A uniqueness and regularity criterion for Q-tensor models with Neumann boundary conditions

A uniqueness and regularity criterion for Q-tensor models with Neumann boundary conditions. Francisco Guill´en-Gonz´alez and Mar´ıa ´ Angeles Rodr´ıguez-Bellido Dpto. de Ecuaciones Diferenciales y An´alisis Num´erico and IMUS, Universidad de Sevilla Aptdo. de Correos 1160 41080 Sevilla, Spain [email protected] and [email protected] January 29, 2015 Abstract We give a regularity criterion for a Q-tensor system modeling a nematic Liquid Crystal, under homogeneous Neumann boundary conditions for the tensor Q. Starting of a criterion only imposed on the velocity field utwo results are proved; the uniqueness of weak solutions and the global in time weak regularity for the time derivative (∂tu,∂ tQ). This paper extends the work done in [8] for a nematic Liquid Crystal model formulated in (u,d), where ddenotes the orientation vector of the liquid crystal molecules. Key words: Nematic liquid crystal, Q-tensor model, regularity criteria, uniqueness criteria, Neumann boundary conditions. AMS 2010 subject classification: 35B65, 35K51, 35Q35, 76A15, 76D03 The authors have been partially supported by MINECO Project MTM2012-32325 of Spain Government. 1 Abbreviated version of the title for the running head: Uniqueness-regularity Q-tensor criterion Corresponding author: Mar´ıa ´ Angeles Rodr´ıguez-Bellido Dpto. de Ecuaciones Diferenciales y An´alisis Num´erico, Facultad de Matem´aticas, Universidad de Sevilla Aptdo. de Correos 1160 41080 Sevilla, Spain e-mail: [email protected] 2 1 Introduction 1.1 The model. Liquid crystals are intermediate phases of matter with properties from both solid and liquid states. The macroscopic properties come from the liquid behavior and are modeled by using the velocity and pressure (u,p). The microscopic structure enters into the model through the molecules of liquid crystals. These molecules influence the behavior of the matter and such an influence can be modeled by using different kinds of unknowns, which depend on different theories and the type of liquid crystal. In the nematic case, where the molecules are arranged by layers and every molecule is oriented equally, two main theories appear: the Oseen-Frank theory, where the microscopic structure is modeled by using a director vector d, which is the average of the main orientation of the rod-like molecules of liquid crystals, and the Landau-De Gennes theory, where the vector dis replaced by a tensor Q. The tensor Qis related to the second moment of a probability measure µ(x,·): L(S2)→[0,1] for each x∈Ω, being L(S2) the family of Lebesgue measurable sets on the unit sphere. For any A⊂S2,µ(x,A) is the probability that the molecules with centre of mass in a very small neighborhood of the point x∈Ωare pointing in a direction contained in A. From a physical point of view, this probability (cf. [5, 10]) must satisfy µ(x,A)=µ(x,−A) in order to reproduce the so-called “head-to-tail” symmetry. As a consequence, the first moment of the probability measure vanishes, that is �p�(x)=�S2 pidµ(x,p)=0. Then, the main information on µcomes from the second moment tensor M(µ)ij =�S2 pipjdµ(p),i,j=1,2,3. As a consequence, M(µ)=M(µ)tand tr(M) = 1. If the orientation of the molecules is equally distributed, then the distribution is isotropic and µ=µ0,dµ0(p)= 1 4πdA and M(µ0)=1 3I.The deviation of the second moment tensor from its isotropic value is therefore measured as: Q=M(µ)−M(µ0)=�S2�p⊗p−1 3I�dµ(p), which is the definition for the tensor Q, and from which the symmetry and traceless for Qis deduced. By following the Landau-De Gennes theory, a model to study the behavior of nematic liquid crystals filling a bounded domain Ω⊂R3, with boundary ∂Ω, is given by: �Dtu−ν∆u+∇p=∇·τ(Q)+∇·σ(H,Q),∇·u=0 inΩ×(0,T), DtQ−S(∇u,Q)=−γH(Q)inΩ×(0,T),(1) 3 where u:(0,T)×Ω→R3is the velocity field, p:(0,T)×Ω→Ris the pressure and Q: (0,T)×Ω→R3×3is a symmetric and traceless tensor. The operator Dt(·)=∂t(·)+(u·∇)(·)is the material derivative, ν>0 is the viscosity coefficient, and γ>0 is a material-dependent elastic constant. The tensors τ=τ(Q), σ=σ(H,Q) are defined by: �τij(Q)=−ε(∂jQ:∂iQ)=−ε∂ jQkl ∂iQkl (ε>0), σ(H,Q)=HQ−QH, (2) where H=H(Q)=−ε∆Q+f(Q) and f(Q)=aQ−b(Q2−1 3tr(Q2)) + c|Q|2Qwith a,b∈Rand c>0,(3) hereafter, |Q|2=Q:Q= 3 � i,j=1 Qij Qij denotes the tensor euclidean norm. The stretching term S(∇u,Q) is given by S(∇u,Q)=WQt−QtW(4) where W=1 2(∇u−(∇u)t) is the antisymmetric part of the gradient of u. In [11, 1] the problem is solved in the space of traceless and symmetric matrices (in the whole R3and in a bounded domain Ω⊂R3, respectively). In [11], the existence of global in time weak solutions in 3D, and strong regularity and weak-strong uniqueness results in 2Dare proved. A generalized model considering more stretching effects is studied in [12]. The analysis of a model considering a more general expression for H(Q) (modifying the ∆Q-term of H(Q)) can be seen in [9]. However, a similar study can be made for a system that generalizes (1)-(4), in which the definition of S(∇u,Q) is replaced by: S(∇u,Q)=∇uQt−Qt∇u(5) and the function f(Q) in (3) is replaced by: f(Q)=aQ−b 3�Q2+QQt+QtQ�+c|Q|2Qwith a,b∈Rand c>0.(6) Although this model does not have the restrictions of symmetry and tracelessness, it can be studied in the same way. In fact, the existence of global weak solutions and the weak/strong uniqueness of this model is proved in [7], and the local regularity and uniqueness is obtained in [6]. This study tell us that these (physical) restrictions are not essential in the mathematical analysis. On the other hand, if S(∇u,Q) is taken as in (4), then any weak solution provides a symmetric tensor Q; meanwhile if f(Q) is taken as in (3), any weak solution provides a traceless tensor Q([7]). The PDE system is enclosed with the following initial and boundary conditions: u|t=0 =u0,Q|t=0 =Q0in Ω,(7) 4 u|∂Ω=0,∂ nQ|∂Ω=0 in(0,T). (8) The compatibility condition ∂nQ0|∂Ω= 0 must be satisfied. In the present work, we focus on the problem without restrictions of symmetry and tracelessness (1)-(2), (5)-(8). At the end of the work, the results proven for this model are extended to the model with tracelessness and symmetry (1)-(4), (7)-(8). The manuscript is organized as follows: Next subsection summarizes the results on regularity and uniqueness for the problem (1)-(2), (5)-(8). In Subsection 1.3, the new results in this paper are stated: Theorems 1.3 and 1.4, and Corollaries 1.5 and 1.6. Section 2 is devoted to the proof of Theorem 1.3 where the spatial treatment of the boundary term is made in Subsection 2.1. The proof of Theorem 1.4 is made in Section 3. The last section analyzes how to extend the results in Subsection 1.3 to the problems imposing symmetry and/or tracelessness. 1.2 Known results on regularity and uniqueness The weak solution for these models is defined extending the Navier-Stokes case. Specifically, (u,Q) is said a weak solution in (0,T) of (1) if u∈L∞(0,T;L2(Ω)) ∩L2(0,T;H1(Ω)),Q∈L∞(0,T;H1(Ω)) ∩L2(0,T;H2(Ω)), satisfying the u-system (1)1in a variational setting and the Q-system (1)2point-wisely. The existence of weak solutions is proved in [11, 12] for the whole R3and in [1, 6] for a bounded domain of R3, considering Dirichlet boundary condition for utogether with Dirichlet or Neumann boundary conditions for Q, or periodic boundary conditions for (u,Q). These results are based on the following energy equality (see [6]): d dt �1 2�u�2 L2(Ω)+�Ω E(Q)dx�+ν�∇u�2 L2(Ω)+γ�H�2 L2(Ω)=0, where the free energy is E(Q)=�Ω�ε 2|∇Q|2+F(Q)�with the functional F(Q) defined as F(Q)=a 2|Q|2−b 3(Q2:Q)+c 4|Q|4.(9) Note that, in the case of f(Q) defined by (6), it is easy to check that F�(Q)=f(Q) and the tensor His the variational derivative in L2(Ω) of E(Q), that is H=δE(Q) δQ(see [6]). On the other hand, in the case of bounded domains, the Q-system (1)2satisfies a maximum principle (see [6]), hence in particular Q∈L∞(0,T;L∞(Ω)).(10) However, the uniqueness of weak solutions needs additional regularity for ∇uand ∆Q,which corresponds to the criterion proved by Berselli in [3] for the Navier-Stokes system. Concretely, the following result is proved in [6]: 5 Theorem 1.1 (Uniqueness criteria) Assume (u0,Q 0)∈L2(Ω)×H1(Ω). Let (u,Q)be a weak solution in (0,T)of problem (1) such that ∇uand ∆Qhave the additional regularity ∇u∈L2q/(2q−3)(0,T;Lq(Ω)) for 2≤q≤3,(11) ∆Q∈L2s/(2s−3)(0,T;Ls(Ω)) for 2≤s≤3.(12) Then, this solution coincides in (0,T)with any weak solution associated to the same data. We can study two types of regularity for weak solutions (u,Q) of (1): •Strong regularity (as in the Navier-Stokes framework): (St)    u∈L∞(0,T;H1(Ω)) ∩L2(0,T;H2(Ω)),∂ tu∈L2(0,T;L2(Ω)), Q∈L∞(0,T;H2(Ω)) ∩L2(0,T;H3(Ω)),∂ tQ∈L∞(0,T;L2(Ω)) ∩L2(0,T;H1(Ω)). •Weak regularity for (∂tu,∂ tQ) or “weak-t” solution: (w-t)    ∂tu∈L∞(0,T;L2(Ω)) ∩L2(0,T;H1(Ω)),u∈L∞(0,T;H1(Ω)), ∂tQ∈L∞(0,T;H1(Ω)) ∩L2(0,T;H2(Ω)),Q∈L∞(0,T;H2(Ω)). Under homogeneous Neumann or Dirichlet boundary conditions for Q, the existence and uniqueness of global in time strong solutions are obtained in [7] if the viscosity νis large enough. Otherwise, when Q|∂Ω= 0 is imposed, and the stretching term S(·,·) is defined as in (4), then the existence and uniqueness of strong solutions are obtained in [7]; either local in time, or global one if ∇u satisfies the regularity criterion given below in (13) (but regularity criterion for ∆Qgiven in (14) can be avoided). The existence of local in time strong solutions when Q|∂Ω= 0 are also obtained via a fixed-point argument in [2]. Remark 1.1 When the model (1)-(7) is considered in the whole space Ω=R3, some regularity criteria to obtain global in time strong solution (in the sense of (St)) are given in [4]. One of them imposes that ∇uhas the regularity appearing in (11) for 2<q≤3. Following Remark 3.2 in [7], if we consider the model (1)-(6) with space-periodic boundary conditions for (u,Q), the regularity criterion only given for ∇uin (11) implies the global in time strong solution in the sense of (St) (no hypothesis for the symmetry of Sis assumed). Observe that in [7] the case q=2in (11) is also considered. However, as far as we know, the previous strong regularity results when Q|∂Ω= 0 cannot be extended to either non-homogeneous Dirichlet or Neumann boundary conditions for Q.Inthese cases, some boundary integrals appearing in the strong estimates argument do not vanish and it is not clear how to bound them. 6 In order to circumvent this difficulty, the weak-t concept (w-t) is considered in [1, 7], obtaining weak-t solutions either local in time for any data or global in time assuming some regularity criteria [7], as can be summarized in the following result: Theorem 1.2 (Regularity criteria for global in time weak-t solution) Assume (u0,Q 0)∈ H2(Ω)×H3(Ω)and let (u,Q)be a weak solution in (0,T)of problem (1) having the additional regularity: ∇u∈L2q/(2q−3)(0,T;Lq(Ω)),3/2≤q≤3,(13) ∆Q∈L2s/(2s−3)(0,T;Ls(Ω)),3/2≤s≤3.(14) Then, this weak solution (u,Q)is a weak-t solution in (0,T)and coincides with any weak solution in (0,T)associated to the same data. For some models of nematic liquid crystals without stretching terms (see [8]), the criteria (11) and (12) to obtain uniqueness of weak solutions or (13) and (14) to obtain strong regularity could be replaced by the following criteria of Serrin’s type ([13]): u∈L2p/(p−3)(0,T;Lp(Ω)) and ∇Q∈L2r/(r−3)(0,T;Lr(Ω)),3≤p, r ≤+∞.(15) Moreover, when periodic boundary conditions for (u,Q) are considered, the previous regularity criteria imposed for ∆Qor ∇Qare not necessary. Remark 1.2 When Q≡0the Q-tensor model reduces to the classical Navier-Stokes problem. In this case, the regularity hypotheses for u(13) and (15)1correspond to Berselli’s and Serrin’s regularity criteria for the Navier-Stokes equations (see [3] and [13], respectively). Indeed, the upper constraint q≤3in (13) comes from the terms depending on Q(see [6, 7]), and therefore they do not appear if Q≡0. In this work, under homogeneous Neumann boundary conditions for Q, a non-hilbertian regularity for ∇Qis obtained only imposing a regularity criteria for ∇u. This non-hilbertian regularity argument for ∇Qfollows the steps given in [8] for nematic liquid crystals without stretching terms and periodic boundary conditions. Afterwards, we use this new regularity for ∇Q(and the regularity criterion for ∇u) to deduce the additional regularity for ∆Qconsidered in (12) and (14) for the particular index s=5/2. This regularity for ∆Qis based on a non-hilbertian regularity result for a heat-Neumann problem (see [14]) and the L∞-regularity for Qgiven by a maximum principle result ([6]). 7 1.3 The new results of this paper Theorem 1.3 Let (u,Q)be a weak solution of (1)-(2), (5)-(8) such that ∇usatisfies (11) and Q satisfies (10). Then, ∇Q∈L∞(0,T;L3(Ω)) ∩L3(0,T;L9(Ω)). The proof is given in Section 2. Theorem 1.4 Let Ω⊂R3be a bounded domain with boundary ∂Ωof class C2+�for some �>0. Let (u,Q)be a weak solution of (1)-(2), (5)-(8) with Qsatisfying (10). Assume (u0,Q 0)∈L2(Ω)× B2−2/γ γ(Ω)and hypotesis (13) for ∇u, where γ=min{q,2q/(2q−3)}and 3/2≤q≤3is the exponent given in (13). Moreover, under the following hypothesis for the tensor: ∇Q∈L∞(0,T;L3(Ω)), the solution (u,Q)satisfies the additional regularity Q∈Lγ(0,T;W2,γ(Ω)) and ∂tQ∈Lγ(0,T;Lγ(Ω)).(16) Note that, since 3/2≤q≤3, then 3/2≤γ≤5/2 and 2/3≤2−2/γ≤6/5. Remark 1.3 Here, B2−2/γ γ(Ω)is a Besov space. In fact, Besov spaces Bl γ(Ω)coincides with the well-known Sobolev spaces Wl,γ(Ω)when the derivability exponent lis non-integer ([14]). In order to define a Besov space Bl γ(Ω)for integer l, it suffices to start with a one-variable function f:R→R where the norm in Bl γ(R)is given by �f�Bl γ(R)=�f�Lγ(R)+��∞ −∞ �∞ 0���� ∂l−1f(x+2h) ∂xl−1−2∂l−1f(x+h) ∂xl−1+∂l−1f(x) ∂xl−1���� γ1 h1+γdh dx�1/γ . Then, the generalization to Bl γ(Ω)for a n-dimensional domain Ωcan be made via a partition of unity argument associated to overlapping subsets of Ω, more details can be found in [14]. The choice of the initial data Q0in the Besov space B2−2/γ γ(Ω)responds to the use of the result given below in Theorem 3.1. We give the proof of Theorem 1.4 in Section 3. Theorems 1.3 and 1.4 imply, in particular, that ∆Q∈Lγ(0,T;Lγ(Ω)) for 3/2≤γ≤5/2.(17) Regularity appearing in (17) can be seen as the regularity criteria of (12) or (14) if and only if γ=5/2, because this is the unique case in (12) or (14) where s=2s/(2s−3) = 5/2. Therefore, since in Theorem 1.4 γ=min{q,2q/(2q−3)},thenγ=5/2 corresponds with q=5/2 in the regularity criterion for ∇ugiven in (11) or (13), that is ∇u∈L5/2(0,T;L5/2(Ω)). As a consequence, by using q=γ=5/2 the following corollaries are deduced, improving Theorems 1.1 and 1.2, respectively: 8 Corollary 1.5 (Uniqueness criteria) Assume (u0,Q 0)∈L2(Ω)×W2−2/(5/2),5/2(Ω). Let (u,Q) be a weak solution of (1)-(2), (5)-(8) such that ∇usatisfies the regularity criterion: ∇u∈L5/2(0,T;L5/2(Ω)).(18) Then, this solution coincides in (0,T)with any weak solution associated to the same data. Corollary 1.6 (Regularity criteria) Let (u,Q)be a weak solution in (0,T)of (1)-(2), (5)-(8). If (u0,Q 0)∈H2(Ω)×H3(Ω)and ∇uhas the additional regularity (18), then (u,Q)is the unique weak-t solution of (1)-(2), (5)-(8) in (0,T). 2 Proof of Theorem 1.3. We can argue as in [8] for a Nematic Liquid Crystal model (without stretching term and periodic boundary conditions). Indeed, by taking −∇·�|∇Q|p−2∇Q�,p≥2, as test function in the Q-system of (1), using the incompressibility equation ∇·u= 0 and the non-slip boundary condition u|∂Ω= 0, we obtain: −�Ω (∂tQ+(u·∇)Q−γε∆Q):[∇·(|∇Q|p−2∇Q)]dx =1 p d dt�∇Q�p Lp(Ω)+γε�Ω |∇Q|p−2|D2Q|2dx +�Ω |∇Q|p−2(∇u·∇)Q:∇Qdx+1 p�Ω (u·∇)(|∇Q|p)dx +γε�2 p�2 (p−2) �Ω���∇�|∇Q|p/2���� 2dx−γε�∂Ω |∇Q|p−2(∂k(∇Q)nk):∇Qdσ x To treat the potential term F�(Q)=f(Q), we use the argument given in [6] splitting f(Q)= F� c(Q)+F� e(Q), where F� c(Q)=c|Q|2Q is the derivate of the convex part Fc(Q)=c 4|Q|4(see (9)) and F� e(Q)=aQ−b 3(Q2+QQ t+QtQ) is the derivate of the rest of F(Q). Thus, −�Ω F� c(Q):�∇·�|∇Q|p−2∇Q��dx=�Ω |∇Q|p−2∇�F� c(Q)�:∇Qdx =c�Ω |∇Q|p−2�|Q|2|∇Q|2+2|(Q:∇Q)|2�dx≥0, 9 [4] J. Fan & T. Ozawa. Regularity criteria for a coupled NavierStokes and Q-tensor system,Int. J. Anal., (2013) (pp. Art. ID 718173, 5). [5] P. de Gennes & J. Prost. The Physics of Liquid Crystals. International Series of Monographs on Physics, 83. 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