Convergence to equilibrium of global weak solutions for a Cahn-Hilliard-Navier-Stokes vesicle model

arXiv:1505.04127v2 [math.AP] 9 Oct 2017 Convergence to equilibrium of global weak solutions for a Cahn-Hilliard-Navier-Stokes vesicle model Blanca Climent-Ezquerra, Francisco Guill´ en-Gonz´ alez Dpto. Ecuaciones Diferenciales y An´alisis Num´erico, Universidad de Sevilla, Aptdo. 1160, 41080 Sevilla, Spain. E-mails: bclimen[email protected], [email protected] July 10, 2018 Abstract In this paper, we introduce a model describing the dynamic of vesicle membranes within an incompressible viscous fluid in 3Ddomains. The system consists of the Navier- Stokes equations, with an extra stress tensor depending on the membrane, coupled with a Cahn-Hilliard phase-field equation associated to a bending energy plus a penalization term related to the area conservation. This problem has a dissipative in time free-energy which leads, in particular, to prove the existence of global in time weak solutions. We analyze the large-time behavior of the weak solutions. By using a modified Lojasiewicz-Simon’s result, we prove the convergence as time goes to infinity of each (whole) trajectory to a single equilibrium. Finally, the convergence of the trajectory of the phase is improved by imposing more regularity on the domain and initial phase. Keywords: Vesicle membranes, Navier-Stokes equations, Cahn-Hilliard equation, energy dissipation, convergence to equilibrium, Lojasiewicz-Simon’s inequalities. 1 Introduction A type of differential equations modeling the dynamic of vesicle membranes within an incompressible viscous fluids was introduced by Helfrich [13]. The membranes are composed by lipid bilayers, which under appropriate conditions, withdraw into themselves forming a sort of bag, named vesicle. The static equilibrium configurations of the vesicle membranes can be obtained minimizing the Helfrich bending elastic energy, under the constraints of fixed area and volume. 1 Numerous studies have been devoted to this type of models and a detailed description of them can be seen in [9] and references therein. A phase function is also used in subsequent papers to model vesicle membranes as diffuse interfaces. In [15] and [22], a coupled Allen- Cahn and Navier-Stokes problem is studied approaching both constrains, area and volume, via a penalization functional. On the other hand, without taking into account the vesiclefluid interaction, a Cahn-Hilliard phase-field model for vesicle membranes is introduced in [2], and the well-posedness of a phase-field approximation satisfying area and volume constraints pointwisely is established in [7]. In this paper, a Cahn-Hilliard-Navier-Stokes model will be considered. Since the volume constraint is implicitly satisfied for the Cahn-Hilliard equation, only the surface area constraint will be approximated via penalization. Moreover, the resulting problem will be thermodynamically consistent because there exists a free-energy (kinetic plus bending plus penalized one) dissipative in time along the trajectories. This fact is used to prove the existence of global in time weak solutions. On the other hand, the large-time behavior of the solutions will be analyzed following the way of [12], [16], [4]. Firstly, we prove that the ω-limit set for weak solutions is composed by critical points of the free-energy. After that, by using a modified Lojasiewicz-Simon’s result we demonstrate the convergence of the whole trajectory to a single equilibrium. Finally, the convergence is improved for the phase assuming more regular data, but without using strong regularity for the velocity and pressure variables. The main novelties in this paper are the following: •The introduction of a new Navier-Stokes-Cahn-Hilliard problem modeling vesicle-fluid interactions. •The proof of a modified Lojasiewicz-Simon’s result associated to weak solutions of a fourth order elliptic problem, which is adapted to the exigences of this new model. In fact, it was not possible to apply any already known Lojasiewicz-Simon’s result to this problem. •Given a global weak solution, we choose a special regularized energy satisfying the energy’s law inequality either in a integral version for all time interval and in a differential version a.e. in time (see (40) and (41) below). •The convergence of each trajectory of weak solutions to a unique equilibrium point, imposing that the regularized energy coincides a.e. in time with the energy evaluated in the weak solution. 2 The current paper is organized as follows. We explain the model in Section 2and give some preliminary results in Section 3. Section 4is devoted to state and prove a new Lojasiewicz- Simon’s result. In Section 5we obtain, via a Galerkin method, the existence of global in time weak solutions satisfying an integral energy’s law a.e. time interval. Moreover, we found a function defined for all time tequal to the free-energy a.e. that satisfies the integral energy’s law for all time and also a differential version of the energy inequality a.e. time. Section 6is devoted to the study of convergence at infinite time for global weak solutions. In fact, we prove that the ω-limit set consists of critical points. After that, by using the modified Lojasiewicz- Simon’s result, we demonstrate that each trajectory converges to a single equilibrium. In Section 7, some global in time strong estimates are obtained for the phase variable, which allow to improve the norm where the phase trajectory converges. Notations •In general, the notation will be abridged. We set Lp=Lp(Ω), p≥1, H1 0=H1 0(Ω), etc. If X=X(Ω) is a space of functions defined in the open set Ω, we denote by Lp(0, T;X) the Banach space Lp(0, T;X(Ω)). Also, boldface letters will be used for vectorial spaces, for instance L2=L2(Ω)N. •The Lp-norm is denoted by |·|p, 1 ≤p≤ ∞, the Hm-norm by k · km(in particular | · |2=k · k0). The inner product of L2(Ω) is denoted by (·,·). •We set Vthe space formed by all fields u∈C∞ 0(Ω)Nsatisfying ∇ · u= 0. We denote H (respectively V) the closure of Vin L2(respectively H1). Hand Vare Hilbert spaces for the norms | · |2and k · k1, respectively. Furthermore, H={u∈L2;∇ · u= 0,u·n= 0 on ∂Ω}, V={u∈H1;∇ · u= 0,u= 0 on ∂Ω}. •From now on, C > 0 will denote different constants, depending only on the data of the problem. •We consider the following zero-mean spaces: L2 ∗=ψ∈L2;ZΩ ψ= 0, H1 ∗=H1∩L2 ∗, Hk 1=ψ∈Hk∩L2 ∗;∂nψ|∂Ω= 0, k = 2,3, and the dual space H−1 ∗= (H1 ∗)′. 3 2 The model Vesicle membranes within incompressible viscous fluids in a bounded 3Ddomain Ω ⊂R3 during a time interval [0,+∞), are modeled via a phase-field function φ:Ω×[0,+∞)→R such that two stable values φ= 1 and φ=−1 represent the exterior and interior of vesicle membranes, respectively, and −1< φ < 1 in the interface. Then, we will analyze the case where the so-called bending energy Eb(φ) is given by a simplified elastic Willmore energy plus a penalization of the surface area constraint [10]: Eb(φ) = 1 2εZΩ (−ε∆φ+1 εF′(φ))2dx+M 2(A(φ)−α)2(1) where F′(φ) = (φ2−1)φdenotes the derivative of the Ginzburg-Landau potential F(φ) = 1 4(φ2−1)2, M > 0 is a penalization constant, ε > 0 is related to the interface width, and A(φ) = ZΩε 2|∇φ|2+1 εF(φ)dx, is an approximation of the surface area. Remark 1 Other possible approximation of the surface area is to consider A(φ) = ZΩ ε 2|∇φ|2dx (see [2]). The same results of this paper may be extended to this case. To model the dynamic in time of the vesicle-fluid interactions, we will introduce the following Navier-Stokes-Cahn-Hilliard equations in Q= Ω ×(0,+∞): ∂tu+ (u· ∇)u−ν∆u−λw∇φ+∇q= 0,(2) ∇ · u= 0,(3) ∂tφ+u· ∇φ−γ∆w= 0,(4) where w:= δEb(φ) δφ is the chemical potential, see (9) below. The coefficients ν > 0, λ > 0 and γ > 0 depend on viscosity, elasticity and mobility, respectively. The system (2)-(4) is completed with the boundary conditions u|∂Ω= 0, ∂nφ|∂Ω= 0, ∂n(∆φ)|∂Ω= 0, ∂nw|∂Ω= 0,(5) 4 and the initial conditions u|t=0 =u0, φ|t=0 =φ0in Ω.(6) For compatibility, we will assume u0|∂Ω= 0 with ∇ · u0= 0 in Ω and ∂nφ0|∂Ω= 0. By integrating the phase-equation (4), using the free-divergence ∇ · u= 0, the non-slip condition u|∂Ω= 0, and the last boundary condition of (5), ∂nw|∂Ω= 0, it is easy to deduce that the total volume of φin Ω is conserved: d dt ZΩ φ(x, t)dx= 0.(7) In order to complete the problem (2)-(4), we compute the variational derivative w=δEb(φ) δφ . On the one hand, for all φ, φ∈H1, DδA(φ) δφ ,φE=ZΩ ε∇φ· ∇φ+1 εF′(φ)φ, hence if φ∈H2and ∂nφ|∂Ω= 0, integrating by parts, we can identify µ(φ) := δA(φ) δφ =−ε∆φ+1 εF′(φ).(8) Note that, if φ∈H3and ∂nφ|∂Ω= 0, then ∂n∆φ|∂Ω= 0 is equivalent to ∂nµ(φ)|∂Ω= 0. On the other hand, by using (8), the bending energy (1) can be rewritten as Eb(φ) = 1 2εZΩ µ(φ)2dx+1 2M(A(φ)−α)2. Then, for all φ, φ∈H2, DδEb(φ) δφ ,φE=1 εZΩ µ(φ)(−ε∆φ+1 εF′′(φ)φ) + M(A(φ)−α)DδA(φ) δφ ,φE. If φ∈H4 2and φ∈H2 1, after some integrations by parts, using ∇µ(φ)·n|∂Ω= 0 and ∇φ·n|∂Ω= 0, we can identify w=δEb(φ) δφ =−∆µ(φ) + 1 ε2F′′(φ)µ(φ) + M(A(φ)−α)µ(φ).(9) Remark 2 The variational derivatives w=δEb(φ) δφ and µ(φ) = δA(φ) δφ given in (8) and (9) have been identified as L2(Ω)-functions via the L2(Ω) scalar product. 5 From (8) and (9), we can decompose was w=ε∆2φ+G(φ)(10) where G(φ) := −1 ε∆(F′(φ)) + 1 ε2F′′(φ)µ(φ) + M(A(φ)−α)µ(φ) =−2 εF′′(φ)∆φ−1 εF′′′(φ)|∇φ|2+1 ε3F′′(φ)F′(φ) +M(A(φ)−α)(−ε∆φ+1 εF′(φ)). (11) 2.1 A equivalent “zero-mean” problem With the aim to use the conservative property given in (7), if we define m0=hφ0i:= 1 |Ω|ZΩ φ0(x)dx, we can introduce the following zero-mean variables: ψ(x, t) := φ(x, t)−m0and z:= w− hG(φ)i. Observe that, ZΩ ψ(t, ·)dx= 0 and ZΩ z(t, ·)dx= 0. In fact, since ∂nφ|∂Ω= 0, in particular ∂n(F′(φ))|∂Ω= 0, hence ZΩ −∆(F′(φ)) dx= 0. Therefore, integrating (10) and (11), hwi=hG(φ)i=1 ε2hF′′(φ)µ(φ)i+M(A(φ)−α)hµ(φ)i.(12) Reciprocally, given (ψ, z) we can recover (φ, w) as φ=ψ+m0and w=z+hG(φ)i. By rewriting the equations (2)-(4) and (10) in these new variables (ψ, z) we arrive at ∂tu+ (u· ∇)u−ν∆u−λ z ∇ψ+∇eq= 0,(13) ∇ · u= 0,(14) ∂tψ+u· ∇ψ−γ∆z= 0,(15) ε∆2ψ+G(ψ)−z= 0,(16) where G(ψ) := G(ψ+m0)− hG(ψ+m0)i,eq:= q−λhG(ψ+m0)iψ. 6 By using (11) and (12), G(ψ) = −2 εF′′(ψ+m0)∆ψ−1 εF′′′(ψ+m0)|∇ψ|2+1 ε3F′′(ψ+m0)F′(ψ+m0) +M(A(ψ+m0)−α)−ε∆ψ+1 εF′(ψ+m0) −1 ε2 1 |Ω|ZΩ−ε∆ψ+1 εF′(ψ+m0)F′′(ψ+m0) −M(A(ψ+m0)−α)1 |Ω|ZΩ−ε∆ψ+1 εF′(ψ+m0). (17) System (13)-(16) is completed with the boundary and initial conditions u|∂Ω= 0, ∂nψ|∂Ω= 0, ∂n(∆ψ)|∂Ω= 0, ∂nz|∂Ω= 0,(18) u|t=0 =u0, ψ|t=0 =ψ0:= φ0− hφ0iin Ω.(19) Consequently, problem (2)-(6) is equivalent to the “zero-mean” problem (13)-(19). Note that, by denoting the bending energy with respect to the zero-mean unknown ψas Eb(ψ) = Eb(ψ+m0),(20) then it is easy to check that z=δEb(ψ) δψ =ε∆2ψ+G(ψ)∀ψ∈H4 2,(21) where the identification (21) has been computed via the L2 ∗(Ω)-scalar product. Latter, we will give a weak sense to this identification in order to define the concept of weak solution. 3 Some preliminary results We assume Ω sufficiently regular (for instance ∂Ω∈C3) in order to have the H3-regularity of the following second order elliptic Poisson-Neumann problem: (P0)   −∆v=fin Ω ∂nv|∂Ω= 0,ZΩ v dx= 0, where f∈L2 ∗(Ω). In fact, the H2and H3-regularity of (P0) provide, respectively, the existence of some constants C > 0, such that: kvk2≤C|∆v|2∀v∈H2 1,kvk3≤Ck∆vk1∀v∈H3 1.(22) In particular, since ∆v∈L2 ∗, by applying the Poincare inequality for zero-mean functions, we have that kvk3≤C|∇∆v|2∀v∈H3 1.(23) 7 In order to define the inner product in H−1 ∗, observe that if r∈H−1 ∗, then applying the Lax-Milgram’s Lemma, there exists a unique ur∈H1 ∗such that hr, viH−1 ∗,H1 ∗= (ur, v)H1 ∗= (∇ur,∇v)L2for all v∈H1 ∗. Therefore, krkH−1 ∗=k∇urkL2and (r, s)H−1 ∗= (∇ur,∇us)L2for all s∈H−1 ∗. In order to give a weak sense to ∆2vtogether with the boundary condition ∂n(∆v)|∂Ω= 0 for any v∈H3 1, we introduce the operator A:D(A) = H3 1⊂H−1 ∗7→ H−1 ∗as follows: hAψ, e ψiH−1 ∗,H1 ∗=−(∇∆ψ, ∇e ψ)L2∀ψ∈H3 1,∀e ψ∈H1 ∗.(24) In particular, denoting r=Aψ and s=Ae ψ, then ur=−∆ψand ur=−∆e ψ, hence the inner product in H−1 ∗remains (Aψ, A e ψ)H−1 ∗= (∇∆ψ, ∇∆e ψ)L2. Observe that A∈ L(H3 1, H−1 ∗) and it is self-adjoint and positive definite. Indeed, Ais continuous because kAψkH−1 ∗= sup e ψ hAψ, e ψiH−1 ∗,H1 ∗ ke ψkH1 ∗ = sup e ψ −(∇∆ψ, ∇e ψ)L2 |∇ e ψ|2 In particular, taking e ψ= ∆ψ∈H1 ∗, one has kAψkH−1 ∗=k∇∆ψkL2≤Ckψk3.(25) On the other hand, for all ψ, e ψ∈H3 1, since ∂nψ|∂Ω=∂ne ψ|∂Ω= 0, we have that hAψ, e ψiH−1 ∗,H1 ∗=−(∇∆ψ, ∇e ψ)L2= (∆ψ, ∆e ψ)L2=−(∇ψ, ∇∆e ψ)L2=hAe ψ, ψiH−1 ∗,H1 ∗. Therefore, Ais symmetric. In particular, hAψ, ψiH−1 ∗,H1 ∗=k∆ψk2 L2≥Ckψk2 2for all ψ∈ H3 1, hence Ais positive definite. Finally, we are going to prove that there exists A−1∈ L(H−1 ∗, H3 1) by applying the Banach-Necas-Babuska’s Theorem (see for instance [11]). For this, we consider the following continuous bilinear form a(·,·) : H3 1×H1 ∗7→ R: a(ψ, e ψ) = hAψ, e ψiH−1 ∗,H1 ∗=−(∇∆ψ, ∇e ψ)L2∀ψ∈H3 1,∀e ψ∈H1 ∗. Owing to (23) and (25), there exists a constant β > 0 such that sup e ψ∈H1 ∗ a(ψ, e ψ) |∇ e ψ|2 =kAψkH−1 ∗≥βkψk3. On the other hand, if we assume a(ψ, e ψ) = 0 for all ψ∈H3 1, then taking in particular ψ∈H3 1 the solution of problem (P0) for f=e ψ, then −∆ψ=e ψand 0 = a(ψ, e ψ) = −(∇∆ψ, ∇e ψ)L2=|∇ e ψ|2 2 8 hence e ψ≡Cte. But, since RΩe ψ= 0, then e ψ≡0. Therefore, the Banach-Necas-Babuska’s Theorem implies that there exists A−1∈ L(H−1 ∗, H3 1). Moreover, Ais self-adjoint. The following lemma shows two compactness results of Aubin-Lions type, see [20]. Lemma 3 Let us consider T > 0and three Banach spaces such that X⊂B⊂Ywith continuous imbedding B→Yand continuous and compact imbedding X7→ B. •If the set Fis bounded in Lp(0, T;X)where p < ∞and ∂tF={∂tf:f∈F}is bounded in L1(0, T;Y), then Fis relatively compact in Lp(0, T;B). •If the set Fis bounded in L∞(0, T;X)and ∂tFis bounded in Lq(0, T;X)with q > 1, then Fis relatively compact in C([0, T]; B). Along this paper, we will use repeatly the following classical interpolation and Sobolev inequalities (for 3Ddomains): |v|6≤Ckvk1,|v|3≤C|v|1/2 2kvk1/2 1∀v∈H1 and the Gagliardo-Nirenberg inequality |v|∞≤Ckvk1/2 1kvk1/2 2∀v∈H2. For the last inequality, see for example ([17], p. 334). The following Lemma gives some global Lipschitz properties of F(φ) its derivatives and A(φ) into H2-bounded sets. Lemma 4 Let us consider K > 0a constant and any functions φi∈H2(Ω), with kφik2≤K, for i= 1,2, then the following inequalities are fulfilled for any pwith 1≤p≤ ∞: |F(φi)|p≤C(K),|F′(φi)|p≤C(K),|F′′(φi)|p≤C(K),|F′′′(φi)|p≤C(K)i= 1,2, |F(φ1)−F(φ2)|p≤C(K)|φ1−φ2|p,|F′(φ1)−F′(φ2)|p≤C(K)|φ1−φ2|p, |F′′(φ1)−F′′(φ2)|p≤C(K)|φ1−φ2|p,|F′′′(φ1)−F′′′(φ2)|p≤C(K)|φ1−φ2|p, |A(φ1)− A(φ2)| ≤ C(K)kφ1−φ2k1, where C(K)>0are different constants depending on K. Proof. Let us remember that F(φ) = 1 4(φ2−1)2,F′(φ) = (φ2−1)φ,F′′(φ) = 3φ2−1, F′′′(φ) = 6φand A(φ) = ZΩε 2|∇φ|2+1 εF(φ)dx. We prove the first inequality: |F(φi)|p=1 4|φ4+ 2φ2−1|p≤C(|φ|4 ∞+ 2|φ|2 ∞+ 1) ≤C(kφk4 2+ 2kφk2 2+ 1) ≤C(K), 9 In particular, interpolating H2 1between H1 ∗and H3 1, we obtain that C|G(ψ)−G(ψ∗)|2≤C1(K)kψ−ψ∗k1/2 1kψ−ψ∗k1/2 3≤ε 2Mkψ−ψ∗k3+M 2εC1(K)2kψ−ψ∗k1. Let β > 0, θ∈(0,1/2) given in Step 1, by choosing β1>0 and β2>0, both sufficiently small, such that M 2εC1(K)2β1≤εβ 4Mand β1−θ 2≤εβ 4M, then, for any ψ∈H3 1(Ω) satisfying kψk2≤K,kψ−ψ∗k1≤β1and |Eb(ψ)− Eb(ψ∗)| ≤ β2, we have C|G(ψ)−G(ψ∗)|2≤ε 2Mkψ−ψ∗k3+εβ 4M(31) and |Eb(ψ)− Eb(ψ∗)|1−θ≤εβ 4M.(32) There are only two possibilities: either kψ−ψ∗k3≤βand then (28) holds by using Step 1, or kψ−ψ∗k3> β. In this latter case, from (30) and (31) kz(ψ)kH−1 ∗≥ε 2Mkψ−ψ∗k3−εβ 4M>εβ 2M−εβ 4M=εβ 4M. On the other hand, from (32), εβ 4M≥ |Eb(ψ)− Eb(ψ∗)|1−θ hence (28) holds. 5 Weak Solutions We define the total free-energy as E(u, ψ) = Ek(u) + λEb(ψ),where Ek(u) = 1 2ZΩ |u|2is the kinetic energy, and Eb(ψ) is the bending energy defined in (20). That is, E(u, ψ) = 1 2ZΩ |u|2+λ 2εZΩ (−ε∆ψ+1 εF′(ψ+m0))2dx+M 2(A(ψ+m0)−α)2.(33) Definition 11 Let u0∈Hand ψ0=φ0−m0∈H2 1, we say that (u, ψ, z)is a global weak solution of (13)-(19)in (0,+∞), if u∈L∞(0,+∞;H)∩L2(0,+∞;V), ψ∈L∞(0,+∞;H2 1)∩L2 loc([0,+∞); H3 1), z ∈L2(0,+∞;H1 ∗),(34) 16 satisfying the variational formulation a.e. t∈(0,+∞): h∂tu,ui+ ((u· ∇)u,u) + ν(∇u,∇u)−λ(z∇ψ, u) = 0,∀u∈V(35) h∂tψ, zi+ (u· ∇ψ, z) + γ(∇z, ∇z) = 0,∀z∈H1 ∗(36) −ε(∇∆ψ, ∇ψ) + (G(ψ), ψ)−(z, ψ) = 0,∀ψ∈H1 ∗(37) the initial conditions (19), and the energy inequality (in integral version) E(u(t1), ψ(t1)) − E(u(t0), ψ(t0)) + Zt1 t0 (ν|∇u(s)|2 2+λγ|∇z(s)|2 2)ds ≤0,(38) for a.e. t1, t0:t1≥t0≥0. Energy inequality (38) shows the dissipative character of the model with respect to the total free-energy E(u(t), ψ(t)). Observe that, using the operator Adefined in (24), then (37) can be rewritten as z(t) = εAψ(t) + G(ψ(t)).(39) On the other hand, from (34), (35) and (36), one can deduce that ∂tu∈L4/3 loc ([0,+∞); V′) and ∂tψ∈L2 loc([0,+∞); H−1 ∗), hence, the following time-continuity can be deduced (see Lemma 1.4 of [21]), u∈C([0,+∞); V′)∩Cw([0,+∞); H), ψ∈C([0,+∞); H−1 ∗)∩Cw([0,+∞); H2 1). Here, the weak continuity u∈Cw([0,+∞); H) means that u(t)∈Hfor all t∈[0,+∞) and u(s)→u(t) weak in Has s→t. In particular, the initial conditions (19) have a sense. The following lemma gives two improved energy inequalities. Lemma 12 Given (u, ψ, z)a global weak solution of (13)-(19)in (0,+∞), there exists a regularized energy function e E=e E(t)∈Rdefined for all t≥0, which satisfies the following integral inequality for all t1, t0with t1≥t0≥0: e E(t1)−e E(t0) + Zt1 t0 (ν|∇u(s)|2 2+λγ|∇z(s)|2 2)ds ≤0,(40) and the following differential inequality for a.e. t≥0: d dt e E(t) + ν|∇u(t)|2 2+λγ|∇z(t)|2 2≤0,a.e. t≥0.(41) 17 Proof. From the weak continuity u∈Cw([0,+∞); H) and ψ∈Cw([0,+∞); H2 1), in particular the energy evaluated in the trajectory E(u(t), ψ(t)) exits for all t. Moreover, since the inequality (38) is satisfied for all t0, t1∈[0,+∞)\N, where Nis a set of null Lebesgue measure, then E(u(t), ψ(t)) is a real decreasing function in [0,+∞)\N, and E(u(t), ψ(t)) ∈L∞(0,+∞). Therefore, we can define a regularized function e Erelated to Efor all t≥0 as: e E(0) := E(u0, ψ0),e E(t) := lim s→t− s∈[0,+∞)\N E(u(s), ψ(s)). Then, this function e E(t) is “continuous from the left” and is decreasing for all t≥0. Indeed, for any t1, t2∈[0,+∞), for instance t1< t2, we can choose sequences {s1 n},{s2 n} ⊂ [0,+∞)\N such that s1 n→t− 1,s2 n→t− 2and, s1 n≤s2 nfor all n≥n0. Since s1 nand s2 nare not in N, we know that E(u(s1 n), ψ(s1 n)) ≥ E(u(s2 n), ψ(s2 n)). By taking limit as s1 n→t− 1and s2 n→t− 2, we obtain that e E(t1)≥e E(t2). Since e E(t) is decreasing for all t∈[0,+∞), it is derivable (and continuous) almost everywhere t∈(0,+∞) (see for example Theorem 3 of [18] p. 100). On the other hand, since the inequality (38) is satisfied for all t0, t1∈[0,+∞)\N, given any t0≤t1, we can take δn>0 and ηn>0 such that t0−δn, t1−ηn6∈ Nand δn→0 and ηn→0. Then E(u(t1−δn), ψ(t1−δn)) − E(u(t0−δn), ψ(t0−δn)) + Zt1−ηn t0−δn (ν|∇u(s)|2 2+λγ|∇z(s)|2 2)ds ≤0. By taking δn→0 and ηn→0, and using the definition of e E(t), we obtain (40) for any t0≤t1. In particular, by choosing t0=tand t1=t+hin (40), we obtain e E(t+h)−e E(t) h+1 hZt+h t (ν|∇u|2 2+λγ|∇z|2 2)ds ≤0,∀t, h ≥0.(42) Observe that, since the map t∈[0,+∞)→ν|∇u(t)|2 2+λγ|∇z(t)|2 2∈Rbelongs to L1(0,+∞), then lim h→0 1 hZt+h t (ν|∇u|2 2+λγ|∇z|2 2)ds =ν|∇u(t)|2 2+λγ|∇z(t)|2 2a.e. t≥0. By taking h→0 in (42), we obtain (41) a.e. t≥0. Remark 13 The fact that the energy is decreasing is essential in Lemma 12. For general problems with nonzero external forces, the thesis of this Lemma is not clear. 18 5.1 Formal energy equality and large-time weak regularity We will arrive at the energy equality related to (41) in a formal manner (assuming a sufficiently regular solution). A rigorous demonstration of existence of weak solutions (in particular, satisfying (38)) will be obtained in Section 5.2 via a Galerkin method, in the same way as was done in [8, pp. 71-73] for the Navier-Stokes problem. By assuming that (u, ψ, z) is a sufficiently regular solution of (13)-(19), and taking u=u,z=zand ψ=∂tψas test function in (35), (36) and (37) respectively, one arrives at the equalities: 1 2 d dt|u|2 2+ν|∇u|2 2−λ(z∇ψ, u) = 0, (∂tψ, z) + (u· ∇ψ, z) + γ|∇z|2 2= 0, εd dt 1 2|∆ψ|2 2+ (G(ψ), ∂tψ)−(z, ∂tψ) = 0. Adding the first equality plus the second and third ones multiplied by λ, the term (z, ∂tψ) cancels, as well as the nonlinear convective term (u· ∇ψ, z) with the elastic term −(z∇ψ, u), arriving at 1 2 d dt(|u|2 2+λε|∆ψ|2 2) + λ(G(ψ), ∂tψ) + ν|∇u(t)|2 2+λγ|∇z(t)|2 2= 0.(43) Since δEb(ψ) δψ =z=ε∆2ψ+G(ψ), in particular d dtEb(ψ(t)) = DδEb(ψ) δψ , ∂tψE= (z, ∂tψ) = ε1 2 d dt|∆ψ|2 2+ (G(ψ), ∂tψ). Then, equality (43) can be rewriten as the following energy equality (which is the equality version of (41)): d dtE(u(t), ψ(t)) + ν|∇u(t)|2 2+λγ|∇z(t)|2 2= 0,(44) From (44), assuming the initial regularity (u0, ψ0) in H×H2 1hence E(u(0), ψ(0)) is finite, we obtain that u∈L∞(0,+∞;H)∩L2(0,+∞;V), z∈L2(0,+∞;H1 ∗), and Eb(ψ(t)) ∈ L∞(0,+∞). In particular, ZΩ (−ε∆ψ+1 εF′(ψ+m0))2dxand A(ψ) = ZΩε 2|∇ψ|2+1 εF(ψ+m0)dx belong to L∞(0,+∞). From A(ψ)∈L∞(0,+∞) one has that ∇ψ∈L∞(0,+∞;L2) and, since ZΩ ψ= 0, also ψ∈L∞(0,+∞;H1 ∗). Secondly, from ZΩ (−ε∆ψ+1 εF′(ψ+m0))2∈L∞(0,+∞), we have that ψ∈L∞(0,+∞;H2 1). 19 In summary, the following global in time “weak” regularity (in the time interval (0,+∞)) hold: u∈L∞(0,+∞;H)∩L2(0,+∞;V), ψ ∈L∞(0,+∞;H2 1), z ∈L2(0,+∞;H1 ∗).(45) By applying Remark 6to G(ψ(t)) defined in (17), since kψ(t) + m0k2≤K, we obtain |G(ψ(t))|2≤ |G(ψ(t) + m0)|2+|hG(ψ(t) + m0)i|2≤C. (46) From (23), (25) and (39), we obtain kψk3≤C|∇∆ψ|2=CkAψkH−1 ∗≤C(|z|2+|G(ψ)|2)≤C(1 + |z|2) (47) From this estimate and the regularity of zgiven in (45), we have ψ∈L2 loc([0,+∞); H3 1).(48) 5.2 Existence of weak solutions satisfying energy inequality (38) For instance, fixed the initial datum (u0, ψ0)∈H×H2 1, the existence of weak solutions of (13)-(19) in (0,+∞), furnished by a limit of an adequate Galerkin approximate solutions can be proved analogously to [3] or [4]. Moreover, these Galerkin solutions satisfy the corresponding energy equality (44) which suffices to prove rigorously the previous estimates (45) and (48). Finally, following similar arguments to those used in [8] for the Navier-Stokes equations, these weak solutions will satisfy the energy inequality (38). Indeed, let {wi}n≥1 and {φi}n≥1 “special” basis of Vand H2 1, respectively, formed by eigenfunctions of the Stokes problem wi∈Vsuch that Swi=λiwi,with kwikL2= 1, λiր+∞ and of the Poisson-Neumann problem φi∈H2 1such that −∆φi=µiφi,with kφikL2= 1, µiր+∞. Here, we consider the Stokes operator Sdefined as Sw∈Vsuch that (∇(Sw),∇v) = (w,v) for all v∈V. Let Vmand Wmbe the finite-dimensional subspaces spanned by {w1,w2,...,wn}and {φ1, φ2,...,φn}respectively. Note that if φ∈Wmthen also ∆φ∈Wmand ∆2φ∈Wm. For each m≥1, we say that (um, ψm) is a Galerkin solution, if um: [0,+∞)7→ Vmand ψm: [0,+∞)7→ Wm, and satisfy              h∂tum,umi+ ((um· ∇)um,um) + ν(Sum,um) −λ(zm∇ψm,um) = 0,∀um∈Vm,a.e. t > 0, (∂tψm, zm) + (um· ∇ψm, zm) + γ(∇zm,∇zm) = 0,∀zm∈Wm,∀t > 0, um(0) = u0m:= Pm(u0), ψm(0) = ψ0m=Qm(ψ0) in Ω, (49) 20 where zm: [0,+∞)7→ Wmis defined as zm:= Qm(ε∆2ψm+G(ψm)). Here, Pm:L27→ Vm denotes the projection from L2onto Vmand Qm:L27→ Wmthe projection from L2onto Wm. Since ∆2ψm∈Wmthen zm=ε∆2ψm+Qm(G(ψm)) = εAψm+Qm(G(ψm)),in Ω ×(0,+∞). (50) Since (∂tum, ∂tψm)∈Vm×Wmand (Sum,−∆zm)∈Vm×Wm, the variational formulation (49) yields to the point-wise equalities ∂tum+νSum+Pm((um· ∇)um−λ zm∇ψm) = 0,in Ω ×(0,+∞). (51) ∂tψm−γ∆zm+Qm(um· ∇ψm) = 0,in Ω ×(0,+∞). (52) The existence and uniqueness of local in time solution of (49)-(50) in Ω ×(0, T), for any T > 0 sufficiently small, can be proved using the Leray-Schauder’s Theorem in the same way as the Theorem 5.2 of [3]. By following the argument given in Subsection 5.1, the energy equality E(um(t), ψm(t)) − E(um(t0), ψm(t0)) + Zt t0 (ν|∇um|2 2+λγ|∇zm|2 2)dτ = 0 (53) holds for all t, t0:t≥t0≥0. Therefore, the solution of (49)-(50) can be extended to the whole time interval (0,+∞), and the following uniform in time estimates (independent of m) can be obtained as in Subsection 5.1: umin L∞(0,+∞;H)∩L2(0,+∞;V), ψmin L∞(0,+∞;H2 1), zmin L2(0,+∞;H1 ∗). By applying these estimates in (51)-(52) one has the estimates for the time derivatives ∂tumin L4/3 loc ([0,+∞); V′) and ∂tψmin L2 loc([0,+∞); H−1 ∗). By following the argument given in Subsection 5.1 to prove (46) and (48), one has the estimates G(ψm) in L∞((0,+∞)×Ω) and ψmin L2 loc([0,+∞); H3 1). By using compactness Lemma 3for the spaces V⊂H⊂V′and H2 1⊂W1,p ∗(Ω) ⊂H−1 ∗with p < 6, and considering the continuous embedding W1,p(Ω) ⊂C(Ω) (p > 3), we have that umis relatively compact in L2(0, T;H), ψmis relatively compact in C([0, T]×Ω) and ∇ψm in C([0, T]; Lp(Ω)), with p < 6 for any T > 0. In particular, from Lemma 4,F(ψm) (and F′(ψm), F′′(ψm)) is relatively compact in C([0, T]×Ω). Finally, by using again Lemma 3for the spaces H3 1⊂H2 1⊂H−1 ∗then ψmis relatively compact in L2(0, T;H2 1). In order to prove the existence of a weak solution (u, ψ, z) of the variational problem (35)-(37) a.e. t∈(0,+∞), obtained as the limit of a (subsequence) of (um, ψm, zm), it suffices 21 to pass to the limit in (50)-(52), which will be possible due to the previous estimates and compactness. For this, given any T > 0 and (u, w)∈V×H2 1we take (um, wm)∈Vm×Wm with (um, wm)→(u, w) in V×H2 1strongly. We rewrite (50) as (zm,wm) = εhAψm, wmi+ (G(ψm), wm),∀t∈[0, T].(54) To prove the weak convergence G(ψm)→G(ψ) weakly in L2((0, T)×Ω), (55) we analyze for example, the term F′′(ψm+m0)∆ψm. Since F′′(ψm+m0) converges to F′′(ψ+m0) in C([0, T ]×Ω) and ∆ψmconverges weakly to ∆ψin L2((0, T)×Ω), hence F′′(ψm+m0)∆ψ→F′′(ψ+m0)∆ψweakly in L2((0, T)×Ω). The proof for the rest of the terms of G(ψm) is easier, and then, one has the convergence of (54) towards the limit equation (37). For example, the convergence of the term (zm∇ψm,um) of (49)1towards (z∇ψ, u) is deduced from the strong convergence of ∇ψmin L∞(0, T;L2(Ω)) and the weak convergence of zmin L2(0, T;H1 ∗). The convergence of the rest of terms of (49)1is rather standard in the Navier-Stokes setting, see [8], hence (35) holds. Finally, the limit of (49)2is standard, arriving at (36). In order to prove that the weak solutions furnished by previous Galerkin method satisfy the energy inequality (38), we follow similar arguments used in [8] for the Navier-Stokes equations. Firstly, we are going to prove that E(um(t), ψm(t)) → E(u(t), ψ(t)) a.e. t≥0.(56) In fact, to show the convergence of the most difficult term RΩµ(ψm+m0)2dxin (56), it suffices to prove that µ(ψm+m0)→µ(ψ+m0) strongly in L2(0, T;L2),∀T > 0.(57) For this, we use that ψmis bounded in L2(0, T ;H3 1), and in particular ∆ψmis bounded in L2(0, T;H1 ∗). Moreover, ∇F′(ψm+m0) = F′′(ψm+m0)∇ψmis bounded in L∞(0, T;L2). Therefore, µ(ψm+m0) is bounded in L2(0, T;H1).(58) On the other hand, by using the parabolic equation (52), ∂tµ(ψm+m0) = −ε∆∂tψm+1 εF′′(ψm+m0)∂tψm =ε∆(Qm(um· ∇ψm)) −γε∆2zm+1 εF′′(ψm+m0)(γ∆zm−Qm(um· ∇ψm)), 22 Here, the second term at the right hand side has the following sense −h∆2zm,e ψi(H3 1)′,H3 1= (∇zm,∇∆e ψ)L2,∀e ψ∈H3 1. Hence, the estimates already obtained for um,ψmand zmimply that ∂tµ(ψm+m0) is bounded in L2(0, T ; (H3 1)′),(59) where the dual space (H3 1)′appears due to the term ∆2zm. Then, (58) and (59) and Lemma 3(for the spaces H1 ∗⊂L2 ∗⊂(H3 1)′) imply (57). On the other hand, since um→ustrongly in L2(0, T;H) then (56) holds. Secondly, since um→uand zm→z, both weakly in L2(0, T;H1), we have that lim inf m→+∞Zt t0 (ν|∇um|2 2+λγ|∇zm|2 2)dτ ≥Zt t0 (ν|∇u|2 2+λγ|∇z|2 2)dτ for all t≥t0≥0.(60) Thirdly, by taking lim inf in (53), we obtain that for all t≥t0≥0, lim inf m→+∞E(um(t), ψm(t)) + lim inf m→+∞Zt t0 (ν|∇um|2 2+λγ|∇zm|2 2)dτ ≤lim sup m→+∞ E(um(t0), ψm(t0)). (61) Finally, by using (56) and (60) in (61), we obtain the energy inequality (38) for a.e. t, t0: t≥t0≥0. 6 Convergence at infinite time. Let (u, ψ, z) be a weak solution of (13)-(19) in (0,+∞) associated to an initial data (u0, ψ0)∈H×H2 1(see Definition 11). From the energy inequality (38), there exists a real number E∞≥0 such that the total energy evaluated in the trajectory (u(t), ψ(t)), satisfies E(u(t), ψ(t)) ցE∞in Ras t↑+∞.(62) Let us define the ω-limit set of this global weak solution (u, ψ) as follows: ω(u, ψ) = {(u∞, ψ∞)∈H×H2 1:∃{tn} ↑ +∞s.t. (u(tn), ψ(tn)) →(u∞, ψ∞) weakly in L2×H2 1}.(63) Let Sbe the set of critical points of the bending energy defined in (27), that is S={ψ∈H3 1:−ε(∇∆ψ, ∇e ψ) + (G(ψ),e ψ) = 0,∀e ψ∈H1 ∗}.(64) Theorem 14 Assume that (u0, ψ0)∈H×H2 1. Fixed (u, ψ, z)a weak solution of (13)-(19) in (0,+∞), then ω(u, ψ)is nonempty and ω(u, ψ)⊂ {0} × S. Moreover, for any ψ∞∈ S such that (0, ψ∞)∈ω(u, ψ), it holds Eb(ψ∞) = E∞. In particular, u(t)→0weakly in L2and E(u(t), ψ(t)) → Eb(ψ∞)in Ras t↑+∞. 23 Proof. Since (u, ψ)∈L∞(0,+∞;H×H2 1), there exists {tn} ↑ +∞and suitable limit functions (u∞, ψ∞)∈H×H2 1, such that u(tn)→u∞weakly in H, ψ(tn)→ψ∞weakly in H2 1.(65) We consider the initial and boundary-value problem associated to (13)-(19) restricted on the time interval [tn, tn+ 1] with initial values u(tn) and ψ(tn). If we define un(s) := u(s+tn), ψn(s) := ψ(s+tn) and zn(s) := z(s+tn) for a.e. s∈[0,1], then, (un, ψn, zn) is a weak solution to the problem (13)-(19) in the time interval [0,1]. From the energy inequality (38), we have that Z1 0 (ν|∇un(s)|2 2+λγ|∇zn(s)|2 2)ds =Ztn+1 tn (ν|∇u(t)|2 2+λγ|∇z(t)|2 2)dt ≤E(u(tn), ψ(tn)) − E(u(tn+ 1), ψ(tn+ 1)) −→ 0 as n→ ∞, hence, ∇un→0 strongly in L2(0,1; L2) and ∇zn→0 strongly in L2(0,1; L2). In particular, by using Poincar´e inequality, one has un→0 strongly in L2(0,1; V) and zn→0 strongly in L2(0,1; H1 ∗).(66) Moreover, since unand ∂tunare bounded in L∞(0,1; H) and L4/3(0,1; V′) respectively, then using Lemma 3,un→0 in C([0,1]; V′). In particular, u(tn) = un(0) →0 in V′, hence u∞= 0 (owing to (65)). Consequently, the whole trajectory u(t)→0 as t→+∞. On the other hand, from the large time regularity of (u, ψ, z) given in (34), unis bounded in L2(0,1; H1), znin L2(0,1; H1 ∗) and ψnin L∞(0,1; H2 1). Moreover, using (47), one also has that ψnis bounded in L2(0,1; H3 1) and from the ψ-equation (15)∂tψnis bounded in L2(0,1; H−1 ∗). Therefore, owing to Lemma 3, there exists a subsequence of ψn(equally denoted) and a limit function ψsuch that ψn→ψstrongly in C0([0,1] ×Ω) ∩L2(0,1; H2 ∗) and weakly in L2(0,1; H3 1). (67) In particular, ψ(tn) = ψn(0) →ψ(0) in C0(Ω), hence ψ(0) = ψ∞(owing to (65)). On the other hand, ∂tψnconverges weakly to ∂tψin L2(0,1; H−1 ∗), hence taking limits in the variational formulation (∂tψn,ez) + (un· ∇ψn,ez) + γ(∇zn,∇ez) = 0 for all ez∈H1 ∗and using convergences (66)-(67), we have that ∂tψn→0 in L2(0,1; H−1 ∗). Therefore, ∂tψ= 0 and ψ(t) is a constant function of H3 1for all t∈[0,1], hence since ψ(0) = ψ∞, one has ψ(t) = ψ∞∈H3 1for all t∈[0,1].(68) 24 Finally, by using convergences (66)-(67) one also has arguing as in (55) that G(ψn)→G(ψ) weakly in L2(0,1; L2), hence taking limit as n→+∞in the variational formulation (zn,e ψ) = −ε(∇∆ψn,∇e ψ) + (G(ψn),e ψ) for all e ψ∈H1 ∗, we deduce −ε(∇∆ψ, ∇e ψ) + (G(ψ),e ψ) = 0,∀e ψ∈H1 ∗,a.e. t∈(0,1). Then, from (68), ψ∞∈H3 1and −ε(∇∆ψ∞,∇e ψ) + (G(ψ∞),e ψ) = 0 for all e ψ∈H1 ∗and the proof is finished. Theorem 15 Assume that e E(t)belongs to the equivalence class of E(u(t), ψ(t)), that is, e E(t) = E(u(t), ψ(t)) a.e. t≥0. Under the hypotheses of Theorem 14, there exists a unique limit ψ∞∈ S such that ψ(t)→ψ∞in H2-weakly as t↑+∞, i.e. ω(u, ψ) = {(0, ψ∞)}. Proof. Let ψ∞∈ S such that (0, ψ∞)∈ω(u, ψ), i.e. there exists tn↑+∞such that u(tn)→0 weakly in L2and ψ(tn)→ψ∞weakly in H2 1(and strongly in H1 ∗). Without loss of generality, it can be assumed that e E(t)> E∞for all t > 0, because otherwise, if it exists some e t > 0 such that e E(e t) = E∞, then the energy inequality (40) implies e E(t) = E∞,|∇u(t)|2 2= 0 and |∇z(t)|2 2= 0,∀t≥e t. In particular, u(t) = 0 and z(t) is constant for all t≥e t, and by using the z-equation (36), ∂tψ(t) = 0, hence ψ(t) = ψ∞for all t≥e t. In this setting the convergence of the whole ψ-trajectory towards ψ∞is trivial. Therefore, we can assume that e E(t)> E∞for all t≥0. In this case, the proof will be divided into three steps. Step 1: Assuming that there exists t1>0such that kψ(t)−ψ∞k1≤β1and |Eb(ψ(t)) − Eb(ψ∞)| ≤ β2 for a.e. t≥t1≥0, where β1>0, β2>0are the constants appearing in Lemma 10 (of Lojasiewicz-Simon’s type), then the following inequalities hold: d dt(E(u(t), ψ(t)−E∞)θ+C θ (|∇u(t)|2+|∇z(t)|2)≤0, a.e. t ∈(t1,∞) (69) Zt2 t1 k∂tψkH−1 ∗≤C θ(E(u(t1), ψ(t1)) −E∞)θ,∀t2∈(t1,∞),(70) where θ∈(0,1/2] is the constant appearing in Lemma 10. Since E∞is constant, we can rewrite the energy inequality (41) as d dt(e E(t)−E∞) + C|∇u(t)|2 2+|∇z(t)|2 2≤0,a.e. t≥0. 25 By using (90) in (83), we obtain d dtkψk2 3+C0(kψk2 6+|∂tψ|2 2)≤C1 + kuk2 1+|z|2 2.(91) By integrating in time in (91), since the initial phase ψ0∈H3 1, we obtain ψ∈L2 loc(0,+∞;H6 2). Finally, from (16), z∈L2 loc(0,+∞;H2 1). In particular, using these improved estimates and the phase equations (15)-(16), we have that the following sixth order equation ∂tψ+u· ∇ψ−γε∆3ψ−γ∆(G(ψ)) = 0, is satisfied point-wisely a.e. t∈(0,+∞). 8 Conclusions and Perspectives For the Navier-Stokes-Cahn-Hilliard model introduced in this paper we have proved the convergence of the trajectory of any global weak solution to a single equilibrium point. Moreover, the regularity for the phase is improved without the need of more regularity for the velocity and pressure variables, and therefore it is not necessary to impose large time or large viscosity constraints. Bearing in mind the results obtained in this paper, it seems achievable to obtain (rational) convergence rate estimates of the convergence of trajectories in a similar way to [12]. Finally, it would be interesting to study if local minimizers of the elastic bending energy are stable, as was proved in [22] for a Navier-Stokes-Allen-Cahn problem modeling vesicle membranes. 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