Global solution of nematic liquid crystals models

Global solution of nematic liquid crystals models Francisco GUILL´ EN-GONZ´ ALEZ a, Marko ROJAS-MEDAR b Abstract. We prove existence of a global weak solution for a nematic liquid crystal problem by means of a penalization method using a simplified Ericksen-Leslie model and a new compactness property for the gradient of the director field. 1 Introduction In this Note, we establish the well-posedness in the large of a nematic liquid crystals model (formulated for instance in [7]) by means of a penalisation argument using a simplified Ericksen-Leslie model with the Ginzburg-Landau approximation [1, 2]. Let us consider a simplified version of the Ericksen-Leslie model, introduced by Lin in [4] and analysed by Lin and Liu [5, 6] who used a modified Galerkin approach, and by Shkoller [8] who relied on a contraction mapping argument coupled with appropriate energy estimates. This model is a modified Navier-Stokes system that take into account of the liquid crystallinity, coupling with the GinzburgLandau equations. A full version of this Ericksen-Leslie model has been recently studied by Coutand and Shkoller in [3], where local well-posedness (global for small enough data) is proved. Now, we are interested in the asymptotic behaviour respect to the penalisation parameter. The unknowns are the time-dependent divergence-free velocity field u(t, x) and pressure p(t, x) of the fluid and the director field d(t, x) representing the orientation of the liquid crystals molecules. The fluid is confined in an open bounded domain Ω ⊂IRn(n= 2 or 3) with boundary ∂Ω of C2type. In the penalised model one verifies the constraint |d| ≤ 1 as consequence of a maximum principle for the Ginzburg-Landau equation where the approximation fε(d) = ε−2(|d|2−1)d is considered (ε > 0). Here |d|=|d(t, x)|denotes the punctual Euclidean norm in IRn. This penalisation function exhibits potential structure, i.e. there exists a potential function Fε(d) = ε−2(|d|2−1)2 such that fε(d) = ∇d(Fε(d)) for all d∈IRn. Accordingly, we consider the penalised model in (0, T)×Ω as follows |d| ≤ 1, ∂td+u· ∇d+γ(fε(d)−∆d) = 0 (1) ∂tu+u· ∇u−ν∆u+∇p+λ∇ · (∇d¯ ∇d) = 0 (2) ∇· u= 0 (3) u|∂Ω= 0,d|∂Ω=h(4) u|t=0 =u0,d|t=0 =d0(5) 1 Here, u0and d0are, respectively, the initial velocity and director fields. In order to obtain the dissipativity of the model, we shall assume (as in all previous works) time independent (Dirichlet) boundary data for the director field d, given by h:∂Ω→IRn. Concerning the coefficients, ν > 0 represents the viscosity of the fluid, λ > 0 is an elasticity constant, ε > 0 is a penalisation parameter (with respect to the unitary constraint), and γ > 0 is a relaxation-time constant. We have used the tensorial notation (∇d¯ ∇d)ij = n X k=1 ∂xidk∂xjdk In the ε-limit model we will find the restriction |d|= 1 and the Lagrange multiplier associated |∇d|2d. Indeed, when ε→0 we will find a limit problem, where it changes (1) by |d|= 1, ∂td+u· ∇d−γ∆d−γ|∇d|2d= 0.(6) Let us introduce the following space of functions H1 h={d∈H1(Ω)n/d=hon ∂Ω} H={u∈L2(Ω)n/∇ · u= 0,u·n= 0 on ∂Ω} V={u∈H1 0(Ω)n/∇ · u= 0}. For simplicity, let us denote L2,H1instead of L2(Ω)n, H1(Ω)netc. Our main result is the following Theorem 1.1 Let T > 0and Ω⊂IRnbe an open, bounded and C2domain. Let us assume u0∈H, d0∈H1 hand h∈H2such that |d0|= 1 in Ωand |h|= 1 on ∂Ω. Then, there exists a global weak solution u∈L2(0, T;V)∩L∞(0, T;H),d∈L∞(0, T;H1 h)of the limit problem (2)-(6) obtained as a limit of “semi-strong” solutions uε∈L2(0, T;V)∩L∞(0, T ;H),dε∈L2(0, T;H2∩H1 h)∩L∞(0, T;H1 h)of the coupled Navier-Stokes and Ginzburg-Landau model (1)-(5) as εgoes to zero. Remark 1 Up to our known, this theorem is the first result of existence of a global in time solution (without restrictions on the data) of the limit problem (2)-(6). In [7], Prohl proves existence (and uniqueness) of a local in time strong solution of (2)-(6). On the other hand, Lin and Liu studied in [6] the asymptotic behaviour of (1)-(5) when εgoes to zero, but the limit of the (strongly) nonlinear terms ∇dε¯ ∇dεis only obtained towards a measure valued tensor M. The main contribution in this note is to identify M with the limit tensor ∇d¯ ∇d. Remark 2 Notice that in the ε-limit problem (2)-(6) one loses the H2-regularity for the director field d, hence although (uε,dε)verifies (1) point-wise a.e. (t, x), their limit (u,d)verifies (6) only in a distributional sense. 2ε-approximate solutions and dissipativity For each ε > 0, let us consider a “semi-strong” solution (uε,dε) of the ε-approximate problem (1)-(5), that is uε∈L2(0, T;V)∩L∞(0, T;H),dε∈L2(0, T;H2∩H1 h)∩L∞(0, T;H1 h),(7) 2 verifying the u-system (2) in the distributional sense and the d-system (1) point-wise a.e. The boundary conditions are verified in the trace sense. Finally, the initial conditions have classical sense; indeed uε and dεare time-continuous functions, as consequence of the additional regularity ∂tuε∈Lp(0, T;V0) and ∂tdε∈Lp(0, T;L2) (p= 2 if n= 2 or p= 4/3 if n= 3) that can be obtained applying the previous regularity (7) to the equations (1) and (2). The existence of (uε,dε) can be proved ([5]) by means of three main arguments: a semi-galerkin method (space-discretization of the u-system (2), remaining the d-system (1) in the continuous sense), a maximum principle for the d-system in order to obtain the constraint |dε| ≤ 1 and the following energy inequality, d dt µ1 2kuεk2+λ 2k∇dεk2+λZΩ Fε(dε)dx¶+νk∇uεk2+λγkfε(dε)−∆dεk2≤0,(8) obtained taking respectively λ(fε(dε)−∆dε) and uεas test functions in (1)-(2) and using the equality ∇ · (∇d¯ ∇d) = ∇(|∇d|2/2) + ∇dt∆d. In (8), k·kdenotes the L2(Ω)-norm. 3ε-independent estimates For simplicity, let us denote L2(H1) instead of L2(0, T;H1), etc. From the maximum principle in (2) |dε| ≤ 1 a.e. (t, x). On the other hand, from the energy inequality (8), one has the following (ε-independent) estimates: uεis bounded in L∞(H)∩L2(V),(9) dεis bounded in L∞(0, T;H1 h),(10) wε:= fε(dε)−∆dεis bounded in L2(L2),(11) Fε(dε) is bounded in L∞(L1).(12) Moreover, applying estimates (9)-(11) in equations (1)-(2), ∂tuεis bounded in L2(V0) + L∞((W1,r ∩V)0), r > n, (13) ∂tdεis bounded in L2(L2) + Lq(L4/3), q = 4 if n= 2 or q= 8/3 if n= 3. (14) Finally, (10) implies in particular Mε:= ∇dε¯ ∇dεis bounded in L∞(L1).(15) 3 4ε-convergence From previous estimates (9)-(15) and compactness results of Aubin-Lions type [9], there exists subsequences (equally denoted) dε,uε,wε, Mεand their respective limit functions d,u,w, M, such that uε→uin L∞(H) weak?, L2(V) weak, L2(H) strong dε→din L∞(H1) weak?, C(L2) strong, ∂tuε→∂tuin L2(V0) + L∞((W1,r ∩V)0) weak, ∂tdε→∂tdin L2(L2) + Lp(L4/3) weak, wε→win L2(L2) weak, ∇dε¯ ∇dε→Min the measure sense. Consequently, taking limits as ε→0 and using De Rham’s lemma [10], we arrive at ∂td+u· ∇d+γw= 0 (16) ∂tu+u· ∇u−ν∆u+∇p+λ∇ · M= 0 (17) ∇· u= 0 (18) u|∂Ω= 0,d|∂Ω=h(19) u|t=0 =u0,d|t=0 =d0(20) On the other hand, since ε−1(|dε|2−1) is bounded in L∞(L2) (using (12)) and dε→dpoint-wise a.e. (t, x), one get the unity constraint |d(t, x)|= 1 a.e. (t, x). Therefore, in order to finish the proof of Theorem 1.1, we have to identify wwith −∆d− |∇d|2dand Mwith ∇d¯ ∇d. 5 Identification of w =−∆d− |∇d|2d. The argument of this section is based in the known literature on harmonic functions with values in the unit sferic surface (see for instance [1, 2] and references therein cited). In particular, we will use the following result, which is a slightly modification (introducing the convection terms) of Lemma 2.2 in [2] (see also Lemma 7.1 in [6]): Lemma 5.1 The following two systems are equivalent: ∂td+u· ∇d−γ∆d=γ|∇d|2d and |d|= 1,(∂td+u· ∇d)∧d−γ∇ · (∇d∧d) = 0.(21) Since we already have that |d|= 1, it suffices to verify the equation of (21). Indeed, making the vectorial product of equation (2) by dε, taking into account that fε(dε)∧dε= 0 and −∆dε∧dε=−∇·(∇dε∧dε), one has (∂tdε+uε· ∇dε)∧dε−γ∇ · (∇dε∧dε) = 0.(22) Making ε→0, we can deduce (21), using the strong convergences of uεand dεand the weak convergences of ∂tdεand ∇dε. 4 6 Identification of M=∇d¯ ∇d. The key of this proof is to obtain L2-compactness for ∇dε, using some ideas of the optimisation framework. Indeed, since wε=−∆dε+fε(dε), then dε(t) can be viewed as a solution of the optimisation (without constraints) problem Jε(dε(t)) = min d∈H1 h(Ω) Jε(d)·=ZΩµ1 2|∇d|2+Fε(d)−wε(t)·d¶¸. On the other hand, a.e. tlet us define ˜ d(t) as a solution of the optimisation (with constraints) problem J(˜ d(t)) = min d∈H1 h(Ω) |d|=1 J(d)·=ZΩµ1 2|∇d|2−w(t)·d¶¸. Obviously, Jε(dε(t)) ≤Jε(˜ d(t)).In particular, ZT 0ZΩµ1 2|∇dε(t)|2+Fε(dε(t)) −wε(t)·dε(t)¶≤ZT 0ZΩµ1 2|∇˜ d(t)|2−wε(t)·˜ d(t)¶.(23) Taking limit inf as ε→0 in (23), bounding previously Fε(dε(t)) ≥0, ZT 0 J(d(t)) ≤lim inf ε→0ZT 0ZΩµ1 2|∇dε(t)|2−wε(t)·dε(t)¶ ≤lim ε→0ZT 0ZΩµ1 2|∇˜ d(t)|2−wε(t)·˜ d(t)¶=ZT 0 J(˜ d(t)), hence one has the equality RT 0J(d(t)) = RT 0J(˜ d(t)) (the opposite inequality is easy to deduce since |d(t)|= 1). Consequently, all the previous inequalities are equalities, and in particular ∃lim ε→0ZT 0ZΩµ1 2|∇dε(t)|2−wε(t)·dε(t)¶=ZT 0 J(d(t)) = ZT 0ZΩµ1 2|∇d(t)|2−w(t)·d(t)¶ Since ∃lim ε→0RT 0RΩwε(t)·dε(t) = RT 0RΩw(t)·d(t) (using the strong convergence of dε), one has ∃lim ε→0k∇dεk2 L2(L2)= k∇dk2 L2(L2)therefore, dε→din L2(H1)−strong. In particular, taking into account the estimates of ∇dε, ∇dε→ ∇din Lp(Lq)−strong,∀p, q :p < ∞, q < 2, hence we can identify M=∇d¯ ∇d. Acknowledgements. The first author has been partially financed by the projet BFM2000-1317, and the second author by the projects CNPq-Brasil 300116-93-4 and Fapesp-Brasil 01/07557-3 5 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] Y. Chen,The weak solutions to the evolution problems of harmonic maps, Math. Z. 201 (1989) 69-74. [3] D. Coutand, S. Shkoller,Well-posedness of the full Ericksen-Leslie model of nematic liquid crystals, Note C.R.A.S., t. 333, S`erie I (2001) 919-924. [4] F.H. Lin,Nonlinear theory of defects in nematic liquid crystals: phase transition and flow phenomena, Comm. Pure Appl. Math. 42 (1989) 789-814. [5] F.H. Lin, C. Liu,Non-parabolic dissipative systems modelling the flow of liquid crystals, Comm. Pure Appl. 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