A review on mathematical analysis for nematic and smectic-A liquid crystal models

A REVIEW ON MATHEMATICAL ANALYSIS FOR NEMATIC AND SMECTIC-A LIQUID CRYSTAL MODELS∗ Blanca CLIMENT-EZQUERRA, Francisco GUILL´ EN-GONZ´ ALEZ Departamento de Ecuaciones Diferenciales y An´alisis Num´erico Universidad de Sevilla, Spain. bclimen[email protected], [email protected] Abstract We review the mathematical analysis of some uniaxial, liquid crystal phases. First, we state the models for the two different studied phases: nematic and smectic-A liquid crystals. The spatial and temporal profiles of the liquid crystal configurations will be described by means of strongly nonlinear parabolic partial differential systems, which are presented at the same time. Then, we will state some results about existence, regularity, time-periodicity and stability of solutions at infinite time for both models. It is our aim to show that, although nematic and smectic-A phases have different physical properties and are modeled by different nonlinear parabolic problems, there exists a common mathematical machinery to rewrite the models and to obtain the analytical results. Keywords: Liquid crystals, nematic phase, smectic-A phase, Navier-Stokes equations, GinzburgLandau penalization, global in time solutions, time-periodic solutions, regularity, stability. 2010 Mathematics Subject Classification: 35Q35, 35K55, 76D03, 76A15. 1 Introduction Liquid crystals (LCs) are substances which exhibit an intermediate phase of matter that has properties between those of a conventional liquid and those of a solid crystal. For instance, a LC may flow like a liquid, but its molecules may be oriented in a crystal-like way, see Figure 1. On the other hand, they have (anisotropic) optical and electro-magnetics characteristics like a solid. There are many different types of LC phases (mesophases), which can be distinguished by their different optical properties. The local average order of molecules ∗This work has been partially financed by DGI-MEC (Spain), Grant MTM2009-12927. 1 Figure 1: [http://moebius.physik.tu-berlin.de/lc/lcs.html’06] characterize the LC phases, see Figure 1. Some of them are nematic or smectic phases. In the nematic phase molecules have no positional order, but they have long-range orientational order, the molecules flow and their center of mass positions are randomly distributed as in a liquid, but they point towards a preferred direction (in a local average). In the smectic phase, we can distinguish positional order (molecules form well-defined layers) and orientational order. There are different smectic phases. In the Smectic-A phase, the molecules are oriented perpendicularly to the layers, while in the Smectic-C phase, they are tilted away from the normal directions to the layers, see Figure 2. Historically, the first smectic phase observed Figure 2: [http://atom.physics.calpoly.edu’09] was the cholesteric phase, which exhibits a twisting of the molecules perpendicular to the director, see Figure 3. However, this type of phase will not be analyzed in our review. 2 Figure 3: [http://www.doitpoms.ac.uk’09] For more information about liquid crystals, see [de Gennes, Prost’93] or [Collings’02]. We will focus on nematic (N) and smectic-A (SmA) phases from a mathematical point of view, analyzing some initial-boundary or periodic-time nonlinear parabolic systems modeling these phases. We do not actually make an attempt to signpost the physical implications or applications of these problems. For this, the reader can see, for instance, [Stewart’04] in the nematic case or [Stewart’07] in the smectic-A case. The main results presented here are the existence and uniqueness of global in time weak and regular solutions of initial-boundary or time-periodic problems, as well as the convergence of the trajectories of strong solutions to equilibrium solutions. Moreover, in this review we give some improvements for the nematic case and some simplifications for the smectic-A case, with regard to the results given previously in the literature. Our aim is to show that, although nematic and smectic-A phases are modeled by different nonlinear parabolic problems, there exists a common mathematical machinery to obtain all the analytical results presented in this paper. 1.1 The Models A simplified model from the original Ericksen-Leslie equations in the continuum theory of nematic LC, due to Ericksen in [Ericksen’61, Ericksen’87] and Leslie in [Leslie’68, Leslie’79], was introduced by Lin in [Lin’89] and studied (from a mathematical point of view) by Lin and Liu in [Lin,Liu’95, Lin,Liu’00] and by Coutand and Shkoller in [Coutand,Shkoller’01]. A model for Smectic-A LC was proposed by E in [E’97] and studied analytically by Liu [Liu’00]. We assume a liquid crystal confined in an open bounded domain Ω ⊂IR3with (regular) boundary ∂Ω, which is thermally isolated during the time interval [0,+∞). Then, the 3 dynamic can be described by the velocity-pressure variables u: Ω ×[0,+∞)7→ R3and p: Ω ×[0,+∞)7→ Rrespectively. For isotropic fluids, these variables are governed by the Navier-Stokes equations, but in LC the anisotropic configurations modify the dynamic, and reciprocally, the movement has an influence on the orientation of the molecules. The following variables can be considered in LC models: •The so-called director is a unitary vectorial function d: Ω×[0,+∞)7→ R3, with |d|= 1, modeling an average of the orientation of the molecules (this vector in Figure 2 is called n). Owing to the head-to-tail symmetry [Collings’02], equations must be invariant changing dby −d. •In smectic LC, the vectorial function n: Ω ×[0,+∞)7→ R3pointing to the single optical axis (in some monographs, this vector is called aand in Figure 2 is called z) is perpendicular to the layers. It is usual to impose the assumption ∇ × n= 0 in order to have a potential function ϕ: Ω ×[0,+∞)7→ R, called the layer variable, such that n=∇ϕand the level sets of ϕindicate the layer structure. •Moreover, in smectic-A LC the preferential direction of molecules is perpendicular to layers, hence dis proportional to n. Also, it is assumed |n|= 1 (and therefore n=d), i.e. |∇ϕ|= 1, because the layers are incompressible. The static equilibria are related to the (elastic) Oseen-Frank energy, which in the more simple case (of equal elasticity constants) can be reduced [de Gennes, Prost’93] to the convex functional ZΩ |∇d|2(called Dirichlet energy). In uniaxial nematic LC phases and under certain circumstances, the energy minimizers of the Oseen-Frank functional approximate the Landau-de Gennes minimizers [Majumdar,Zarnescu’10]. But, this might not always be the case and indeed, it can be significant differences between the classical Oseen-Frank theory and the Landau-de Gennes theory, see [Ball,Zarnescu’11]. See also [Lin,Liu’01] for a review on the static and dynamic theory of nematic and smectic-A LC, with some connections between Oseen-Frank and Landau-de Gennes theories. To minimize the Oseen-Frank functional, the non-convex constraint |d|= 1 is considered. In order to avoid this constraint, one possibility is to consider an approximation of the OseenFrank functional by a penalization functional of Ginzburg-Landau type [Bethuel et al.’93]. Indeed, by using the function f(d) = 1 ε2(|d|2−1)d and the corresponding potential function F(d) = 1 4ε2(|d|2−1)2(i.e. f(d) = ∇dF(d)), 4 the minimization problem under the constraint |d|= 1 is replaced by a minimization problem without constraints but applied to the non-convex penalized elastic energy: (N) ZΩ Ee(d)dx=ZΩ (1 2|∇d|2+F(d))dx,(1) (SmA) ZΩ Ee(ϕ)dx=ZΩ (1 2|∆ϕ|2+F(∇ϕ))dx.(2) It should be noticed that the balance between the Oseen-Frank energy and the penalized part is realized by the penalization parameter ε > 0, which appears in F. Then, the associated Euler-Lagrange systems (i.e. the critical point equations related to each functional given in (1) or (2)) are: (N) ω(d)≡ −∆d+f(d)=0,(SmA) ω(ϕ)≡∆2ϕ−∇·f(∇ϕ) = 0.(3) Note that ω(d)∈R3and ω(ϕ)∈R. Now, we will introduce equations governing the dynamics of LC. The conservation of angular momentum is related to the proportionality between the material time derivative of the order parameter (din (N) or ϕin (SmA)) and the Euler-Lagrange equations given by ω(d) in (N) or ω(ϕ) in (SmA) (see [Lin’89] for (N) or [E’97] for (SmA)). It writes as the following equations: (N) ∂td+ (u· ∇)d+γ ω(d)=0,(SmA) ∂tϕ+u· ∇ϕ+γ ω(ϕ) = 0 (4) in (0, T)×Ω, where the positive proportionality constant γis an elastic relaxation time. In [Climent,Guillen’12], we study a smectic-A model written in the vectorial variable n(or d), using a time material derivative with respect to nand not considering ϕ. The conservation of linear momentum and the incompressibility of the fluid (assuming constant density, ρ0= 1) are written as: ∂tu+ (u· ∇)u−∇·(σd+λ σe) + ∇p= 0,∇ · u= 0,in (0, T)×Ω,(5) where the shear stress tensor has been split, in a dissipative tensor σdplus an elastic tensor σemultiplied by an elastic constant, λ > 0. For instance, we will take the simplified tensors given in [Lin’89] for (N), and the more general tensors given in [E’97], for (SmA): (N) σd=µ4D(u), σe=−(∇d)t∇d(6) (SmA)    σd=µ1(ntD(u)n)n⊗n+µ4D(u) + µ5(D(u)n⊗n+n⊗D(u)n), σe=−f(n)⊗n+∇(∇ · n)⊗n−(∇ · n)∇n. (7) Here, µ1≥0, µ4>0, µ5≥0 are dissipative constant coefficients, D(u)=(∇u+ (∇u)t)/2 denotes the deformation rate tensor (symmetrized velocity gradient) and (a⊗b)i,j =aibjis 5 the tensorial product. We are dealing with a coupling system in which σedepends on din (N) (or nin (SmA)) and uappears in the convection term u· ∇din (N) (or u· ∇ϕin (SmA))). It should be noticed that, a simplified version of the dissipative tensor has been considered in (N) only by simplicity, because the analytical results of the posed problems can be extended to more general tensors [Lin,Liu’00]. Then, the models consist of (3), (4), (5) and (6) for (N) or (7) for (SmA), completed with the (Dirichlet) boundary conditions: (N) u|∂Ω= 0,d|∂Ω=hon ∂Ω×(0, T) (8) (SmA) u|∂Ω= 0, ϕ|∂Ω=ϕ1, ∂nϕ|∂Ω=ϕ2on ∂Ω×(0, T) (9) where T > 0 is a given final time (T < +∞or T= +∞), and •either the initial conditions: (N) u|t=0 =u0d|t=0 =d0in Ω (10) (SmA) u|t=0 =u0ϕ|t=0 =ϕ0in Ω (11) •or the time-periodic conditions (fixed T < +∞): (N) u|t=0 =u|t=T,d|t=0 =d|t=Tin Ω (12) (SmA) u|t=0 =u|t=T, ϕ|t=0 =ϕ|t=Tin Ω.(13) In general, we consider time-depending boundary data hor ϕ1,ϕ2. In the first case, (10) or (11), the compatibility condition d0|∂Ω=h(0) or ϕ0|∂Ω=ϕ1(0) must be assumed. In the last case, (12) or (13), it is assumed h|t=0 =h|t=Tor ϕ1|t=0 =ϕ1|t=Tand ϕ2|t=0 =ϕ2|t=Tin Ω. (14) Finally, all the mathematical results that we will present in this paper are dependent of the penalization parameter ε. Up to known, there are very few results taking limits as ε→0. 1.1.1 Reformulation of Nematic Model Taking into account that ∇ · ((∇d)t∇d) = ∇|∇d|2 2+F(d)+ (∇d)t(∆d−f(d)) = ∇Ee(d)−(∇d)tω(d) 6 and ∇·(µ4D(u)) = µ4 2∆u(since ∇·u= 0), (4), (5) and (6) can be rewritten as the following PDE system in (0, T)×Ω:    ∂tu+ (u· ∇)u−ν∆u+∇p−λ(∇d)tω(d) = 0,∇ · u= 0, ∂td+ (u· ∇)d+γω(d)=0,−∆d+f(d) = ω(d), (15) where ν=µ4/4 and pis a reformulated potential function equal to p+λEe(d). 1.1.2 Reformulation of Smectic-A Model By splitting the symmetric dissipative tensor into the linear and nonlinear part σd=µ4D(u) + σd nl(D(u),∇ϕ), where σd nl := µ1(ntD(u)n)n⊗n+µ5(D(u)n⊗n+n⊗D(u)n), notice that (again for ν=µ4/2) −∇ · σd=−ν∆u−∇·σd nl. On the other hand, by decomposing the penalization term in the elastic tensor σeas follows σe=−f(n)⊗n+σe np(n), where σe np(n) := ∇(∇·n)⊗n−(∇·n)∇nis the non-penalized tensor, and taking into account that ∇ · (f(n)⊗n)=(∇ · f(∇ϕ))∇ϕ+fi(∇ϕ)∂i∇ϕ= (∇ · f(∇ϕ))∇ϕ+∇F(∇ϕ) and (∇ · σe np(n))j= (∇ · (∇(∇ · n)⊗n−(∇ · n)∇n))j= (∇ · (∇(∆ϕ)⊗ ∇ϕ−∆ϕ∇2ϕ))j =∂i(∂i(∆ϕ)∂jϕ−∆ϕ∂2 ijϕ)=∆2ϕ∂jϕ+∂i(∆ϕ)∂2 ijϕ−∂i(∆ϕ)∂2 ijϕ−∆ϕ∂i∂2 ijϕ = ∆2ϕ∂jϕ−∆ϕ∂j∆ϕ= ∆2ϕ∂jϕ−1 2∂j(|∆ϕ|2), we have −∇ · σe= (∇ · f(∇ϕ))∇ϕ+∇F(∇ϕ)−∆2ϕ∇ϕ+∇|∆ϕ|2 2 = (−∆2ϕ+∇ · f(∇ϕ))∇ϕ+∇F(∇ϕ) + |∆ϕ|2 2=−ω(ϕ)∇ϕ+∇Ee(ϕ). Then, (4), (5) and (7) can be rewritten as the following PDE system in (0, T )×Ω:    ∂tu+ (u· ∇)u−ν∆u−∇·σd nl −λ ω(ϕ)∇ϕ+∇p= 0, ∂tϕ+u· ∇ϕ+γ ω(ϕ) = 0,∆2ϕ−∇·f(∇ϕ) = ω(ϕ), (16) where pis a reformulated potential function equal to p+λEe(ϕ). 7 1.2 Notation •In general, the notation will be abridged. We set Lp=Lp(Ω), p≥1, H1 0=H1 0(Ω), etc. H−1=H−1(Ω) is the dual space of H1 0. 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)3. •The Lpnorm is denoted by | · |p, 1 ≤p≤ ∞, the Hmnorm by k · km(in particular |·|2=k · k0), k·k−1denotes the usual norm in H−1and the product norm in Hn×Hm by k·kn×m. The inner product of L2(Ω) is denoted by (·,·). •We will consider Ω regular enough to have the following equivalent norms: kvk1≈ |∇v|2in H1 0,kvk2≈ |∆v|2in H1 0∩H2, kvk3≈ |∇(∆v)|2+|∆v|2=k∆vk1in H1 0∩H3,kvk4≈ |∆2v|2in H1 0∩H4. •We set Vthe space formed by all fields u∈C∞ 0(Ω)3satisfying ∇ · 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, Ci, D > 0, for i≥0, will denote different constants, depending only on the fixed data of the problem, as Ω, εand boundary data: hor ϕ1, ϕ2(and u0,d0 or ϕ0for the initial-value problem). 1.3 Some comments about time-independent boundary data Assuming time-independent boundary data, an important fact of both models is their dissipative character, because they admit (at least for regular solutions) the following energy equalities (see [Lin,Liu’95] and [Liu’00] respectively): (N) d dt 1 2|u|2 2+λZΩ Ee(d)+ν|∇u|2 2+λγ |ω(d)|2 2= 0,(17) (SmA) d dt 1 2|u|2 2+λZΩ Ee(ϕ)+ν|∇u|2 2+ZΩ σd nl :D(u) + λγ |ω(ϕ)|2 2= 0,(18) for any time t∈(0,+∞). Equality (17) is obtained from (15), by taking uas test function in the u-system, ω(d) in the d-system and adding up. The equality (18) is obtained from (16) by taking uas test function in the u-system, ω(ϕ) in the ϕ-equation and adding up. Note that the time-independent boundary data is applied to vanish boundary integrals (arising 8 after integrating by parts) using that ∂td|∂Ω= 0 for (N) or ∂tϕ|∂Ω= 0 and ∂t(∂nϕ)|∂Ω= 0 for (SmA). Equalities (17) and (18) imply that the total free energy (that is, the kinetic energy 1 2|u|2 2 plus the elastic energy λRΩEe(d) for (N) or λRΩEe(ϕ)) for (SmA) decreases with respect to time, with a rate proportional to the dissipative part |∇u|2 2and |ω(d)|2 2for (N) or |ω(ϕ)|2 2 for (SmA). Certainly, the existence of weak and strong solutions and the long-time convergence to no-flow states, for both nematic and smectic cases, are reassuring in terms of the model verification. If solutions did not behave like this, then the model would be incorrect. On the other hand, it is important to remark that, like boundary data hor ϕ1, ϕ2are time-independent, the following (static) critical points are steady solutions (and in particular time-periodic solutions): (N)        u= 0, d: any solution of the problem: −∆d+f(d) = 0 in Ω, d=hon ∂Ω, p=−λEe(d). (SmA)        u= 0, ϕ: any solution of ∆2ϕ−∇·f(∇ϕ) = 0 in Ω, ϕ=ϕ1,∂nϕ=ϕ2on ∂Ω, p=−λEe(ϕ). Therefore, in order to consider nontrivial time-periodic problems, it will be essential to assume time-dependent and time-periodic boundary data (satisfying (14)). This situation occurs, for example, when an external magnetic or electric force is acting such that the molecules return to their initial position periodically. Other interesting situation is to consider timeindependent boundary data but assuming a time-periodic force g(t) (for instance electrical periodic impulses) acting on the system. In this case, the same results about time-periodic solutions could be obtained. 1.4 Some simplifications •In the following, to make the exposition clearer and without loss of generality, we fix the constants of the problem, excepting the viscosity µ4, taking λ=γ=µ1=µ5= 1,(recall ν=µ4/2). •The case of time-dependent boundary data requires to introduce a lifting function, getting energy equalities like (17) and (18), where source terms depending on the time derivative of the boundary data appear, see (21) below. For clarity in the statement of 9 Let dbe a critical point of Ee(d)subject to d∈H1(Ω) with the boundary condition d|∂Ω=h. There exist constants θ∈(0,1/2) and β > 0depending on dsuch that for any d∈H1(Ω) satisfying d|∂Ω=hand kd−dk1< β, there holds k − ∆d+f(d)k−1≥ |Ee(d)−Ee(d)|1−θ, one can obtain that for all tsuch that kd(t)−dk1< β, the following differential inequality holds: C θ d dt((E(u(t),d(t)) −Ee(d))θ) + G(u(t),d(t))1/2≤0.(29) Step 3: By an argument of contradiction, one deduces that there exists n0big enough such that kd(t)−dk1< β for all t≥tn0 Step 4: From (29) for all t≥tn0one gets Z+∞ tn0 G(u(t),d(t))1/2≤C θ(E(u(tn0),d(tn0)) −Ee(d))θ≤C. Hence, in particular, Z+∞ tn0 |∂td|2≤C. This last bound implies that (d(t))t≥tn0is a Cauchy sequence in L2(Ω) as t↑+∞, hence d(t)→din L2(Ω). Finally, this strong convergence also can be proved in H2(Ω) Remark: The argument of Step 1 is done in [Climent et al.’10] for a model with stretching terms and periodic boundary conditions for d. The arguments of Step 2, 3 and 4 are done in [Liu,Wu,Xu’09] (for periodic boundary conditions) and [Wu’10] (for Dirichlet boundary conditions). Theorem 7 (Stability) Under conditions of Theorem 6, for each ε > 0, there exists δ(ε)> 0such that if E(u0,d0)−E∞≤δ(ε) (30) and G(u0,d0) := ν|∇u0|2 2+|∆d0−f(d0)|2 2≤ε 3,(31) then for each t≥0, one has: G(u(t),d(t)) := ν|∇u(t)|2 2+|∆d(t)−f(d(t))|2 2≤ε. Remark: Hypothesis (31) means that (u0,d0) is near to an equilibrium state (0,d?) and (30) means that the total decay of the energy is small enough. Arguing as in [Climent et al.’10], it is not difficult to prove that hypothesis (30) implies in particular the global regularity in (0,+∞) without imposing large viscosity ν. Moreover, this hypothesis (30) includes the 16 particular case where the initial data (u0,d0) is near to a global minimizer (0,d?) (because E∞=E(0,d?)), where the global regularity can also be proved without imposing large viscosity ν, see [Lin,Liu’95]. In the recent paper [Petzeltova et al.], there are some more specific stability results than in Theorem 7, for instance, changing hypothesis (30) by the more general assumption that d0is near to a local minimizer of the elastic energy Ee(d) (and u0near of zero). 2.3 Time-periodic problem Let T > 0 a finite fixed number which states the time period, and a boundary data h(t) for d(t) time-dependent and time-periodic, i.e. h(0) = h(T). Definition 8 (u,d)is said a weak time-periodic solution in (0, T)of (15),(8) and (12) if u∈L∞(0, T;H)∩L2(0, T ;V),d∈L∞(0, T;H1)∩L2(0, T;H2) satisfying (15) and boundary conditions (8) as in Definition 1 and time-periodic conditions u(0) = u(T),d(0) = d(T)in the sense of spaces L2and H1respectively. Definition 9 A weak time-periodic solution in (0, T)of (15),(8) and (12) is said a strong solution if u∈L∞(0, T;H1)∩L2(0, T ;H2),d∈L∞(0, T;H2)∩L2(0, T;H3) and verifying point-wise the fully differential system (15). Theorem 10 (Existence of weak time-periodic solutions) Let Ωand hbe regular enough with h(0) = h(T)on ∂Ωand such that the steady lifting function e ddefined in (19) satisfies e d∈L∞(0, T;H1). Then, there exists a weak time periodic solution (u,d)of problem (15), (8) and (12). To prove this theorem, a fully Galerkin discretization is introduced (approximating in finite dimension both variables uand d), proving existence and uniqueness of approximate solution associated to arbitrary initial conditions. The finite-dimensional Galerkin problem let us find time-periodic approximate solutions via a fixed-point argument (Leray-Shauder’s Theorem) applied to the operator mapping the initial and final time values. This allows us to obtain a time-periodic Galerkin solution, which converges towards a time-periodic solution of the continuous problem. Reasoning in the same way that in Therorem 5.2 of [Climent,Guillen’10], also in this case, the use of the Maximum Principle is not necessary. 17 Theorem 11 (Existence of strong time-periodic solutions for νlarge) Under conditions of Theorem 10, if moreover e d∈L∞(0, T;H3)and ∂te d∈L∞(0, T;H1)then, for each ν≥ν0for a certain ν0=ν0(T, ∂teϕ), there exists a strong time-periodic solution of (15),(8) and (12). To prove this theorem is suffices to use the existence of weak time-periodic solutions, and for the initial-valued problem, the weak/strong uniqueness and the existence of global strong solution for big enough viscosity ν(see [Climent,Guillen’10]). 3 Smectic-A Problem The initial-boundary value problem (16) with time-independent boundary data has been studied in [Liu’00] obtaining existence of weak solutions in [0, T ] for all T > 0, existence of regular solution for big enough viscosity and uniqueness of weak/regular solution. Here, we are going to show some results concerning time-dependent boundary data that are developed in [Climent,Guillen’10], although some of them (as the proof of Theorem 16) are slightly simplified. We define the lifting function eϕ=eϕ(t) as the weak solution of the problem    ∆2eϕ= 0 in Ω, eϕ=ϕ1(t), ∂neϕ=ϕ2(t) on ∂Ω. (32) In the time-periodic case, since by hypothesis ϕ1(0) = ϕ1(T) and ϕ2(0) = ϕ2(T) on ∂Ω, then eϕ(0) = eϕ(T) in Ω. If we define bϕ(t) = ϕ(t)−eϕ(t), then ∆2bϕ= ∆2ϕin Qand bϕ=∇bϕ= 0 on Σ. In the time-periodic case, one has ϕ(0) = ϕ(T) if and only if bϕ(0) = bϕ(T). Then, we can rewrite the problem (16) respect to the variables (u,bϕ) (with bϕ(t) = ϕ(t)−eϕ(t)) as follows (recall that all coefficients have been taken equal to one, excepting viscosity ν=µ4/2):                  ∂tu+ (u· ∇)u−ν∆u−∇·σd nl −(∆2bϕ−∇·f(∇ϕ))∇ϕ+∇p= 0, ∇ · u= 0, ∂tbϕ+u· ∇ϕ+ ∆2bϕ−∇·f(∇ϕ) = ∂teϕ, u|∂Ω= 0,bϕ|∂Ω= 0, ∂nbϕ|∂Ω= 0 on ΣT (33) jointly with either initial conditions u(0) = u0,bϕ(0) = ϕ0−eϕ(0) or time-periodic conditions u(0) = u(T), bϕ(0) = bϕ(T). 18 If (u, ϕ) is a regular enough solution of (33), the following energy equality holds: d dt 1 2|u|2 2+1 2|∆bϕ|2 2+ZΩ F(∇ϕ)+|∇ϕTD(u)∇ϕ|2 2+|D(u)∇ϕ|2 2 +ν|∇u|2 2+|∆2bϕ−∇·f(∇ϕ)|2 2= (∂teϕ, ∆2bϕ−∇·f(∇ϕ)) + (∂t∇eϕ, f(∇ϕ)) (34) for t∈(0,+∞). Here, the auxiliary variable w= ∆2bϕ−∇·f(∇ϕ) has been used. To obtain the strong estimates (see (38), (39) below), we consider an other auxiliary variable bw:= w−∂teϕ (because bw|∂Ω= 0), eϕregular enough and the weak estimate ϕ∈L∞(0,+∞;H2), then, we can deduce kbϕk4≤C(|bω|2+ 1),kϕk4≤C(|bω|2+ 1), and, for t∈(0,+∞): d dt(kuk2 1+|bω|2 2) + ν 2kuk2 2+kbωk2 2≤C(|bω|2 2+ 1) +C νkbωk2 2(1 + |bω|2+|bω|2 2) + kuk2 2(kuk2 1+|bω|2 2+ 1). (35) Inequality (35) is slight different from the inequality stated in [Climent,Guillen’10] because in that paper the auxiliary variable bw=∂tbϕ+u· ∇bϕ=−∆2bϕ+∇ · f(∇ϕ) + ∂teϕ−u· ∇eϕ was used. 3.1 The initial value problem up to infinite time Definition 12 We say that (u, ϕ)is a weak solution of (16)-(11) in (0,+∞)if ∇ · u= 0 in Q, u|Σ= 0, ϕ|Σ=ϕ1, ∂nϕ|Σ=ϕ2a.e. t∈(0,+∞), ku(t), ϕ(t)k0×2≤C1∀t≥0 (36) ∀γ > 0, e−γt Zt 0 eγsku(s), ϕ(s)k2 1×4ds ≤C21 + 1 ν,∀t≥0,(37) where C1, C2>0are constants independent of ν, verifying h∂tu,vi+ ((u· ∇)u,v) + ν(∇u,∇v)+(σd nl,∇v) −((∆2ϕ−∇·f(∇ϕ))∇ϕ, v)=0 in D0(0,+∞),∀v∈V, ∂tϕ+ (u· ∇)ϕ+ ∆2ϕ−∇·f(∇ϕ)=0,a.e. in (0,+∞)×Ω u(0) = u0, ϕ(0) = ϕ0in Ω. In the finite time case (T < ∞), (37) holds even when γ= 0, i.e. (u, ϕ)∈L2(0, T;H1×H4). Definition 13 We say that a weak solution (u, ϕ)of (16)-(11) is a strong solution if ku(t), ϕ(t)k1×4≤C3∀t≥0,(38) 19 ∀γ > 0, e−γt Zt 0 eγsku(s), ϕ(s)k2 2×6ds ≤C4,∀t≥0 (39) and verifying point-wise the fully differential system (16). Now, we state three results given in [Climent,Guillen’10]. Theorem 14 (weak/strong uniqueness) If (u1, ϕ1)and (u2, ϕ2)are respectively a weak and a strong solution of (16)-(11), then u1=u2and ϕ1=ϕ2. Theorem 15 (Existence of weak solutions) Let u0∈Hand ϕ0∈H2. Let Ω,ϕ1and ϕ2be regular enough, verifying the compatibility conditions ϕ0|∂Ω=ϕ1(0),∂nϕ0|∂Ω=ϕ2(0) and such that the lifting function eϕdefined in (32) satisfies eϕ∈L∞(0,+∞;H4(Ω)) and ∂teϕ∈L∞(0,+∞;W1,4(Ω)). Then, there exists a weak solution (u, ϕ)of (16)-(11) in (0,+∞). The proof is based on a semi-Galerkin method as in [Liu’00]. The novelty respect to [Liu’00] is that in [Climent,Guillen’10] we will find a weak solution bounded up to infinity time, even imposing time-dependent boundary conditions for the layer variable ϕ. Theorem 16 (Existence of strong solutions for νlarge) In the conditions of Theorem 15, if moreover (u0, ϕ0)∈H1×H4with ku0k1≤R1,kϕ0k4≤R2, ∂teϕ∈L∞(0,+∞;W1,4(Ω)) and ∂tt eϕ∈L∞(0,+∞;L2(Ω)), then there exists ν0=ν0(R1, R2, ∂teϕ, ∂tt eϕ)such that for each ν≥ν0, there exists a unique strong solution of (16)-(11) in (0,+∞), which satisfies (38) and (39) with constants C3and C4depending on ν0(but independent of ν). The proof of this theorem is based in the following inequality obtained from (35): d dt(Φ1+ Φ2) + ν 2−C ν(Φ1+ Φ2+ 1)Ψ1 +1 2−C ν(1 + Φ1/2 2+ Φ2)Ψ2≤C(Φ2+ 1) (40) for t∈(0,+∞), where Φ1(t) = kuk2 1,Φ2(t) = |bw|2 2,Ψ1(t) = kuk2 2,Ψ2(t) = kbwk2 2. Again, inequality (40) is rather similar to inequality obtained in [Climent,Guillen’10]. 20 3.2 Behavior at infinite time In this section, we assume time-independent boundary data. We will present two results for the Smectic-A case corresponding with Theorems 6 and 7 for the Nematic case. In fact, the proof follows the same lines as in the nematic case. Theorem 17 (Asymptotic stability) Under conditions of Theorem 16 the energy E(u(t), ϕ(t)) = Ek(u(t)) + Ee(ϕ(t)) = 1 2|∇u(t)|2+ZΩ (1 2|∆ϕ|2+F(∇ϕ)) (sum of kinetic and elastic energies), satisfies E(u(t), ϕ(t)) &E∞ when t↑+∞and the strong solution (u, ϕ)satisfies u(t)→0in H1 0(Ω),(∆2ϕ−∇·f(∇ϕ))(t)→0in L2(Ω) when t↑+∞. Moreover, for each sequence tj→+∞, there exists a subsequence tjk →+∞such that ϕ(tjk)→ϕin H4(Ω)-weak, where ϕis a critical point of the elastic energy, that is, a solution of the stationary problem (∆2ϕ−∇·f(∇ϕ)=0 in Ω, ϕ|∂Ω=ϕ1, ∂nϕ|∂Ω=ϕ2 and any posible critical point limit ϕmust have the same elastic energy equal to the limit of the total energy, that is, Ee(ϕ) = E∞ Remark: In smectic-A case, the uniqueness of the critical point ϕ(as limit of the trajectory at infinity time) remains open. This problem is considered in a submitted paper [Segatti,Wu’10], where a specific Lojasiewicz-Simon inequality is proved giving a relation between the residual ∆2ϕ−∇·f(∇ϕ)) and the energies Ee(ϕ)−Ee(ϕ), for any ϕnear from ϕ. Theorem 18 (Stability) In the conditions of theorem 17, ∀ε > 0, there exists δ(ε)such that if E(u0, ϕ0)−E∞≤δ(ε)and ν|∇u0|2 2+|∆2ϕ0−∇·f(∇ϕ0)|2 2≤ε 3,then for each t≥0, one has: ν|∇u(t)|2 2+|∆2ϕ(t)−∇·f(∇ϕ(t))|2 2≤ε. 21 3.3 The time-periodic problem Let T > 0 a finite fixed number and a boundary data for ϕtime-dependent and timeperiodic, i.e. ϕ1(0) = ϕ1(T) and ϕ2(0) = ϕ2(T). The results of this section can be found in [Climent,Guillen’10]. Definition 19 We say that (u, ϕ)is a weak time-periodic solution of (16),(9) and (13) if u∈L∞(0, T;H)∩L2(0, T ;H1), ϕ ∈L∞(0, T;H2)∩L2(0, T ;H4) satisfying (16) and boundary conditions (9) as in Definition 12 and time-periodic conditions u(0) = u(T),ϕ(0) = ϕ(T)in the sense of spaces L2and H2respectively. Definition 20 We say that a weak time periodic solution of (16),(9) and (13) is a strong solution if u∈L∞(0, T;H1)∩L2(0, T ;H2), ϕ ∈L∞(0, T;H4)∩L2(0, T ;H6) and verifying point-wise the fully differential system (16). We only present the two main results of existence of (weak and strong) time-periodic solutions. Theorem 21 (Existence of weak time-periodic solutions) Let Ω,ϕ1and ϕ2be regular enough with ϕ1(0) = ϕ1(T),ϕ2(0) = ϕ2(T), and such that the lifting function eϕdefined in (32) satisfies eϕ∈L∞(0, T;H4(Ω)), ∂teϕ∈L∞(0, T;W1,4(Ω)). Then, there exists a weak time-periodic solution of (16),(9) and (13). Theorem 22 (Existence of regular time-periodic solutions for νlarge) Under conditions of previous theorem, if moreover ∂teϕ∈L∞(0,+∞;W1,4(Ω)) and ∂tt eϕ∈L∞(0,+∞;L2(Ω)), then there exists ν0=ν0(∂teϕ, ∂tt eϕ)such that, for each ν≥ν0, there exists a strong timeperiodic solution of (16),(9) and (13). References [http://moebius.physik.tu-berlin.de/lc/lcs.html’06] http://moebius.physik.tuberlin.de/lc/lcs.html, (n.d.) Retrieved July 11, 2006. 22 [http://www.doitpoms.ac.uk’09] http://www.doitpoms.ac.uk, (n.d.) Retrieved Septembre 2, 2009 [http://atom.physics.calpoly.edu’09] http://atom.physics.calpoly.edu, (n.d.) Retrieved Septembre 2, 2009. [Ball,Zarnescu’11] J.M. Ball, A. Zarnescu, Orientability and energy minimization in liquid crystal models, Arch. Ration. Mech. Anal. 202 (2011), no. 2, 493-535. [Bethuel et al.’93] F. Bethuel, H. Brezis, F. H´elein, Asymptotics for the minimization of a Ginzburg-Landau functional, Calc. Var. 1 (1993), 123–148. [Climent et al.’06] B. Climent-Ezquerra, F. Guill´en-Gonz´alez, M.A. Rojas-Medar. Reproductivity for a Nematic Liquid Crystal Model, Z. Angew. Math. Phys., 576 no. 6 (2006), 984-998. [Climent et al.’09] B. Climent-Ezquerra, F. Guill´en-Gonz´alez, M.J. Moreno-Iraberte. Regularity and Time-periodicity for a Nematic Liquid Crystal model, Nonlinear Analysis Theory, Methods and Applications. Vol. 71. (2009) Pag. 539-549 [Climent,Guillen’10] B. Climent-Ezquerra, F. Guill´en-Gonz´alez. Global in time solution and time-periodicty for a Smectic-A Liquid Crystal Model, Commun. Pure Appl. Anal. 9 (2010), no. 6, 1473-1493. [Climent et al.’10] B. Climent-Ezquerra, F. Guill´en-Gonz´alez, M.A. Rodr´ıguez-Bellido. Stability for Nematic Liquid Crystal with Stretching Terms, International Journal of Bifurcations and Chaos, Vol. 20, No. 9 (2010) 2937-2942. [Climent,Guillen’12] B. Climent-Ezquerra, F. Guill´en-Gonz´alez. On a double penalized smectic-A model, Discrete Continuous Dynamical Systems, Vol. 32, No. 12 (2012) 41714182. [Collings’02] P.J. Collings, Liquid Crystals: Nature’s Delicate Phase of Matter, Princeton University Press, 2002 [Coutand,Shkoller’01] D. Coutand, S. Shkoller. Well posedness of the full Ericsen-Leslye model of nematic liquid crystals, C.R.Acad.Sci.Paris, Ser. I 333 (2001), 919-924. [de Gennes, Prost’93] P.G. de Gennes, J. Prost The Physics of Liquid Crystals, Oxford University Press (1993). [Du et al.’07] Q. Du, M. Li, C. Liu, Analysis of a phase field Navier-Stokes vesicle-fluid interaction model, Discrete Contin. Dyn. Syst. Ser. B 8 (2007), no. 3, 539-556. 23 [E’97] W. E. Nonlinear Continuum Theory of Smectic-A Liquid Crystals, Arch. Rat. Mech. Anal., 137, 2 (1997), 159-175. [Ericksen’61] J. Ericksen, Conservation laws for liquid crystals, Trans. Soc. Rheol. 5 (1961) 22-34. [Ericksen’87] J. Ericksen, Continuum theory of nematic liquid crystals, Res. Mechanica 21 (1987) 381-392. [Haraux’91] A. Haraux, Syst`emes dynamiques dissipatifs et applications, Mason, Paris (1991). [Kinderlehrer,Liu’96] D. Kinderlehrer, C. Liu. Revisiting the Focal Conic Structures in Smectic A. Proc. Symposium on Elasticity to honor Professor J. L. Ericksen 1996, ASME Mechanics and Materials Conference. [Lai,Liu, Wenston’06] M.J. Lai, C. Liu, P. Wenston, On Two Nonlinear Biharmonic Evolution Equations: Existence, Uniqueness and Stability, Z. Angew. Math. Phys., 576 no. 6 (2006), 984-998. [Leslie’68] F. Leslie, Some constitutive equations for liquid crystals., Arch. Rational Mech. Anal. 28 (1968) 265-283. [Leslie’79] F. Leslie, Theory of flow phenomenon in liquid crystal, 4, Brown Ed., Academic Press. New York, (1979), 1-81. [Lin’89] F.H. Lin Nonlinear theory of defects in nematic liquid crystals: phase transition and flow phenomena, Comm. Pure Appl. Math., 42 (1989), 789-814. [Lin,Liu’95] F.H. Lin, C. Liu. Non-parabolic dissipative systems modelling the flow of liquid crystals, Comm. Pure Appl. Math., 4 (1995), 501-537. [Lin,Liu’00] F.H. Lin, C. Liu. Existence of solutions for the Ericksen-Leslie system, Arch. Rat. Mech. Anal., 154 (2000), 135-156. [Lin,Liu’01] F.H. Lin, C. Liu. Static and Dynamic Theories of Liquid Crystals, Journal of Partial Differential Equations, Vol 14, No. 4, 289-330 (2001) [Liu,Shen’03] C. Liu and J. Shen. A phase field model for the mixture of two incompressible fluids and its approximation by a fourier-spectral method, Physica D, 179 (2003), 211-228. [Lions’96] P.L. Lions. Mathematical Topics in Fluids Mechanics, Clarendon Press, Oxford, 1996. 24 [Liu’00] C. Liu. Dynamic Theory for Incompressible Smectic Liquid Crystals: Existence and Regularity, Discrete and Continuous Dynamical Systems 6, 3 (2000), 591-608. [Liu,Wu,Xu’09] C. Liu, H. Wu, X. Xu. Asymptotic Behavior for a Hydrodynamic System in the Nematic Liquid Cristal Flows, Arxiv preprint arXiv:0901.1751, (2009). [Majumdar,Zarnescu’10] A. Majumdar, A. Zarnescu. Landau-de Gennes theory of nematic liquid crystals: the Oseen-Frank limit and beyond, Arch. Ration. Mech. Anal., 196 (2010), no.1, 227-280 [Paicu,Zarnescu’12] M. Paicu, A. Zarnescu, Energy dissipation and regularity for a coupled Navier-Stokes and Q-tensor system, Arch. Ration. Mech. Anal. 203 (2012), no. 1, 45-67. [Petzeltova et al.] H. Petzeltov´a, E. Rocca, G, Schimperna, On the long-time behavior of some mathematical models for nematic liquid crystals, Calc. Var. DOI 10.1007/s00526012-0496-1. [Segatti,Wu’10] A. Segatti, H. Wu. Finite dimensional reduction and convergence to equilibrium for incompressible Smectic-A liquid crystal flows, Arxiv preprint arXiv:1011.0358v1, (2010). [Stewart’04] I.W. Stewart, The Static and Dynamic Continuum Theory of Liquid Crystals: A Mathematical intrtoduction, Taylor & Francis, London & New-York (2004). [Stewart’07] I.W. Stewart, Dynamic theory for smectic A liquid crystals, Continuum Mech. Therm., 18 (2007) 343-360. [Wu’10] H. Wu. Long-Time Behavior for Nonlinear Hydrodynamic System Modeling the Nematic Liquid Cristal Flows, Discrete and Continuous Dynamical Systems 26, 1 (2010), 379-396. 25