Sufficient conditions for regularity and uniqueness of a 3D nematic liquid crystal model

Sufficient conditions for regularity and uniqueness of a 3D nematic liquid crystal model. F. Guill´en-Gonz´alez1, M. A. Rodr´ıguez-Bellido1, M. A. Rojas-Medar2 5th July 2005 1Dpto. de Ecuaciones Diferenciales y An´alisis Num´erico, Universidad de Sevilla, Aptdo. 1160, 41080 Sevilla, Spain. e-mails: [email protected], [email protected] 2Dpto. de Matem´atica Aplicada, IMECC-UNICAMP, C.P. 6065, 13083-970, Campinas- SP, Brazil. e-mail: mark[email protected] Abstract In [3], L. Berselli showed that the additional regularity hypothesis for the velocity gradient ∇u∈L2q 2q−3(0, T;Lq(Ω)), for some q∈(3/2,+∞], implies the strong regularity for the weak solutions of the Navier-Stokes equations. In this work, we prove that such hypothesis is also sufficient in order to obtain the strong solution for a nematic Liquid Crystal model (a coupled system of velocity uand orientation crystals vector d) when periodic boundary conditions for dare considered. For Neumann and Dirichlet boundary conditions, we obtain the same result only for the cases of q∈[2,3], whereas in the cases q∈(3/2,2) ∪(3,+∞], we also need to impose an additional regularity hypothesis for d(either on ∇dor ∆d). On the other hand, when the following hypothesis for uof Serrin’s type is imposed ([18]): u∈L2p p−3(0, T;Lp(Ω)) for some p∈(3,+∞], we can obtain strong regularity only in the case of periodic boundary conditions for d. For Neumann or Dirichlet boundary conditions, additional regularity for dmust be imposed in all cases. 1 21. Introduction and main results Palabras clave: Liquid Crystal system, sufficient hypothesis of regularity, strong solution, uniqueness, regularity criterion. AMS subject classification: 35Q30. 1 Introduction and main results 1.1 The model In this work, we are interested in the study of regularity for a Navier-Stokes type model for incompressible fluids that takes into account the crystallinity of the fluid molecules. Thus, the Navier-Stokes equations are coupled with the Ginzburg-Landau equations, obtaining a nematic Liquid Crystal model. The unknowns are 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 three-dimensional smooth enough domain Ω. Here, we present an Ericksen-Leslie type formulation, introduced by F. H. Lin in [11] and analyzed by F. H. Lin & C. Liu in [12]. In fact, the model is a penalized one where the restriction |d| ≤ 1 appears as a consequence of a maximum principle for the Ginzburg-Landau equations, which is approximated by: fδ(d) = 1 δ2¡|d|2−1¢d, where |d|denotes the euclidean norm in R3and δ > 0 is a penalization parameter. This penalization function has a potential structure, i. e. there exists a potential function Fδ(d) = 1 δ2¡|d|2−1¢2such that fδ(d) = ∇d(Fδ(d)) for all d∈R3. Denoting Q= (0, T)×Ω and Σ = (0, T)×∂Ω, where Ω a smooth enough domain and ∂Ω its boundary, 1.2 Known results 3 the model can be written as: (LC)                                                    ∂tu−ν∆u+u· ∇u+∇p=−λ(∇d)t∆din Q, ∇ · u= 0 in Q, ∂td+γ(fδ(d)−∆d) + u· ∇d=0in Q, u=0on Σ, d=hor ∂d ∂n= 0 or dis space-periodic on Σ, u|t=0 =u0,d|t=0 =d0in Ω, As for the coefficients, ν > 0 is the fluid viscosity, λ > 0 is the elasticity constant and γ > 0 is a relaxation in time constant. Now we explain the boundary conditions: for the velocity uwe always consider the non-slip boundary condition (u=0on Σ); for the orientation vector d, the condition used for the model derivation as a penalized model [12] is of non-homogeneous Dirichlet type: d=hon Σ such that |h| ≤ 1 in Σ. (1) There exists some others works where the boundary condition considered is of homogeneous Neumann type [15] : ∂nd=0on Σ. (2) jointly with the following free-slip boundary conditions for u: u·n= 0,(∇ × u)×n=0on Σ. Finally, we will also consider periodic boundary conditions for d. In this case, the domain Ω is a square (n= 2) or a cube (n= 3). 1.2 Known results For the Navier-Stokes equations, several works have studied the influence of the additional hypothesis over the velocity uin order to turn a weak solution uinto a strong solu- 41. Introduction and main results tion, or to obtain uniqueness. Serrin proved in 1962 that if uis a Leray-Hopf weak solution belonging to Lα(0, T;Lγ(R3)) where 2/α + 3/γ ≤1 for 2 < α < +∞and 3 < γ < +∞, then the solution u∈C∞((0, T)×R3). The same tesis was improved by other authors imposing weaker hypothesis, for instance Giga supposing u∈C([0, T]; L3(R3)), [7]. There are also works were the weak regularity of the velocity is improved imposing regularity criteria over the gradient of velocity (the case of the whole space was proved in [1] and the bounded domain case in [3]) or the pressure (see for example, [2, 4]). Or even, only supposing that one component of the velocity field belongs to L∞((0, T)×R3) (see [10]). Recently, in [17], several different criteria on uor over some components of uare considered with the same objective. In our case, a coupled system of uand dis treated in a bounded domain. Here, we analyze the cases where only additional hypothesis on uare necessary to provide the strong regularity for (u,d). Sometimes, this will be possible (see results below when imposing (H2) hypothesis for ∇uwith q∈[2,3]). Otherwise, an additional regularity on d(the crystal orientation field) will be imposed. We denote by Hand Vthe space of type L2and H1respectively, associated to the incompressibility and adherence velocity conditions: H={u∈L2(Ω) : ∇ · u= 0,u·n|∂Ω= 0} V={u∈H1(Ω) : ∇ · u= 0,u|∂Ω= 0} Observe that the boldface is used to denote the vectorial spaces in order to distinguish from the scalar spaces. On the Liquid Crystal model (LC), under compatibility hypothesis: |d0| ≤ 1,|h| ≤ 1 a. e. in Σ (3) and (weak) regularity hypothesis for the data: u0∈H,d0∈H1(Ω),h∈L∞(0, T;H1/2(∂Ω)) ∩L2(0, T;H3/2(∂Ω)),(4) it is known ([12]) the existence of weak solution global in time, that is a solution with the 1.2 Known results 5 following regularity:        u∈L∞(0, T;H)∩L2(0, T;V) d∈L∞(0, T;H1(Ω)) ∩L2(0, T;H2(Ω)),|d| ≤ 1. In a previous work, some of the authors of this work obtained in [9] the existence of a weak solution global in time (without restrictions for the data) for the limit problem (when δ→0 in (LC)). Note that if Ω ⊆R2, we know the existence and uniqueness of smooth solution global in time, see [12]. For this, we focus our study in Ω ⊆R3. On the other hand, under the hypothesis of strong regularity for the data: u0∈V, d0∈H2(Ω),h∈L∞(0, T;H3/2(∂Ω)) ∩L2(0, T;H5/2(∂Ω)),(5) it is known the existence and uniqueness of strong solution with restrictions (global in time for small data or local in time for any data), that is, the solution has the following regularity:        u∈L∞(0, T∗;V)∩L2(0, T∗;H2(Ω)) d∈L∞(0, T∗;H2(Ω)) ∩L2(0, T∗;H3(Ω)), with T∗≤T(small enough) or T∗=Tfor small data. Finally, from the previous regularity and under the hypothesis: ∂th∈L∞(0, T;H−1/2(∂Ω)) ∩L2(0, T;H1/2(∂Ω)),(6) it is easy to obtain (a posteriori) the following additional regularity (see Lemma 2.3):        ∂tu∈L2(0, T;H), p ∈L2(0, T;H1(Ω)) ∂td∈L∞(0, T;L2(Ω)) ∩L2(0, T;H1(Ω)). 61. Introduction and main results 1.3 Main new results Our aim is to elucidate sufficient regularity hypothesis for uin order to obtain strong regularity global in time (and uniqueness) for the solution of (LC), without imposing additional regularity hypothesis for d(if it is possible). In this way, we impose two kinds of conditions for u: (H1) ∇u∈L2q 2q−3(0, T;Lq(Ω)) for some q: 3/2< q ≤ ∞ or (H2) u∈L2p p−3(0, T;Lp(Ω)) for some 3 < p ≤+∞. When necessary, we will impose the following two kinds of additional regularity for d: (H3) ∇d∈L2p p−3(0, T;Lp(Ω)) for some p: 3 ≤p≤+∞, or (H4) ∆d∈L2q 2q−3(0, T;Lq(Ω)) for some q: 3/2< q ≤+∞. Observe that hypothesis (H2) is the Serrin’s condition (see [18]) and hypothesis (H1) was introduced by Berselli in [3] when q≥3 (the case q∈(3/2,3) is implied by the Serrin’s condition). Both hypothesis guarantee the strong regularity and uniqueness for the Navier-Stokes equations. Next, we state the main results, indicating the sections where each result will be proved. Always, the hypothesis (3), (5) and (6) are imposed on the data. Theorem 1.1 (Non-homogeneous Dirichlet or homogeneous Neumann cases) Let (u,d)be a weak solution of (LC)in (0, T)such that ∇uverifies (H1) with exponent q: 2 ≤q≤3. Then, (u,d)is the unique strong solution of (LC)in (0, T). The proof will be made using Prodi’s type estimates in a decoupled manner, first for d, and afterwards using this regularity, for u(see Section 2). Theorem 1.2 (Other cases) If (u,d)is a weak solution of (LC)such that uverifies either (H1) or (H2), and dverifies either (H3) or (H4), then (u,d)is the unique strong solution of (LC). 1.3 Main new results 7 In this case (see Section 3), we use Ladyzhenskaya’s type estimates for uand din a coupled manner. Theorem 1.3 (Periodic boundary conditions for d) Let (u,d)be a weak solution of (LC)with periodic boundary conditions for dsuch that uverifies either (H2) for p > 3 or (H1). Then, (u,d)is the unique strong solution of (LC). The proof (see Section 4) is based on the obtention of the following non-hilbertian regularity for d,∇d∈L3(0, T;L9(Ω)), and the application of Theorem 1.2. Theorem 1.4 (Uniqueness) Suppose that the data verify the weak regularity given by (4) and the constraints (3), and (u,d)is a weak solution of (LC)such that uverifies either (H2) or (H1), and dverifies either (H3) or (H4). Then, this solution coincides with any weak solution (associated to the same data). Proof: See Section 5 when (H1) condition is imposed, and Section 6 for (H2) condition. Remark 1.1 Observe that in Theorems 1.1,1.2and 1.3, strong regularity is obtained assuming strong regularity on the data (u0,d0,h). But the uniqueness result (Theorem 1.4) is obtained only assuming weak regularity on the data (and additional regularity on weak solutions). However, Theorems 1.1and 1.3give uniqueness results (a posteriori) under less additional regularity hypothesis over the weak solution than in Theorem 1.4 (only with additional regularity for uand not for d). Remark 1.2 (Comparison between the hypothesis (H1) and (H2) for u, and (H3) and (H4) for d) Thanks to the Sobolev’s embeddings, the hypothesis (H1) (resp. (H4)) for 3/2< q < 3 implies the hypothesis (H2) (resp. (H3)) for 3< p < +∞. In particular, the hypothesis (H2) guarantees the strong regularity for the solution of the Navier-Stokes equations, see for instance [18]. 82. Proof of Theorem 1.1 2 Proof of Theorem 1.1 We want to use the additional regularity hypothesis (H1) for ∇u(with 2 ≤q≤3), in order to obtain strong regularity for (LC) without additional regularity hypothesis for d. Due to the particular structure of the equations for u, specially the strongly nonlinear term (∇d)t∆din the u-system, now it will be necessary to obtain greater regularity for d. An essential characteristic of the problem for d(given u) is the following weak maximum principle: Lemma 2.1 (Maximum principle) Given u∈L2(0, T;V)∩L∞(0, T;H)and d0∈ L2(Ω) with |d0| ≤ 1a. e. in Ω, let dbe any weak solution for the d-problem (either in the non-homogeneous Dirichlet case, d|Σ=hwith |h| ≤ 1a. e. on Σ, or the homogeneous Neumann case, ∂nd|Σ=0or the space-periodic case), i.e. d∈L2(0, T;H1(Ω)) ∩ L∞(0, T;L2(Ω)) such that for all e∈H1 0(Ω) (or H1(Ω)): h∂td,ei+ ((u· ∇)d,e)+(fδ(d),e) + (∇d,∇e) = 0,(7) d(0) = d0.(8) Then, |d(x, t)| ≤ 1a.e. in Q. Proof: See [5]. For reader’s convenience completeness, a proof is included in Appendix A.1. In particular, |fδ(d)| ≤ Cand |∇fδ(d)| ≤ C|∇d|, where Cis a constant (depending on 1/δ2). In the proof of Theorem 1.1, we distinguish between two kinds of boundary conditions: homogeneous Neumann boundary conditions, that is, (∂nd)|∂Ω=0, and nonhomogeneous Dirichlet boundary conditions, that is, d|∂Ω=h(the space-periodic case will be easier). 2.1 Homogeneous Neumann boundary conditions 9 2.1 Homogeneous Neumann boundary conditions Notice that for homogeneous Neumann boundary conditions, we are going to consider the following equivalent norms: kdkH1(Ω) ∼³k∇dk2 L2(Ω) +kdk2 L2(Ω)´1/2 kdkH2(Ω) ∼³k∆dk2 L2(Ω) +kdk2 L2(Ω)´1/2 kdkH3(Ω) ∼³k∇(∆d)k2 L2(Ω) +kdk2 H2(Ω)´1/2 The proof of existence of weak solution ([12]) is based on the restriction |d| ≤ 1 in Q (consequence of a weak maximum principle, see Theorem 2.1 for the equation verified by |d|2), and the following inequalities: 1 2 d dtkdk2 L2+γk∇dk2 L2≤C(9) 1 2 d dt³kuk2 L2+λγk∇dk2 L2+λγFδ(d)´ +νk∇uk2 L2+λγk(fδ(d)−∆d)k2 L2≤0 (10) If we take ∆2das a test function in the d-system, integrating by parts in every term and a second time in the derivative in time term ∂td, the boundary integrals cancel using that ∂n∂td= 0 on Σ): 1 2 d dtk∆dk2 L2(Ω) +γk∇(∆d)k2 L2(Ω) =γZΩ ∇fδ(d)· ∇(∆d)dΩ + ZΩ ∇[(u· ∇)d]∇(∆d)dΩ ≤γZΩ |∇(fδ(d))| |∇(∆d)|dΩ + ZΩ |∇u||∇d||∇(∆d)|dΩ +ZΩ |∇u||D2d|2dΩ, where the last term on the right hand side is obtained as follows: ¯¯¯ZΩ (u· ∇)∇d· ∇(∆d)dΩ¯¯¯=¯¯¯−ZΩ (∇u· ∇)∇d:D2ddΩ−1 2ZΩ (u· ∇)(|D2d|2)dΩ¯¯¯ ≤ZΩ |∇u| |D2d|2dΩ 16 3. Proof of Theorem 1.2 (when (H1) is imposed) 3 Proof of Theorem 1.2 (when (H1) is imposed). We want to analyze those cases from the precedent Section where it could not be deduced the strong regularity for d. For this aim, we will use Ladyzhenskaya’s type of estimates. We study again separately the homogeneous Neumann boundary conditions for d, i. e., ∂nd|∂Ω=0, and the non-homogeneous Dirichlet boundary conditions, that is, d|∂Ω=h. The periodic boundary conditions for dwill be treated in the next Section. 3.1 Homogeneous Neumann boundary conditions Remember that we use the following boundary condition for d: ∂nd=0on Σ,(23) Deriving (LC) with respect to the time, and taking first ∂tdas test function for (LC)3, we obtain (using |∂tfδ(d)| ≤ C|∂td|): 1 2 d dtk∂tdk2 L2(Ω) +k∇(∂td)k2 L2(Ω) ≤C³k∂tuk2 L2(Ω) +k∂tdk2 L2(Ω)´, (where we have integrated by parts the term ZΩ (∂tu· ∇)d·∂tddΩ). On the other hand, taking λ−1∂tuand −∆(∂td) as test functions, we obtain: 1 2 d dt ³λ−1k∂tuk2 L2(Ω) +k∇(∂td)k2 L2(Ω)´+ν λk∇(∂tu)k2 L2(Ω) +γk∆(∂td)k2 L2(Ω) ≤ZΩ³∂tfδ(d)·∂t(∆d)+(u· ∇)(∂td)·∂t(∆d)−1 λ(∂tu· ∇)u·(∂tu)´dΩ −ZΩ (∇(∂td))t·∆d·∂tudΩ ≤C½ZΩ |∂td||∆(∂td)|+ZΩ |∇u||∇(∂td)|2dΩ + λ−1ZΩ |∂tu|2|∇u|dΩ¾ −ZΩ (∇(∂td))t·∆d·∂tudΩ = 4 X i=1 Ji. (24) In fact, we proceed as follows: J1≤Ck∂tdkL2(Ω)k∆(∂td)kL2(Ω) ≤¯εk∇(∂td)k2 H1(Ω) +C¯εk∂tdk2 L2(Ω) 3.1 Homogeneous Neumann boundary conditions 17 The estimate for J2is similar to (13): J2≤ k∇ukLq(Ω)k∇(∂td)k2 L 2q q−1(Ω) ≤ k∇ukLq(Ω)k∇(∂td)k 2q−3 q L2(Ω)k∇(∂td)k 3 q H1(Ω) ≤¯εk∇(∂td)k2 H1(Ω) +C¯εk∇uk 2q 2q−3 Lq(Ω)k∇(∂td)k2 L2(Ω), where the additional regularity (H1) for ∇uappears. The term J3can be bound as in Berselli’s article [3]: J3≤λ−1k∂tuk2 L 2q q−1(Ω) k∇ukLq(Ω) ≤Ck∂tuk 2q−3 q L2(Ω)k∇(∂tu)k 3 q L2(Ω)k∇ukLq(Ω) ≤¯εk∇(∂tu)k2 L2(Ω) +C¯εk∇uk 2q 2q−3 Lq(Ω)k∂tuk2 L2(Ω) Summary, for ¯εsmall enough, we have the following inequality: d dt ³λ−1k∂tuk2 L2(Ω) +k∂tdk2 H1(Ω)´+ν λk∂tuk2 H1(Ω) +γk∂tdk2 H2(Ω) ≤Cµk∇uk 2q 2q−3 Lq(Ω) + 1¶³k∂tuk2 L2(Ω) +k∂tdk2 H1(Ω)´ −ZΩ (∂tu· ∇)(∂td)·∆ddΩ (25) Finally, there is left the last term to bound. We can choose two possible strategies, depending on the way we write such term: 1. If we try to integrate the term directly, we need to impose the additional regularity (H4) for ∆d, estimating as follows: ZΩ |∇∂tdt||∆d||∂tu|dΩ≤ k∂tukL 2q q−1(Ω)k∆dkLq(Ω)k∇(∂td)kL 2q q−1(Ω) ≤ k∂tukα L2(Ω)k∇(∂tu)k1−α L2(Ω)k∆dkLq(Ω)k∇(∂td)kα L2(Ω)k∇(∂td)k1−α H1(Ω) ≤¯ε³k∇(∂tu)k2 L2(Ω) +k∇(∂td)k2 H1(Ω)´+C¯εk∆dk1/α Lq(Ω) ³k∂tuk2 L2(Ω) +k∇(∂td)k2 L2(Ω)´ with 1 α=2q 2q−3. We can conclude the following inequality: d dt ³λ−1k∂tuk2 L2(Ω) +k∂tdk2 H1(Ω)´+ν λk∂tuk2 H1(Ω) +γk∂tdk2 H2(Ω) ≤C½µk∇uk 2q 2q−3 Lq(Ω) +k∆dk 2q 2q−3 Lq(Ω) + 1¶³k∂tuk2 L2(Ω) +k∂tdk2 H1(Ω)´¾(26) 18 3. Proof of Theorem 1.2 (when (H1) is imposed) 2. If we integrate by parts, we will have to impose the hypothesis (H3) on ∇d. Indeed, −ZΩ (∂tu· ∇)(∂td)·∆ddΩ = ZΩ (∂tu· ∇)∇(∂td)· ∇ddΩ +ZΩ (∇(∂tu)· ∇)(∂td) : ∇ddΩ, because of the boundary term vanishes due to ∂tu|∂Ω=0, independently from the boundary conditions imposed on d. The remaining terms can be bounded as follows: ZΩ |D2(∂td)||∇d||∂tu|dΩ≤ kD2(∂td)kL2(Ω)k∇dkLp(Ω)k∂tukL 2p p−2(Ω) ≤ k∂tdkH2(Ω)k∇dkLp(Ω)k∂tuk p−3 p L2(Ω)k∇(∂tu)k 3 p L2(Ω) ≤¯ε³k∂tdk2 H2(Ω) +k∂tuk2 H1(Ω)´+C¯εk∇dk 2p p−3 Lp(Ω)k∂tuk2 L2(Ω) And similarly, ZΩ |∇(∂td)| |∇d| |∇(∂tu)|dΩ≤ k∇(∂td)kL 2p p−2(Ω)k∇dkLp(Ω)k∇(∂tu)kL2(Ω) ≤ k∇(∂td)k p−3 p L2(Ω)k∇(∂td)k 3 p H1(Ω)k∇dkLp(Ω)k∂tukH1(Ω) ≤¯ε³k∂tdk2 H2(Ω) +k∂tuk2 H1(Ω)´+C¯εk∇dk 2p p−3 Lp(Ω)k∇(∂td)k2 L2(Ω) Therefore, we obtain the following inequality: d dt ³λ−1k∂tuk2 L2(Ω) +k∂tdk2 H1(Ω)´+ν λk∂tuk2 H1(Ω) +γk∂tdk2 H2(Ω) ≤Cµk∇dk 2p p−3 Lp(Ω) +k∇uk 2q 2q−3 Lq(Ω) + 1¶³k∂tuk2 L2(Ω) +k∂tdk2 H1(Ω)´(27) Thus, using the Gronwall’s Lemma, we can conclude (as in [3]) that: ∂tu∈L∞(ε, T;L2(Ω)) ∩L2(ε, T;H1(Ω)), ∂td∈L∞(ε, T;H1(Ω)) ∩L2(ε, T;H2(Ω)) (28) In particular, from uand ∂tu∈L2(ε, T;H1(Ω)), we obtain u∈C([ε, T]; H1(Ω)) (dand ∂td∈L2(ε, T;H2(Ω)) implies that d∈C([ε, T ]; H2(Ω)). Therefore, using the local in time strong regularity as in [3], we can conclude that: ∇u∈L∞(0, T;L2(Ω)) y ∆d∈L∞(0, T;L2(Ω)). 3.2 Nonhomogeneous Dirichlet boundary condition case 19 Then, d∈L∞(0, T;H2(Ω)) and u∈L∞(0, T;H1(Ω)). Thus, it is only necessary to obtain d∈L2(0, T;H3(Ω)) and u∈L2(0, T;H2(Ω)). To this aim, we use the following result: Lemma 3.1 Suppose that d∈L∞(0, T;H2(Ω)) and u∈L∞(0, T;H1(Ω)), then d∈ L2(0, T;H3(Ω)) and u∈L2(0, T;H2(Ω)). Moreover, ∂td∈L∞(0, T;L2(Ω))∩L2(0, T;H1(Ω)), ∂tu∈L2(0, T;L2(Ω)) and p∈L2(0, T;H1(Ω)). Proof: See Appendix A.4. 3.2 Nonhomogeneous Dirichlet boundary condition case In this case, we derive the equation for uin (LC) and the system for b d(19) with respect to the time, and take λ−1∂tuand −∆(∂tb d), respectively, as test functions. Thus we obtain an inequality similar to (25) (with three additional terms (on the right)). Concretely: 1 2 d dt ³λ−1k∂tuk2 L2(Ω) +k∂tb dk2 H1(Ω)´+ν λk∂tuk2 H1(Ω) +γk∂tb dk2 H2(Ω) ≤¯εk∆(∂tb d)k2 L2(Ω) +C¯εZΩ³|∂tb d|2+|∂te d|2´dΩ + λ−1ZΩ |∂tu|2|∇u|dΩ −ZΩ (∂tu· ∇)∂td·∆ddΩ−ZΩ (∂tu· ∇)d·∆(∂td)dΩ +ZΩ (∂tu· ∇)d·∆(∂tb d)dΩ + ZΩ (u· ∇)(∂td)·∆(∂tb d)dΩ := 7 X i=1 e Ji (29) The estimate of e J3is identical to those of J3(in the Neumann case). The term e J4can be bounded in two different ways (as the J4-term in the homogeneous Neumann boundary case), depending on te hypothesis imposed on d. That is: •if the hypothesis (H4) on ∆dis considered, then: e J4≤¯ε³k∂tuk2 H1(Ω) +k∂tb dk2 H2(Ω) +k∂te dk2 H2(Ω)´ +C¯εk∆dk 2q 2q−3 Lq(Ω) ³k∂tuk2 L2(Ω) +k∂tb dk2 H1(Ω) +k∂te dk2 H1(Ω)´ 20 3. Proof of Theorem 1.2 (when (H1) is imposed) •if the hypothesis (H3) on ∇dis considered, then: e J4≤¯ε³k∂tuk2 H1(Ω) +k∂tb dk2 H2(Ω) +k∂te dk2 H2(Ω)´ +C¯εk∇dk 2p p−3 Lp(Ω) ³k∂tuk2 L2(Ω) +k∂tb dk2 H1(Ω) +k∂te dk2 H1(Ω)´ Adding e J5and e J6: e J5+e J6=−ZΩ (∂tu· ∇)d·∆(∂te d)dΩ. If we consider (H3) on ∇d, we bound as follows: e J5+e J6≤¯ε³k∂tuk2 H1(Ω) +k∂te dk2 H2(Ω)´+C¯εk∇dk 2p p−3 Lp(Ω)k∂tuk2 L2(Ω) If we consider (H4) hypothesis for ∆d, we can bound as follows: e J5+e J6≤ k∂tukL 2p p−2(Ω)k∇dkLp(Ω)k∆(∂te d)kL2(Ω) ≤Ck∂tukβ L2(Ω)k∂tuk1−β H1(Ω) ³k∆dkLq(Ω) +kdkLq(Ω)´k∂te dkH2(Ω) ≤¯εk∂tuk2 H1(Ω) +C¯εnk∂te dk2 H2(Ω) +³k∆dk 2 β Lq(Ω) +kdk 2 β H1(Ω)´k∂tukL2(Ω)o where p=3q 3−q,2 β=2p p−3=2q 2q−3if q < 3. For q > 3, we integrate by parts and: e J5+e J6≈ZΩ |∂tu| |∇(∂te d)| |D2d|dΩ + ZΩ |∇(∂tu)| |∇(∂te d)| |∇d|dΩ = A+B, then, bounding Aas in J4(for the (H4) condition): A≤¯εk∂tuk2 H1(Ω) +Ck∂te dk2 H2(Ω) +CkD2dk 2q 2q−3 Lq(Ω) ³k∂tuk2 L2(Ω) +k∂te dk2 H1(Ω)´ In order to bound B, by simplicity we do not use (H4), but we impose additional regularity on e d, as follows B≤ k∇(∂tu)kL2(Ω)k∇(∂te d)kL3(Ω)k∇dkL6(Ω) ≤¯εk∂tuk2 H1(Ω) +C¯εkdk2 H1(Ω)k∂te dk2 W1,3(Ω) 3.2 Nonhomogeneous Dirichlet boundary condition case 21 hence ∂te d∈L2(0, T;W1,3(Ω)) will be imposed. Integrating by parts the term e J7, e J7=ZΩ (u· ∇)(∂tb d)·∆(∂tb d)dΩ + ZΩ (u· ∇)(∂te d)·∆(∂tb d)dΩ =−ZΩ (∇u· ∇)(∂tb d)· ∇(∂tb d)dΩ + ZΩ (u· ∇)(∂te d)·∆(∂tb d)dΩ Observe that the first term of e J7is similar to J4, replacing dby b d, hence ZΩ |∇u| |∇(∂tb d)|2dΩ≤¯εk∂tb dk2 H2(Ω) +C¯εk∇uk 2q 2q−3 Lq(Ω)k∂tb dk2 H1(Ω), The second term can be estimated differently depending on the range of q: •If 3 2≤q < 3, then: ZΩ (u· ∇)(∂te d)·∆(∂tb d)dΩ≤ kukL 3q 3−q(Ω)k∇(∂te d)kL 6q 5q−6(Ω)k∆(∂tb d)kL2(Ω) ≤Ck∇ukLq(Ω)k∇(∂te d)kα L2(Ω)k∇(∂te d)k1−α H1(Ω)k∂tb dkH2(Ω) ≤¯εk∂tb dk2 H2(Ω) + ¯εnk∂te dk2 H2(Ω) +k∇uk 2 α Lq(Ω)k∇(∂te d)k2 L2(Ω)o where 2 α=2q 2q−3. •If q≥3, only supposing that ∇u∈L2q 2q−3(0, T;Lq(Ω)), we cannot bound this term. By simplicity, we will impose some additional regularity for ∂te din order to get this aim. ZΩ (u· ∇)∇(∂te d)· ∇(∂tb d)dΩ≤ kukL6(Ω)kD2(∂te d)kL3/2(Ω)k∇(∂tb d)kL6(Ω) ≤¯εk∂tb dk2 H2(Ω) +C¯εkuk2 H1(Ω)k∂te dk2 W2,3/2(Ω) (hence ∂te d∈L∞(0, T;W2,3/2(Ω)) will be imposed). 22 4. Regularity for the periodic boundary condition case And therefore, the inequality obtained in the nonhomogeneous Dirichlet boundary case is: d dt ³λ−1k∂tuk2 L2(Ω) +k∂tb dk2 H1(Ω)´+ν λk∂tuk2 H1(Ω) +k∂tb dk2 H2(Ω) ≤Cµk∇uk 2q 2q−3 Lq(Ω) +hk∇dk 2p p−3 Lp(Ω) or k∆dk 2q 2q−3 Lq(Ω) +kdk 2q 2q−3 L2(Ωi¶ ׳k∂tuk2 L2(Ω) +k∂tb dk2 H1(Ω) +k∂te dk2 H1(Ω)´ +Cnkuk2 H1(Ω)k∂te dk2 W2,3/2(Ω) +k∂tb dk2 L2(Ω) +k∂te dk2 L2(Ω) +µkdk2 H1(Ω)k∂te dk2 W1,3(Ω)o, (30) where µ= 0 if (H3) is supposed and µ= 1 if (H4). 4 Regularity for the periodic boundary condition case (proof of Theorem 1.3 with (H1)) Attending to the development of the previous section, if we only impose the regularity hypothesis (H1) on ∇ubut not hypothesis (H3) for ∇dnor hypothesis (H4) for ∆d, for the exponent q∈(3/2,2) ∪(3,+∞) we cannot assure that d∈L∞(0, T;H2(Ω)) ∩ L2(0, T;H3(Ω)), nor that u∈L∞(0, T;H1(Ω))∩L2(0, T;H2(Ω)). In a attempt to improve the regularity for d(without imposing additional regularity hypothesis for d) and in the case of periodic boundary conditions for d, we will study a certain non-hilbertian regularity. To this aim, we take −∇ · (|∇d| ∇d) as test function in the system for d, where separating term by term and integrating by parts (the boundary integral vanish thanks to the periodic boundary conditions on d), we obtain: −ZΩ ∂td·[∇ · (|∇d| ∇d)]dΩ = ZΩ ∂t(∇d)· ∇d|∇d|dΩ = 1 3 d dtk∇dk3 L3(Ω), 5. Uniqueness 23 ZΩ ∆d·[∇ · (|∇d| ∇d)] dΩ = −ZΩ ∇(∆d)∇d|∇d|dΩ = ZΩ D2d∇(∇d|∇d|)dΩ =ZΩ |D2d|2|∇d|dΩ + µ2 3¶2ZΩ |∇ ¡|∇d|3/2¢|2dΩ ZΩ (u· ∇)d·∇·(|∇d| ∇d)dΩ = −ZΩ ∇u· ∇d|∇d| ∇ddΩ−ZΩ (u· ∇)|∇d|2 2|∇d|dΩ ≤CZΩ |∇u| ∇d|3dΩ−1 3ZΩ (u· ∇)¡|∇d|3¢dΩ ≤CZΩ |∇u| ∇d|3dΩ We arrive to the inequality: 1 3 d dtk∇dk3 L3(Ω) +γZΩ |D2d|2|∇d|dΩ + γµ2 3¶2 k∇(|∇d|3/2)k2 L2(Ω)dΩ ≤CZΩ |∇u| |∇d|3dΩ−γZΩ fδ(d)∇ · (|∇d|∇d)dΩ (31) The bound for the first term on the right hand side is: ZΩ |∇u||∇d|3dΩ≤ k∇ukLq(Ω)k∇dkL 3q q−1(Ω) ≤ k∇ukLq(Ω)k∇dk3α L3(Ω)k∇dk3(1−α) L9(Ω) ≤ k∇ukLq(Ω)k∇dk3α L3(Ω)k∇(|∇d|3/2)k2(1−α) L2(Ω) ≤¯εk∇(|∇d|3/2)k2 L2(Ω) +C¯εk∇uk1/α Lq(Ω)k∇dk3 L3(Ω) with 1 α=2q 2q−3. Here, we have used the inequality ([4]): k∇dk9 L9(Ω) ≤ k|∇d|3/2k6 L6(Ω) ≤Ck∇ ¡|∇d|3/2¢k6 L2(Ω) (32) The bound we get for the term fδ(d) is: ZΩ fδ(d)∇ · (|∇d|∇d)dΩ≤CZΩ |d|3|∇d||D2d|dΩ≤CZΩ |∇d| |D2d|dΩ ≤¯εZΩ |∇d| |D2d|2dΩ + C¯εZΩ |∇d|dΩ 24 5. Uniqueness Therefore, only imposing the additional regularity (H1) for ∇u, we can conclude that:                    ∇d∈L∞(0, T;L3(Ω)), ∇(|∇d|3/2)∈L2(0, T;L2(Ω)), ZΩ |∇d||∆d|2∈L1(0, T) (33) Moreover, from ∇(|∇d|3/2)∈L2(0, T;L2(Ω)) and the inequality (32), one can be deduced that ∇d∈L3(0, T;L9(Ω)). In consequence, in the periodic boundary condition case, the additional regularity (H3) which was necessary in order to obtain the strong regularity for (LC), is verified for the exponent p= 9. Therefore, the only additional hypothesis to impose is (H1), that is, ∇u∈L2q 2q−3(0, T;Lq(Ω)) for any q > 3/2. 5 Uniqueness (proof of Theorem 1.4 with (H1)) We consider the systems verified by ¯ u=u1−u2,¯ d=d1−d2and ¯π=π1−π2, where (u1,d1, π1) and (u2,d2, π2) are two solutions of the Nematic Liquid Crystal system (LC), respectively. Then:                                                              ∂t¯ u−ν∆¯ u+u2· ∇¯ u +λ(∇d2)t∆¯ d+∇¯p=−¯ u· ∇u1−λ(∇¯ d)t∆d1in Q, ∇ · u=0in Q, ∂t¯ d−γ∆¯ d+¯ u· ∇d2=−u1· ∇¯ d−γhfδ(d2)−fδ(d1)iin Q, ¯ u=0on Σ, ¯ d=0or ∂n¯ d=0or dis space-periodic on Σ, ¯ u|t=0 =0,¯ d|t=0 =0in Ω. (34) 5. Uniqueness 25 Arguing in a formal way, we take ¯ das test functions in the system for ¯ dand use the equality: fδ(d2)−fδ(d1) = −1 δ2h(|d1|2−1) d+³(d1+d2)·d´d2i, obtaining: 1 2 d dtk¯ dk2 L2(Ω) +γk∇¯ dk2 L2(Ω) =−ZΩ (¯ u· ∇)d2·¯ ddΩ −γZΩ (|d1|2−1) |¯ d|2dΩ−γZΩ [(d1+d2)·¯ d] (d2·¯ d)dΩ which can be rewritten as (using that |d1|and |d2| ≤ 1): 1 2 d dtk¯ dk2 L2(Ω) +γk∇¯ dk2 L2(Ω) ≤ZΩ |¯ u| |∇¯ d| |d2|dΩ + Ck¯ dk2 L2(Ω) ≤¯εk∇¯ dk2 L2(Ω) +C¯εk¯ uk2 L2(Ω) +Ck¯ dk2 L2(Ω) Thus, we obtain: d dtk¯ dk2 L2(Ω) +γk∇¯ dk2 L2(Ω) ≤C³γk¯ dk2 L2(Ω) +k¯ uk2 L2(Ω)´(35) Now, we add the estimates obtained taking λ−1¯ uas test functions in the system for ¯ u, and −∆¯ das test functions in the system for ¯ d. Observe that the terms ZΩ ¯ u·∇d2·∆¯ d cancel. Moreover, adding (35) to the resulting estimate, we obtain: d dt ³λ−1k¯ uk2 L2(Ω) +k¯ dk2 H1(Ω)´+ν λk¯ uk2 H1(Ω) +γk∇¯ dk2 H1(Ω) ≤ −λ−1ZΩ ¯ u· ∇u1¯ udΩ−ZΩ ¯ u· ∇¯ d∆d1dΩ + ZΩ u1· ∇¯ d∆¯ ddΩ +γZΩ³fδ(d2)−fδ(d1)´·∆¯ ddΩ + C³k¯ uk2 L2(Ω) +k¯ dk2 L2(Ω)´ (36) The terms on the right hand side of (36) can be bound as follows: −ZΩ ¯ u· ∇u1¯ udΩ≤ZΩ |¯ u|2|∇u1|dΩ≤ k¯ uk2 L 2q q−1(Ω)k∇u1kLq(Ω) ≤ k¯ uk 2q−3 q L2(Ω)k∇¯ uk 3 q L2(Ω)k∇u1kLq(Ω) ≤¯εk¯ uk2 H1(Ω) +C¯εk∇u1k 2q 2q−3 Lq(Ω)k¯ uk2 L2(Ω) 32 A AUXILIARY RESULTS Finally, fδ(d)∈L∞(0, T;H2(Ω)) ∩L2(0, T;H3(Ω)), because of:                  kfδ(d)k2 L2(Ω) ≤Ckdk2 L2(Ω) k∇fδ(d)k2 L2(Ω) ≤Ck∇dk2 L2(Ω) kD2fδ(d)k2 L2(Ω) ≤CkD2dk2 L2(Ω) Therefore, we can conclude that ∂td∈L∞(0, T;L2(Ω)) ∩L2(0, T ;H1(Ω)). Now, we focus our attention on ∂tu. Taking ∂tuas a test function in the system for u, we obtain: k∂tuk2 L2(Ω) +νd dtk∇uk2 L2(Ω) ≤C·ZΩ |u|2|∇u|2dΩ + ZΩ |∇d|2|∆d|2dΩ¸ We bound the terms on the right hand side as follows: ZΩ |u|2|∇u|2dΩ≤ kuk2 L3(Ω)k∇uk2 L6(Ω) ≤ kukL2(Ω)k∇ukL2(Ω)k∆uk2 L2(Ω) ∈L1(0, T) ZΩ |∇d|2|∆d|2dΩ≤ k∇dk2 L3(Ω)k∆dk2 L6(Ω) ≤ k∇dkL2(Ω)k∆dkL2(Ω)k∆dk2 H1(Ω) ∈L1(0, T) Then, applying the Gronwall’s Lemma, we conclude that ∂tu∈L2(0, T;L2(Ω)).Finally, we look again at the equation for uand we can easily deduce that ∇p∈L2(0, T;L2(Ω)), and, therefore, that p∈L2(0, T;H1(Ω)). A.4 Proof of Lemma 3.1: Thanks to the Lemmas 2.1 and 2.2, it suffices to prove that u∈L2(0, T;H2(Ω)) and d∈L2(0, T;H3(Ω)). To this aim, we argue as follows: 1. We obtain the regularity ∆d∈L2(0, T;H1(Ω)) using the Prodi’s estimates for d. 2. Once the regularity for dis known, we obtain u∈L2(0, T ;H2(Ω)), using also the Prodi’s estimates for u. A.4 Proof of Lemma 3.1:33 Estimates for ∆d: In the homogeneous Neumann boundary condition case, we take ∆2das test functions in the d-system, obtaining: 1 2 d dtk∆dk2 L2(Ω) +γk∇(∆d)k2 L2(Ω) ≤ZΩ |∇u||∇d||∇(∆d)|dΩ +ZΩ |∇u||∆d|2dΩ + γZΩ ∇(fδ(d)) · ∇(∆d)dΩ (39) For the first term we have (using hypothesis k∇ukL2(Ω) ≤Cand k∇dkH1(Ω) ≤C): ZΩ |∇u||∇d||∇(∆d)|dΩ≤Ck∇ukL2(Ω)k∇dkL∞(Ω)k∇(∆d)kL2(Ω) ≤Ck∇dk1/2 H1(Ω)k∇dk1/2 H2(Ω)k∇(∆d)kL2(Ω) ≤Ck∇dk3/2 H2(Ω) ≤¯εk∇dk2 H2(Ω) +C¯ε. In a similar way, we obtain for the second term: ZΩ |∇u||∆d|2dΩ≤ k∇ukL2(Ω)k∆dk2 L4(Ω) ≤Ck∆dk1/2 L2(Ω)k∆dk3/2 H1(Ω) ≤Ck∆dk3/2 H1(Ω) ≤¯εk∆dk2 H1(Ω) +C¯ε And the last term can be bound as follows: γZΩ ∇(fδ(d)) · ∇(∆d)dΩ≤¯εk∇(∆d)k2 L2(Ω) +C¯εk∇dk2 L2(Ω) Therefore, integrating in time (39), we arrive at d∈L2(0, T;H3(Ω)). The estimates for the nonhomogeneous Dirichlet boundary condition case do not present any additional difficulty. Estimates for u: In the case of Prodi’s estimates for u, we take Auas test functions, obtaining : d dtk∇uk2 L2(Ω) +kAuk2 L2(Ω) ≤ZΩ |u|2|∇u|2dΩ + ZΩ |∇d|2|∆d|dΩ The terms on the right hand side can be bound in the following way: ZΩ |u|2|∇u|2dΩ≤ kuk2 L∞(Ω)k∇uk2 L2(Ω) ≤CkukH1(Ω)kukH2(Ω) ≤CkAukL2(Ω) ≤¯εkAuk2 L2(Ω) +C¯ε 34 REFERENCES ZΩ |∇d|2|∆d|2dΩ≤ k∇dk2 L∞(Ω)k∆dk2 L2(Ω) ≤Ck∇dkH1(Ω)k∇dkH2(Ω) ≤Ck∇dkH2(Ω) ≤CkdkH3(Ω) Using the regularity obtained previously, we can conclude that u∈L2(0, T;H2(Ω)). Acknowledgements The authors have been partially supported by the project BFM2003-06446-C02-01, and the third author by the project CNPq-Brasil 3013541-03-0. References [1] H. Beir˜a da Veiga, A new regularity class for the Navier-Stokes equations in Rn. Chinese Ann. Math. 16 (1995), no. 4, 407–412. [2] H. Beir˜a da Veiga, A sufficient condition on the pressure for the regularity of weak solutions to the Navier-Stokes equations. J. Math. Fluid Mech. 2 (2000), 99–106. [3] L. C. 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