall the elementary inequality for q < x + y ) q ≤ x q + y q , ∀ x > , y > . Choosing q = p , with p ≥ , we find k F ′ ( φ ( t )) k L p (Ω) ≤ k F ′ ( φ ( σ ′ )) k L p (Ω) e − θ ( t − σ ′ ) p + (cid:16) C pθ (cid:17) p e C , ∀ t ≥ σ ′ . Recalling (5.61) and taking the limit as p → + ∞ , we deduce that k F ′ ( φ ( t )) k L ∞ (Ω) ≤ k F ′ ( φ ( σ ′ )) k L ∞ (Ω) + e C , ∀ t ≥ σ ′ . As a result, there exists δ = δ ( σ ) > such that − δ ≤ φ ( x, t ) ≤ − δ, ∀ x ∈ Ω , t ≥ σ ′ . The proof is complete. (cid:3) IFFUSE INTERFACE MODELS FOR INCOMPRESSIBLE BINARY FLUIDS 55 6. M ASS - CONSERVING E ULER -A LLEN -C AHN S YSTEM IN T WO D IMENSIONS In this section, we study the dynamics of ideal two-phase flows in a bounded domain Ω ⊂ R withsmooth boundary, which is described by the mass-conserving Euler-Allen-Cahn system: ∂ t u + u · ∇ u + ∇ P = − div ( ∇ φ ⊗ ∇ φ ) , div u = 0 ,∂ t φ + u · ∇ φ + µ = µ,µ = − ∆ φ + Ψ ′ ( φ ) , in Ω × (0 , T ) . (6.1)The above system corresponds to the inviscid NS-AC system (4.1) (i.e. ν ≡ ) with matched densities(i.e. ρ ≡ ). The system is subject to the following boundary conditions u · n = 0 , ∂ n φ = 0 on ∂ Ω × (0 , T ) , (6.2)and initial conditions u ( · , 0) = u , φ ( · , 0) = φ in Ω . (6.3)The main result of this section is as follows: Theorem 6.1. Let Ω be a smooth bounded domain in R . Assume that u ∈ H σ ∩ H (Ω) , φ ∈ H (Ω) such that F ′ ( φ ) ∈ L (Ω) , k φ k L ∞ (Ω) ≤ , | φ | < and ∂ n φ = 0 on ∂ Ω . Then, there exists a global solution ( u , φ ) which satisfies theproblem (6.1) - (6.3) in the sense of distribution on Ω × (0 , ∞ ) and, for all T > , u ∈ L ∞ (0 , T ; H σ ∩ H (Ω)) , φ ∈ L ∞ (0 , T ; H (Ω)) ∩ L (0 , T ; W ,p (Ω)) ,∂ t φ ∈ L ∞ (0 , T ; L (Ω)) ∩ L (0 , T ; H (Ω)) ,φ ∈ L ∞ (Ω × (0 , T )) : | φ ( x, t ) | < a.e. in Ω × (0 , T ) , where p ∈ (2 , ∞ ) . Moreover, ∂ n φ = 0 on ∂ Ω × (0 , ∞ ) . Assume that u ∈ H σ ∩ W ,p (Ω) , p ∈ (2 , ∞ ) , φ ∈ H (Ω) such that F ′ ( φ ) ∈ L (Ω) , F ′′ ( φ ) ∈ L (Ω) , k φ k L ∞ (Ω) ≤ , | φ | < , ∂ n φ = 0 on ∂ Ω , and in addition ∇ µ = ∇ ( − ∆ φ + F ′ ( φ )) ∈ L (Ω) . Then, there exists a global solution ( u , φ ) which satisfies theproblem (6.1) - (6.3) almost everywhere in Ω × (0 , ∞ ) and, for all T > , u ∈ L ∞ (0 , T ; H σ ∩ W ,p (Ω)) , φ ∈ L ∞ (0 , T ; W ,p (Ω)) ,∂ t φ ∈ L ∞ (0 , T ; H (Ω)) ∩ L (0 , T ; H (Ω)) ,φ ∈ L ∞ (Ω × (0 , T )) : | φ ( x, t ) | < a.e. in Ω × (0 , T ) . In addition, for any σ > , there exists δ = δ ( σ ) > such that − δ ≤ φ ( x, t ) ≤ − δ, ∀ x ∈ Ω , t ≥ σ. To prove Theorem 6.1, we first derive formal estimates leading to the required estimates of solu-tions. Then the existence results can be proved by a suitable approximation scheme with fixed pointarguments and then passing to the limit, which is standard owing to uniform estimates obtained in thefirst step. Hence, here below we only focus on the a priori estimates and omit further details. Case 1. Let us first consider initial datum ( u , φ ) such that u ∈ H σ ∩ H (Ω) , φ ∈ H (Ω) , ∂ n φ = 0 on ∂ Ω , with k φ k L ∞ (Ω) ≤ , | φ | < and F ′ ( φ ) ∈ L (Ω) . Lower-order estimate. As in the previous section, we have the conservation of mass φ ( t ) = φ , ∀ t ≥ . By the same argument for (4.9), we deduce the energy balance dd t E ( u , φ ) + k ∂ t φ + u · ∇ φ k L (Ω) = 0 . (6.4)Integrating the above relation on [0 , t ] , we find E ( u ( t ) , φ ( t )) + Z t k ∂ t φ + u · ∇ φ k L (Ω) d τ = E ( u , φ ) , ∀ t ≥ . This implies that u ∈ L ∞ (0 , T ; H σ ) , φ ∈ L ∞ (0 , T ; H (Ω)) , ∂ t φ + u · ∇ φ ∈ L (0 , T ; L (Ω)) , (6.5)where the last property also implies µ − µ ∈ L (0 , T ; L (Ω)) . In addition, it follows from the estimates(4.14) and (4.21) that φ ∈ L (0 , T ; H (Ω)) , µ ∈ L (0 , T ; L (Ω)) and F ′ ( φ ) ∈ L (0 , T ; L (Ω)) . The latter entails that φ ∈ L ∞ (Ω × (0 , T )) such that | φ ( x, t ) | < almost everywhere in Ω × (0 , T ) .We remark that in comparison with the viscous case, it is not possible at this stage to prove that ∂ t φ ∈ L (Ω × (0 , T )) . Higher-order estimates. In the two dimensional case, it is convenient to consider the equation forthe vorticity ω = ∂u ∂x − ∂u ∂x that reads as follows ∂ t ω + u · ∇ ω = ∇ µ · ( ∇ φ ) ⊥ , (6.6)where v ⊥ = ( v , − v ) for any v = ( v , v ) . Multiplying (6.6) by ω and integrating over Ω , we obtain 12 dd t k ω k L (Ω) = Z Ω ∇ µ · ( ∇ φ ) ⊥ ω d x. (6.7)On the other hand, differentiating (6.1) with respect to time, multiplying by ∂ t φ and integrating over Ω , we find 12 dd t k ∂ t φ k L (Ω) + Z Ω ∂ t u · ∇ φ ∂ t φ d x + k∇ ∂ t φ k L (Ω) + Z Ω F ′′ ( φ ) | ∂ t φ | d x = θ k ∂ t φ k L (Ω) . (6.8)Here we have used the following equalities Z Ω u · ∇ ∂ t φ ∂ t φ d x = Z Ω u · ∇ (cid:16) | ∂ t φ | (cid:17) d x = 0 and Z Ω ∂ t φ d x = 0 . We now define H ( t ) = 12 k ω k L (Ω) + 12 k ∂ t φ k L (Ω) . IFFUSE INTERFACE MODELS FOR INCOMPRESSIBLE BINARY FLUIDS 57 By adding together (6.7) and (6.8), we infer from the convexity of F (i.e. F ′′ > ) that dd t H ( t ) + k∇ ∂ t φ k L (Ω) ≤ Z Ω ∇ µ · ( ∇ φ ) ⊥ ω d x − Z Ω ∂ t u · ∇ φ ∂ t φ d x + θ k ∂ t φ k L (Ω) . (6.9)Before proceeding to control the terms on the right-hand side of (6.9), we rewrite the second one usingthe Euler equation. We first observe that ∂ t u = P (cid:0) µ ∇ φ − u · ∇ u (cid:1) , where P is the Leray projection operator. Thus, we write Z Ω ∂ t u · ∇ φ ∂ t φ d x = Z Ω P (cid:0) µ ∇ φ − u · ∇ u (cid:1) · ∇ φ ∂ t φ d x = Z Ω µ ∇ φ · P (cid:0) ∇ φ ∂ t φ (cid:1) d x − Z Ω ( u · ∇ u ) · P (cid:0) ∇ φ ∂ t φ (cid:1) d x = − Z Ω µ ∇ φ · P (cid:0) φ ∇ ∂ t φ (cid:1) d x − Z Ω div ( u ⊗ u ) · P (cid:0) ∇ φ ∂ t φ (cid:1) d x = − Z Ω µ ∇ φ · P (cid:0) φ ∇ ∂ t φ (cid:1) d x + Z Ω ( u ⊗ u ) : ∇ P (cid:0) ∇ φ ∂ t φ (cid:1) d x − Z ∂ Ω u ⊗ u P (cid:0) ∇ φ ∂ t φ (cid:1) · n d σ = − Z Ω µ ∇ φ · P (cid:0) φ ∇ ∂ t φ (cid:1) d x + Z Ω ( u ⊗ u ) : ∇ P (cid:0) ∇ φ ∂ t φ (cid:1) d x − Z ∂ Ω ( u · n ) (cid:0) u · P (cid:0) ∇ φ ∂ t φ (cid:1)(cid:1) d σ = − Z Ω µ ∇ φ · P (cid:0) φ ∇ ∂ t φ (cid:1) d x + Z Ω ( u ⊗ u ) : ∇ P (cid:0) ∇ φ ∂ t φ (cid:1) d x. Here we have used that P ( ∇ v ) = 0 for any v ∈ H (Ω) , the relation div ( S t v ) = S t : ∇ v + div S · v for any d × d tensor S and vector v , and the no-normal flow condition u · n = 0 at the boundary. As aconsequence, we rewrite (6.9) as follows dd t H ( t ) + k∇ ∂ t φ k L (Ω) ≤ Z Ω ∇ µ · ( ∇ φ ) ⊥ ω d x + Z Ω µ ∇ φ · P (cid:0) φ ∇ ∂ t φ (cid:1) d x − Z Ω ( u ⊗ u ) : ∇ P (cid:0) ∇ φ ∂ t φ (cid:1) d x + θ k ∂ t φ k L (Ω) . (6.10)We now turn to estimate the right-hand side of (6.10). By H¨older’s inequality, we have Z Ω ∇ µ · ( ∇ φ ) ⊥ ω d x ≤ k∇ µ k L (Ω) k∇ φ k L ∞ (Ω) k ω k L (Ω) . (6.11)By taking the gradient of (6.1) , we observe that k∇ µ k L (Ω) ≤ k∇ ∂ t φ k L (Ω) + k∇ φ u k L (Ω) + k∇ u ∇ φ k L (Ω) . Recalling the elementary inequality k v k H (Ω) ≤ C (cid:16) k v k L (Ω) + k div v k L (Ω) + k curl v k L (Ω) + k v · n k H ( ∂ Ω) (cid:17) , ∀ v ∈ H (Ω) , and exploiting Lemma 2.1 as well as (2.10), we find that k∇ µ k L (Ω) ≤ k∇ ∂ t φ k L (Ω) + C k u k H (Ω) k∇ φ k L (Ω) log (cid:16) C k∇ φ k L p (Ω) k∇ φ k L (Ω) (cid:17) + k∇ u k L (Ω) k∇ φ k L ∞ (Ω) ≤ k∇ ∂ t φ k L (Ω) + C (1 + k ω k L (Ω) ) k∇ φ k L (Ω) log (cid:16) C k∇ φ k L p (Ω) k∇ φ k L (Ω) (cid:17) + C (1 + k ω k L (Ω) ) k∇ φ k H (Ω) log (cid:16) C k∇ φ k W ,p (Ω) k∇ φ k H (Ω) (cid:17) , for some p > . Using (5.7), we rewrite the above estimate as follows k∇ µ k L (Ω) ≤ k∇ ∂ t φ k L (Ω) + C (1 + k ω k L (Ω) ) (cid:16) k∇ φ k H (Ω) log (cid:0) C k∇ φ k W ,p (Ω) (cid:1) + 1 (cid:17) . Then, using again the inequality (2.10), (6.11) can be controlled as follows Z Ω ∇ µ · ( ∇ φ ) ⊥ ω d x ≤ k∇ ∂ t φ k L (Ω) k ω k L (Ω) k∇ φ k H (Ω) log (cid:16) C k∇ φ k W ,p (Ω) k∇ φ k H (Ω) (cid:17) + C k ω k L (Ω) (1 + k ω k L (Ω) ) (cid:16) k∇ φ k H (Ω) log (cid:0) C k∇ φ k W ,p (Ω) (cid:1) + 1 (cid:17) × k∇ φ k H (Ω) log (cid:16) C k∇ φ k W ,p (Ω) k∇ φ k H (Ω) (cid:17) ≤ k∇ ∂ t φ k L (Ω) k ω k L (Ω) (cid:16) k∇ φ k H (Ω) log (cid:0) C k∇ φ k W ,p (Ω) (cid:1) + 1 (cid:17) + C (1 + k ω k L (Ω) ) (cid:16) k∇ φ k H (Ω) log (cid:0) C k∇ φ k W ,p (Ω) (cid:1) + 1 (cid:17) ≤ k∇ ∂ t φ k L (Ω) + C (1 + k ω k L (Ω) ) (cid:16) k φ k H (Ω) log (cid:0) C k φ k W ,p (Ω) (cid:1) + 1 (cid:17) , for some p > . Next, since φ is globally bounded, we have Z Ω µ ∇ φ · P (cid:0) φ ∇ ∂ t φ (cid:1) d x ≤ C k µ k L (Ω) k∇ φ k L ∞ (Ω) k φ ∇ ∂ t φ k L (Ω) ≤ C k µ k L (Ω) k∇ φ k H (Ω) log (cid:16) C k∇ φ k W ,p (Ω) k∇ φ k H (Ω) (cid:17) k φ k L ∞ (Ω) k∇ ∂ t φ k L (Ω) ≤ C k µ k L (Ω) (cid:16) k φ k H (Ω) log (cid:0) C k φ k W ,p (Ω) (cid:1) + 1 (cid:17) k∇ ∂ t φ k L (Ω) , for some p > . In order to estimate the L -norm of µ , we notice that k µ − µ k L (Ω) ≤ k ∂ t φ k L (Ω) + k u · ∇ φ k L (Ω) ≤ k ∂ t φ k L (Ω) + k u k L (Ω) k∇ φ k L (Ω) ≤ k ∂ t φ k L (Ω) + C k u k L (Ω) k u k H (Ω) k∇ φ k L (Ω) k φ k H (Ω) ≤ k ∂ t φ k L (Ω) + C (1 + k ω k L (Ω) ) (1 + k µ − µ k L (Ω) ) ≤ k ∂ t φ k L (Ω) + C (1 + k ω k L (Ω) ) + 12 k µ − µ k L (Ω) . IFFUSE INTERFACE MODELS FOR INCOMPRESSIBLE BINARY FLUIDS 59 Here we have used the equation (6.1) , the Ladyzhenskaya inequality, and the estimates (4.14), (6.5).Since k µ k L (Ω) ≤ C (1 + k µ − µ k L (Ω) ) (recalling (4.21)), we then infer that k µ k L (Ω) ≤ C (1 + k ∂ t φ k L (Ω) + k ω k L (Ω) ) . Thus, we can deduce that Z Ω µ ∇ φ · P (cid:0) φ ∇ ∂ t φ (cid:1) d x ≤ k∇ ∂ t φ k L (Ω) + C (cid:0) k ∂ t φ k L (Ω) + k ω k L (Ω) (cid:1)(cid:16) k φ k H (Ω) log (cid:0) C k φ k W ,p (Ω) (cid:1) + 1 (cid:17) . Recalling that P is a bounded operator from H (Ω) to H σ ∩ H (Ω) , and using the inequalities (2.4),(2.10), Poincar´e’s inequality and Lemma 2.1, we have − Z Ω ( u ⊗ u ) : ∇ P (cid:0) ∇ φ ∂ t φ (cid:1) d x ≤ k u k L (Ω) k P ( ∇ φ ∂ t φ ) k H (Ω) ≤ C k u k L (Ω) k u k H (Ω) k∇ φ ∂ t φ k H (Ω) ≤ C (1 + k ω k L (Ω) ) (cid:16) k∇ φ ∂ t φ k L (Ω) + k∇ φ ∂ t φ k L (Ω) + k∇ φ ∇ ∂ t φ k L (Ω) (cid:17) ≤ C (1 + k ω k L (Ω) ) h k∇ φ k L ∞ (Ω) k∇ ∂ t φ k L (Ω) + k∇ ∂ t φ k L (Ω) k∇ φ k L (Ω) log (cid:16) C k∇ φ k L p (Ω) k∇ φ k L (Ω) (cid:17)i ≤ C (1 + k ω k L (Ω) ) k∇ ∂ t φ k L (Ω) (cid:16) k∇ φ k H (Ω) log (cid:16) C k∇ φ k W ,p (Ω) k∇ φ k H (Ω) (cid:17) + k∇ φ k L (Ω) log (cid:16) C k∇ φ k L p (Ω) k∇ φ k L (Ω) (cid:17)(cid:17) ≤ C (1 + k ω k L (Ω) ) k∇ ∂ t φ k L (Ω) (cid:16) k φ k H (Ω) log (cid:0) C k φ k W ,p (Ω) (cid:1) + 1 (cid:17) ≤ k∇ ∂ t φ k L (Ω) + C (cid:0) k ω k L (Ω) (cid:1)(cid:16) k φ k H (Ω) log (cid:0) C k φ k W ,p (Ω) (cid:1) + 1 (cid:17) , for some p > .Combining the above estimates together with (6.10), we arrive at the differential inequality dd t H ( t ) + 12 k∇ ∂ t φ k L (Ω) ≤ C (1 + H ( t )) (cid:16) k φ k H (Ω) log (cid:0) C k φ k W ,p (Ω) + 1 (cid:17) . (6.12)In order to close the estimate, we are left to absorb the logarithmic term on the right-hand side of theabove differential inequality. To this aim, we first multiply µ = − ∆ φ + Ψ ′ ( φ ) by | F ′ ( φ ) | p − F ′ ( φ ) , forsome p > , and integrate over Ω . After integrating by parts and using the boundary condition for φ ,we obtain Z Ω ( p − | F ′ ( φ ) | p − F ′′ ( φ ) |∇ φ | d x + k F ′ ( φ ) k pL p (Ω) = Z Ω ( µ + θ φ ) | F ′ ( φ ) | p − F ′ ( φ ) d x. By Young’s inequality and the fact that F ′′ > , we deduce k F ′ ( φ ) k L p (Ω) ≤ C (1 + k µ k L p (Ω) ) . Using a well-known elliptic regularity result, together with the above inequality and (6.5), we obtainthat (cf. (5.12)) k φ k W ,p (Ω) ≤ C (1 + k µ k L p (Ω) ) . On the other hand, we infer from equation (6.1) that k µ − µ k L p (Ω) ≤ k ∂ t φ k L p (Ω) + k u · ∇ φ k L p (Ω) . Then by Poincar´e’s inequality and a Sobolev embedding theorem, we find k µ k L p (Ω) ≤ C k µ − µ k L p (Ω) + C | µ |≤ C k∇ ∂ t φ k L (Ω) + C k u k H (Ω) k φ k H (Ω) + C (1 + k µ − µ k L (Ω) ) ≤ C k∇ ∂ t φ k L (Ω) + C (1 + k ω k L (Ω) )(1 + k µ − µ k L (Ω) ) ≤ C k∇ ∂ t φ k L (Ω) + C (1 + k ω k L (Ω) )(1 + k ∂ t φ k L (Ω) + k ω k L (Ω) ) . Thus, for p > , we reach k φ k W ,p (Ω) ≤ C (1 + k∇ ∂ t φ k L (Ω) + H ( t )) , which, in turn, allows us to rewrite (6.12) as dd t H ( t ) + 12 k∇ ∂ t φ k L (Ω) ≤ C (1 + H ( t )) (cid:16) k φ k H (Ω) log (cid:16) C (cid:0) k∇ ∂ t φ k L (Ω) + H ( t ) (cid:1)(cid:17) + 1 (cid:17) . (6.13)We now observe that, for any ε > , the following inequality holds x log( Cy ) ≤ εy + x log (cid:16) Cxε (cid:17) ∀ x, y > . By using the above inequality with x = 1 + H ( t ) , y = 1 + k∇ ∂ t φ k L (Ω) + H ( t ) and ε = 1 , we deducethat dd t H ( t ) + 12 k∇ ∂ t φ k L (Ω) ≤ k∇ ∂ t φ k L (Ω) k φ k H (Ω) + C (1 + k φ k H (Ω) )(1 + H ( t )) log (cid:0) C (1 + H ( t )) (cid:1) . By Young’s inequality, we obtain dd t H ( t ) + 14 k∇ ∂ t φ k L (Ω) ≤ k φ k H (Ω) + C (1 + k φ k H (Ω) )(1 + H ( t )) log (cid:0) C (1 + H ( t )) (cid:1) . Recalling that k φ k H (Ω) ≤ C (1 + H ( t )) , we are finally led to the differential inequality dd t H ( t ) + 14 k∇ ∂ t φ k L (Ω) ≤ C (1 + k φ k H (Ω) )(1 + H ( t )) log (cid:0) C (1 + H ( t )) (cid:1) . (6.14)Since φ ∈ L (0 , T ; H (Ω)) , then applying the generalized Gronwall lemma B.2, we find the doubleexponential bound sup t ∈ [0 ,T ] (cid:16) k ∂ t φ ( t ) k L (Ω) + k ω ( t ) k L (Ω) (cid:17) ≤ C (cid:0) k u k H (Ω) k φ k H (Ω) + k φ k H (Ω) + k Ψ ′ ( φ ) k L (Ω) + k u k H (Ω) (cid:1) e R T k φ ( s ) k H s , IFFUSE INTERFACE MODELS FOR INCOMPRESSIBLE BINARY FLUIDS 61 for some constant C > . Here we have used that k ∂ t φ (0) k L (Ω) ≤ C k u k H (Ω) k φ k H (Ω) + C k φ k H (Ω) + C k Ψ ′ ( φ ) k L (Ω) . Hence, we get ∂ t φ ∈ L ∞ (0 , T ; L (Ω)) ∩ L (0 , T ; H (Ω)) , ω ∈ L ∞ (0 , T ; L (Ω)) , (6.15)which, in turn, entail that u ∈ L ∞ (0 , T ; H (Ω)) , φ ∈ L ∞ (0 , T ; H (Ω)) ∩ L (0 , T ; W ,p (Ω)) , (6.16)for any p ∈ [2 , ∞ ) .6.2. Case 2. We now consider an initial condition ( u , φ ) such that u ∈ H σ ∩ W ,p (Ω) , φ ∈ H (Ω) , ∂ n φ = 0 on ∂ Ω , for p ∈ (2 , ∞ ) , with k φ k L ∞ (Ω) ≤ , | φ | < and F ′ ( φ ) ∈ L (Ω) , F ′′ ( φ ) ∈ L (Ω) , ∇ µ = ∇ ( − ∆ φ + F ′ ( φ )) ∈ L (Ω) . Thanks to the first part of Theorem 6.1, we have a solution ( u , φ ) satisfying (6.15) and (6.16). More-over, repeating the same argument performed in Section 5, we have (cf. (5.35)) dd t Z Ω F ′′ ( φ ) d x + 14 Z Ω F ′′′ ( φ ) F ′ ( φ ) d x ≤ C, for some positive constant C only depending on Ω and the parameters of the system. Since F ′′ ( φ ) ∈ L (Ω) , we learn, in particular, that (cf. (5.43)) Z t +1 t Z Ω | F ′′ ( φ ) | log(1 + F ′′ ( φ )) d x d τ ≤ C, ∀ t ≥ . (6.17)Multiplying (6.6) by | ω | p − ω ( p > ) and integrating over Ω , we obtain p dd t k ω k pL p (Ω) = Z Ω ∇ µ · ( ∇ φ ) ⊥ | ω | p − ω d x. By H¨older’s inequality, we easily get p dd t k ω k pL p (Ω) ≤ k∇ µ · ( ∇ φ ) ⊥ k L p (Ω) k ω k p − L p (Ω) , which, in turn, implies 12 dd t k ω k L p (Ω) ≤ k∇ µ · ( ∇ φ ) ⊥ k L p (Ω) k ω k L p (Ω) . Next, differentiating (6.1) with respect time, then multiplying the resultant by − ∆ ∂ t φ and integratingover Ω , we obtain dd t k∇ ∂ t φ k L (Ω) + k ∆ ∂ t φ k L (Ω) = θ k∇ ∂ t φ k L (Ω) + Z Ω F ′′ ( φ ) ∂ t φ ∆ ∂ t φ d x + Z Ω ( ∂ t u · ∇ φ )∆ ∂ t φ d x + Z Ω ( u · ∇ ∂ t φ )∆ ∂ t φ d x. Here we have used the fact that ∆ ∂ t φ = 0 since ∂ n ∂ t φ = 0 on ∂ Ω . Collecting the above two estimates,we find that dd t (cid:16) k ω k L p (Ω) + 12 k∇ ∂ t φ k L (Ω) (cid:17) + k ∆ ∂ t φ k L (Ω) ≤ k∇ µ · ( ∇ φ ) ⊥ k L p (Ω) k ω k L p (Ω) + θ k∇ ∂ t φ k L (Ω) + Z Ω F ′′ ( φ ) ∂ t φ ∆ ∂ t φ d x + Z Ω ( ∂ t u · ∇ φ )∆ ∂ t φ d x + Z Ω ( u · ∇ ∂ t φ )∆ ∂ t φ d x. Notice that, by (6.1) , we have the relation ∇ µ = ∇ ∂ t φ + ( ∇ u ) t ∇ φ + ( u · ∇ ) ∇ φ . By exploiting thisidentity, we obtain k∇ µ · ( ∇ φ ) ⊥ k L p (Ω) k ω k L p (Ω) ≤ (cid:0) k∇ ∂ t φ k L p (Ω) + k ( ∇ u ) t ∇ φ k L p (Ω) + k ( u · ∇ ) ∇ φ k L p (Ω) (cid:1) k∇ φ k L ∞ (Ω) k ω k L p (Ω) . Using the Gagliardo-Nirenberg inequality (2.5) and the following inequality for divergence free vectorfields satisfying the boundary condition (6.2) k∇ u k L p (Ω) ≤ C ( p ) k ω k L p (Ω) , p ∈ [2 , ∞ ) , (6.18)we deduce that k∇ µ · ( ∇ φ ) ⊥ k L p (Ω) k ω k L p (Ω) ≤ C k∇ ∂ t φ k p L (Ω) k ∆ ∂ t φ k − p L (Ω) k∇ φ k L ∞ (Ω) k ω k L p (Ω) + C k∇ u k L p (Ω) k∇ φ k L ∞ (Ω) k ω k L p (Ω) + k u k L ∞ (Ω) k φ k W ,p (Ω) k∇ φ k L ∞ (Ω) k ω k L p (Ω) ≤ k ∆ ∂ t φ k L (Ω) + C k∇ φ k pp +2 L ∞ (Ω) k∇ ∂ t φ k p +2 L (Ω) k ω k pp +2 L p (Ω) + C (cid:0) k∇ φ k L ∞ (Ω) + k φ k W ,p (Ω) k∇ φ k L ∞ (Ω) (cid:1)(cid:0) k ω k L p (Ω) (cid:1) . Next, using (6.1) together with the bounds (6.15), we have Z Ω ∂ t u · ∇ φ ∆ ∂ t φ d x ≤ Z Ω P (cid:0) − u · ∇ u − ∆ φ ∇ φ (cid:1) · ∇ φ ∆ ∂ t φ d x ≤ C k P (cid:0) u · ∇ u (cid:1) k L (Ω) k∇ φ ∆ ∂ t φ k L (Ω) + C k P (cid:0) ∆ φ ∇ φ (cid:1) k L (Ω) k∇ φ ∆ ∂ t φ k L (Ω) ≤ k ∆ ∂ t φ k L (Ω) + C k u k L ∞ (Ω) k∇ u k L (Ω) k∇ φ k L ∞ (Ω) + C k ∆ φ k L (Ω) k∇ φ k L ∞ (Ω) ≤ k ∆ ∂ t φ k L (Ω) + C (1 + k ω k L p (Ω) ) k∇ φ k L ∞ (Ω) + C k∇ φ k L ∞ (Ω) . Arguing as for (5.57) and (5.58), we have Z Ω F ′′ ( φ ) ∂ t φ ∆ ∂ t φ d x ≤ k ∆ ∂ t φ k L (Ω) + C k F ′′ ( φ ) k L (Ω) log (cid:0) C k F ′′ ( φ ) k L (Ω) (cid:1) k∇ ∂ t φ k L (Ω) , IFFUSE INTERFACE MODELS FOR INCOMPRESSIBLE BINARY FLUIDS 63 Z Ω ( u · ∇ ∂ t φ )∆ ∂ t φ d x ≤ k ∆ ∂ t φ k L (Ω) + C k∇ ∂ t φ k L (Ω) . Collecting the above estimates and using Young’s inequality, we arrive at the differential inequality dd t (cid:16) k ω k L p (Ω) + k∇ ∂ t φ k L (Ω) (cid:17) + k ∆ ∂ t φ k L (Ω) ≤ R ( t ) (cid:16) k ω k L p (Ω) + k∇ ∂ t φ k L (Ω) (cid:17) + R ( t ) , where R = C (cid:16) k∇ φ k L ∞ (Ω) + k F ′′ ( φ ) k L (Ω) log (cid:0) C k F ′′ ( φ ) k L (Ω) (cid:1)(cid:17) and R = C k φ k W ,p (Ω) + C (cid:0) k∇ φ k L ∞ (Ω) + 1) . By using (2.10), and recalling (5.7), we see that k∇ φ k L ∞ (Ω) ≤ C k∇ φ k L (Ω) log (cid:0) k∇ φ k L p (Ω) (cid:1) + 1 ≤ C log (cid:0) k φ k W ,p (Ω) (cid:1) + 1 , for p > . In light of (6.16), we infer that both R and R belong to L (0 , T ) . Thanks to Gronwall’slemma, we obtain k ω ( t ) k L p (Ω) + k∇ ∂ t φ ( t ) k L (Ω) ≤ (cid:16) k ω (0) k L p (Ω) + k∇ ∂ t φ (0) k L (Ω) + Z T R ( τ ) d τ (cid:17) e R T R ( τ ) d τ , for any t ∈ [0 , T ] . Since k ω (0) k L p (Ω) ≤ k∇ u k L p (Ω) and k∇ ∂ t φ (0) k L (Ω) ≤ k ( ∇ u ) t ∇ φ k L (Ω) + k ( u · ∇ ) ∇ φ k L (Ω) + k∇ µ k L (Ω) ≤ C k∇ u k L p (Ω) k φ k H (Ω) + C k u k L ∞ (Ω) k φ k H (Ω) + k∇ µ k L (Ω) ≤ C k u k W ,p (Ω) k φ k H (Ω) + k∇ µ k L (Ω) , we deduce that for any p ∈ (2 , ∞ ) ω ∈ L ∞ (0 , T ; L p (Ω)) , ∂ t φ ∈ L ∞ (0 , T ; H (Ω)) ∩ L (0 , T ; H (Ω)) . This, in turn, implies that u ∈ L ∞ (0 , T ; W ,p (Ω)) , φ ∈ L ∞ (0 , T ; W ,p (Ω)) . As a consequence, the above estimates yield that e µ = − ∆ φ + F ′ ( φ ) ∈ L (0 , T ; L ∞ (Ω)) . The rest part of the proof is the same as the proof of Theorem 5.2 with the choice σ > .The proof of Theorem 6.1 is complete.7. C ONCLUSIONS AND F UTURE D EVELOPMENTS In this paper we present mathematical analysis of some Diffuse Interface models that describe theevolution of incompressible binary mixture having (possibly) different densities and viscosities. Wefocus on the mass-conserving Allen-Cahn relaxation of the transport equation with the physically rel-evant Flory-Huggins potential. We show the existence of global weak solution in three dimension andof global strong solutions in two dimensions. For the latter, we discuss additional properties, such asuniqueness, regularity and the separation property. On the other hand, several still unsolved questionsconcern the analysis of the complex fluid, Navier-Stokes-Allen-Cahn and Euler-Allen-Cahn systems in the three dimensional case, which will be the subject of future investigations. We conclude bymentioning some interesting open problems related to the results proved in this work: • An important possible development of this work is to show the existence of global solutions to thecomplex fluids system (3.1)-(3.3) originating from small perturbation of some particular equilibriumstates. We mention that some remarkable results in this direction have been achieved in [51, 52, 65](see also [50] and the references therein). In addition, it would be interesting to study the globalexistence of weak solutions as in [41] and to generalize Theorem 3.1 to the case with zero viscosity(cf. [51, Theorem 3.1]). • Two possible improvements of this work concern the Navier-Stokes-Allen-Cahn system (4.1)-(4.3). The first question is whether the entropy estimates in Theorem 5.1 can be achieved for strongsolutions with small initial data, but without restrictions on the parameters of the system, or evenwithout any condition on the initial data. The second issue is to show the uniqueness of strong solutionsgiven from Theorem 5.1-(1), without relying on the entropy estimates in Theorem 5.1-(2). Also, wemention the possibility of considering moving contact lines for the Navier-Stokes-Allen-Cahn system(see [55] for numerical). • Interesting open issues regarding the Euler-Allen-Cahn system (6.1)-(6.3) are the existence andthe uniqueness of solutions corresponding to an initial datum ω ∈ L ∞ (Ω) as well as the study of theinviscid limit on arbitrary time intervals (cf. [81] for short times).A CKNOWLEDGMENTS Part of this work was carried out during the first and second authors’ visit to School of MathematicalSciences of Fudan University whose hospitality is gratefully acknowledged. M. Grasselli is a memberof the Gruppo Nazionale per l’Analisi Matematica, la Probabilit e le loro applicazioni (GNAMPA) ofthe Istituto Nazionale di Alta Matematica (INdAM). H. Wu is partially supported by NNSFC grant No.11631011 and the Shanghai Center for Mathematical Sciences at Fudan University.C OMPLIANCE WITH E THICAL S TANDARDS The authors declare that they have no conflict of interest. The authors also confirm that the manuscripthas not been submitted to more than one journal for simultaneous consideration and the manuscripthas not been published previously (partly or in full).A PPENDIX A. S TOKES S YSTEM WITH V ARIABLE V ISCOSITY We prove an elliptic regularity result for the following Stokes problem with concentration dependingviscosity − div ( ν ( φ ) D u ) + ∇ P = f , in Ω , div u = 0 , in Ω , u = 0 , on ∂ Ω . (A.1)This result is a variant of [2, Lemma 4]. Theorem A.1. Let Ω be a bounded domain of class C in R d , d = 2 , . Assume that ν ∈ W , ∞ ( R ) such that < ν ∗ ≤ ν ( · ) ≤ ν ∗ in R , φ ∈ W ,r (Ω) with r > d , and f ∈ L p (Ω) with < p < ∞ if IFFUSE INTERFACE MODELS FOR INCOMPRESSIBLE BINARY FLUIDS 65 d = 2 and ≤ p < ∞ if d = 3 . Consider the (unique) weak solution u ∈ V σ to (A.1) such that ( ν ( φ ) D u , ∇ w ) = ( f , w ) for all w ∈ V σ . We have: If p = + r then there exists C = C ( p, Ω) > such that k u k W ,p (Ω) ≤ C k f k L p (Ω) + C k∇ φ k L r (Ω) k D u k L (Ω) . (A.2)2. Suppose that u ∈ V σ ∩ W ,s (Ω) with s > such that p = 1 s + 1 r , r ≥ ss − . Then, there exists C = C ( s, p, Ω) > such that k u k W ,p (Ω) ≤ C k f k L p (Ω) + C k∇ φ k L r (Ω) k D u k L s (Ω) . (A.3) Proof. We denote by B the Bogovskii operator. We recall that B : L q (0) (Ω) → W ,q (Ω) , < q < ∞ ,such that div Bf = f . It is well-known (see, e.g., [31, Theorem III.3.1]) that, for all < q < ∞ , k Bf k W ,q (Ω) ≤ C k f k L q (Ω) , (A.4)In addition, by [31, Theorem III.3.4], if f = div g , where g ∈ L q (Ω) , < q < ∞ , is such that div g ∈ L q (Ω) , and g · n = 0 on ∂ Ω , we have k Bf k L q (Ω) ≤ C k g k L q (Ω) . (A.5)For the sake of simplicity, we start proving the second part of Theorem A.1, and then we show thefirst part. Case 2 . Let us take v ∈ C ∞ ,σ (Ω) . As in [2, Lemma 4], we define w = v ν ( φ ) − B [div (cid:0) v ν ( φ ) (cid:1) ] . We observethat w ∈ W ,r (Ω) with div w = 0 . In particular, w ∈ V σ . Taking w in the weak formulation, we obtain ( D u , ∇ v ) = (cid:16) f , v ν ( φ ) − B h div (cid:16) v ν ( φ ) (cid:17)i(cid:17) − (cid:16) ν ( φ ) D u , v ⊗ ∇ (cid:16) ν ( φ ) (cid:17)(cid:17) + (cid:16) ν ( φ ) D u , ∇ B h div (cid:16) v ν ( φ ) (cid:17)i(cid:17) Since ss − ≤ r , we deduce that r ≥ p ′ ( p ′ = 1 − p ) . This implies that div (cid:0) v ν ( φ ) (cid:1) ∈ L p ′ (Ω) . By usingthe assumptions on ν and the estimate (A.5) with q = p ′ , we find (cid:12)(cid:12)(cid:12)(cid:16) f , v ν ( φ ) − B h div (cid:16) v ν ( φ ) (cid:17)i(cid:17)(cid:12)(cid:12)(cid:12) ≤ k f k L p (Ω) (cid:16) ν ∗ k v k L p ′ (Ω) + (cid:13)(cid:13)(cid:13) B h div (cid:16) v ν ( φ ) (cid:17)i(cid:13)(cid:13)(cid:13) L p ′ (Ω) (cid:17) ≤ C k f k L p (Ω) (cid:16) ν ∗ k v k L p ′ (Ω) + (cid:13)(cid:13)(cid:13) v ν ( φ ) (cid:13)(cid:13)(cid:13) L p ′ (Ω) (cid:17) ≤ C k f k L p (Ω) k v k L p ′ (Ω) . Also, we have (cid:12)(cid:12)(cid:12)(cid:16) ν ( φ ) D u , v ⊗ ∇ (cid:16) ν ( φ ) (cid:17)(cid:17)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:16) D u , ν ′ ( φ ) ν ( φ ) v ⊗ ∇ φ (cid:17)(cid:12)(cid:12)(cid:12) ≤ C k D u k L s (Ω) k∇ φ k L r (Ω) k v k L p ′ (Ω) . Recalling that div v = 0 and r > s ′ , by using (A.4) we obtain (cid:12)(cid:12)(cid:12)(cid:16) ν ( φ ) D u , ∇ B h div (cid:16) v ν ( φ ) (cid:17)i(cid:17)(cid:12)(cid:12)(cid:12) ≤ k D u k L s (Ω) (cid:13)(cid:13)(cid:13) ∇ B h ∇ (cid:16) ν ( φ ) (cid:17) · v i(cid:13)(cid:13)(cid:13) L s ′ (Ω) ≤ C k D u k L s (Ω) (cid:13)(cid:13)(cid:13) ∇ (cid:16) ν ( φ ) (cid:17) · v (cid:13)(cid:13)(cid:13) L s ′ (Ω) ≤ C k D u k L s (Ω) k∇ φ k L r (Ω) k v k L p ′ (Ω) . Therefore, by the Riesz representation theorem and a density argument, we find ( D u , ∇ v ) = ( e f , v ) ∀ v ∈ V σ , where k e f k L p (Ω) ≤ C k f k L p (Ω) + C k D u k L s (Ω) k∇ φ k L r (Ω) , for some C depending on s, p and Ω . By the regularity of the Stokes operator (see, e.g., [31, TheoremIV.6.1]), the claim easily follows. Case 1 . We consider φ n ∈ C ∞ c (Ω) such that φ n → φ in W ,r (Ω) as n → ∞ . For any n ∈ N , we define u n as the solution to ( ν ( φ n ) D u n , ∇ w ) = ( f , w ) ∀ w ∈ V σ . Since ν ( · ) ≥ ν ∗ > , by taking w = u n , it is easily seen that { u n } n ∈ N is bounded in V σ independentlyof n . In addition, recalling that W ,r (Ω) ֒ → L ∞ (Ω) , we have ν ( φ n ) → ν ( φ ) in L ∞ (Ω) . By uniquenessof the weak solution u to (A.1), we deduce that u n ⇀ u weakly in V σ .Let us take w = v ν ( φ n ) − B [div (cid:0) v ν ( φ n ) (cid:1) ] with v ∈ C ∞ ,σ (Ω) . Then we find ( D u n , ∇ v ) = (cid:16) f , v ν ( φ n ) − B h div (cid:16) v ν ( φ n ) (cid:17)i(cid:17) − (cid:16) ν ( φ n ) D u n , v ⊗ ∇ (cid:16) ν ( φ n ) (cid:17)(cid:17) + (cid:16) ν ( φ n ) D u n , ∇ B h div (cid:16) v ν ( φ n ) (cid:17)i(cid:17) Note that, by construction, v ν ( φ n ) ∈ W ,q (Ω) for all q ∈ [1 , ∞ ] . Therefore, by repeating the samecomputations carried out above with s = 2 , we arrive at ( D u n , ∇ v ) = ( e f , v ) ∀ v ∈ V σ , where k e f k L p (Ω) ≤ C k f k L p (Ω) + C k D u n k L (Ω) k∇ φ n k L r (Ω) , for some C depending on p and Ω . By the regularity theory of the Stokes operator, we infer k u n k W ,p (Ω) ≤ C k f k L p (Ω) + C k D u n k L (Ω) k∇ φ n k L r (Ω) . Since { u n } n ∈ N is bounded in V σ and φ n → φ in W ,r (Ω) , u n is bounded in W ,p (Ω) independently of n . By the choice of the parameters r > d and p = + r , W ,p (Ω) ∩ V σ is compactly embedded in V σ .In particular, k D u n k L (Ω) → k D u k L (Ω) as n → ∞ . As a consequence, by the lower semi-continuityof the norm with respect to the weak topology, the conclusion follows. The proof is complete. (cid:3) IFFUSE INTERFACE MODELS FOR INCOMPRESSIBLE BINARY FLUIDS 67 A PPENDIX B. S OME L EMMAS ON ODE I NEQUALITIES For convenience of the readers, we collect some useful results concerning ODE inequalities thathave been used in this paper. First, we report the Osgood lemma. Lemma B.1. Let f be a measurable function from [0 , T ] to [0 , a ] , g ∈ L (0 , T ) , and W a continuousand nondecreasing function from [0 , a ] to R + . Assume that, for some c ≥ , we have f ( t ) ≤ c + Z t g ( s ) W ( f ( s )) d s, for a.e. t ∈ [0 , T ] . - If c > , then for almost every t ∈ [0 , T ] −M ( f ( t )) + M ( c ) ≤ Z T g ( s ) d s, where M ( s ) = Z as W ( s ) d s. - If c = 0 and R a W ( s ) d s = ∞ , then f ( t ) = 0 for almost every t ∈ [0 , T ] . Next, we report two generalizations of the classical Gronwall lemma and of the uniform Gronwalllemma. Lemma B.2. Let f be a positive absolutely continuous function on [0 , T ] and g , h be two summablefunctions on [0 , T ] that satisfy the differential inequality dd t f ( t ) ≤ g ( t ) f ( t ) ln (cid:0) e + f ( t ) (cid:1) + h ( t ) , for almost every t ∈ [0 , T ] . Then, we have f ( t ) ≤ (cid:0) e + f (0) (cid:1) e R t g ( τ ) d τ e R t e R tτ g ( s ) d s h ( τ ) d τ , ∀ t ∈ [0 , T ] . Lemma B.3. Let f be an absolutely continuous positive function on [0 , ∞ ) and g , h be two positivelocally summable functions on [0 , ∞ ) which satisfy the differential inequality dd t f ( t ) ≤ g ( t ) f ( t ) ln (cid:0) e + f ( t ) (cid:1) + h ( t ) , for almost every t ≥ , and the uniform bounds Z t + rt f ( τ ) d τ ≤ a , Z t + rt g ( τ ) d τ ≤ a , Z t + rt h ( τ ) d τ ≤ a , ∀ t ≥ , for some positive constants r , a , a , a . Then, we have f ( t ) ≤ e (cid:0) r + a r + a (cid:1) e a , ∀ t ≥ r. R EFERENCES [1] H. 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