# Nematic Liquid Crystals. Existence and Uniqueness of Periodic Solutions to Ericksen-Leslie Equations

код для вставкиСкачатьGenerated by Foxit PDF Creator © Foxit Software http://www.foxitsoftware.com For evaluation only. Nematic Liquid Crystals. Existence and Uniqueness of Periodic Solutions to Ericksen-Leslie Equations G.A. Chechkin, Department of Differential Equations Faculty of Mechanics and Mathematics M.V. Lomonosov Moscow State University T.S. Ratiu, École polytechnique f édérale de Lausanne, CH-1015 Lausanne, Switzerland & IMISS M.V. Lomonosov Moscow State University M.S. Romanov, Department of Differential Equations Faculty of Mechanics and Mathematics M.V. Lomonosov Moscow State University V.N. Samokhin, Moscow State University of Printing Art 2A, Pryanishnikova ul. Introduction Liquid crystals were discovered in 1888 by an Austrian botanist Friedrich Reinitzer, but the debates about their existence and properties have continued until the mid XX. Only in 1940, V.N. Tsvetkov formulated modern concepts of physics of liquid crystals, using viscosity, dielectric and diamagnetic anisotropy of liquid crystals. Tsvetkov created a general theory of the mesomorphic state. He created methods allowed to understand the dynamic properties of the mesophases. In 1963, American James Fergason used liquid crystals in practical purposes, for detecting thermal fields. After that, interest in liquid crystals has increased dramatically. Scientists say thermotropic liquid cristals (formed by heating the solid) or liotropic liquid cristals (formed in mixtures of solids with solvents). The thermotropic liquid cristals may be nematics or smectics (see Fig. 1). In all these types of LC the orientation of dipole molecules in some direction defined by a unit vector which is called "director". The study of liquid crystals nematodynamics (hydrodynamics of nematic liquid crystals) is of our interest. 139 Generated by Foxit PDF Creator © Foxit Software http://www.foxitsoftware.com For evaluation only. Fig. 1. The structure of Nematic (left) and Smectic (right) liquid cristals Applied questions of the theory of liquid crystals occur in many fields, including the printing industry. Visualization of products, improving of image resolution are connected in particular with behavior of liquid crystals. 1. Problem setup Subject of our research is Ericksen-Leslie system = p ( n) x F f , divu = 0 u u j n x j (1) J q g G 2 = , | |= 1 n n h n Here u is the velocity vector, n is the director; > 0 is the viscosity, J > 0 is the moment of inertia; F( x, t ), G( x, t ) are external forces. Terms f and g corresponds the dissipative part of the stress tensor and dissipative part of internal body force respectively and depends on The function defined as u , n and their derivatives. (n, n ) is called Oseen-Z öcher-Frank free energy and is (n, n) = K1n = curln 1 K11 (divn) 2 K 22 (n curln) 2 K 33 | n curln |2 ; 2 h= ( )x . n n x j j 140 Generated by Foxit PDF Creator © Foxit Software http://www.foxitsoftware.com For evaluation only. The pressure p determine by condition divu = 0 and the lagrangian multiplier 2 q by condition | n |= 1. We are interested in non-dissipative case, i.e. g = 0, f = 0. Also K1 = 0, thus the fluid is supposed to be nematic one, and for simplicity we study one-constant approximation K11 = K 22 = K 33 = K > 0. In this case system (1) could be replaced with the system u u = p ( Kn x n ) x F , divu = 0, (2) J = K n n G , (3) n = n (4) j j where is a new unknown vector function (see [1]). This system is connected with Ericsen-Leslie one in following way: if the initial conditions satisfies identities | n( x, 0) |= 1, n( x, 0) ( x, 0) then for any t >0 | n | 1, = n n , 2q = n h J | |2 and (2) – (4) turns into (1). Consider a domain QT = (0, T ) , where = 2 /[0,1]2 is two-dimensional torus. In this domain we study system (2) – (4) with initial conditions (5) u(0, x ) = u 0 , (0, x ) = 0 , n(0, x) = n 0 . Here u , , n are supposed to be unknown vector-functions, an unknown scalar function; J, K, are fixed positive numbers. p is 2. Notations and definitions Here and later we use the following notations: the full time derivative of f ; bold b stands for a 3- dimensional vector (b1; b 2 ; b3 ); ai bi means the sum 2 ab; i =1 i i Sol () = {v : 3 such as QT = (0, T ) 141 f f t u j f x is j v C (), div v = 0}; Generated by Foxit PDF Creator © Foxit Software http://www.foxitsoftware.com For evaluation only. Sol (QT ) = {v C (QT ) : t v(t , ) Sol ()}; Sol2 () is a closure of Sol () in the norm L2 (); Sol2m () is a closure of Sol () in the norm W2m (); Definition 1. Quadruple (u, , n, p) is a strong solution of Q = (0, T ) if u is a vector field of the class L2 (0, T ; Sol ()), ut L2 (QT ); n is a vector field from L (0, T ;W21 ()), t L (0, T ; L2 ); n is a vector field from L (0, T ;W22 ()); p L2 (QT ); u, n, n satisfies initial conditions (5), i.e. (u, n, ) (u 0 , n 0 , 0 ) weakly in L2 () as t 0, and the equations (2)–(4) take place almost everywhere. problem (2)–(5) in domain 2 2 (2)–(5). Our aim is to prove the existence of the solution for the problem Theorem 1. Suppose Let F, G be equal to zero . u 0 Sol22 ( ), 0 W21 ( ), n 0 W22 (). Then for some T > 0 the solution (in terms of Definition 1) of problem (2)–(4) does exist. 3. Galerkin-type approximations Select two sequences of subspaces E 1 E 2 and 2 k F 1 F 2 such as E k is dense in Sol2 ( ) and F is dense in W22 ( ) (and, consequently, in W21 () ). Since embeddings W21 () L2 () and Sol21 () Sol2 () are both compact, : Sol () Sol () and is a self-adjoint operator in these spaces, the eigenbases of the operator in Sol2 () and L2 () do exist. Denote E k and F k to be a linear span of first k eigenfunctions in corresponding spaces. Constructed subspaces consist of smooth functions and thus are dense in 142 Sol22 ( ) and W22 ( ) respectively. Generated by Foxit PDF Creator © Foxit Software http://www.foxitsoftware.com For evaluation only. Proposition 1. projection of initial data Suppose k is fixed and (u 0 k , 0, k , n 0,k ) is a (u 0 , 0 , n 0 ) on E k F k F k . Then for some T > 0 a solution (u k , k , n k ) C (0, T ; E k F k F k ) of the problem (u k , )t = (ukl u k , x , ) ( u k , ) K ( n k n k , ), (6) l J ( k , ) t = ( Ju kl n k , x , ) K ( n k n k , ), (7) (nk , )t = (ukl n k , x , ) ( k n k , ), (8) (u k , k , n k ) |t =3 D 0 = 3(u0 k , 0 k , n 0 k ) (9) l l does exist, where the identities above take place for all (u, v) = u v dx is a scalar multiplication in E k , , F k , L2 (). The system above could be regarded as a Cauchy problem for ordinary differential equation X t = 3Df ( X ) in 3k-dimensional space with continuous function in the right-hand side. Due to Cauchy-Peano theorem the problem has a solution for T > 0 small enough. (u k , n k , n k ) in The next step is to get uniform estimation of some appropriate norm. 4. Energy conservation First of all we need the following proposition For all t > 0 we have Lemma 1. t |u 2 k 2 2 (t ) | J | k (t ) | K | nk (t ) | dx 2 | u |2 dxdt = 0 2 2 2 | u 0 k | | 0 k | | n 0 k | dx. Proof . In this section we will use u instead of u k for more comfortable notification. 143 Generated by Foxit PDF Creator © Foxit Software http://www.foxitsoftware.com For evaluation only. In equations (6)–(7) we substitute ( , , ) = (u k , k , K n k ). Since : F k F k , this substitution is legal. Taking the integral over the interval of all functions we obtain with respect to periodicity t 1 2 t ( | u |2 Knxs nxs u ix ) dxdt . 2 | u | |0 dx = j i j 0 (10) The sum of the integrals of (7) and (8) reads t 1 0 = J | |2 |t0 dx ((h n) ( n) h 2 0 Knti nix x Ku j nix nix x )dxdt = j s s s s t 1 j = ( J | |2 Knix nix )dx |t0 Ku x nix nix dxdt. s s s s j 2 Q 0 (11) t Taking the sum of (10) and (11) we obtain t 1 | u |2 J | |2 K | n |2 dx |t0 | u |2 dxdt = 0. 2 0 The proof is complete. Corollary 1. Problem (6)–(9) has a solution for every T > 0. 5. Estimates of higher derivatives Regretful, results of Lemma 1 are not sufficient to prove the convergence of estimates. (u k , k , n k ) to the solution of (2)–(5). We need more precise Theorem 2. initial data, J, K, such as || u k || L2 || u k ,t || There exists T > 0 and C > 0 depending only on (0,T ;W22 ) L2 (QT ) ,|| k || ,|| k ,t || L2 L (QT ) (0,T ;L2 ) ,|| n k || ,|| n k ,t || L2 L (QT ) (0,T ;W22 ) C C Proof . Set in the equation (7) = and integrate over the domain (0, T) with respect to Green's identities. We have 144 Generated by Foxit PDF Creator © Foxit Software http://www.foxitsoftware.com For evaluation only. J J j | |2 dx |T0 = ( u x xi xi K (n n x ) x l k 2 2 l j l Q T (12) K (n x n) x )dxdt. l k The integral of (8) with K 2n could be written as K i n x x nix x dx |T0 = 2 T k j k j j = K (u j nix x nix u x nix nix k k j k k j QT ( n x x n) n x )dxdt = k k k j = K (2u x n x x n u j n x n k j k j QT ( x n x ) n (n x ) n x dxdt. k k k k (13) To estimate the second derivatives of u we should set w = u and take an integral over (0, T): 1 | u |2 dx |T0 | u |2 dxdt = QT 2 j (u x u x u x Kn x nu i )dxdt. k QT j k i The sum of (12)–(14) reads 2 1 2 ( J | | K | n x x |2 | u |2 )dx |T0 j k 2 j ,k =3 D1 | u |2 dxdt = QT = ( Ju j xk x j xk 2 Ku xjk n x x n ( xk n xk ) j k QT j n u x u x u x )dxdt. k 145 j k (14) Generated by Foxit PDF Creator © Foxit Software http://www.foxitsoftware.com For evaluation only. We can estimate the integral in the right-hand side as 2 1 2 ( J | | K | n x x |2 | u |2 )dx |T0 | u |2 dxdt j k 2 j , k =1 Q T T 2 C1 (esssup | u(t ) | (|| (t ) ||22 || n x x (t ) ||= 22 || u ||22 ) i i , j =1 0 j (15) esssup | n(t ) | (|| (t ) ||2 || n(t ) ||2 ))dt . Due to the fact that ij || u x x ||22 =|| u ||2 (see [2, III, Section i j u in L2 (0, T ;W ) -norm. 2 2 8]), we have estimated We need to estimate esssup | n (t ) | via highest derivative. Regretful, W22 - norm is not enough, so we repeat similar procedure for ( , , ) ( 2u, 2 , K 3n). We have 2 1 2 2 J u K ( | | | | | (n |2 )dx |t0 | (u) |2 dxdt QT 2 j , k =3 D1 j u x (2 J x xk 2u x xk u K n x n xk 2n xi x j x n xi ) k j j j k Q T u j ( J x u x u Kn x j x n xk ) Ku xjk xi n x j x n xi j j k k K u xjk (n x j n xk n xk n x j n x j xk n) K ( 2n n 2(n xk n xk )) K ( n) 2n K (2 x j xk n x j 2 x j n x j xk xk n n xk )n xk dxdt T C (esssup | u(t ) | (|| (t ) ||W2 2 || u(t ) ||W2 2 || n(t ) ||W2 3 ) 2 2 2 0 (|| (t ) ||W2 1 || u(t ) ||W2 2 )(|| (t ) ||W 1 || u(t ) ||W 1 || n(t ) ||W 2 ) 4 4 4 4 4 (|| (t ) ||W 2 || u (t ) ||W 2 || n (t ) ||W 3 ) || u (t ) ||W 3 || n (t ) ||W2 2 2 2 2 2 4 esssup | u(t ) | (|| (t ) ||W 2 || n(t ) ||W 3 ) 2 2 W23 esssup | (t ) | || n (t ) || ) dt. 146 2 (16) Generated by Foxit PDF Creator © Foxit Software http://www.foxitsoftware.com For evaluation only. Denote (t ) =|| u(t ) ||W2 2 J || (t ) ||W2 2 K || nx x (t ) ||W2 3 . 2 i j 2 2 Since || u(t ) ||L C (|| u(t ) ||L || u(t ) ||L ), as p 2 p< 2 d , d 2 and max | u | C (|| u (t ) ||L || u (t ) ||L ), as p p d, p we have from (14), (15) and (16) is followed by the inequality t (t ) (0) C4 2 (t )dt , as t < T0 , (17) 0 where T0 depends only on j, K, . Here we need some simple lemma. Lemma 2. Suppose that for almost all t we have t 0 Y (t ) Y (0) k Y 2 dt. 0 Then Y (t ) Y (0) . 1 ktY (0) Proof . Consider the function X( t ) of the form t X (t ) = Y (0) k X 2 dt <=> 0 i.e. X = 3D The difference X = kX 2 , X (0) = Y (0) , Y (0) . 1 kt (Y (0) ) W = Y X satisfies the inequality t W (t ) k ( X Y )Wdt. 0 Since Y and X are measurable functions, the function t f (t ) = ( X Y )Wdt is a continuous one. Suppose 0 of f . Since X Y > 0, the condition on positive measure subset of (0, t < t0 , consequently 147 t 0 is the least zero f (t0 ) = 0 means that W is non-negative t 0 ). But it contradicts with f (t ) < 0 for all t > 0. f (t ) < 0 as Generated by Foxit PDF Creator © Foxit Software http://www.foxitsoftware.com For evaluation only. Tending to zero, we prove the lemma. Thus for any t < min{T0 ,1/(C4 (0))} (t ) C5 , (16) where C5 depends only on k,, J, (0). Estimate time-derivative Qt u t . Set = u t and rewrite (6) as 1 | u t |2 || u || dx |t0 = (u j u x u t nn x utj )dxdt . Qt j j 2 Since u j u x and nn x j are uniformly bounded in j L2 (Qt ) (it follows from (16) and the the embedding theorems), we derive Qt | ut |2 || u || dx |t0 || u j u x ut ||22 || nn ||2 . j The same inequalities could be obtained for t and n t . The theorem is proved 6. Convergence of the approximations Theorem 2 provides the existence of measurable functions u, , n such that u k u weakly in L2 (0, T ; Sol22 ()), u k ,t ut * k n * nk weakly in Sol2 (QT ), * weakly in L (0, T ;W21 ()), * weakly in L (0, T ;W22 ()). u k ,t u t , k ,t t , = n k ,t n t weakly in Moreover, due to the embedding theorems u k u strongly in Sol2 (QT ), k strongly in L2 (QT ), and n k n strongly in L2 (QT ). 148 L2 (QT ). Generated by Foxit PDF Creator © Foxit Software http://www.foxitsoftware.com For evaluation only. C1 (0, T ; E k ), (t), (t) C 1 (0, T ; F k ) integrate (6)–(8) over (0,T). Passing to the limits as k we have Fix (u u nn) dxdt = 0, ( J/K (n n) )dxdt = 0, = n ( n) dxdt = 0. QT QT QT Since C 1 (0, T ; E k ) and and (17) C 1 (0, T ; F k ) are dense in the corresponding spaces, these equations give us (2)–(4). Denote p to be a projection of u u nn L2 (QT ) L2 (0, T ; Sol2 ()). It's obvious that on the orthogonal complement of the p L2 (QT ). Verify initial conditions (5). Fix of functions f k (t ) = (uk (t ), ) L2 ( ) Sol2 () and consider a family . Since f k = (uk ,t (t ), ) L () (ut , ) L () 2 2 weakly in L2 (0, T ), f k tends to (u, ) L2 () in C(0,T). Thus u (t , ) tends to lim u0 k = u0 weakly in L2 (). k Other initial conditions are cheking in the same way. Q.e.d. 7. Uniqueness Theorem 3. Suppose (u1 , 1 , n1 , p1 ) and (u 2 , 2 , n 2 , p2 ) are solutions of the problem (2)–(5) in the domain Then for some QT . T0 > 0 (u 2 , 2 , n 2 , p2 ) = (u1 , 1 , n1 , p1 ) almost everywhere in Qt . 0 Proof . First of all, every solution of the problem satisfies identities (17) for all W21 (QT ), W21 (Qt ), L2 (0, T ;W22 ). 149 Generated by Foxit PDF Creator © Foxit Software http://www.foxitsoftware.com For evaluation only. Denote = u1 u 2 , f = 1 2 , g = n1 n 2 and set ( , , ) = ( w , f , K g ). The identities (17) becomes the following: (w t w u2i w x w w i u1, x w ww i i Qt K n1g x w i K gn 2,= x w j ) dxdt = 0, j j i 2 xi ( J (f f u f t f w i 1, x f ) K (g n1 n 2 g ) f )dxdt = 0, i Qt K (g t g u1i g x g w i n 2, x g i i Qt (18) ( 1 g f n 2 )g )dxdt = 0 and 1 2 | w (t ) | 2 dx | w |2 dxdt = Qt ( w i u1, x w K (n1g x w i gn 2,= x w j )) dxdt , i j j Qt J | f (t ) |2 dx = ( Jw i 1, x f K (g n 2 n1 g ) f )dxdt , i 2 Q t K | g (t ) |2 dx = K (w i n 2, x g u1,i x g x g x i k i k 2 Q t (19) ( 1, x g ) g x (f n 2 ) g )dxdt. k k The sum of the last three identities reads 1 || w (t ) ||2 J/K || f (t ) ||2 K || g (t ) ||2 dx || w ||2 dxdt = 2 = w j u1,i x w i K n1g x w i Jw i 1, x f K (n1 g ) f j j i Qt Ku1,i x g x g x K ( 1, x g) g x dxdt. k 150 i k k k Generated by Foxit PDF Creator © Foxit Software http://www.foxitsoftware.com For evaluation only. Due to embedding theorems and H ölder inequalities 1 || w (t ) ||2 J/K || f (t ) ||2 K || g (t ) ||2 dx || w ||2 dxdt 2 1 Ct 2 esssup (|| u1 (t ) || L2 () || n1 || L2 () (esssup (|| w (t ) ||22 || f ||22 || g ||22 ) || w ||2 L2 || 1 || (Qt ) L2 () || w (t ) ||2 ) L2 (Qt ) ). Thus if t is sufficiently small, then (w, f , K g) = (0, 0, 0), consequently (u1 , 1 , n1 ) = (u 2 , 2 , n 2 ) and, since p i is a u nn, we have projection of u p1 p2 . 8. The presence of external forces The same results take place for Theorem 4. F , G 0. Suppose u 0 Sol22 (), 0 W21 (), n 0 W22 () and F L2 (QT ), G L1 (0, T ; L2 (QT )). Then for some 0 < T0 < T the solution (in terms of Definition 1) of problem (2)–(4) exists and is unique in QT . 0 The proof is similar to the proof of Theorems 1 and 3. Acknowledgments The work was supported by the Government grant of the Russian Federation under the Resolution No. 220 ``On measures designed to attract leading scientists to Russian institutions of higher education'' according to the Agreement No. 11.G34.31.0054, signed by the Ministry of education and science of the Russian Federation, the leading scientist, and Lomonosov Moscow State University (on the basis of which the present research is organized). The work of the first author was also supported in part by RFBR grant (12-01-00445). References [1] François Gay-Balmaz, Tudor S. Ratiu. The geometric structure of complex fluids. Advances in Applied Mathematics 42 (2009) 176–275. [2] O. Ladyzhenskaya, N. Uraltseva, Linear and Quasilinear Elliptic Equations, Academic Press, New York (1968). 151

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