Stability for nematic liquid crystals with stretching terms

STABILITY FOR NEMATIC LIQUID CRYSTALS WITH STRETCHING TERMS B. CLIMENT-EZQUERRA, F. GUILL´ EN-GONZ´ ALEZ, M. A. RODR´ IGUEZ-BELLIDO Department of Ecuaciones Diferenciales y An´alisis Num´erico, Universidad de Sevilla, Aptdo. 1160, 41080 Sevilla, SPAIN. E-mail: b[email protected], guil[email protected], [email protected] February 27, 2009 Dedicated to the memory of Valery S. Melnik We study a nematic crystal model appearing in [Liu et al.,2007] modeling stretching effects depending on the different shape of microscopic molecules of the material, under periodic boundary conditions. The aim of the present article is twofold: to extend the results given in [Sun & Liu, 2009], to a model with more complete stretching terms and to obtain some stability and asymptotic stability properties for this model. Keywords: Nematic Liquid Crystal system, asymptotic stability, stability, stretching effects, existence, regularity. Mathematics Subject Classifications (2000): 35B65, 35Q35, 76D03 1. Introduction The Nematic Liquid Crystal system is a Navier- Stokes type model for incompressible fluids respect to the macroscopic variables, that takes into account the crystallinity of the microscopic molecules of the material. It can be obtained coupling Navier- Stokes equations with the Ginzburg-Landau equations, being its unknowns the solenoidal velocity u(t, x), the pressure of the fluid p(t, x), and the director field d(t, x), that represents the orientation of the liquid crystal molecules. Moreover, we suppose that the fluid is confined in a domain Ω ⊂R3. We deal with an Ericksen-Leslie type formulation. A simplified model was analyzed by F. H. Lin & C. Liu in [Lin & Liu, 1995]. In fact, this model is a penalized one depending on the Ginzburg-Landau function: f(d) = 1 2|d|2−1d, where |d|denotes the euclidean norm in R3and  > 0 is a penalization parameter. This penalization function has a potential structure, i. e. there exists the function F(d) = 1 42|d|2−12such that f(d) = ∇d(F(d)) for all d∈R3. We denote Q= (0,+∞)×Ω and Σ = (0,+∞)× ∂Ω, where Ω ⊂R3is a smooth enough domain and ∂Ω its boundary. We consider the EDP system appearing in [Liu et al.,2007; system (1.9), p. 1187], that reads as: (LC)     Dtu−ν∆u+∇p−λ∇ · σe=0in Q, ∇ · u= 0 in Q, Dtd+γw=0in Q, 1 2B. Climent-Ezquerra et al. being Dtu=∂tu+ (u· ∇)u the material derivative of u, σe=−(∇d)t∇d−βw dt−(1 + β)d wt(1) the elastic stress tensor (β∈R) and w=−∆d+f(d) the Euler-Lagrange system derived from the minimization problem respect to the elastic energy Ee(d) = 1 2ZΩ |∇d|2+ZΩ F(d).(2) The term Dtd=Dtd+C(d,∇u) describes a general derivative containing the material derivative Dtd=∂td+ (u· ∇)dand the quadratic term C(d,∇u) = β(∇u)d+ (1 + β)(∇u)td modeling the so-called stretching effects, depending on the form of the molecules [Liu et al., 2007]. In fact, the constant β=−αis associated with the aspect ratio rof the ellipsoid particles. The case of αnear to 1 corresponds to rod like particles (then the transport is purely covariant stretching), the case of αnear to 0 corresponds to disc like particles (then the transport is anti-stretching) and the case of αnear to 1/2 corresponds to the spherical shape (the transport is the rigid rotation of the center of the mass). Finally, ν > 0 is the fluid viscosity, λ > 0 is the elasticity constant and γ > 0 is a relaxation in time constant. The theoretical analysis of a simplified model without stretching effects, i.e for C(d,∇u) = 0 and the corresponding elastic tensor σe=−(∇d)t∇d, was made in [Lin & Liu, 1995] obtaining existence of global weak solution, i. e. u∈L∞(0, T;L2(Ω)) ∩L2(0, T ;H1(Ω)), d∈L∞(0, T;H2(Ω)) ∩L2(0, T ;H3(Ω)), for all T > 0, and the existence (and uniqueness) of local strong solution, i. e. u∈L∞(0, T∗;H1(Ω)) ∩L2(0, T∗;H2(Ω)), d∈L∞(0, T∗;H2(Ω)) ∩L2(0, T∗;H3(Ω)), with T∗≤T(small enough) or T∗=T(for each T > 0) for big enough viscosity coefficient νor for two-dimensional domains. All these previous results are given for the time-independent Dirichlet boundary data: u=0,d=hon Σ, (h6=h(t)) and for the initial-value boundary problem with initial condition: u|t=0 =u0,d|t=0 =d0in Ω. (3) When time-dependent Dirichlet data for dis considered (h=h(t)), the existence of weak timeperiodic solution, that is solutions obtained by changing (3) by u(0) = u(T) and d(0) = d(T), is obtained in [Climent-Ezquerra et al.,]. The strong regularity up to infinite time for big enough viscosity νjointly with the strong regularity of time-periodic solutions are obtained in [Climent-Ezquerra et al.,]. The results corresponding to the initialvalue boundary problem are extended in [Lin & Liu, 2000] to a much more complete model respect to the dissipative tensor and considering the particular stretching effects for the case of spherical molecules, i.e. taking β=−1/2 in (1). Recently, a liquid crystal model with a stretching term for the case of rod like particles (taking β=−1 in (1)) and periodic boundary conditions for both uand dhas been studied in [Sun & Liu, 2009], obtaining global weak solution and local strong solution (which is global for large enough viscosity) The aim of the present article is twofold: to extend the last results of [Sun & Liu, 2009] to a model with more complete stretching terms and to obtain some stability and asymptotic stability properties for this model. 2. General Framework. Assume that we have the following situation, a.e. t∈(t0,+∞): E(t), F(t)≥0, E0(t) + F(t)≤0.(4) Then, E∈Cb[t0,+∞), is a decreasing function and there exists lim t→+∞E(t) = E∞≥0. Stability for Nematic Liquid Crystals with Stretching Terms 3 On the other hand, F∈L1(t0,+∞), that is, Z+∞ t0 F(t)dt < +∞. In this case, for any δ > 0, there exists a large enough time t∗ 1=t∗ 1(δ)≥t0such that: Z+∞ t∗ 1 F(t)dt ≤δ. (5) In particular, we can say that for each δ > 0 there exists a large enough time t∗ 1(δ)≥t0such that 1 τZt+τ t F(t)dt ≤δ τ,∀τ > 0,∀t≥t∗ 1(δ).(6) Lemma 2.1. Let F∈L1(t0,+∞),F≥0in (t0,+∞), satisfying (6). Then, ∀δ > 0,∀t≥t∗ 1(δ) and ∀τ > 0there exists a time ¯ t∈[t, t +τ]such that: F(¯ t)≤2δ τ.(7) Indeed, the set of points ¯ t∈[t, t +τ]satisfying (7) has measure ≥τ/2. Proof. We focus on the proof in the interval [t∗ 1, t∗ 1+ τ]. The proof for another interval of length τcontained in [t∗ 1,+∞) is similar. Indeed, we define: A={s∈[t∗ 1, t∗ 1+τ]/ F(s)≥2δ τ}. Therefore, ZA F(t)dt +ZAc F(t)dt ≤δ, and thus 2δ τ|A| ≤ δ⇒ |A| ≤ τ 2. That is, |Ac| ≥ τ/2. Now, we assume that the following differential inequality for F(t) holds: F0(t)≤C2(F(t)3+ 1).(8) Lemma 2.2. Let F∈L1(t0,+∞)be a function satisfying the differential inequality (8). For any ε < 1, if F(t0)≤ε/3, then F(t)≤ε∀t∈ [t0, t0+T∗(ε)], where T∗(ε) = ε 3C2 . Proof. We argue by contradiction: Suppose that there exists a time t1∈[t0, t0+T∗(ε)] such that F(t)< ε in [t0, t1) and F(t1) = ε. Then, from eq. (8) we obtain that F0<2C2en [t0, t1]. Integrating in [t0, t1], we get: F(t1)< F(t0)+2C2(t1−t0) ≤F(t0)+2C2T∗(ε) ≤ε 3+ 2C2 ε 3C2 =ε. This fact contradicts the starting hypothesis. 2.1. Asymptotic stability. Theorem 2.3. Let ε < 1, and F∈L1(0,+∞), F≥0, such that both (8) and inequality (6) hold for δ=ε2 36C2 ,t∗ 1=t∗ 1(δ)and τ=T∗(ε) 2. Then, F(t)≤ε, ∀t≥t∗ 2=t∗ 1+T∗(ε) 2=t∗ 1(δ)+ ε 6C2 .(9) Remark 2.4. In particular, F∈W1,1(t∗ 2,+∞),→ C[t∗ 2,+∞). Proof. We argue by contradiction: Assume that there exists a time et > t∗ 2such that F(t)> ε. We consider the interval [et−T∗(ε)/2,et]⊂[t∗ 1,+∞). From Lemma 2.1 we conclude that for each interval of length T∗(ε)/2 contained in [t∗ 1,+∞) and ∀t≥t∗ 1there exists a time ¯ t1∈[et−T∗(ε)/2,et] such that: F(¯ t1)≤2δ τ=ε2/(18C2) ε/(6C2)=ε 3. Thus, applying Lemma 2.2, one verifies: F(t)≤ε, ∀t∈[¯ t1,¯ t1+T∗(ε)]. Observe that et∈[¯ t1,¯ t1+T∗(ε)], which gives us to contradiction. Corollary 2.5. Let F∈L1(t0,+∞)be a function satisfying eq. (8). Then, F(t)is a function asymptotically stable to 0, that is, lim t→+∞F(t)=0. 4B. Climent-Ezquerra et al. 2.2. Stability until infinite time. If we assume that: (H1) E(t0)≤δ(ε) = ε2 36C2 , then, from (4) we get: ∀t1> t0 Zt1 t0 F(t)dt ≤E(t0)−E(t1)≤δ(ε) In fact, one has (5) for t∗ 1(ε) = t0. Then, applying Theorem 2.3, we obtain: F(t)≤ε∀t≥t0+ε 6C2=t0+T∗(ε) 2(10) If, moreover, (H2) F(t0)≤ε 3, then applying Lemma 2.2, we get: F(t)≤ε∀t∈[t0, t0+T∗(ε)] (11) In summary, assuming (H1) and (H2), one has: E(t)≤δ(ε), F(t)≤ε, ∀t≥t0. 3. The Nematic Liquid Crystal Model 3.1. Weak estimates If we consider both u(t) and w(t) as test functions in the u-system and d-system of (LC) respectively, taking into account the equality: ∇ · ((∇d)t∇d) = −(∇d)tw+∇Ee(d), we obtain: 1 2 d dtku(t)k2 L2(Ω) +νk∇u(t)k2 L2(Ω) +−λZΩ [(u· ∇)d]·wdx −λZΩβwtd+ (1 + β)dtw:∇udx= 0 (12) and d dt 1 2k∇d(t)k2 L2(Ω) +F(d)(t)+γkwk2 L2(Ω) +ZΩ [(u· ∇)d]·wdx+ZΩ C(d,∇u)·wdx= 0 (13) for any boundary conditions for (u,d) given in the Introduction (that is, Dirichlet, Neumann or periodic for d). Then, adding (12) to (13) multiplied by λ, the last two terms of (12) and (13) cancel and the so-called energy equality holds: d dt 1 2ku(t)k2 L2(Ω) +λEe(d(t)) +νk∇u(t)k2 L2(Ω) +λγkw(t)k2 L2(Ω) = 0 (14) (recall that Ee(d) is given in (2)). Note that, defining the time functions E(t) and F(t) as: E(t) = 1 2ku(t)k2 L2(Ω) +λEe(d(t)),(15) F(t) = νk∇u(t)k2 L2(Ω) +γλkw(t)k2 L2(Ω), (16) the inequality (4) is deduced from (14). As a consequence, we can deduce the existence of weak solutions of the problem. Moreover, by using this weak regularity and the H2(Ω) and H3(Ω) regularity of the elliptic problem −∆d+f(d) = w with appropriate boundary conditions, we can deduce [Climent-Ezquerra et al.,]: kdkH2(Ω) ≤CkwkL2(Ω) + 1, kdkH3(Ω) ≤CkwkH1(Ω) + 1. (17) In the next section, we will use repeatedly these estimates. 3.2. Strong estimates In this section, we only consider the periodic boundary conditions case for all variables (u, p, d). Taking both −∆uand −λ∆was test functions in the u-system and in the d-system of (LC) respectively, one can obtain ([Sun & Liu, 2009]): d dtk∇uk2 L2(Ω) +νk∆uk2 L2(Ω) ≤ k(u· ∇)uk2 L2(Ω) +Cλ2 νk(∇d)twk2 L2(Ω) +λZΩβwdt+ (1 + β)dwt∇(∆u)dx (18) Stability for Nematic Liquid Crystals with Stretching Terms 5 and λZΩ ∇∂td:∇wdx+λγk∇wk2 L2(Ω) +λZΩ ∇((u· ∇)d) : ∇wdx +λZΩ ∇C(d,∇u) : ∇wdx= 0 (19) Observe that the first term of (19) can be rewritten as: ZΩ ∇∂td:∇wdx=−ZΩ ∆(∂td)wdx =ZΩ ∂tw w dx−ZΩ f0 (d)(∂td)wdx =1 2 d dtkwk2 L2(Ω) −ZΩ f0 (d)(∂td)wdx (20) Now, using the d-system of (LC), that is, ∂td= −(u· ∇)d−C(d,∇u)−γw, one has: −ZΩ f0 (d)(∂td)wdx =ZΩ f0 (d) ((u· ∇)d+C(d,∇u) + γw)wdx ≤εk∆uk2 L2(Ω) +k∇wk2 L2(Ω) +CεF3(t)+1(21) for εsmall enough. The more nonlinear terms of the last term of (19) are manipulated as follows (here, the periodic boundary conditions are again applied): λZΩβ∇(∇u)d+ (1 + β)∇(∇u)td:∇wdx ≤ −λZΩ ∇(∆u) : βwdt+ (1 + β)dwtdx +CkD2ukL2(Ω)k∇dkL6(Ω)kwkL3(Ω) (22) Note that the first term on the right-hand side of (22) cancels with the last term in (18). The remaining part of the last term of (19) can be written as: λZΩ (β(∇d· ∇)u:∇w +(1 + β)(∇w· ∇)u:∇d)dx ≤C(λ, β)k∇wkL2(Ω)k∇dkL6(Ω)k∇ukL3(Ω) (23) being C(λ, β) a constant depending on λand β. Therefore, adding (18) to (19) and taking into account estimates (20)-(23), we obtain: d dt k∇uk2 L2(Ω) +λkwk2 L2(Ω) +νk∆uk2 L2(Ω) +λγk∇wk2 L2(Ω) ≤ k(u· ∇)uk2 L2(Ω) +Cλ2 νk(∇d)twk2 L2(Ω) +λZΩ ∇((u· ∇)d) : ∇wdx +CF3(t)+1 +CkD2ukL2(Ω)k∇dkL6(Ω)kwkL3(Ω) +C(λ, β)k∇wkL2(Ω)k∇dkL6(Ω)k∇ukL3(Ω) =P6 i=1 Ii (24) Using Sobolev’s inequalities, the estimates for the Ii-terms can be summarized as follows: I1≤ kuk2 L6(Ω)k∇uk2 L3(Ω) ≤εk∆uk2 L2(Ω) +Cεk∇uk6 L2(Ω) I2≤ k∇dk2 L6(Ω)kwk2 L3(Ω) ≤εk∇wk2 L2(Ω) +Cεkwk6 L2(Ω) I3≤ k∇ukL3(Ω)k∇dkL6(Ω)k∇wkL2(Ω) +kukL6(Ω)k∆dkL3(Ω)k∇wkL2(Ω) =I31 +I32 where term I31 as the same type of estimates as term I6. I32 ≤ kukL6(Ω)k∆dkL3(Ω)k∇wkL2(Ω) ≤εk∇wkL2(Ω) +Cεk∇uk4 L2(Ω)kwk2 L2(Ω) I5≤εk∆uk2 L2(Ω) +k∇wk2 L2(Ω)+Cεkwk6 L2(Ω) I6≤εk∆uk2 L2(Ω) +k∇wk2 L2(Ω) +Cεkwk4 L2(Ω)k∇uk2 L2(Ω) Now, we introduce function G(t) defined as: G(t) = νk∆uk2 L2(Ω) +λγk∇wk2 L2(Ω) (25) Observe that, finally, (24) can be written in the following form: F0(t) + G(t)≤C1 + F3(t), 6B. Climent-Ezquerra et al. being F(t) and G(t) the functions defined in (16) and (25) respectively. Therefore, function F(t) satisfies (8). 4. Applications of the General Framework to Nematic Liquid Crystal. 4.1. Asymptotic stability. Let (u0,d0)∈H1(Ω) ×H2(Ω) be two given functions and (u(t),d(t)) a weak solution of system (LC) with periodic boundary conditions and initial data (u0,d0). Since (4) and (8) hold, applying the results from Section 2, one has: (E(t)↓E∞(≥0) in R, t ↑+∞ F(t)→0 in R, t ↑+∞ hence:          E0(t)→0 in Rt↑+∞ u(t)→0 in H1 0(Ω) t↑+∞ w(t)→0 in L2(Ω) t↑+∞ Moreover, for each subsequence tj↑+∞, there exists a subsequence (tjk)⊂(tj) such that: d(tjk)*¯ din H2(Ω)-weak for k↑+∞. being ¯ da critical point of the elastic energy Ee(d), that is, a solution for the stationary problem: −∆¯ d+fε(¯ d) = 0in Ω, with periodic boundary conditions on ∂Ω. Note that, E∞=λ 2|∇¯ d(t)|2 2+ 2 ZΩ F(¯ d(t))=λEe(¯ d) that is, every possible limit of the director field ¯ d when t↑+∞is a critical point of the elastic energy and all these possible limits have the same elastic energy E∞. 4.2. Stability for constant director fields. If (u0,d0) are such that: (H1) 1 2ku0k2 L2(Ω) +λ 2k∇d0k2 L2(Ω) +λZΩ F(d0)≤δ(ε) (in particular, kdcte −d0k2 H1(Ω) ≤C ε2for a constant vector ¯ dcte with |¯ dcte|= 1) and (H2) νk∇u0k2 L2(Ω) +γλkw0k2 L2(Ω) ≤ε 3, where w0=−∆d0+f(d0) then for each t≥t0, applying the results from Section 2 one has: 1 2ku(t)k2 L2(Ω)+λ 2k∇d(t)k2 L2(Ω)+λZΩ F(d(t)) ≤δ(ε) and νk∇u(t)k2 L2(Ω) +γλkw(t)k2 L2(Ω) ≤ε 3. Acknowledgements The authors have been partially supported by Projects MTM2006-07932 and P06-FQM-02373. References Climent-Ezquerra, B., Guill´en-Gonz´alez, F. & Rojas-Medar, M.A. [2006] “Reproductivity for a nematic liquid crystal model,” Z. Angew. Math. Phys. 576, No. 6, 984-998. Climent-Ezquerra, B., Guill´en-Gonz´alez, F. & Moreno-Iraberte, M. J. [] “Regularity and Time-periodicity for a Nematic Liquid Crystal model,” Accepted in Nonlinear Analysis , http://dx.doi.org/10.1016/j.na.2008.10.092. Lin, F. H.& Liu, C. [1995] “Nonparabolic Dissipative Systems Modeling the Flow of Liquid Crystals,” Comm. Pure Appl. Math. 48, 501–537. Lin, F. H. & Liu, C. [2000] “Existence of Solutions for the Ericksen-Leslie System,” Arch. Rational Mech. Anal. 154, 135–156. Liu, C., Shen, J. & Yang, X. [2007] “Dynamics of Defect Motion in Nematic Liquid Crystal Flow: Modeling and Numerical Simulation,” Commun. Comput. Phys. 2-No. 6, 1184–1198. Sun, H. & Liu, C. [2009] “On Energetic Variational Approaches in Modeling the Nematic Liquid Crystal Flows,” Discrete and Continuous Dynamical Systems 23-No.1&2, 455-475.