Global in time solution and time-periodicity for a smectic-A liquid crystal model

Global in time solution and time-periodicity for a smectic-A liquid crystal 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] November 5, 2012 Abstract In this paper some results are obtained for a smectic-A liquid crystal model with timedependent boundary Dirichlet data for the so-called layer variable ϕ(the level sets of ϕ describe the layer structure of the smectic-A liquid crystal). First, the initial-boundary problem for arbitrary initial data is considered, obtaining the existence of weak solutions which are bounded up to infinity time. Second, the existence of time-periodic weak solutions is proved. Afterwards, the problem of the global in time regularity is attacked, obtaining the existence and uniqueness of regular solutions (up to infinity time) for both problems, i.e. the initial-valued problem and the time-periodic one, but assuming a dominant viscosity coefficient in the linear part of the diffusion tensor. Keywords: solution bounded up to infinity time, time-periodic solutions, global in time regular solutions, Navier-Stokes equations, Smectic-A liquid crystal, coupled non-linear parabolic system. 1 Introduction In this work, we study the time evolution of a smectic-A liquid crystal model proposed in [E’97]. Smectic crystals are in a liquid-crystalline phase, where the molecules of the liquid crystal not only have a certain orientational order (as in the nematic case) but also have a ∗This work has been partially financed by DGI-MEC (Spain), Grant MTM2006–07932 and by the project P06-FQM-02373 (Junta de Andaluc´ıa) 1 certain positional order (layer structure); the molecules are arranged in almost incompressible layers of almost constant width. Within each layer, the smectic-A system consists of a single optical axis nperpendicular to the layer such that ∇ × n= 0. In this case, n=∇ϕfor a potential function ϕ, and the level sets of ϕwill represent the layer structure in the sample. This study is motivated by the following problem in liquid crystals. The usual nematic molecule configuration is determined by minimizing the Oseen Frank energy, which in the more simple case of equal constants derives to Dirichlet energy ZΩ |∇d|2. Here, the unit vector dstands for the orientation of liquid crystal molecules. Now, in the smectic-A case, this orientation dcoincides with the normal vector nof each layer. Then, in order to study the energy ZΩ |∇(∇ϕ)|2under the constraint |∇ϕ|= 1, it is natural to introduce the penalized energy ZΩ 1 2|∆ϕ|2+f(∇ϕ) where fis the Ginzburg-Landau penalization function f(n) = 1 ε2(|n|2−1)n, which has the potential function F(n) = 1 4ε2(|n|2−1)2 verifying f(n) = ∇nF(n) for each n∈IRN. As ε→0, one can hope that the minimizer of the penalized energy, or the solution of the corresponding Euler-Lagrange equation ∆2ϕ−∇·f(∇ϕ) = 0, will be convergent to the minimizer of the energy RΩ 1 2|∆ϕ|2with the non-convex constraint |∇ϕ|= 1 (cf. [Kinderlehrer,Liu’96] and [E’97]). Thus, it is important to study the asymptotic behavior as ε→0 (cf. [Guill´en,Rojas’02] for nematic crystal models). However, very little is known about this. We assume the smectic-A liquid crystal confined in an open bounded domain Ω ⊂IRN (N= 2 or 3) with regular boundary ∂Ω. We consider the following PDE system in Ω × (0,+∞):    ρ(∂tu+ (u· ∇)u)−∇·(σd+λ σe) + ∇p= 0,∇ · u= 0, ∂tϕ+u· ∇ϕ+γ(∆2ϕ−∇·f(∇ϕ)) = 0, (1) where u: Ω ×[0,+∞)7→ RNis the flow velocity, p: Ω ×[0,+∞)7→ Rdescribes the fluid pressure and ϕ: Ω×[0,+∞)7→ Ris the layer variable. The constants ρ,λ, and γare positive, representing respectively, the density of the fluid, the ratio between the kinetic energy and 2 the elastic one, and the elastic relaxation time. Moreover, we consider the same constitutive laws for the dissipative stress tensor σdand the elastic stress tensor σeas in [Liu’00]: 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 where µ1≥0, µ4>0, µ5≥0 are dissipative constant coefficients, n=∇ϕand D(u) denotes the symmetric tensor of the velocity gradient: D(u) = 1 2(∇u+∇tu). The problem (1) is completed with the (Dirichlet) boundary conditions u|∂Ω= 0, ϕ|∂Ω=ϕ1, ∂nϕ|∂Ω=ϕ2(2) (assuming time-depending boundary data ϕ1, ϕ2:∂Ω×(0,+∞)7→ IRN) and one of the following conditions: •either the initial conditions u(0) = u0ϕ(0) = ϕ0in Ω (3) •or the time-periodic conditions: u(0) = u(T), ϕ(0) = ϕ(T) in Ω,(4) where T > 0 is a given final time. In the first case, the compatibility condition ϕ0|∂Ω=ϕ1(0) must be assumed. In this last case, one assumes ϕ1(0) = ϕ1(T) and ϕ2(0) = ϕ2(T). 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 −∇ · σd=−µ4∇ · D(u)−∇·σd nl =−µ4 2∆u−∇·σd nl since ∇ · u= 0. By decomposing the term due to the penalization to the other terms in the elastic tensor as follows σe=−f(n)⊗n+σe np(n), where σe np := ∇(∇ · 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) 3 and (∇ · σe np)j= (∇ · (∇(∇ · n)⊗n−(∇ · n)∇n))j= (∇ · (∇(∆ϕ)⊗ ∇ϕ−∆ϕ∇2ϕ))j =∂i(∂i(∆ϕ)∂jϕ−∆ϕ∂ijϕ)=∆2ϕ∂jϕ+∂i(∆ϕ)∂i∂jϕ−∂i(∆ϕ)∂ijϕ−∆ϕ∂i∂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. Then, joining together all the gradient terms, the momentum system of (1) can be written as: ρ(∂tu+ (u· ∇)u)−µ4 2∆u−∇·σd nl −λ(∆2ϕ−∇·f(∇ϕ))∇ϕ+∇q= 0 (5) where qis the potential function q=p+λF(∇ϕ) + λ|∆ϕ|2 2. One observes that the liquid crystal model (1)-(2) lacks of maximum or comparison principles (for ∇ϕ) so one looses one of the strongest tools in analyzing nonlinear pdes. Assuming time-independent boundary data ϕ1, ϕ2, an important fact of the model (1)-(2) is its dissipative character, because this system admits (at least formally) the following energy equality: d dt ZΩ1 2|u|2+λ1 2|∆ϕ|2+F(∇ϕ) +ZΩµ4 2|∇u|2+σd nl :D(u) + λγ|∆2ϕ−∇·f(∇ϕ)|2= 0. (6) This equality is obtained multiplying the ϕ-equation by −λ(∆2ϕ− ∇ · f(∇ϕ)), the u-system (5) by uand integrating by parts, because all the nonlinear convective and elastic terms cancel and the boundary terms vanish by using that u|∂Ω= 0, ∂tϕ|∂Ω= 0 and ∂t∂nϕ|∂Ω= 0 (see (20) below for an energy equality related to a system with time-dependent boundary data). In particular, since RΩσd nl :D(u)≥0 (see (22) below), this equality implies that the total energy (that is, the kinetic energy 1 2RΩ|u|2plus the elastic energy λRΩ 1 2|∆ϕ|2+F(∇ϕ)) decreases respect to the time. Now, since time-dependent boundary data ϕ1, ϕ2will be considered, (6) must be modified with a right hand side depending on time derivatives of ϕ1, ϕ2which act as force terms, see (20). If we consider again time-independent boundary data, it is important to remark that the following (static) critical points are particular solutions of the time-periodic problem: u= 0, ϕ: any solution of the problem: ∆2ϕ−∇·f(∇ϕ) = 0 in Ω, ϕ=ϕ1,∂nϕ=ϕ2on ∂Ω, p=−λF(∇ϕ)−λ|∆ϕ|2 2. 4 Therefore, in order to consider a nontrivial time-periodic problem, it will be essential to assume time-dependant boundary data for ϕ. Definition 1 We say that (u, ϕ)is a weak solution of (1)-(3) in [0, T),0< T < +∞if ∇ · u= 0 in Q, u|Σ= 0, ϕ|Σ=ϕ1, ∂nϕ|Σ=ϕ2 u∈L∞(0, T;L2(Ω)) ∩L2(0, T;H1(Ω)), ϕ ∈L∞(0, T;H2(Ω)) ∩L2(0, T;H4(Ω)),(7) verifying h∂tu,vi+ ((u· ∇)u,v) + ((µ4/2)∇u+σd nl,∇v)−λ((∆2ϕ−∇·f(∇ϕ))∇ϕ, v) = 0 ∀v∈V, ∂tϕ+ (u· ∇)ϕ+γ(∆2ϕ−∇·f(∇ϕ)) = 0,a.e. in Q u(0) = u0, ϕ(0) = ϕ0in Ω. Definition 2 We say that a weak solution (u, ϕ)is a strong solution of (1)-(3) in [0, T)if u∈L∞(0, T;H1(Ω)) ∩L2(0, T;H2(Ω)), ϕ ∈L∞(0, T;H4(Ω)) ∩L2(0, T;H6(Ω)),(8) verifying point-wise the fully differential system (1). In [Liu’00], considering a problem like (1)-(3), with variable density and time-independent boundary conditions for ϕ, author proves the existence of weak solutions using a semi-Galerkin procedure keeping the transport equation for density and the ϕ-equation at infinity dimension. Moreover, the global regularity of weak solutions (for big enough µ4if N= 3) is deduced in [Liu’00], and a preliminary analysis about the asymptotic behavior in time is made (see also [Lai,Liu’06] for other asymptotic behavior study for a related model). The main results of present paper are the following, always for boundary data ϕ1and ϕ2 depending on the time: 1. the uniqueness of weak/strong solutions of the initial-value problem (1)-(3), 2. the existence of global weak solutions of problem (1)-(3), which is bounded up to infinity time (with an exponential weighted norm for the L2(0,+∞)-norm, see (31)), 3. the existence of weak time-periodic solutions, 4. the existence of regular solutions for both previous cases, the initial-valued problem and the time-periodic one, but assuming a dominant viscosity coefficient µ4in the linear part of the diffusion tensor. 5 The results obtained in this paper are in a certain sense similar to the results presented in [Climent et al.’06] and [Climent et al.] for the weak solutions and the regular solutions of a nematic liquid crystal model, respectively. The paper is organized as follows. In Section 2, some differential inequalities are deduced, which will be used in the rest of the paper. In Section 3, the uniqueness of weak/strong solutions of the initial-value problem (1)-(3) is analyzed. In Section 4, the global in time solution of the initial valued problem is studied at infinity time and the existence of weak time-periodic solutions is obtained in Section 5. Finally, under the constraints of viscosity coefficient µ4big enough, the existence and uniqueness of global regular solutions of the initial valued problem is proved in Section 6 and the existence of regular time-periodic solutions is deduced in Section 7. For simplicity we fix the constants excepting the viscosity µ4, taking ρ=λ=γ=µ1=µ5= 1, ν =µ4/2. Notation •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·kn×m. 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, C1, C2>0 will denote different constants, depending only on the fixed data of the problem, as Ω, ϕ1, ϕ2,ε(and u0, ϕ0for the initial-value problem). 6 2 Preliminaries 2.1 A lifting function We define eϕ=eϕ(t) as the weak solution of the problem    −∆2eϕ= 0 in Ω, eϕ=ϕ1(t)∂neϕ=ϕ2(t) on ∂Ω. (9) 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. Therefore, 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 (1)-(2) 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+∇q= 0 in QT, ∇ · u= 0 in QT, ∂tbϕ+u· ∇ϕ+ ∆2bϕ−∇·f(∇ϕ) = ∂teϕin QT, u= 0,bϕ= 0, ∂nbϕ= 0 on ΣT (10) jointly with either the initial conditions u(0) = u0,bϕ(0) = ϕ0−eϕ(0) or the time-periodic conditions u(0) = u(T), bϕ(0) = bϕ(T). Since u∈H1 0and bϕ∈H2 0, the following norms are equivalents: kuk1≈ |∇u|2,kbϕk2≈ |∆bϕ|2kbϕk4≈ |∆2bϕ|2. 2.2 Some inequalities We will give two inequalities in the next two lemmas, relating the elliptic operator ∆2bϕ− ∇ · f(∇ϕ) and the penalized energy ZΩ F(∇ϕ) with some norms. Lemma 3 The following inequality holds: |∆bϕ|2 2+1 2ε2|∇bϕ|4 4≤1 2|∆2bϕ−∇·f(∇ϕ)|2 2+C1,(11) where C1>0is a constant depending on ε,|Ω|, and k∇eϕkL∞(L4). Proof. We denote ω= ∆2bϕ−∇·f(∇ϕ). Testing this equality by bϕone has: (ω, bϕ) = (∆2bϕ, bϕ)−(∇ · f(∇ϕ),bϕ) = |∆bϕ|2 2+ (f(∇ϕ),∇bϕ).(12) 7 The last term on the right hand side of (12) can be written as (f(∇ϕ),∇bϕ)=(f(∇ϕ)−f(∇bϕ),∇bϕ)+(f(∇bϕ),∇bϕ) = (f(∇ϕ)−f(∇bϕ),∇bϕ) + 1 ε2|∇bϕ|4 4−1 ε2|∇bϕ|2 2 (13) From (12) and (13), one has |∆bϕ|2 2+1 ε2|∇bϕ|4 4= (ω, bϕ) + 1 ε2|∇bϕ|2 2−(f(∇ϕ)−f(∇bϕ),∇bϕ).(14) The first term on the right hand side of (14) can be bounded as |(ω, bϕ)|≤|ω|2|bϕ|2≤1 2|ω|2 2+C 2|∇bϕ|2 4≤1 2|ω|2 2+1 6ε2|∇bϕ|4 4+C ε2(15) where the constant Cdepends on εand |Ω|. On the other hand, the second term on the right hand side of (14) will be bounded as: 1 ε2|∇bϕ|2 2≤1 6ε2|∇bϕ|4 4+C ε2.(16) Now, we are going to bound the third term of (14). Taking into account that f(a)−f(b) = 1 ε2|a|2+|b|2+a·b−1(a−b)∀a,b∈IRN, in particular f(∇ϕ)−f(∇bϕ) = 1 ε2|∇ϕ|2+|∇bϕ|2+∇ϕ· ∇bϕ−1∇eϕ. (17) Consequently, by using that ∇eϕ∈L∞(L4), the H¨older and Young’s inequalities, the last term on the right hand side of (14) can be bounded as follows |(f(∇ϕ)−f(∇bϕ),∇bϕ)| ≤ 1 ε2||∇ϕ|2+|∇bϕ|2+∇ϕ· ∇bϕ−1|2|∇eϕ|4|∇bϕ|4 ≤C ε2|∇ϕ|2 4+|∇bϕ|2 4+ 1|∇bϕ|4 ≤C ε2|∇bϕ|2 4+ 1|∇bϕ|4≤1 6ε2|∇bϕ|4 4+C ε2. (18) where Cdepends on |Ω|and k∇eϕkL∞(L4). Finally, from (14)-(18), the inequality (11) is deduced. Lemma 4 The following inequality holds: ZΩ F(∇ϕ)≤1 2ε2|∇bϕ|4 4+C2 ε2(19) where C2depends on |Ω|and k∇eϕkL∞(L4) 8 Proof. Since F(∇ϕ) = 1 4ε2|∇ϕ|2−12, one has ZΩ F(∇ϕ) = 1 4ε2ZΩ |∇ϕ|4+1 2ε2ZΩ |∇ϕ|2+1 4ε2|Ω| ≤1 2ε2ZΩ |∇ϕ|4+C ε2≤1 2ε2ZΩ |∇bϕ|4+C2 ε2 where Cdepends on |Ω|and C2depends, moreover, on k∇eϕkL4(L4). Therefore, (19) holds. 2.3 Energy Inequality Lemma 5 (Energy equality) If (u, ϕ)is a regular enough solution of (10), 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+|ω|2 2= (∂teϕ, ω)+(∂t∇eϕ, f(∇ϕ)). (20) where ω= ∆2bϕ−∇·f(∇ϕ). Proof. Taking uas test function in the u-system of (10), one has 1 2 d dt|u|2 2+ν|∇u|2 2+ (σd nl,∇u)−(ω· ∇ϕ, u)=0.(21) The nonlinear dissipative tensor σd nl verifies: (σd nl,∇u) = (σd nl, D(u)) = |∇ϕTD(u)∇ϕ|2 2+|D(u)∇ϕ|2 2,(22) since (n⊗n) : D(u) = ninjD(u)ij =nTD(u)n and (D(u)n⊗n+n⊗D(u)n) : D(u) = 2(D(u)n⊗n) : D(u) = D(u)iknknjD(u)ij =|D(u)n|2. Therefore, from (21) we obtain 1 2 d dt|u|2 2+ν|∇u|2 2+|∇ϕTD(u)∇ϕ|2 2+|D(u)∇ϕ|2 2+|ω|2 2−(ω· ∇ϕ, u) = 0.(23) On the other hand, by taking ωas test function in the ϕ-equation of (10), one has 1 2 d dt|∆bϕ|2 2−(∂tbϕ, ∇ · f(∇ϕ)) + (u· ∇ϕ, ω) + |ω|2 2= (∂teϕ, ω).(24) The second term on the left hand side of (24) can be written as −(∂tbϕ, ∇ · f(∇ϕ)) = (∂t∇ϕ, f(∇ϕ)) −(∂t∇eϕ, f(∇ϕ)) = d dt ZΩ F(∇ϕ)−(∂t∇eϕ, f(∇ϕ)).(25) By adding (23) and (24) and into account (25) we obtain (20). 9 5 Weak time-periodic solutions In this section, let T > 0 a finite fixed number which states the time period. Definition 11 We say that (u, ϕ)is a weak time-periodic solution of (1),(2) and (4) if u∈L∞(0, T;H)∩L2(0, T;H1), ϕ ∈L∞(0, T;H2)∩L2(0, T;H4) satisfying (1) and boundary conditions (2) as in Definition 9 and time-periodic conditions u(0) = u(T),ϕ(0) = ϕ(T)in the sense of spaces L2and H2respectively. Theorem 12 (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 (9) satisfies eϕ∈L∞(0, T ;H4(Ω)), ∂teϕ∈L∞(0, T ;W1,4(Ω)). Then, there exists a weak time-periodic solution of (1),(2) and (4). Proof. In the proof of this theorem, a fully Galerkin method (approximating in finite dimension both variables uand ϕ) will be used. The reason is that this finite-dimensional Galerkin problem let us to find time-periodic approximate solutions via a fixed-point argument for the operator mapping the initial and final time values. Firstly, we consider the initial-boundary Galerkin problem associated to any arbitrary finite-dimensional initial data. Afterwards, the key is to find an initial data at t= 0 which will be “reproduced” at final time t=T. Finally, by means of a pass to the limit procedure, a weak time-periodic solution will be found. We divide the proof in several steps. Step 0: Existence of local in time Galerkin solution. Let {wi}n≥1 and {φi}n≥1 “special” basis of Vand H2 0(Ω), respectively, formed by eigenfunctions of the Stokes problem (∇wi,∇v) = λi(wi,v)∀v∈V,wi∈V,con kwikL2= 1, λi%+∞ and of the bilaplacian problem (∆φi,∆e) = µi(φi, e)∀e∈H2 0, φi∈H2 0,con kφikL2= 1, µi%+∞. Let Vmand Wmbe the finite-dimensional subspaces spanned by {w1,w2,...,wn}and {φ1, φ2, . . . , φn}respectively. Given u0∈Hand ϕ0∈H2 0, for each m≥1, we seek an approximate solution (um, ϕm), with um: [0, T]7→ Vmand ϕm=bϕm+eϕ, with bϕm: [0, T ]7→ Wm, verifying the following variational formulation a.e. t∈(0, T): 16                          (∂tum(t),vm) + ((um(t)· ∇)um(t),vm) + ν(∇um(t),∇vm)+(σd,m nl (t), Dvm) −(Qm∆2bϕm(t)−∇·f(∇ϕm(t))∇ϕm(t),vm)=0 ∀vm∈Vm, (∂tbϕm(t), em) + ((um(t)· ∇)ϕm(t), em) + (∆2bϕm(t)−∇·f(∇ϕm(t)), em) = (∂teϕ(t), em),∀em∈Wm, um(0) = u0m=Pm(u0), ϕm(0) = ϕ0m=Pm(ϕ0) in Ω. (42) Here, Pm:H7→ Vmdenotes the usual orthogonal projector from Honto Vm, and Qm: L27→ Wmthe orthogonal projector from L2onto Wm. In particular, u0m→u0in L2and ϕ0m→ϕ0in H2(as m→0). If we write um(t) = m X j=1 ξi,m(t)φiand bϕm(t) = m X j=1 ζi,m(t)ϕi, (42) can be rewritten as a first order ordinary differential system (in normal form) associated to the unknowns (ξi,m(t), ζi,m(t)). Then, one has existence of a maximal solution (defined in some interval [0, τm)⊂[0, T]) of the related Cauchy problem. Moreover, from a priori estimates (independent on m) which will be obtained below, in particular one has that τm=T. Step 1: Energy estimates. By taking in (42) vm=um∈Vmand em=Qm(∆2bϕm−∇·f(ϕm(t))) ∈Wmas test functions in (42), one can arrives at a similar inequality to (26) changing (u,bϕ) by (um,bϕm) and |∆2bϕm−∇·f(ϕm(t))|2 2by |Qm(∆2bϕm−∇·f(ϕm(t)))|2 2. That is, one has d dt |um|2 2+|∆bϕm|2 2+ 2 ZΩ F(∇ϕm)+ 2ν|∇um|2 2+|Qm(∆2bϕm−∇·f(∇ϕm))|2 2 ≤1 4ε2|∇bϕm|4 4+C. (43) On the other hand, the proof of Lemma 3 can be mimic for the case of Galerkin solutions, obtaining the following inequality (similar to (11)) |∆bϕm|2 2+1 2ε2|∇bϕm|4 4≤1 2|Qm(∆2bϕm−∇·f(∇ϕm))|2 2+C1.(44) Following the same argument of the proof of Theorem 10, from (43) and (44) one has (35) and (36). Since now the final time T > 0 is finite, in particular, the following estimates holds: umis bounded in L∞(0, T;H)∩L2(0, T;V), 17 ϕmis bounded in L∞(0, T;H2) and ωm:= Qm(∆2bϕm−∇·f(∇ϕm)) is bounded in L2(0, T ;L2). Step 2: ϕmis bounded in L2(0, T;H4). We have defined ωm=Qm(∆2bϕm−∇·f(∇ϕm)), namely, ωm∈Wm,(ωm, em) = ∆2bϕm−∇·f(∇ϕm), em∀em∈Wm.(45) By taking em= ∆2bϕm∈Wmas test function in (45) (that is possible because a spectral basis of the eigenfunctions of the bilaplacian has been considered), one obtains |∆2bϕm|2 2≤ |∇ · f(∇ϕm))|2|∆2bϕm|2+|ωm|2|∆2bϕm|2, hence kbϕmk4≤C|∇ · f(∇ϕm)|2+|ωm|2.(46) From (39) and by using the bound of ϕmin L∞(0,+∞;H2) and the interpolation inequality kϕk3≤Ckϕk1/2 2kϕk1/2 4, one has |∇ · f(∇ϕm))|2 2≤C|∇ϕm|4 6|∇∇ϕm|2 6+|∆ϕm|2 2≤Ckϕmk4 2kϕmk2 3+kϕmk2 2 ≤C(kϕmk4kϕmk2+ 1) ≤δkbϕmk2 4+δkeϕk2 4+C≤δkbϕmk2 4+C. (47) By using this last inequality for δsmall enough in (46) we obtain kbϕmk2 4≤C+C|ωm|2 2. As ωmis bounded in L2(Q), integrating in [0, T] we have that ϕmis bounded in L2(0, T;H4). Step 3: Uniqueness of Galerkin solution By applying the arguments given in Theorem 8 to (um, ϕm), we can obtain the uniqueness of Galerkin solution. Notice that this is possible because ∆2ϕm∈Wm(and um∈Vm). Step 4: Existence of time-periodic Galerkin solution Given (um 0, ϕm 0)∈Vm×Wm, we define the map Lm: [0, T]7→ IRm×IRm t7→ (ξ1m(t), ..., ξmm(t), ζ1m(t), ..., ζmm(t)) where (ξ1m(t), ..., ξmm(t)) and (ζ1m(t), ..., ζmm(t)) are the coefficients of um(t) and bϕm(t) respect to Vmand Wmrespectively, being (um(t),bϕm(t)) the (unique) approximate solution of (42) corresponding to the initial data (um 0, ϕm 0). 18 Now, varying the initial data (um 0, ϕm 0), we are going to define a new map Φm: IRm×IRm7→ IRm×IRmas follows: given Lm 0∈IRm×IRm, we define Φm(Lm 0) = Lm(T), where Lm(t) is related to the solution of problem (42) with initial data Lm 0(= Lm(0)). By uniqueness of approximate solution of (42), this map is well-defined. Moreover, using regularity of the corresponding ordinary differential system (equivalent to (42)), this map is continuous. In order to prove existence of fixed point of Φm, we will use Leray-Schauder’s Theorem. Consequently, we have to prove that for all λ∈[0,1], solutions Lm 0(λ) of Lm 0(λ) = λΦm(Lm 0(λ)) are uniformly bounded (independent of λ). Since Lm 0(0) = {0}, it is suffices to analyze λ∈(0,1] and the equation 1 λLm 0(λ)=Φm(Lm 0(λ)). Since we have considered the eigenfunctions of ∆2to furnish Wmand (45), it is easy to justify the computations of lemma 3, lemma 4 and Corollary 6, in order to arrive at (35). Considering the norm kLm(t)kIRm×IRm=kum(t)k2 L2+k∆bϕm(t)k2 L21/2in IRm×IRm, inequality (35) yields k1 λLm 0(λ)k2 IRm×IRm≤e−C0TkLm 0(λ)k2 IRm×IRm+C(1 −eC0T). Since λ∈(0,1], one has kLm 0(λ)k2 IRm×IRm≤C(1 −eC0T) eC0T−1 which is a bound independent of λ(and m). Consequently, Leray-Schauder Theorem implies the existence of fixed point of Φm, and therefore the existence of time-periodic Galerkin solutions. Moreover, for each time-periodic Galerkin solution (um, ϕm), their corresponding initialend data (um(0), ϕm(0)) = (um(T), ϕm(T)) is bounded in the L2×H2-norm, i.e k(um,bϕm)(0)kL2×H2≤C(Cindependent of m). Step 5: Pass to the limit in time-periodic Galerkin solutions The pass to the limit in variational formulation (42) can be done using estimations (independents of m) and compactness obtained in order to control nonlinear terms. Consequently, here we will only write the pass to the limit in time-periodic conditions. From estimations of (ϕm) in L∞(H2) and (∂tϕm) in L2(L3/2) and using the triplet of spaces H2,→H1,→L3/2, one has that (ϕm) is relatively compact in C([0, T]; H1), hence 19 ϕm(T)→ϕm(T) and ϕm(0) →ϕ(0) in H1(Ω). Since ϕm(T) = ϕm(0), then ϕ(T) = ϕ(0) in H1(Ω). Moreover, it is easy to see that ϕ∈Cw([0, T]; H2) (i.e. ϕis continuous from [0, T] onto H2, respect to the weak topology in H2), therefore ϕ(T) = ϕ(0) in H2(Ω). The argument for uis similar. Consequently, we have found a weak time-periodic solution of problem (1)-(2), (4) and the proof of Theorem 12 is finished. 6 Regularity for the initial-value problem The idea now is to obtain regularity for the weak solutions of the initial-value problem (1)-(3) (see [Lin,Liu’95], [Lin,Liu’00] for a nematic liquid crystal case and [Liu’00] for the smectic-A case, imposing time-independent boundary data in all these previous cases). In this sense, we will see that a global regularity result hold but only for the case of dominant viscosity, that is for νbig enough. In our opinion, the global regularity imposing constraints of initial data near of special equilibrium solutions is an interesting problem, which up to our knowledge remains as an open problem. Definition 13 We say that a weak solution (u, ϕ)of (1)-(3) is a strong solution if k(u(t), ϕ(t))k1×4≤C3∀t≥0,(48) ∀γ > 0, e−γt Zt 0 eγsk(u(s), ϕ(s))k2 2×6ds ≤C4,∀t≥0 (49) and verifying point-wise the fully differential system (1). Theorem 14 In the conditions of theorem 10, if moreover (u0, ϕ0)∈H1×H4with ku0k1≤ R1,kϕ0k4≤R2, ∂teϕ∈L∞(0,+∞;W1,4(Ω)) and ∂tt eϕ∈L∞(0,+∞;L2(Ω)), then for each ν≥ν0, with ν0=ν0(R1, R2, ∂teϕ, ∂tt eϕ), there exists a unique strong solution of (1)-(3) in [0,+∞), which verifies (48) and (49) with constants C3and C4depending on ν0 (but independent of ν). Proof. We define bω=−∂tbϕ−(u· ∇)bϕ. (50) By owing to bϕ|Σ= 0, ∇bϕ|Σ= 0 and u|Σ= 0, we have bω|Σ= 0,∇bω|Σ= 0.(51) 20 On the other hand, we are going to obtain the following inequalities: kϕk4≤C(|bω|2+|u|2+ 1),kϕk6≤C(kbωk2+kuk2+ 1).(52) Indeed, as ω= ∆2ϕ−∇·f(∇ϕ) = −∂tϕ−(u· ∇)ϕ=bω−∂teϕ−(u· ∇)eϕ, one has ∆2bϕ= ∆2ϕ=ω+∇ · f(∇ϕ) = bω−u· ∇eϕ−∂teϕ+∇ · f(∇ϕ).(53) Hence kbϕk4≤ |bω|2+|∇eϕ|∞|u|2+|∂teϕ|2+|∇ · f(∇ϕ)|2. Proceeding in the analogous way that in (47) to bound the term |∇ · f(∇ϕ)|2and using the regularity of eϕone arrives at the bound of kbϕk4given in (52). The bound for kbϕk6given in (52) can be obtained in a similar way. Notice that, 1 2 d dt|bω|2 2= (bω, ∂tbω)=(bω, ∂t(ω+u· ∇eϕ+∂teϕ)) = (bω, ∂t(∆2bϕ−∇·f(∇ϕ) + u· ∇eϕ+∂teϕ)) = (∆bω, ∂t∆bϕ)+(∇bω, ∂tf(∇ϕ)) + (bω, ∂tu· ∇eϕ+u·∂t∇eϕ+∂tt eϕ). By using (50), one has ∂t∆bϕ=−∆bω−∆((u· ∇)bϕ) = −∆bω− ∇2u∇bϕ− ∇u∇2bϕ−(u· ∇)∆bϕ and ∂tf(∇ϕ) = (3|∇ϕ|2−1)∂t∇ϕ= (3|∇ϕ|2−1)(−∇bω− ∇((u· ∇)bϕ) + ∇∂teϕ) = (3|∇ϕ|2−1)(−∇bω− ∇u∇bϕ−(u· ∇)∇bϕ+∇∂teϕ),. Therefore, we obtain that 1 2 d dt|bω|2 2+|∆bω|2 2=−(∆bω, ∇2u∇bϕ)−(∆bω, ∇u∇2bϕ)−(∆bω, (u· ∇)∆bϕ)−(∇bω, 3|∇ϕ|2∇bω) −(∇bω, 3|∇ϕ|2∇u∇bϕ)−(∇bω, 3|∇ϕ|2(u· ∇)∇bϕ)) + (∇bω, ∇bω)+(∇bω, ∇u∇bϕ) +(∇bω, (u· ∇)∇bϕ)) + (bω, ∂tu· ∇eϕ)+(bω, u·∂t∇eϕ)+(bω, ∂tt eϕ)+(∇bω, (3|∇ϕ|2−1)∂t∇eϕ). By bounding the terms on the right hand side of previous equality one arrives at d dt|bω|2 2+kbωk2 2≤ν 2kuk2 2+1 2|∂tu|2 2+C νkbωk2 2(1 + kbϕk3+kbϕk2 3) + C|bω|2 2+C, (54) where C > 0 may denote different constants, always independent of ν. 21 On the other hand, taking Au+∂tuas test functions in the u-system (Abeing the Stokes operator, i.e. A=−P∆ with Pthe Leray projector onto H) it is easy to obtain d dt((ν+ 1)kuk2 1) + νkuk2 2+|∂tu|2 2≤1 2kbωk2 2+C+C νkuk1kuk2 2 +C+C ν(|bω|2 2+kuk1) + C νkbϕk4 3+kbϕk2 3+ 1kuk2 2, (the last term on the right hand side of previous inequality is a bound of |∇ · σd nl|2 2). Since we want to choose νbig enough, for instance we assume ν0≥1. Then, for each ν > ν0≥1, we get d dt((ν+ 1)kuk2 1) + νkuk2 2+|∂tu|2 2≤1 2kbωk2 2+Ckuk1kuk2 2+|bω|2 2+kuk1 +C νkbϕk4 3+kbϕk2 3+ 1kuk2 2. (55) Adding (54) and (55) we have d dt((ν+ 1)kuk2 1+|bω|2 2) + ν 2kuk2 2+1 2|∂tu|2 2+1 2kbωk2 2≤C νkbωk2 2(1 + kbϕk3+kbϕk2 3) +D(kuk1kuk2 2+|bω|2 2+kuk1) + C νkbϕk4 3+kbϕk2 3+ 1kuk2 2+E (56) where C,Dand Eare constants independent of ν≥1. On the other hand, using (52), the regularity of u,bϕ,eϕand the interpolation inequality kbϕk3≤Ckbϕk1/2 2kbϕk1/2 4we get kbϕk3≤C(1 + |bω|1/2 2). Hence, from (56) one has the following inequality: d dt((ν+ 1)kuk2 1+|bω|2 2) + ν 2kuk2 2+1 2|∂tu|2 2+1 2kbωk2 2≤C νkbωk2 2(1 + |bω|1/2 2+|bω|2) +D(kuk1kuk2 2+|bω|2 2+kuk1) + C ν|bω|2 3+|bω|3+ 1kuk2 2+E. (57) If we denote Φ1(t) = kuk2 1,Φ2(t) = |bω|2 2,Ψ1(t) = kuk2 2,Ψ2(t) = kbωk2 2, we obtain from (57), d dt((ν+ 1)Φ1+ Φ2) + ν 2−DΦ1/2 1−C νΦ2+ Φ1/2 2+ 1Ψ1 +1 2−C ν(1 + Φ1/4 2+ Φ1/2 2)Ψ2≤D(Φ2+ Φ1/2 1) + E. (58) Let R1,R2,Mand ν0≥1 some positive constants that we will specify below, such that if Φ1(0) ≤R1and Φ2(0) ≤R2, we will prove that (ν+ 1)Φ1(t)+Φ2(t)≤M∀t∈[0,+∞),(59) 22 for any ν≥ν0. Indeed, by contradiction, let t∗>0 the first value such that (ν+ 1)Φ1(t∗) + Φ2(t∗) = M, hence (ν+ 1)Φ1(t∗)+Φ2(t∗) = Mand (ν+ 1)Φ1(t)+Φ2(t)< M ∀t∈[0, t∗). Then, Φ1(t)≤M ν+ 1 and Φ2(t)≤M∀t∈[0, t∗]. Assume that there exists ν0big enough such that, for each ν≥ν0 ν 2−DM ν+ 11/2 −C ν(M+M1/2+ 1) ≥ν+ 1 4 and 1 2−C ν1 + M1/4+M1/2≥1 4.(60) Then, for each t∈[0, t∗] d dt((ν+ 1)Φ1+ Φ2) + ν+ 1 4Ψ1+1 4Ψ2≤D(Φ2+ Φ1/2 1) + E. (61) We define P= min{P1, P2}where 1/P1and 1/P2are the Poincar´e constants that verify Φ1≤1 P1 Ψ1and Φ2≤1 P2 Ψ2respectively. Therefore, d dt((ν+ 1)Φ1+ Φ2) + P 4((ν+ 1)Φ1+ Φ2)≤D(Φ2+ Φ1/2 1) + E. (62) Multiplying (62) by eP t/4and integrating in [0, t∗] we deduce (ν+ 1)Φ1(t∗)+Φ2(t∗)≤((ν+ 1)Φ1(0) + Φ2(0))e−P t∗/4 +e−Pt∗/4Zt∗ 0 (D(Φ2(s)+Φ1/2 1(s)) + E)ePs/4ds. (63) By (53), bω= ∆2bϕ+u· ∇eϕ+∂teϕ−∇·f(∇ϕ), hence we get Φ2=|bω|2 2≤C(kbϕk2 4+|u|2+1+kbϕk4)≤C(kbϕk2 4+|u|2+ 1). Therefore, taking into account weak estimates (31), the second term on the right hand side of (63) is bounded by a constant Cwindependent of ν(in fact, Cwdepends on the constant C2given in (31)) and (ν+ 1)Φ1(t∗)+Φ2(t∗)≤((ν+ 1)Φ1(0) + Φ2(0)) + Cw1 ν+ 1≤((ν+ 1)R1+R2)+2Cw. Hence, if we choose M > (ν+ 1)R1+R2+ 2Cw,(64) 23 then we arrives at a contradiction. Therefore, we could get the estimate (59) whether there exists big enough constants Mand ν0such that (60) and (64) hold, for each ν≥ν0. Indeed, if we choose M=λ ν then (64) holds for any λ > 2R1+R2+ 2Cw. If we fix λwith this condition, then the two conditions given in (60) hold if λ1/2 ν≤εand 1 ν(1 + λ1/4ν1/4+λ1/2ν1/2)≤ε for ε > 0 small enough. But these conditions hold for each ν≥ν0with ν0big enough respect to λ. Therefore, we get estimates (59). From (59), we obtain u∈L∞(0,+∞;H1) and bω∈L∞(0,+∞;L2). Recalling (52) we also obtain ϕ∈L∞(0,+∞;H4). By going back to (61), multiplying by eγt for any γ > 0 and integrating in [0, t] we deduce ∀γ > 0, e−γt Zt 0 eγsk(u(s),bω(s))k2 2×2ds ≤C2,∀t≥0. Again, by applying (52) we get (49). 7 Regularity for the time-periodic problem The results obtained up to now allow us to obtain, for big enough ν, the regularity given in Definition 13 also for the time-periodic problem. Indeed, arguing as in [Climent et al.] for a nematic crystal model, to prove that weak time-periodic solution is regular it suffices to use the following three results: 1. the existence of the weak time-periodic solution (proved in Section 4), 2. the weak/strong uniqueness of the initial-valued problem (proved in Section 3), 3. the existence of global strong solution for big enough viscosity of the initial-valued problem (proved in Section 6). Consequently, the regularity for time-periodic solutions can be deduced. References [Climent et al.’06] B. Climent-Ezquerra, F. Guill´en-Gonz´alez, M.A. Rojas-Medar. 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