Regularity and time-periodicity for a nematic liquid crystal model

Regularity and time-periodicity for a nematic liquid crystal model Blanca Climent-Ezquerra∗ , Francisco Guill´ en-Gonz´ alez∗, M. Jesus Moreno-Iraberte Dpto. Ecuaciones Diferenciales y An´alisis Num´erico, Universidad de Sevilla, Aptdo. 1160, 41080 Sevilla, Spain. E-mails: bclimen[email protected], [email protected], [email protected] Abstract In this paper two main results are obtained for a nematic liquid crystal model with timedependent boundary Dirichlet data for the orientation of the crystal molecules. First, the initial-boundary problem is considered, obtaining the existence of global in time (up to infinity time) weak solution, the existence of global regular solution for viscosity coefficient big enough, and the weak/strong uniqueness. Second, using these previous results and the existence of time-periodic weak solutions proved in [2], the regularity of any time-periodic weak solution is deduced for viscosity coefficient big enough. Keywords: solution up to infinity time, time-periodic solutions, uniqueness, Navier-Stokes equations, Nematic liquid crystal models, coupled non-linear parabolic system. 1 Introduction In this work, a simplified Ericksen-Leslie version for a nematic liquid crystal model is considered; see for instance [8] for a formulation of a more complete liquid crystal problem. This model can be seen as a variant of the Navier-Stokes problem (respect to the velocitypressure unknowns (u, p)) coupled with a convection-diffusion system for a new variable d, which is a unit vectorial function modelling the orientation of the crystal molecules. On the other hand, it is usual to consider an approximation by Ginzburg-Landau penalization ([1]) for the constraint |d|= 1 (|d|=|d(t, x)|denotes the point-wise euclidean norm). This penalized model (where the constraint |d|= 1 is relaxed by |d| ≤ 1) was introduced by Lin in [6] and studied (from a mathematical point of view) by Lin and Liu in [7, 8]. Coutand and Shkoller in [4] also studied this simplified model but including stretching effects. We assume a (newtonian) fluid confined in an open bounded domain Ω ⊂IRN(N= 2 or 3) with regular boundary ∂Ω. In the penalized model the constraint |d|= 1 is partially conserved to ∗First and second authors have been partially financed by the projets P06-FQM-02373 and MTM2006–07932. 1 |d| ≤ 1 as consequence of the maximum principle for the Ginzburg-Landau equation considering the function f(d) = 1 ε2(|d|2−1)dwhere ε > 0 is the penalization parameter. There exists a potential function F(d) = 1 4ε2(|d|2−1)2such that f(d) = ∇dF(d) for each d∈IRN. Then, we consider the following PDE system in (0,+∞)×Ω:    ∂tu+ (u· ∇)u−ν∆u+∇p=−∇dt∆d,∇ · u= 0, ∂td+ (u· ∇)d= ∆d−f(d),|d| ≤ 1, (1) The constants ν,λand γare positive, representing respectively, the fluid viscosity, an elasticity constant and a relaxation time (for simplicity we consider λ=γ= 1 and ν > 0 and for the last result, large enough). The problem (1) is completed with the (Dirichlet) boundary conditions u(x, t)=0,d(x, t) = h(x, t) on ∂Ω×(0,+∞) (2) (assuming as in [2] a time-depending boundary data for dgiven by h:∂Ω×(0,+∞)7→ IRN; in [7, 8] only a time-independent boundary data is considered) and either the initial condition u(x, 0) = u0d(x, 0) = d0in Ω (3) or the time-periodic condition: u(x, 0) = u(x, T),d(x, 0) = d(x, T) in Ω,(4) where T > 0 is a given final time. In this last case, we assume, moreover, that h(0) = h(T). This model has, beside well known difficulties for the Navier-Stokes problem (a nonlinear parabolic system with the free divergence constraint related to the pressure), other different difficulties which come from the strongly nonlinear coupling between the orientation vector dand the velocity-pressure (u, p) and from the constraint |d| ≤ 1. An essential characteristic of the problem for d(given u), either the initial-value problem with (3) or the time-periodic case with (4), is the following weak maximum principle (see [7, 2]): Assume |h| ≤ 1 on ∂Ω×(0, T) and either |d0| ≤ 1 in Ω for the initial-value problem or h(0) = h(T) on ∂Ω for the time-periodic problem. Then, given u∈L2(0, T;V)∩L∞(0, T;H) (see the notations below for the definition of spaces Vand H), any point-wise solution for the d-problem verifies |d(x, t)| ≤ 1 a.e. in Ω ×(0, T). In [7], considering the initial-value problem (1)-(3) with time-independent boundary conditions for d, authors prove existence of global weak solution (with u∈L∞(L2)∩L2(H1), d∈L∞(H1)∩ L2(H2)), existence of global regular solution (with u∈L∞(H1)∩L2(H2), d∈L∞(H2)∩L2(H3)) if νis big enough for N= 3 and uniqueness of regular solutions. However, in these previous results of [7] there is an important simplification; the boundary data hdoes not depend on time. In this case, the time-periodic problem (1),(2),(4) with boundary condition independent of the time (d(x, t)|∂Ω×(0,T )=d0(x)), leads to a trivial problem (see [2]), because all “static” solutions u= 0 and dverifying stationary problem −∆d+f(d) = 0 in Ω, d|∂Ω=d0, are in particular time-periodic solutions. Respect to the nontrivial case of time-dependent boundary condition, the existence of weak time-periodic solutions of (1),(2),(4) is proved in [2]. 2 The main results of the present article are the following: always for boundary data hdepending on the time, we prove existence of global weak solution (defined in [0,+∞)) for the initial value problem (1)-(3), existence of global strong solutions under the constraint of viscosity coefficient νbig enough and uniqueness of strong/weak solutions, that is any weak solution coincides with the strong solution (if this strong solution exists). Moreover, we prove existence of regular timeperiodic solutions under the same type of constraint. A existence result of regular time-periodic solutions for a generalized Boussinesq model can be seen in [3]. The paper is organized as follows. In Section 2, some differential inequalities in weak norms are deduced, whereas Section 3 is devoted to obtain differential inequalities in strong norms. In Section 4, the global in time solution of the initial-value problem is studied (at infinity time), and finally the existence of strong time-periodic solution is obtained in Section 5, using results of Section 4 and the existence of weak time-periodic solution of [2]. Notations •We denote Q= (0,+∞)×Ω, QT= (0, T)×Ω, Σ = (0,+∞)×∂Ω and ΣT= (0, T)×∂Ω. •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(X) the Banach space Lp(0, T;X). Also, boldface letters will be used for vectorial spaces, for instance L2=L2(Ω)N. •The Lpnorm is denoted by |·|p, 1 ≤p≤ ∞, the Hmnorm by k·km(in particular |·|2=k·k0) and the product norm in Hn×Hmby k·km×n. 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} •In the sequel, C, D > 0 will denote different constants, depending only on the fixed data of the problem, as Ω, λ, γ. 2 Differential inequalities in weak norms 2.1 A lifting function We define e d(t) as the weak solution of the Laplace-Dirichlet problem    −∆e d= 0 in Ω, e d=h(t) on ∂Ω. (5) 3 In the time-periodic case, since by hypothesis h(0) = h(T) on ∂Ω, then e d(0) = e d(T) in Ω. Therefore, if we define b d(t) = d(t)−e d(t), then ∆b d= ∆din Ω×(0, T) and b d= 0 on ∂Ω×(0, T). In the time-periodic case, d(0) = d(T) if and only if b d(0) = b d(T). Then, we can rewrite the problem (1)-(2) in the variables (u,b d) (with d=b d+e d) as follows:        ∂tu+ (u· ∇)u−ν∆u+∇p+∇dt∆b d= 0,∇ · u= 0 in QT, ∂tb d+ (u· ∇)d−∆b d+f(d) = −∂te din QT, u= 0,b d= 0 on ∂Ω×(0, T ), (6) jointly with either the initial condition u(0) = u0,b d(0) = d0−e d(0) or the time-periodic conditions u(0) = u(T), b d(0) = b d(T). Remark: The choice of this type of lifting function allows us to obtain estimates up to infinity time, which is not possible with the lifting function that we will consider in Section 3. 2.2 Differential inequalities We will give two different differential equalities in the next two lemmas. Lemma 1 If uand dare regular enough, the following differential inequality holds: d dt |u|2 2+|∇b d|2 2+ 2ν|∇u|2 2+|∆b d|2 2≤2|f(d)|2 2+|∂te d|2 2,(7) Proof: Taking uand −∆b das test functions in (6), adding up, taking into account that ((u· ∇)u,u) = 0 and (∇dt∆b d,u)−((u· ∇)d,∆b d)=0,(8) one arrives (at least formally) at the following energy equality: 1 2 d dt |u|2 2+|∇b d|2 2+ν|∇u|2 2+|∆b d|2 2= (f(d),∆b d)+(∂te d,∆b d). Consequently, applying Young inequality, one has (7). Lemma 2 If uand dare regular enough, the following differential inequality holds: d dt |u|2 2+|∇b d|2 2+ 2 ZΩ F(d)+ 2ν|∇u|2 2+|∆b d−f(d)|2 2≤ |∂te d|2 2(9) Proof: Taking uand −∆b d+f(d) as test functions in (6), adding up and taking into account (8), ∂td·f(d) = ∂tF(d) and ((u· ∇)d,f(d)) = 0,one obtains 1 2 d dt |u|2 2+|∇b d|2 2+ 2 ZΩ F(d)+ν|∇u|2 2+|∆b d−f(d)|2 2= (∂te d,∆b d−f(d)). ****************** By rewriting the second term as (∂te d,∆b d)=(∂te d,∆b d−f(d)) + (∂te d,f(d)), by using |f(d)| ≤ 1 ε2owing to the maximum principle |d| ≤ 1 one has |(∂te d,∆b d)| ≤ 1 2|∂te d|2 2+1 2|∆b d−f(d)|2 2+1 ε2|∂te d|1. Therefore, one arrives (at least formally) at (9). 4 3 Differential inequalities in regular norms 3.1 A lifting function We will consider another suitable lifting function e dfor the boundary data h(that we denote equal) in such a way that we could made estimates of H3-type for the homogeneous variable related to d(see [5]). Concretely, we define e das the solution of the problem: (∂te d−∆e d= 0 in QT, e d=hon ∂Ω×(0, T),(10) jointly e d(0) = d0in Ω for the initial valued problem or e d(0) = e d(T) in Ω for the time-periodic case. Then, the d−problem of (1) can be rewritten as follows:    ∂tb d+ (u· ∇)d−∆b d+f(d) = 0 in QT, b d= 0 on ∂Ω×(0, T ) (11) and b d(0) = 0 in Ω for the initial-valued problem or b d(0) = b d(T) in Ω for the time-periodic case. As consequence of the maximum principle |d| ≤ 1 and |e d| ≤ 1. Although we do not know if |b d| ≤ 1, we have kb dkL∞(L∞)≤ kdkL∞(L∞)+ke dkL∞(L∞)≤2. We are going to consider the following equivalents norms: kuk1≈ |∇u|2,kb dk1≈ |∇b d|2in H1 0, kuk2≈ |∆u|2,kb dk2≈ |∆b d|2in H1 0∩H2 kb dk3≈ |∇(∆b d)|2+|∆b d|2=k∆b dk1in H1 0∩H3 Remark: Owing to the lifting function considered in this section, we have that ∆b d−f(d)|∂Ω= 0 because the rest of the terms in (11) vanish on the boundary. That is not true with the lifting function of the Section 2. 3.2 Differential inequalities In the previous conditions, the following regularity result will be frequently used. Lemma 3 Assume d=b d+e d, with |d| ≤ 1, a) if ∆b d−f(d)∈L2(Ω) and e d∈H2(Ω), then d∈H2(Ω) and kdk2≤ ke dk2+C(1 + |∆b d−f(d)|2), b) if ∆b d−f(d)∈H1(Ω) and e d∈H3(Ω), then d∈H3(Ω) and kdk3≤ ke dk3+C(|∇d|2+|∇(∆b d−f(d))|2). 5 Proof: For the proof, it is fundamental that ∆b d−f(d) = 0 on ∂Ω. We have kdk2≤ kb dk2+ke dk2 and kdk3≤ kb dk3+ke dk3. Then, by adding and subtracting f(d) into the norms kb dk2≈ |∆b d|2and kb dk3≈ k∆b dk1, we obtain the first and second inequality, respectively, using that |f(d)| ≤ Cand |∇f(d)| ≤ C|∇d|. Lemma 4 Assume e d∈L∞(0,+∞;H3(Ω)),u∈L∞(0,+∞;L2(Ω)) and d∈L∞(0,+∞;H1(Ω)). Then, if uand dare regular enough, the following differential inequality holds: d dt kuk2 1+|∆b d−f(d)|2 2+νkuk2 2+ 2|∇(∆b d−f(d))|2 2 ≤D(1 + |∆b d−f(d)|2 2) + E νkuk1kuk2 2+ (1 + |∆b d−f(d)|2 2)|∇(∆b d−f(d))|2 2, (12) where D, E > 0are constants independent of ν. Proof: Taking Auas test functions in the u-system of (1) (Abeing the Stokes operator) and applying adequately H¨older and Young’s inequalities, one obtains: d dtkuk2 1+4 3νkuk2 2≤C ν|(u· ∇)u|2 2+|∇td∆d|2 2 ≤C ν|u|2 3|∇u|2 6+|∇td|2 3|∆d|2 6≤C ν|u|2kuk1kuk2 2+kdk1kdk2kdk2 3 Hence, owing to Lemma 3, weak estimates (|u(t)|2≤C,kd(t)k1≤Ca.e. t∈(0,+∞)) and strong regularity of e d∈L∞(0,+∞;H3(Ω)) (in particular, inequalities of Lemma 3 derive in the simplest inequalities: kdk2≤C(1 + |∆b d−f(d)|2) and kdk3≤C(1 + |∇(∆b d−f(d))|2)), we have: d dtkuk2 1+4 3νkuk2 2≤C νkuk1kuk2 2+ (1 + |∆b d−f(d)|2)|∇(∆b d−f(d))|2 2 +C(1 + |∆b d−f(d)|2). (13) In the last term we have considered that C/ν is uniformly bounded respect to ν, as ν≥ν0. By taking gradient in the b d-system (11), multiplying by −∇(∆b d−f(d)), integrating by parts in the ∂tb d-term (where all the boundary terms vanish owing to the choice of the lifting function d that implies (∆b d−f(d))|∂Ω= 0) and adding both sides the term −(∂tf(d),∆b d−f(d)), we find: 1 2 d dt|∆b d−f(d)|2 2+|∇(∆b d−f(d))|2 2=−(∂tf(d),∆b d−f(d)) −((∇u· ∇)d,∇(∆b d−f(d))) + (u· ∇∇d,∇(∆b d−f(d))) (14) By using the b d-system we have that −∂tf(d) = −∇df(d)∂td=∇df(d)(u· ∇)d−∆b d+f(d)−∆e d hence, the first term on the right hand side of (14) can be written as (∇df(d)(u· ∇)d,∆b d−f(d)) −(∇df(d)(∆b d−f(d)),∆b d−f(d)) −(∇df(d)∆e d,∆b d−f(d)). Taking into account that k∇df(d)kL∞(L∞)≤C, weak estimates (|u|2≤C,kdk1≤C) and the strong regularity for e d∈L∞(0,+∞;H3(Ω)), we can bound these terms by: C(|u|∞|∇d|2|∆b d−f(d)|+|∆b d−f(d)|2 2+|∆e d|2|∆b d−f(d)|2) ≤C(kuk1/2 1kuk1/2 2kdk1|∆b d−f(d)|2+|∆b d−f(d)|2 2+ 1) ≤ν 18kuk2 2+C ν|∆b d−f(d)|2 2+C(|∆b d−f(d)|2 2+ 1). 6 The second term on the right hand side of (14) is estimated by | − ((∇u· ∇)d,∇(∆b d−f(d)))| ≤ C|∇u|6|∇d|3|∇(∆b d−f(d))|2 ≤ν 18kuk2 2+C νkdk1kdk2|∇(∆b d−f(d))|2 2 ≤ν 18kuk2 2+C ν(1 + |∆b d−f(d)|2)|∇(∆b d−f(d))|2 2. Analogously, the last term on the right hand side of (14) is bounded by |(u· ∇2d,∇(∆b d−f(d)))| ≤ C|u|∞|∇2d|2|∇(∆b d−f(d))|2 ≤ν 18kuk2 2+C νkdk2 2|∇(∆b d−f(d))|2 2 ≤ν 18kuk2 2+C ν(1 + |∆b d−f(d)|2 2)|∇(∆b d−f(d))|2 2. Consequently, applying previous estimates in (14) and considering that C/ν is uniformly bounded respect to νas ν≥ν0, we arrive at d dt|∆b d−f(d)|2 2+ 2|∇(∆b d−f(d))|2 2≤ν 3kuk2 2+C(1 + |∆b d−f(d)|2 2) +C ν(1 + |∆b d−f(d)|2 2)|∇(∆b d−f(d))|2 2. (15) From (13) and (15) we obtain (12). 4 Global solution of the initial-value problem Definition 5 We say that (u,d)is a weak solution of (1)-(3) if ∇ · u= 0 in Q, u|Σ= 0,d|Σ=h, k(u(t),d(t))k0×1≤C1∀t≥0i.e. (u,d)∈L∞(0,+∞;L2×H1),(16) ∀γ > 0, e−γt Zt 0 eγsk(u(s),d(s))k2 1×2ds ≤C2,∀t≥0,(17) verifying h∂tu,vi+ ((u· ∇)u,v)+(∇u,∇v)+(∇dt∆d,v)=0 ∀v∈V, ∂td+ (u· ∇)d+f(d)−∆d= 0,|d| ≤ 1a.e. in Q u(0) = u0,d(0) = d0in Ω. In the finite time case (T < ∞), (17) holds even when γ= 0, i.e. (u,d)∈L2(0, T;H1×H2). Remark: (16) and (17) imply that (∂tu, ∂td)∈L4/3 loc ([0,∞); V0×L2). Definition 6 We say that (u, p, d)a weak solution of (1)-(3) is also a strong solution if k(u(t),d(t))k1×2≤C3∀t≥0,(18) 7 ∀γ > 0, e−γt Zt 0 eγsk(u(s),d(s))k2 2×3ds ≤C4,∀t≥0,(19) verifying the following system a.e. in Q:    ∂tu+ (u· ∇)u−ν∆u+∇p=−∇dt∆d,∇ · u= 0, ∂td+ (u· ∇)d= ∆d−f(d),|d| ≤ 1. (20) In the finite time case (T < ∞), again γ= 0 can be taken in (19).tu Remark: (18) and (19) imply that for all γ > 0 and for all t≥0: e−γt Zt 0 eγs|∂tu(s)|2 2ds ≤C5,(21) |∂td(t)|2≤C6, e−γt Zt 0 eγsk∂td(s)k2 1ds ≤C7,(22) and e−γt Zt 0 eγs|∇p(s)|2 2ds ≤C8.(23) Theorem 7 (Existence and uniqueness of the initial-valued problem) (1) Let Ωbe a bounded domain in R3with boundary ∂Ωof class C1,1. Assume (u0,d0)∈ H×H1with |d0| ≤ 1in Ω,h∈L∞(0,+∞;H3/2(∂Ω)) with |h| ≤ 1on Σand ∂th∈ L∞(0,+∞;L2(∂Ω)), verifying the compatibility condition d0|∂Ω=h(0). Then there exists a weak solution (u,d)of (1)-(3) in [0,+∞)which verifies (16)-(17) with constants C1,C2 independent of νfor each ν≥1/2, and the follow energy inequality: |u(t)|2 2+|∇b d(t)|2 2+ 2 Zt 0ν|∇u|2 2+|∆b d|2 2 ≤ |u0|2 2+ 2 Zt 0ZΩf(d)·∆b d−(u· ∇)d·∂te d(24) where the lifting function e dis defined as in Section 3. (2) If moreover, ∂Ωis of class C2,1,(u0,d0)∈H1×H2with k(u0,d0)kH1×H2≤M0,h∈ L∞(0,+∞;H5/2(∂Ω)) and ∂th∈L∞(0,+∞;H1/2(∂Ω)), for each ν≥ν0, with ν0=ν0(M0,h, ∂th), there exists an unique strong solution of (1)-(3) in [0,+∞), which verifies (18) and (19) with constants C3,C4independent of ν. (3) If (u1,d1)is a weak solution of (1)-(3) which verifies the energy inequality (24) and (u2,d2) is a strong solution of (1)-(3), then both solutions coincide. Proof: (1) In the proof of this part a semi-Galerkin method will be used. Let {wi}n≥1 a “special” basis of Vformed by eigenfunctions of the Stokes problem (∇wi,∇v) = λi(wi,v)∀v∈V,wi∈V,with kwikL2= 1, λi%+∞. Let Vmbe the finite-dimensional subspace spanned by {w1,w2,...,wn}. 8 For each m≥1, we say that (um,dm) is an approximate solution, if um: [0,+∞)7→ Vmand dm: [0,+∞)7→ H2with b dm=dm−e dand e dthe lifting function given in Section 3 (in particular, from regularity hypothesis of h, one has that e d∈L∞(H2) and e dt∈L∞(L2)), and the following variational formulation holds:                  (∂tum(t),vm) + ((um(t)· ∇)um(t),vm) + ν(∇um(t),∇vm) +(∇dt m(t)∆dm(t),vm)=0 ∀vm∈Vm,a.e. in t, ∂tb dm(t)+(um(t)· ∇)dm(t)=∆b dm(t)−f(dm(t)),|dm| ≤ 1,a.e. in Q, um(0) = u0m=Pm(u0),dm(0) = d0in Ω. (25) Here, Pm:H7→ Vmdenotes the usual orthogonal projector from Honto Vm. In particular, u0m→u0in L2. The existence and uniqueness of local in time solution of (25) (in QT, for small enough T) is proved in the Appendix. Moreover, one has the estimates (independent of m): umbounded in L∞(0, T;H)∩L2(0, T;V) and dmbounded in L∞(0, T;H1)∩L2(0, T;H2). This suffices to control nonlinear terms and to pass to the limit in (25). Therefore, we get a weak solution of initial-valued problem (1)-(3) in [0, T]. Next, to extend the solution to whole [0,+∞) we will prove that the approximate solutions (um(t),dm(t)) are bounded in [0,+∞). By using the lifting (5) (Section 2), the approximate problem (25) can be rewritten as follows:                  (∂tum(t),vm) + ((um(t)· ∇)um(t),vm) + ν(∇um(t),∇vm) +(∇dt m(t)∆dm(t),vm)=0 ∀vm∈Vm,a.e. t, ∂tb dm(t)+(um(t)· ∇)dm(t)=∆dm(t)−f(dm(t)) −∂te d(t) in Q, um(0) = u0m=Pm(u0),dm(0) = d0in Ω. (26) Notice that b dmand e dare not the same functions in (25) and (26) respectively, since the lifting functions furnished in (5) or (10) are different, but the function dmdoes not change. From (7), one has in particular d dt |um|2 2+|∇b dm|2 2+C0(|um|2 2+|∇b dm|2 2)≤2|f(dm)|2 2+|∂te d|2 2≤C, (27) where C0= min{2ν P,1 P}and Pis a Poincar´e constant (for each ν≥1/2, C0= 1/P a constant independent of ν). In the last estimate we have used that |f(dm)|2 2is bounded in L∞(0,+∞) and |∂te d|2 2∈L∞(0,+∞). Multiplying by eC0t, d dt eC0t(|um|2 2+|∇b dm|2 2)≤CeC0t and integrating in [0, t] we have |um(t)|2 2+|∇b dm(t)|2 2≤e−C0t|u0m|2 2+|∇b d0|2 2+C(1 −eC0t)≤ |u0|2 2+|∇b d0|2 2+C(28) for all t≥0, with C > 0 a constant independent of ν, hence (16) holds with a constant C1 independent of νfor all ν≥1/2. 9 References [1] F. Bethuel, H. Brezis, F. H´elein, Asymptotics for the minimization of a Ginzburg-Landau functional, Calc. Var. 1 (1993), 123–148. [2] B. Climent-Ezquerra, F. Guill´en-Gonz´alez, M.A. Rojas-Medar Reproductivity for a nematic liquid crystal model, Z. Angew. Math. Phys., 576 (2006) no. 6, 984-998. [3] B. Climent-Ezquerra, F. 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