Recent developments of analysis for hydrodynamic flow of nematic liquid crystals
aa r X i v : . [ m a t h . A P ] A ug RECENT DEVELOPMENTS OF ANALYSIS FORHYDRODYNAMIC FLOW OF NEMATIC LIQUID CRYSTALS
FANGHUA LIN AND CHANGYOU WANG
Abstract.
The study of hydrodynamics of liquid crystal leads to many fasci-nating mathematical problems, which has prompted various interesting worksrecently. This article reviews the static Oseen-Frank theory and surveys somerecent progress on the existence, regularity, uniqueness, and large time asymp-totic of the hydrodynamic flow of nematic liquid crystals. We will also proposea few interesting questions for future investigations. Static Theory
Liquid crystal is an intermediate phase between the crystalline solid state andthe isotropic fluid state. It possess none or partial positional order but displays anorientational order at the mean time. There are roughly three types of liquid crys-tals commonly referred in the literature: nematic, cholesteric, and smectic. Thenematic liquid crystals are composed of rod-like molecules with the long axes ofneighboring molecules approximately aligned to one another. The simplest contin-uum model to study the equilibrium phenomena for nematic liquid crystals is theOseen-Frank theory, proposed by Oseen [1] in 1933 and Frank [2] in 1958. A uni-fying program describing general liquid crystal materials is the Landau-de Gennestheory ([3] and [4]), which involves the orientational order parameter tensors.In the absence of surface energy terms and applied fields, the Oseen-Frank theoryseeks unit-vector fields d : Ω Ď R Ñ S “ t y P R : | y | “ u , representing meanorientations of molecule’s optical axis, that minimizes the Oseen-Frank bulk energyfunctional E OF p d q “ ş Ω W p d, ∇ d q dx , where2 W p d, ∇ d q “ k p div d q ` k p d ¨ curl d q ` k | d ˆ curl d | `p k ` k qr tr p ∇ d q ´ p div d q s , (1)where k , k , k ą k ě | k | , and2 k ě k ` k . Utilizing the null-Lagrangian property of last term in W p d, ∇ d q ,it can be shown that under strong anchoring condition (or Dirichlet boundarycondition) d ˇˇ B Ω “ d P H p Ω , S q , there always exists a minimizer d P H p Ω , S q ofthe Oseen-Frank energy functional E OF . Furthermore, such a d solves the Euler-Lagrange equation: δ W δd ˆ d : “ ´ div p B W B ∇ d p d, ∇ d qq ` B W B d p d, ∇ d q ( ˆ d “ . (2)The basic question for such a minimizer d concerns both the regularity propertyand the defect structure. A fundamental result by Hardt, Lin, and Kinderlehrer [5]in 1986’s (see also [6]) asserts1.1. Theorem. — If d P H p Ω , S q is a minimizer of the Oseen-Frank energy E OF , then d is analytic on Ω z sing p d q for some closed subset sing p d q Ď Ω , whoseHausdorff dimension is smaller than one. If, in addition, d P C k,α p Ω , S q for some k P N ` , then there exists a closed subset Σ Ď B Ω , with H p Σ q “ , such that d P C k,α p Ω z p sing p d q Y Σ q , S q . Concerning the interior singular set sing p d q in Theorem 1.1, an outstanding openquestion is1.2. Question. —
How large is the singular set sing p d q for minimizers of Oseen-Frank energy E OF ? or equivalently, what is the optimal estimate of size of sing p d q ? Notice that when k “ k “ k “ k “
0, the Oseen-Frank energy densityfunction W p d, ∇ d q “ | ∇ d | is the Dirichlet energy density. Hence the problemreduces to minimizing harmonic maps into S and (2) becomes the equation ofharmonic maps: δ W δd : “ ∆ d ` | ∇ d | d “ . (3)Harmonic maps have been extensively studied in the past several decades, whichare much better understood. See [7, 8, 9, 10, 11]. In particular, it is well-knownthat in dimension n “
3, the singular set of minimizing harmonic maps is at mosta nonempty set of finitely many points (see [7]). A typical example is the followinghedgehog singularity.1.3.
Example. — ([10] [12]). x | x | : R Ñ S is a minimizing harmonic map.Motivated by the isolated point singularity for minimizing harmonic maps, it isnatural to ask1.4. Question. —
Is the set sing p d q for a minimizer of Oseen-Frank energy E OF in Theorem 1.1 a set of finite points? The main difficulty to investigate the size of singular set of minimizers d to theOseen-Frank energy functional E OF is that it is an open question whether an energymonotonicity inequality, similar to that of harmonic maps [7], holds for d . In fact,the following seemly simple question remains open.1.5. Question. —
Assume k “ ´ k and max ď i ď | k i ´ | ăă . Suppose d P H p Ω , S q is a minimizer of the Oseen-Frank energy E OF . Is the following mono-tonicity inequality true? r ´ ż B r | ∇ d | ` ż B R z B r | x | ´ ˇˇ B d B| x | ˇˇ ď ` ` ω p R ´ r q ˘ R ´ ż B R | ∇ d | (4) for ă r ď R , with lim τ Ó ω p τ q “ . Notice that when k “ ´ k and max ď i ď | k i ´ | is sufficiently close to 0, it hasbeen shown by [6] that sing p d q for any minimizer d of E OF has locally finite points.The readers can consult with the survey article by Hardt-Kinderlehrer [13] forsome earlier developments of Oseen-Frank theory. The book by Virga [14] and thelecture note by Brezis [15] provide detailed exposition of Oseen-Frank theory. Thereis an important survey article by Lin-Liu [16] on the development of mathematicalanalysis of liquid cyrstals up to 2001. Partly motivated by these mathematicalstudies and physical experiments where line and surface defects of nematic liquidcrystals have been observed (see [17]), Ericksen [18] proposed the so-called Ericksenmodel of liquid crystals with variable degree of orientation. It assumes that thebulk energy density depends on both the orientation vector d , with | d | “
1, andorientational order s P r´ , s , which is given by w p s, d, ∇ s, ∇ d q “ w p s q ` w p s, d, ∇ s, ∇ d q , (5)where w p s q is roughly a W-shape potential function with w p´ q “ w p q “ `8 .The term w is given by2 w p s, ∇ s, d, ∇ d q “ W p d, ∇ d q` K | ∇ s ´p ∇ s ¨ d q d ´ α p ∇ d q d | ` K p ∇ s ¨ d ´ β divd q , YDRODYNAMIC FLOW 3 where W p d, ∇ d q is given by (1), and K ą , K ą , α, β are coefficients that maydepend on s . We can view the Oseen-Frank model as a special case of Ericksen’smodel by imposing the constraint on the orientational order s “ s ˚ , with w p s ˚ q “ min t w p s q : s P r´ , su . The simplest form of Ericksen’s bulk energy density is w p s, d q “ k | ∇ s | ` s | ∇ d | ` w p s q . (6)One can translate the problem of minimizers p s, d q of Ericksen’s energy functional(6) into the problem of minimizing harmonic maps p s, u q , with potentials w ,min ż Ω ` p k ´ q| ∇ s | ` | ∇ u | ` w p s q ˘ dx (7)subject to the constraint | s | “ | u | and p s, u q “ p s , u q on B Ω. Then the problem (7)is essentially a minimizing harmonic map into a circular cone C k “ tp s, u q P R ˆ R : | s | “ ? k ´ | u |u Ď R for k ą C k “ tp s, u q P R ˆ R : | s | “ ? ´ k | u |u Ď R , -the Minkowski space for 0 ă k ă
1, Lin has proved in [19] (see also Ambrosio[20, 21], Ambrosio-Virga [22], Hardt-Lin [23], Lin-Poon [24], and Calderer-Golovaty-Lin-Liu [25] for further related works)1.6.
Theorem. —
For any p s , u q P H pB Ω , C k q , there exists a minimizer p s, u q of (6) such that the following properties hold: (1) If k ą , then p s, u q P C α p Ω q for some ă α ă , and p s, u q is analyticaway from s ´ t u . If s ı , then s ´ t u has Hausdorff dimension at most . (2) If ă k ă , then p s, u q P C , p Ω q , and p s, u q is analytic away from s ´ t u .If s ı , then s ´ t u has Hausdorff dimension at most . Since a liquid crystal molecule is indistinguishable between its head and tail,i.e. d « ´ d , its odd order moments must vanish and the second order moment isbelieved to be most important to the energy. In general, one can use a class of 3 ˆ Q to describe the second order moments: M “ Q P R ˆ : Q “ ż S p p b p ´ I q dµ p p q , µ P P p S q ( , where P p S q denotes the space of probability measures on S . It is clear that for anyorder parameter tensor Q P M , its three eigenvalues λ i p Q q P r´ , s (1 ď i ď ÿ i “ λ i p Q q “
1. If Q has two equal eigenvalues, it becomes a uniaxial nematicor cholesteric phase so that it can be expressed as Q “ s ` d b d ´ I q , with | d | “ s P r´ , s . If Q P M has three distinct eigenvalues, then the liquid materialis biaxial. The Landau-de Gennes bulk energy for nematic liquid crystal materialwas derived by de Gennes [3, 4]. A simplified form is given by E DG p Q q “ ż Ω ” ` L | ∇ Q | ` L p div Q q ` L tr p ∇ Q q ` L ∇ Q b ∇ Q : Q ˘ ` ` A tr p Q q ´ B tr p Q q ` C tr p Q q ˘ı . (8)It is well-known that if Q is uniaxial, then the Landau-de Gennes energy can beessentially reduced to the Ossen-Frank energy or the Ericksen energy. We wouldlike to mention that it is not hard to show that there exists at least one minimizer Q P M of the Landau-de Gennes energy functional E DG , provided we remove theconstraint on the eigenvalues ´ ď λ i p Q q ď , 1 ď i ď
3. Moreover, the defect setof biaxial minimizers Q P M of the Landau-de Gennes energy E DG is contained inthe set on which at least two of its eigenvalues are equal. The study of such defectsets is extremely challenging and remains largely open. Here we mention that there FANGHUA LIN AND CHANGYOU WANG have been several recent interesting works on the static case of Landau-de Gennes’model by J. Ball and his collaborators, and others. See Ball-Zarnescu [28, 32], Ball-Majumdar [29], Majumdar [30], Majumdar-Zarnescu [31], Henao-Majumdar [33],Ngugen-Zarnescu [34], Bauman-Park-Phillips [35], and Evans-Kneuss-Tran [36].2.
Hydrodynamic Theory
Ericksen-Leslie system modeling nematic liquid crystal flows.
TheEricksen-Leslie system modeling the hydrodynamics of nematic liquid crystals, re-ducing to the Oseen-Frank theory in the static case, was proposed by Ericksen andLeslie during the period between 1958 and 1968 ([26] and [37]). It is a macroscopiccontinuum description of the time evolution of the materials under the influence ofboth the flow velocity field u p x, t q and the macroscopic description of the micro-scopic orientation configuration d p x, t q of rod-like liquid crystals, i.e., d p x, t q is aunit vector in R . The full Ericksen-Leslie system can be described as follows. Thehydrodynamic equation takes the form B t u ` u ¨ ∇ u ` ∇ P “ ∇ ¨ σ, (9)where the stress σ is modeled by the phenomenological constitutive relation: σ “ σ L p u, d q ` σ E p d q . Here σ L p u, d q is the viscous (Leslie) stress given by σ L p u, d q “ µ p d b d : A q d b d ` µ d b N ` µ N b d ` µ A ` µ d b p A ¨ d q ` µ p A ¨ d q b d, (10)with six Leslie coefficients µ , ¨ ¨ ¨ , µ , and A “ p ∇ u ` p ∇ u q T , ω “ p ∇ u ´ p ∇ u q T q , N “ B t d ` u ¨ ∇ d ´ ω ¨ d representing the velocity gradient tensor, the vorticity field, and rigid rotation partof the changing rate of the director by fluid respectively. Assuming the fluid isincompressible, we have ∇ ¨ u “ . (11)While σ E p d q is the elastic (Ericksen) stress σ E p d q “ ´ B W Bp ∇ d q ¨ p ∇ d q T where W “ W p d, ∇ d q is the Oseen-Frank energy density given by (1). The dynamicequation for the director field takes the form d ˆ ` ´ δ W δd ´ λ N ´ λ A ¨ d ˘ “ , (12)where δ W δd is the Euler-Lagrangian operator given by (2). We also have the com-patibility condition λ “ µ ´ µ , λ “ µ ´ µ , (13)and Parodi’s condition µ ` µ “ µ ´ µ , (14)which provides certain dynamical stability of the whole system (see [38]). YDRODYNAMIC FLOW 5
Energetic Variational Approach.
The derivation of Ericksen-Leslie systemin [26] and [37] is based on the conservation of mass, the incompressibility of fluid,and conservation of both linear and angular momentums. Here we will sketcha modern approach called energy variational approach, prompted by C. Liu andhis collaborators (see [39], [40], and [41]), which is based on both the least actionprinciple and the maximal dissipation principle. In the context of hydrodynamics ofnematic liquid crystals, the basic variable is the flow map x “ x p X, t q : Ω X Ñ Ω xt ,with X the Lagrangian (or material) coordinate and x the Eulerian (or reference)coordinate. For a given velocity field u p x, t q , the flow map x p X, t q solves: x t p x p X, t q , t q “ u p x p X, t q , t q ; x p X, q “ X P Ω X . The deformation tensor F of the flow map x p X, t q is F “ ` B x i B X j ˘ ď i,j ď . By thechain rule, F satisfies the following transport equation: B t F ` u ¨ ∇ F “ ∇ u F . The kinematic transport of the director field d represents the molecules moving inthe flow [43, 41]. It can be expressed by d p x p X, t q , t q “ E d p X q , (15)where the deformation tensor E carries information of micro structures and config-urations. It satisfies the following transport equation: B t E ` u ¨ ∇ E “ ω E ` p α ´ q A E , (16)where 2 α ´ “ r ´ r ` P r´ , s ( r P R ) is related to the aspect ratio of the ellipsoidshaped liquid crystal molecules, see Jeffrey [42]. Notice that if there is no internaldamping, then the transport equation of d from (15) and (16) is given by B t d ` v ¨ ∇ d ´ ω ¨ d ´ p α ´ q Ad “ . (17)The total energy of the liquid crystal system is the sum of kinetic energy andinternal elastic energy and is given by E total “ E kinetic ` E int , E kinetic “ ż | v | , E int “ E OF p d q “ ż W p d, ∇ d q . The action functional A of particle trajectories in terms of x p X, t q is given by A p x q “ ż T ` E kinetic ´ E int ˘ dt. Since the fluid is assumed to be incompressible, the flow map x p X, t q is volumepreserving, which is equivalent to ∇ ¨ u “
0. The least action principle asserts thatthe action functional A minimizes among all volume preserving flow map x p X, t q ,i.e., δ x A “ ∇ ¨ u “
0. To simplify the derivation, weconsider one constant approximation of Oseen-Frank energy density function andrelax the condition | d | “ W p d, ∇ d q “ | ∇ d | ` F p d q for d : Ω Ñ R .Then direct calculations (see [41] 7.1) yield that0 “ δ x A “ ddǫ ˇˇ ǫ “ A p x ǫ q “ ż T ż Ω ` B t u ` u ¨ ∇ u ` ∇ ¨p ∇ d d ∇ d q´ ∇ ¨ r σ ˘ y dxdt, (18)where y “ ddǫ ˇˇ ǫ “ x ǫ satisfies ∇ ¨ y “
0, and r σ “ ´ ` ´ λ λ ˘ p ∆ d ´ f p d qq b d ` ` ` λ λ ˘ d b p ∆ d ´ f p d qq , f p d q “ ∇ F p d q . Hence it follows from (18) that B t u ` u ¨ ∇ u ` ∇ P “ ´ ∇ ¨ p ∇ d d ∇ d q ` ∇ ¨ r σ, (19) ∇ d d ∇ d “ ` x B d B x i , B d B x j y ˘ ď i,j ď FANGHUA LIN AND CHANGYOU WANG where the pressure P serves as a Lagrangian multiplier for the incompressibility offluid.It follows from (16) that the total transport equation of d , without internalmicroscopic damping, is B t d ` u ¨ ∇ d ´ ωd ` λ λ Ad “ . (20)If we take the internal microscopic damping into account, then we have B t d ` u ¨ ∇ d ´ ωd ` λ λ Ad “ λ δ E int δd “ ´ λ p ∆ d ´ f p d qq . (21)It is known that the dissipation functional D to the system (29) is D “ ż Ω “ µ | d T Ad | ` µ | ∇ u | ` p µ ` µ q| Ad | ` λ | N | ` p λ ´ µ ´ µ qx N, Ad y ‰ . (22)By (21) and Parodi’s condition (14), D can be rewritten as D “ ż Ω “ µ | d T Ad | ` µ | ∇ u | ` p µ ` µ ` λ λ q| Ad | ´ λ | N ` λ λ Ad | ‰ . (23)According to the maximal dissipation principle, one has that first order variationof D with respect to the rate function u must vanish, i.e. δ D δu “ ∇ ¨ u “
0. By direct calculations (see [41] 7.2), we have that0 “ δ D δu “ ż Ω x u, ∇ ¨ p ∇ d d ∇ d q ´ ∇ ¨ σ L p u, d qy , where σ L p u, d q is the Leslie stress tensor given by (10). Hence we arrive at0 “ ´ ∇ P ´ ∇ ¨ p ∇ d d ∇ d q ` ∇ ¨ ` σ L p u, d q ˘ . (24)Combining the dissipative part derived from the maximal dissipation principle withthe conservative part derived from the least action principle, we arrive at the equa-tion (9). This, together with (21) and (11), yields the Ginzburg-Landau approxi-mated version of the Ericksen-Leslie system (9)-(11)-(12).2.3. Brief descriptions on relationship between various models.
There arebasically three different kinds of theories to model nematic liquid crystals: Doi-Onsager theory, Landau-de Gennes theory, and Ericksen-Leslie theory. The first isthe molecular kinetic theory, and the latter two are the continuum theory. Kuzzu-Doi [44] and E-Zhang [45] have formally derived the Ericksen-Leslie equation fromthe Doi-Onsager equation by taking small Deborah number limit. Wang-Zhang-Zhang [46] have rigorously justified this formal derivation before the first singulartime of the Ericksen-Leslie equation. Wang-Zhang-Zhang [47] has rigorously de-rived Ericksen-Leslie equation from Beris-Edwards model in the Landau-de Gennestheory. There are several dynamic Q-tensor models describing hydrodynamic flowsof nematic liquid crystals, which are derived either by closure approximations (see[50] [51] and [55]), or by variational methods such as Beris-Edwards’ model [48] andQian-Sheng’s model [49]. See [52] and [53] for the well-posedness of some dynamicQ-tensor models. Furthermore, a systematical approach to derive the continuumtheory for nematic liquid crystals from the molecular kinetic theory in both staticand dynamic cases was proposed in [54] and [55]. We would also like to mention thederivation of liquid crystal theory from the statistical view of points by Seguin-Fried[56, 57].
YDRODYNAMIC FLOW 7
Analytic issues on simplified Ericksen-Leslie system.
When we considerone constant approximation for the Oseen-Frank energy, i.e., W p d, ∇ d q “ | ∇ d | ,the general Ericksen-Leslie system reduces to $’&’% B t u ` u ¨ ∇ u ` ∇ P “ ´ ∇ ¨ p ∇ d d ∇ d q ` ∇ ¨ p σ L p u, d qq , ∇ ¨ u “ , B t d ` u ¨ ∇ d ´ ωd ` λ λ Ad “ | λ | ` ∆ d ` | ∇ d | d ˘ ` λ λ ` d T Ad ˘ d. (25)Direct calculations, using both (13) and (14), show that smooth solutions of (25),under suitable boundary conditions, satisfy the following energy law (see [58]): ddt ż p| u | ` | ∇ d | q ` ż “ µ | ∇ u | ` | λ | | ∆ d ` | ∇ d | d | ‰ “ ´ ż ”` µ ´ λ λ ˘ | A : d b d | ` ` µ ` µ ` λ λ ˘ | A ¨ d | ı . (26)In particular, we have the following energy dissipation property.2.1. Lemma. —
Assume both (13) and (14). If Leslie’s coefficients satisfy thealgebraic condition: λ ă , µ ´ λ λ ě , µ ą , µ ` µ ě ´ λ λ , (27) then any smooth solution ( u, d ) of (25), under suitable boundary conditions, satisfiesthe energy dissipation inequality: ddt ż p| u | ` | ∇ d | q ` ż “ µ | ∇ u | ` | λ | | ∆ d ` | ∇ d | d | ‰ ď . (28)A fundamental question related to the general Ericksen-Leslie system (25) is2.2. Question. —
For dimensions n “ , smooth domains Ω Ď R n (or Ω “ R n ), establish i) the existence of global Leray-Hopf type weak solutions of (25) under genericinitial and boundary values p u , d q P H ˆ H p Ω , S q . ii) partial regularity and uniqueness properties for certain restricted classes ofweak solutions of (25). iii) an optimal global (or local) well-posedness of (25) for rough initial data( u , d ), and the long time behavior of global solutions of (25). While both part i) and part ii) remain open for dimensions n “ n “ n “ p u, d q : Ω ˆ p , `8q Ñ R n ˆ S solves $’&’% B t u ` u ¨ ∇ u ` ∇ P “ µ ∆ u ´ ∇ ¨ p ∇ d d ∇ d q , ∇ ¨ u “ , B t d ` u ¨ ∇ d “ ∆ d ` | ∇ d | d. (29)It is readily seen that under suitable boundary conditions, (29) enjoys the followingenergy dissipation property: ddt ż p| u | ` | ∇ d | q “ ´ ż “ µ | ∇ u | ` | ∆ d ` | ∇ d | d | ‰ ď . (30) Throughout this paper, H “ Closure of v P C p Ω , R n q | div v “ ( in L p Ω , R n q and J “ Closure of v P C p Ω , R n q | div v “ ( in H p Ω , R n q . FANGHUA LIN AND CHANGYOU WANG
Because of the supercritical nonlinearity ∇ ¨ p ∇ d d ∇ d q in (29) , Lin-Liu [63, 64, 65]have studied the Ginzburg-Landau approximation (or orientation of variable degreesin Ericksen’s terminology [27]) of (29): for ǫ ą p u, d q : Ω ˆ p , `8q Ñ R n ˆ R solves $’&’% B t u ` u ¨ ∇ u ` ∇ P “ µ ∆ u ´ ∇ ¨ p ∇ d d ∇ d q , ∇ ¨ u “ , B t d ` u ¨ ∇ d “ ∆ d ` ǫ ` ´ | d | ˘ d. (31)Under the initial and boundary conditions: p u, d q ˇˇ t “ “ p u , d q , x P Ω p u, d q ˇˇ B Ω “ p , d q , t ą , (32)we have the following basic energy law: ż Ω ` | u | ` | ∇ d | ` ǫ p ´ | d | q ˘ p t q ` ż t ż Ω p µ | ∇ u | ` |B t d ` u ¨ ∇ d | qď ż Ω ` | u | ` | ∇ d | ` ǫ p ´ | d | q ˘ . (33)Besides stability and long time asymptotic, the following existence and partialregularity were proven in [63, 64] (see also [66]).2.3. Theorem. —
For any ǫ ą fixed, the following holds: (a) for u P H and d P H p Ω q X H pB Ω q , there exists a global weak solution p u, d q of (31) and (32) satisfying (33) and u P L p , T ; J q X L p , T ; H q ,d P L p , T ; H p Ω qq X L p , T ; H p Ω qq , @ T P p , `8q . (b) there exists a unique global classical solution p u, d q to (31) and (32) provided p v , d q P J ˆ H p Ω q and either n “ or n “ and µ ě µ p u , d q . (c) if p u, d q is a global suitable weak solution of (31), then p u, d q P C p Ω ˆp , `8qz Σ q , with H p Σ q “ . A long outstanding open problem pertaining to the Ginzburg-Landau approx-imation equation (31) and the original nematic liquid crystal flow equation (29)is2.4.
Question. —
Whether weak limits p u, d q of weak solutions p u ǫ , d ǫ q of (31)and (32) are weak solutions of (29) and (32), as ǫ Ñ . It follows from (33) that there exists p u, d q P ` L p , T ; J q X L p , T, H q ˘ ˆ L p , T ; H p Ω , S qq such that, up to a subsequence, $’&’% p u ǫ , d ǫ q á p u, d q in L p , T ; L p Ω qq ˆ L p , T ; H p Ω qq ,e ǫ p d ǫ q dxdt : “ ` | ∇ d ǫ | ` ǫ p ´ | d ǫ | q ˘ dxdt á µ : “ | ∇ d | dxdt ` η, p ∇ d ǫ d ∇ d ǫ q dxdt á p ∇ d d ∇ d q dxdt ` M , for some nonnegative Radon measure η and a positive semi-definite n ˆ n symmetricmatrix-valued function M with each entry being a Radon measure, as convergenceof Radon measures in Ω ˆ r , T s . Both ν and M are called defect measures , and M ăă ν . It is readily seen that $’&’% B t u ` u ¨ ∇ u ` ∇ P “ µ ∆ u ´ ∇ ¨ p ∇ d d ∇ d ` M q , ∇ ¨ u “ , B t d ` u ¨ ∇ d “ ∆ d ` | ∇ d | d. (34)Thus we need to study the following difficult question. YDRODYNAMIC FLOW 9
Question. —
Let η and M be the defect measures. How large are the supports of ν and M ? Under what conditions, ν or M vanishes? Before describing a very recent progress towards Question 2.4 and 2.5, we presentslightly earlier works on the simplified Ericksen-Leslie system (29) in dimension n “ Lemma. —
For n “ and ă r ď , let P r “ B r ˆ r´ r , s be the parabolicball of radius r . There exists ǫ ą such that if u P L t L x X L t H x p P r , R q , d P L t H x p P r , S q , and P P L p P r q solves (29), satisfying Φ p u, d, P, r q : “ } u } L p P r q ` } ∇ u } L p P r q ` } ∇ d } L p P r q `} ∇ d } L p P r q ` } P } L p P r q ď ǫ , (35) then p u, d q P C p P r , R ˆ S q , and ›› p u, ∇ d q ›› C l p P r q ď C p l q ǫ r ´ l , l ě . (36)Approximating initial and boundary data p u , d q by smooth p u ǫ , d ǫ q , and em-ploying Lemma 2.6 to the short time smooth solutions p u ǫ , d ǫ q of (29) under theinitial and boundary value ( u ǫ , d ǫ ), we have established the following result onexistence and uniqueness of (29) in [67] and Lin-Wang [68] respectively.2.7. Theorem. —
For any u P H and d P H p Ω , S q , with d P C ,α pB Ω , S q forsome α P p , q , there exists a global weak solution u P L p , `8 ; H q X L p , `8 ; J q and d P L p , `8 ; H p Ω , S q of (29) and (32) such that the following properties hold: there exists a nonnegative integer L depending only on u , d and ă T 㨠¨ ¨ ă T L ă `8 such that p u, d q P C ` Ω ˆ rp , `8qzt T i u Li “ s ˘ X C , α ` Ω ˆ rp , `8qzt T i u Li “ s ˘ . at each time singularity T j , j “ , ¨ ¨ ¨ , L , it holds lim inf t Ò T j max x P Ω ż Ω X B r p x q p| u | ` | ∇ d | qp y, t q dy ě π, @ r ą . There exist x jm Ñ x j P Ω and t jm Ò T j , r jm Ó , and a nontrivial harmonicmap ω j : R Ñ S with finite energy such that p u jm , d jm q : “ ` r jm u p x jm ` r jm x, t jm ` p r jm q q , d p x jm ` r jm x, t jm ` p r jm q q ˘ Ñ p , ω j q in C p R ˆ r´8 , sq . there exist t k Ò `8 and a harmonic map d P C p Ω , S q X C ,αd p Ω , S q such that u p t k q Ñ in H p Ω q , d p t k q á d in H p Ω q , and there exist t x i u li “ Ď Ω and t m i u li “ Ď N such that | ∇ d p t k q| dx á | ∇ d | dx ` l ÿ i “ πm i δ x i . if either d ě or ş Ω p| u | ` | ∇ d | q ď π , then p u, d q P C p Ω ˆ p , `8qqX C , α p Ω ˆ p , `8qq . Furthermore, there exists t k Ò `8 and a harmonic map d P C p Ω q X C ,αd p Ω , S q such that p u p t k q , d p t k qq Ñ p , d q in C p Ω q . the global weak solution p u, d q obtained in 1) is unique in the same class ofweak solutions, i.e. if p v, e q P ` L p , T ´ ; H q X L p , T ´ ; J q ˘ ˆ L p , T ´ ; H p Ω qq isanother weak solution of (29) and (32), then p v, e q ” p u, d q in Ω ˆ r , T q . Some related works on the existence of global weak solutions of (29) in R havealso been considered by Hong [70], Hong-Xin [71], Xu-Zhang [72], and Lei-Li-Zhang[69]. There are a few interesting questions related to Theorem 2.7, namely,2.8. Question. —
1) Does the global weak solution p u, d q obtained in Theorem 2.7have at most finitely many singularities?2) Does there exist a smooth initial value p u , d q such that the short time smoothsolution p u, d q to (29) and (32) develop finite time singularity?3) Does the solution p u, d q obtained in Theorem 2.7 have a unique limit p , d q as t Ò `8 ? It is not hard to check that the example of finite time singularity of harmonicmap heat flow in dimension two by Chang-Ding-Ye [73] satisfies (29) with u ” t “ 8 for (29) on S , see [74].2.9. Remark. —
For the simplified Ericksen-Leslie system (29) in R n for n ě p u, d q P C pr , T s , L n p R n , R n q ˆ W ,ne p R n , S q , e P S ,by Lin-Wang [68]; andii) the class of weak solutions p u, d q P L p p , T ; L q p R n , R n qqˆ L p p , T ; W ,qe p R n , S qq ,with p ą q ą n satisfying Serrin’s condition p ` nq “
1, by Huang [75](itis interesting to ask whether the uniqueness holds for the end point case p p, q q “p`8 , n q ).It turns out that the ǫ -regularity lemma 2.6 for the simplified Ericksen-Lesliesystem (29) can also be proved by a blow-up type argument. Furthermore, withsome delicate analysis to establish smoothness of the limiting linear coupling systemresulting from the blow-up argument, such a blow up argument also works for thegeneral Ericksen-Leslie system (25). Indeed, it was shown by Huang-Lin-Wang in[58] that Lemma 2.6 also holds for (25). As a consequence, we have extended in[58] Theorem 2.7 to the general Ericksen-Leslie system (25) in R .2.10. Theorem. —
For u P H and d P H e p R , S q , e P S , assume theconditions (13), (14), and (27) hold. Then there exists a global weak solution p u, d q : R ˆ r , `8q Ñ R ˆ S to (25) in R , with u P L p , `8 ; H q X L p , `8 ; J q , d P L p , `8 ; H e p R , S qq , under the initial condition p u, d q| t “ “ p u , d q in R .Furthermore, ( u, d ) enjoys the properties 1a), 1b), 1c), 1d) in Theorem 2.7 ( with Ω replaced by R ) . See also Wang-Wang [59] for some related work. It is an interesting question toask whether the uniqueness theorem for (29) by Lin-Wang [68] also holds for (25)in R .Now we outline a program to utilize the ǫ -version of Ericksen-Leslie system (31)in the study of defect motion of nematic liquid crystals in dimension 2, see Lin [61]and [62]. For a bounded smooth domain Ω Ď R , let p u ǫ , d ǫ q : Ω ˆr , `8q Ñ R ˆ R be solutions of (31), and d ǫ p x, t q “ g p x q : B Ω Ñ S is smooth with deg( g ) “ m P N .Assume the initial data p u ǫ , d ǫ q satisfies ż Ω | u ǫ | ď A , E ǫ p d ǫ q “ ż Ω ` | ∇ d ǫ | ` ǫ p ´ | d ǫ | q ˘ ď πm log 1 ǫ ` B . (37)Then the following facts hold:(a) E ǫ p d ǫ p t qq ` ş Ω | u ǫ p t q| ď πm log ǫ ` A ` B for t ě YDRODYNAMIC FLOW 11 (b) E ǫ p d ǫ p t qq ě πm log ǫ ´ C , ş Ω | u ǫ p t q| ď C p A , B , C q for t ě
0, and ż ż Ω ` µ | ∇ u ǫ | ` | ∆ d ǫ ` ǫ p ´ | d ǫ | q d ǫ | ˘ ď C p A , B , C q . (c) ›› ∇ d ǫ p t q ›› L p p Ω q ď C p p, Ω , g, A , B , C q for 1 ď p ă u ǫ p t q Ñ u p t q in L p Ω q and weakly in H p Ω q , and D h p t q P H p Ω q , with } h p t q} H p Ω q ď C , and t a j p t qu mj “ Ď C , pr , T q , Ω q such that, up to a sub-sequence, d ǫ p t q Ñ Π mj “ x ´ a j p t q| x ´ a j p t q| e ih p x,t q “ e i Θ a ` h in L p Ω q X H p Ω z t a j p t qu mj “ q . (e) ∇ d ǫ d ∇ d ǫ á p ∇ Θ a ` ∇ h qdp ∇ Θ a ` ∇ h q` η in D p Ω q for some positive semi-definite matrix valued function η with each entry being Radon measures inΩ. Moreover, supp p η p t qq Ď t a j p t qu mj “ .Then we have da j dt “ u p a j p t q , t q ; a j p q “ a ,j , j “ , ¨ ¨ ¨ , m, (38)and p u, h, Θ a q satisfies $’&’% h t ` u ¨ ∇ h “ ∆ h ` p u p a p t q , t q ´ u q ¨ ∇ Θ a ,u t ` u ¨ ∇ u ` ∇ P “ µ ∆ u ´ ∇ h ¨ ∆ h ´ ∇ Θ a p ∆ h ´ ∆ h p a p t q , t qq , ∇ ¨ u “ . (39)An important question is to show that the system (38) and (39) admits a smoothsolution p a p t q , u, h, P q . Once this has been established, the consistency of energylaws will yield the defect measure η ” n ě
3. For (29), it is standard (see [76] [77]) thatfor any initial data p u , d q P H p R n , R n q ˆ H e p R n , S q , e P S , there exist T ˚ ą p u, d q : R n ˆ r , T ˚ q Ñ R n ˆ S of (29) and (32), see[78] [79] [80] for related works on the existence of global strong solutions to (29) forsmall initial data in suitable function spaces. Furthermore, a criterion on possiblebreakdown for local strong solutions of (29), analogous to BKM criterion for theNavier-Stokes equation [81], was obtained by Huang-Wang [85]: if 0 ă T ˚ ă `8 isthe maximum time, then ż T ˚ ` } ∇ ˆ u } L p R n q ` } ∇ d } L p R n q ˘ dt “ `8 . (40)See Wang-Zhang-Zhang [87] and Hong-Li-Xin [86] for related works on the generalEricksen-Leslie system (25).For (29), inspired by Koch-Tataru [89] the following well-posedness result hasbeen obtained by Wang [88].2.11. Theorem. —
There exists ǫ ą such that if p u , d q : R n Ñ R n ˆ S , with ∇ ¨ u “ , satisfies “ u ‰ BMO ´ p R n q ` “ d ‰ BMO p R n q ď ǫ , then there exists a unique global smooth solution p u, d q P C p R n ˆ p , `8q , R n ˆ S q of (29), under the initial condition p u , d q , which enjoys the decay estimate t ` } u p t q} L p R n q ` } ∇ d p t q} L p R n q ˘ À ǫ , t ą . (41) Higher order decay estimates for the solution given by Theorem 2.11 have beenestablished by [90] and [91]. Local well-posedness of (29) in R for initial data p u , d q , with p u , ∇ d q P L p R q (the uniformly locally L -space in R ) havingsmall norm }p u , ∇ d q} L p R q , has been established by Hineman-Wang [92]. Wewould like to mention some interesting works on both new modeling and analysisof the hydrodynamics of non-isothermal nematic liquid crystals by Feireisl-Rocca-Schimperna [82], Feireisl-Fr´emond-Rocca-Schimperna [83], and Li-Xin [84].Now we describe a very recent work by Lin and Wang [93] on the existence ofglobal weak solutions of (29) in dimensions three by performing blow up analysis ofthe Ginzburg-Landau approximation system (31). More precisely, we have proved2.12. Theorem. —
For Ω Ď R either a bounded domain or the entire R , assume u P H and d P H p Ω , S q satisfies d p Ω q Ď S ` , the upper half sphere. Then thereexists a global weak solution p u, d q : Ω ˆ r , `8q Ñ R ˆ S to the initial andboundary value problem of (29) and (32) such that(i) u P L p , `8 ; H q X L p , `8 ; J q .(ii) d P L p , `8 ; H p Ω , S qq and d p t qp Ω q Ď S ` for L a.e. t P r , `8q .(iii) p u, d q satisfies the global energy inequality: for L a.e. t P r , `8q , ż Ω p| u | ` | ∇ d | qp t q ` ż t ż Ω ` µ | ∇ u | ` | ∆ d ` | ∇ d | d | ˘ ď ż Ω p| u | ` | ∇ d | q . (42)The proof of Theorem 2.12 in [93] is very delicate, which is based on suitableextensions of the blow-up analysis scheme that has been developed in the context ofharmonic maps by Lin [94] and harmonic map heat flows by Lin-Wang [95, 96, 97].A crucial ingredient is the following compactness result.2.13. Theorem. —
For any ă a ď , L ą , and L ą , set X p L , L , a ; Ω q consisting of maps d ǫ P H p Ω , R q , ă ǫ ď , that are solutions of ∆ d ǫ ` ǫ p ´ | d ǫ | q d ǫ “ τ ǫ in Ω (43) such that the following properties hold: (i) | d ǫ | ď and d ǫ ě ´ ` a for a.e. x P Ω . (ii) E ǫ p d ǫ q “ ż Ω e ǫ p d ǫ q dx ď L . (iii) ›› τ ǫ ›› L p Ω q ď L .Then X p L , L , a ; Ω q is precompact in H p Ω , R q , i.e., if t d ǫ u Ď X p L , L , a ; Ω q ,then there exists d P H p Ω , S q such that, up to a subsequence, d ǫ Ñ d in H p Ω , R q as ǫ Ñ . Sketch of proof of Theorem 2.13 . Assume d ǫ á d in H p Ω q and e ǫ p d ǫ q dx á µ : “ | ∇ d | dx ` ν as convergence of Radon measures for some nonnegative Radon measure ν , calleddefect measure. We claim that ν ”
0. This will be done in several steps.
Step1 (almost monotonicity). d ǫ P X p L , L , a ; Ω q satisfiesΦ ǫ p R q ě Φ ǫ p r q ` ż B R z B r | x | ´ ˇˇ B d ǫ B| x | ˇˇ , @ ă r ď R, (44)where Φ ǫ p r q : “ r ż B r ` e ǫ p d ǫ q ´ x x ¨ ∇ d ǫ , f ǫ y ˘ ` ż B r | x || f ǫ | . YDRODYNAMIC FLOW 13
Step2 ( δ -strong convergence). D δ ą , α P p , q , and C ą d ǫ P X p L , L , a ; Ω q satisfies Φ ǫ p r q ď δ , then | d ǫ | ě in B r , and ” d ǫ | d ǫ | ı C α p B r q ď C . In particular, d ǫ Ñ d in H p B r q and ν “ B r . Step3 (almost monotonicity for µ )Θ p µ, r q : “ r µ p B r q ď Θ p µ, R q ` C p R ´ r q , @ ă r ď R. (45)Thus Θ p µ q “ lim r Ó Θ p µ, r q exists, and is upper semicontinuous. Step4 (concentration set). d ǫ Ñ d in H p Ω z Σ q , whereΣ : “ x P Ω : Θ p µ q ě δ ( is a 1-rectifiable, closed subset, with H p Σ q ă `8 . Moreover,supp p ν q Ď Σ and δ ď Θ p ν, x q “ Θ p µ, x q ď C H a . e . x P Σ . Step5 (stratification and blow-up). If ν ı
0, then H p Σ q ą
0. We can pick up ageneric point x P Σ such thatlim r i Ó r ´ i ż B ri p x q | ∇ d | “ , Θ p ν, ¨q is H approximately continuous at x . Define blow-up sequences d i p x q “ d ǫ i p x ` r i x q and ν i p A q “ r ´ i ν p x ` r i A q for A Ď R . Then ν i á ν : “ θ H L Y, e ǫ i p d i q dx á ν p “ θ H L Y q as convergence of Radon measures, for some θ ą Y “ tp , , x q : x P R u .Applying (44), one has ż B ˇˇ B d i B x ˇˇ Ñ . Now we can perform another round of blow up process to extract a nontrivialsmooth harmonic map ω : R Ñ S ´ ` δ that has finite energy (see [93] for thedetail). This is impossible, since any such a ω has non-zero topological degree. (cid:3) Compressible flow of nematic liquid crystals.
When the fluid is assumedto compressible, the Ericksen-Leslie system becomes more complicate and thereseems very few analytic works available yet. We would like to mention that therehave been both modeling study (see [98]) and numerical study (see [99]) on thehydrodynamics of compressible nematic liquid crystals under the influence of tem-perature gradient or electromagnetic forces. Here we briefly describe some recentstudies on a simplified version of compressible Ericksen-Leslie system in Ω Ď R n ,which is given by $’’’&’’’% B t ρ ` ∇ ¨ p ρu q “ , B t p ρu q ` ∇ ¨ p ρu b u q ` ∇ p P p ρ qq “ L u ´ ∇ ¨ p ∇ d d ∇ d ´ | ∇ d | I n q , B t d ` u ¨ ∇ d “ ∆ d ` | ∇ d | d, ` ρ, u, d ˘ˇˇ t “ “ ` ρ , u , d ˘ , ` u, d ˘ˇˇ B Ω “ ` u , d ˘ , (46)where ρ : Ω Ď R n Ñ R ` is the fluid density and P p ρ q : Ω Ñ R ` denotes thepressure of the fluid, and L denotes the Lam´e operator L u “ µ ∆ u ` p µ ` λ q ∇ p ∇ ¨ u q , where µ and λ are the shear viscosity and the bulk viscosity coefficients of the fluid,satisfying the following physical condition: µ ą , λ ` µ ě . The system (46) is a strong coupling between compressible Navier-Stokes equationand the transported harmonic map heat flow to S . Since it is a challenging questionto establish the existence of global weak solutions to compressible Navier-Stokesequation itself (see [100]), it turns out to be more difficult to show the existence ofglobal weak solutions to (46) than the incompressible nematic liquid crystal flow(29). See [107] for a rigorous proof of convergence of (46) to (29) under suitableconditions. The local existence of strong solutions to (46) can be established undersuitable regularity and compatibility conditions on initial data. For example, wehave proven in [103, 104]2.14. Theorem. —
For n “ , assume P P C , p R ` q , ď ρ P W ,q X H X L for some ă q ă , u P H X H , and d P H p Ω , S q satisfies L u ´ ∇ p P p ρ qq ´ ∆ d ¨ ∇ d “ ρ g, g P L . Then there exist T ą and a unique strong solution p ρ, u, d q to (46) in Ω ˆ r , T q .Furthermore, if T ă `8 is the maximal time interval, then } ∇ u ` p ∇ u q t } L p ,T ; L q ` } ∇ d } L p ,T ; L q “ `8 , or } ρ } L p ,T ; L q ` } ∇ d } L p ,T ; L q “ `8 p if 7 µ ą λ q . (47)The local strong solution established in Theorem 2.14 has been shown to beglobal for small initial-boundary data p ρ , u , d q by [108] and [109].The global existence of weak solutions to (46) is more challenging than (29) indimensions n ě
2, see [101, 102] for n “
1. When relaxing the condition | d | “ d P R by the Ginzburg-Landau approximation, similar to (31), the globalexistence of weak solutions has been established by [105] and [106] in dimensions n “
3. Finally, we would like to mention that by employing the precompactnesstheorem 2.13, the following global existence of weak solutions to (46) has beenestablished by [110] in dimensions n “ , n “ Theorem. —
Assume P p ρ q “ ρ γ for γ ą , ď ρ P L γ , m P L γγ ` and | m | ρ χ t ρ ą u P L , and d P H p Ω , S q with d ě . Then there exists a globalrenormalized weak solution p ρ, u, d q to (46). Acknowledgements . The first author is partially supported by NSF grantsDMS1065964 and DMS1159313. The second author is partially supported by NSFgrant DMS1001115 and DMS1265574, and NSFC grant 11128102 and a SimonsFellowship in Mathematics. The second author wishes to thank his close friend C.Liu for many thoughtful discussions on the subject.
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Courant Institute of Mathematical Sciences, New York University, NY 10012, USAand NYU-ECNU Institute of Mathematical Sciences, at NYU Shanghai, 3663, NorthZhongshan Rd., Shanghai, P. R. China 200062
E-mail address : [email protected] Department of Mathematics, University of Kentucky, Lexington, KY 40506, USA,and Department of Mathematics, Purdue University, 150 N. University Street, WestLafayette, IN 47907, USA
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