REGULARITY AND EXISTENCE OF GLOBAL SOLUTIONS TO THE ERICKSEN-LESLIE SYSTEM IN R 2

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1 REGULARITY AND EXISTENCE OF GLOBAL SOLUTIONS TO THE ERICKSEN-LESLIE SYSTEM IN R JINRUI HUANG, FANGHUA LIN, AND CHANGYOU WANG Abstract. In this paper, we first establish the regularity theorem for suitable weak solutions to the Ericksen- Leslie system in R. Building on such a regularity, we then establish the existence of a global weak solution to the Ericksen-Leslie system in R for any initial data in the energy space, under the physical constraint conditions on the Leslie coefficients ensuring the dissipation of energy of the system, which is smooth away from at most finitely many times. This extends earlier works by Lin-Lin-Wang 3 on a simplified nematic liquid crystal flow to the general Ericksen-Leslie system.. Introduction The hydrodynamic theory of nematic liquid crystals was developed by Ericksen Leslie during the period of 958 through 968 5, 6, 8, 7. It is referred as the Ericksen-Leslie system in the literature. It reduces to the Ossen-Frank theory in the static case, which has been successfully studied see, e.g., Hardt- Lin-Kinderlehrer 9. In R 3, let u = u, u, u 3 T : R 3 R 3 denote the fluid vector field of the underlying incompressible fluid, d = d, d, d 3 T : R 3 S denote the orientation order parameter representing the macroscopic average of nematic liquid crystal molecular directors. Assume that the fluid is homogeneous e.g., the fluid density ρ the inertial constant is zero i.e., ρ = 0, the general Ericksen-Leslie system in R 3 consists of the following equations cf. 7, 7, 8, : t u + u u = σ, u = 0, d where σ = σ ij is the total stress given by g + W d W d = 0,. σ ij = P δ ij W d k,i d k,j + σ L ij,. with P a scalar function representing the pressure, W being the Ossen-Frank energy density function given by W := W d, d k d + k d d + k 3 d d + k + k tr d d.3 for some elasticity constants k, k, k 3, k, σ L = σ L u, d = σij L u, d representing the Leslie stress tensor given by σiju, L d = µ d k d p A kp d i d j + µ N i d j + µ 3 d i N j + µ A ij + µ 5 A ik d k d j + µ 6 A jk d k d i,. k,p for six viscous coefficients µ,, µ 6, called Leslie s coefficients, g i = λ N i + λ d j A ji = λ Ni + λ d j A ji..5 λ Throughout this paper, we use A ij = j uj + u i, Ω ij = uj u i, x i x j x i x j Date: June 6, 03. Key words phrases. Ericksen stress, Leslie stress, nematic liquid crystal flow, Leray-Hopf type weak solution. j k k

2 J. HUANG, F. LIN, AND C. WANG ω i = d i = t d i + u d i, N i = ω i j Ω ij d j, denote the rate of strain tensor, the skew-symmetric part of the strain rate, the material derivative of d the rigid rotation part of the director changing rate by fluid vorticity, respectively. Due to the temperature dependence of the Leslie coefficients, various coefficients have different behavior: µ does not involve the alignment properties hence is a smooth function of temperature, while all the other µ is describe couplings between molecule orientation the flow, might be affected by the change of the nematic order parameter d. In this paper, we only consider the isothermal case in which all µ i s are assumed to be constants. The Leslie coefficients µ i s λ, λ satisfy the following relations: λ = µ µ 3, λ = µ 5 µ 6,.6 µ + µ 3 = µ 6 µ 5..7 The relation.6 is a necessary condition in order to satisfy the equation of motion identically, while.7 is called Parodi s relation. Under the assumption of Parodi s relation, the hydrodynamics of an incompressible nematic liquid crystal flow involves five independent Leslie s coefficients. To avoid the complexity arising from the general Ossen-Frank energy functional, we consider the elastically isotropic case k = k = k 3 = k = 0 so that the Ossen-Frank energy reduces to the Dirichlet energy: W d, d = d. In this case, we have W d d k,j =, d W = d d ij, = d, d k,i x i x j d As a consequence, the Ericksen-Leslie system. can be written as t u + u u + P = d d + σ L u, d, u = 0, t d + u d Ωd + λ λ Ad = λ d + d d + λ λ d T Ad d. W d = 0. Since the general Ericksen-Leslie system is very complicated, earlier attempts of rigorous mathematical analysis of.8 were made for a simplified system that preserve the crucial energy dissipation feature as in.8, pioneered by Lin 9 Lin-Liu 0,. More precisely, by adding the penalty term ɛ d ɛ > 0 in the energy functional W to remove the nonlinearities resulting from the nonlinear constraints d =, Lin Liu have studied in 0, the following Ginzburg-Lau approximate system: t u + u u + P = µ u d d, u = 0, t d + u d = d +.9 ɛ d d. They have established in 0 the existence of global weak solutions in dimensions 3, global strong solutions of.9 in dimension, the local existence of strong solutions in dimension 3, the existence of global strong solutions for large viscosity µ > 0 in dimension 3. A partial regularity for suitable weak solutions of.9 in dimension 3, analogous to Caffarelli-Kohn-Nirenberg on the Naiver-Stokes equation, has been proved in. As already pointed out by 0, it is still a challenging open problem as ɛ tends to zero, whether solutions u ɛ, d ɛ of.9 converge to that of the following simplified Ericksen-Leslie system: t u + u u + P = µ u d d, u = 0,.0 t d + u d = d + d d. Very recently, there have been some important advances on.0. For dimension, Lin-Lin-Wang 3 Lin-Wang have established the existence of a global unique weak solution of.0 in any smooth bounded domain, under the initial boundary conditions, which is smooth away from possibly finitely many times see also Hong 0, Xu-Zhang 35, Hong-Xin, Lei-Li-Zhang6 for related results in R. It is an open problem whether there exists a global weak solution of.0 in dimension 3. There have been some partial results towards this problem. For example, the local existence uniqueness of strong solutions of.0 has been proved by Ding-Wen, the blow-up criterion of local strong solutions of.0,.8

3 THE ERICKSEN-LESLIE SYSTEM IN R 3 similar to Beale-Kato-Majda for Naiver-Stokes equations, has been established by Huang-Wang, the global well-posedness of.0 for rough initial data u 0, d 0 with small BMO BMO -norm has been shown by Wang 3, the local well-posedness of.0 for initial data u 0, d 0 with small L 3 uloc R3 -norm of u 0, d 0 has been proved by Hineman-Wang here L 3 uloc R3 denotes the locally uniform L 3 -space on R 3. For the general Ericksen-Leslie system.8 in R 3, there have also been some recent works. For example, Lin-Liu Wu-Xu-Liu 3 have considered its Ginzburg-Lau approximation: t u + u u + P = d d + σ L u, d, u = 0, t d + u d Ωd + λ λ Ad = λ d + ɛ d d,. have established the existence of global weak solutions, the local existence uniqueness of strong solutions of. under certain conditions on the Leslie coefficients µ i s. In particular, there have been results developed by 3 concerning the role of Parodi s condition.7 in the well-posedness stability of.. Most recently, Wang-Zhang-Zhang 33 have studied the general Ericksen-Leslie system.8 established the local well-posedness, the global well-posedness for small initial data under a seemingly optimal condition on the Leslie coefficients µ i s. In this paper, we are mainly interested in both the regularity existence of global weak solutions of the initial value problem of the general Ericksen-Leslie system.8 in R. In R, however, we need to modify several terms inside the system.8 in order to make it into a closed system. Since u is a planar vector field in R, both Ω A are horizontal -matrices, henceforth we assume that Ωd := Ω d, 0 T, Ad := A d, 0 T, N := t d + u d T Ω d, 0 T,. as vectors in R 3, while N := t d + u d Ω d T is a vector in R. Here d = d, d, 0 T for d = d, d, d 3 T R 3, σ L u, d is a -matrix valued function given by σiju, L d := µ d k d p A kp d i d j + µ N i d j + µ 3 N j d i + µ A ij + µ 5 for i, j, k,p= d T Ad := d T A d A ij d i d j. i,j= A ik d k d j + µ 6 k= A jk d k d i.3 We will consider the initial value problem of.8 in R, i.e., u, d = u 0, d 0 in R. t=0 for any given u 0 H, d 0 H e 0 R, S. Here we denote the relevant function spaces for some constant vector e 0 S, H = the closure of C0 R, R {v v = 0} in L R, R, { } He k 0 R, S = d : R S d e 0 H k R, R 3 k N + J = the closure of C 0 R, R {v v = 0} in H R, R. Definition.. For 0 < T, u L 0, T, H d L 0, T, He 0 R, S is called a weak solution of the Ericksen-Leslie system.8 together with the initial condition. in R, if T T T u, ψ φ u u : ψ φ + σ L u, d : ψ φ 0 R 0 R 0 R T = ψ0 u 0, φ + d d, ψ φ, R 0 R k=

4 J. HUANG, F. LIN, AND C. WANG T d, ψ φ + 0 R = ψ0 d 0, φ + R T 0 T 0 R R u d Ωd + λ Ad, ψ λ φ for any ψ C 0, T with ψt = 0, φ J, φ H R, R 3. λ d, ψ φ + λ d d, ψ φ + λ λ d T Ad d, ψ φ In this paper, we will establish the regularity of suitable weak solutions of.8, the existence uniqueness of global weak solutions to.8. in R. As consequences, these extend the previous works by Lin-Lin-Wang 3 to the general case. For x 0 R, t 0 0, +, z 0 = x 0, t 0 0 < r t 0, denote B r x 0 = {x R x x 0 r}, P r z 0 = B r x 0 t 0 r, t 0. When x 0 = 0, 0 t 0 = 0, we simply denote B r = B r 0, P r = P r 0. Now we introduce the notion of suitable weak solutions of.8. Definition.. For 0 < T < + a domain O R, a weak solution u L O 0, T, R, with u = 0, d L t H xo 0, T, S of.8 is called a suitable weak solution of.8 if, in addition, u L t L x L t H x O 0, T, R, d L t H x L t H x O 0, T, S, P L O 0, T. Theorem.3. For 0 < T < + a domain O R, assume that u L t L x L t H x O 0, T, R, d L 0, T, H O, S L 0, T, H O, S, P L O 0, T is a suitable weak solution of.8. Assume both.6.7 hold. If, in additions, the Leslie coefficients µ i s satisfy then u, d C O 0, T, R S. λ < 0, µ λ λ 0, µ > 0, µ 5 + µ 6 λ λ,.5 Employing the priori estimate given by the proof of Theorem.3, we will prove the existence of global weak solutions of.8. that enjoy partial smoothness properties. Theorem.. For any u 0 H d 0 H e 0 R, S, assume the conditions.6,.7,.5 hold. Then there is a global weak solution u L 0, +, H L 0, +, J d L 0, +, H e 0 R, S of the general Ericksen-Leslie system.8. such that the following properties hold: i There exist a nonnegative integer L, depending only on u 0, d 0, 0 < T < < T L < + such that u, d C R 0, + \ {T i } L i=, R S. ii Each singular time T i, i L, can be characterized by lim inf max u + d y, t 8π, r > 0..6 t T i x R B rx Moreover, there exist x i m x i 0 R, t i m T i, r i m 0 a non-trivial smooth harmonic map ω i : R S with finite energy such that as m, where iii Set T 0 = 0. Then u i m, d i m 0, ω i in C locr, 0, u i mx, t = r i mux i m + r i mx, t i m + r i m t, d i mx, t = dx i m + r i mx, t i m + r i m t. d t, d L j=0 ɛ>0 By using.83, we see that N L t H x is well defined. Similarly, T 0 L R T j, T j+ ɛ T L <T <+ +L t L x. Hence σl u, d L t H x R Ωd + λ Ad, ψ φ T λ 0 L R T L, T. T Hx +L t L x L x σ L u, d : ψ φ 0 R R d T Ad d, ψ φ are also defined as pairs between H H0..

5 THE ERICKSEN-LESLIE SYSTEM IN R 5 iv There exist t k + a smooth harmonic map d C R, S with finite energy such that u, t k 0 in H R d, t k d in H R, there exist a nonnegative integer l, {x i } l i= R, nonnegative integers {m i } l i= such that l d, t k dx d dx + 8π m i δ xi..7 v If either the third component of d 0, d 0 3, is nonnegative, or u 0 + d 0 8π, R then u, d C R 0, +, R S. Moreover, there exist t k + a smooth harmonic map d C R, S with finite energy such that i= u, t k, d, t k 0, d in C locr. An important first step to prove Theorem.3 is to establish the decay lemma 3. under the small energy condition, which is proved by a blow-up argument. Here the local energy inequality. for suitable weak solutions to.8 plays a very important role, which depends on the conditions.6,.7,.5 heavily. In contrast with earlier arguments developed by 3 on the simplified nematic liquid crystal equation.0, where the limiting equation resulting from the blow up process is the linear Stokes equation the linear heat equation, the new linear system 3.9 arising from the blow-up process of the general Ericksen-Leslie system.8 is a coupling system. It is an interesting question to establish its smoothness. The proof of regularity of 3.9 is based on higher order local energy inequalities. The cancelation properties among the coupling terms play critical roles in the argument of various local or global energy inequalities for both the linear system 3.9 the nonlinear system.8. The second step is to establish a higher integrability estimate of suitable weak solutions to.8 under the small energy condition, which is done by employing the techniques of Riesz potential estimates between parabolic Morrey spaces developed by. The third step is to establish an arbitrary higher order energy estimate of.8 under the small energy condition. With the regularity theorem.3, we show the existence theorem. by adapting the scheme developed by 3. Motivated by the uniqueness theorem proved by 3 35 on.0, we believe that the weak solution obtained in Theorem. is also unique in its own class plan to address it in a forthcoming article. The paper is organized as follows. In section two, we derive both local global energy inequality for suitable weak solutions of.8. In section three, we prove an ɛ-regularity theorem for.8 first then prove Theorem.3. In section four, we prove the existence theorem... Global local energy inequalities of Ericksen-Leslie system in R In this section, we will establish both global local energy inequality for suitable weak solutions of.8 in R under the conditions.6,.7,.5. We begin with the global energy inequality. Lemma.. For 0 < T +, assume the conditions.6,.7,.5 hold. If u L t L x L t HxR 0, T, R, d L 0, T, He 0 R, S L 0, T, He 0 R, S, P L R 0, T is a suitable weak solution of the Ericksen-Leslie system.8. Then for any 0 t < t T, it holds t u + d t + µ u + R R λ d + d d u + d t.. R t Proof. Since d =, we have d + d, d = 0. This, combined with the fact d L 0, T, He 0 R, S, implies that d L R 0, T. Since u L t L x L t HxR 0, T, R, it follows from Ladyzhenskaya s inequality that u L R 0, T. For η C0 R or η in R, multiplying.8 by uη, integrating the resulting equation over R, using u = 0, we obtain d u η = η d d : u σ L u, d : u dt R R We remark that this condition was first brought out explicitly by Lei-Li-Zhang 6 for the simplified Ericksen-Leslie system.0.

6 6 J. HUANG, F. LIN, AND C. WANG + u + P u η + d d σ L u, d : u η.. R Using.6,.7, the symmetry of A, the skew-symmetry of Ω, we find η σ L u, d : u R = η µ dk dp A kp di dj + µ Ni dj + µ 3 Nj di + µ A ij + µ 5 A ik dk dj + µ 6 A jk dk di A ij + Ω ij R = η µ A : d d + µ A + µ + µ 3 d A N + µ µ 3 d Ω N R +µ 5 + µ 6 A d + µ 5 µ 6 A dω d = η µ A : d d + µ A + µ 5 + µ 6 A d R +λ N Ω d λ N A d + λ A dω d..3 Putting.3 into., we have d u η = η d d : u µ A : dt d d µ A µ 5 + µ 6 A d R R λ N Ω d + λ N A d λ A dω d + u + P u η + d d σ L u, d : u η.. R Since d+ d d L R 0, T, it follows from the equation.8 3 that t d+u d L R 0, T. Multiplying.8 3 by η d integrating the resulting equation over R yields that d dt B d η + B λ d + d d η = t d, d η + η u d, d + λ A R R λ d Ω d, d + λ λ d T A d d..5 Using integration by parts u = 0, we see η u d, d = R η d d : u + R R d u η d d : u η. Substituting this into.5, we obtain d dt B d η + B λ d + d d η = η λ A R λ d Ω d, d + λ λ d T A d d η d d : u + d u η d d : u η t d, d η..6 R Adding. together with.6, we obtain d u + d η + µ A + dt R R λ d + d d η = µ A : d d + µ 5 + µ 6 A d + λ N Ω d λ N A d R η +λ A dω d λ A λ d Ω d, d λ λ d T A d d + u + P u η + d d σ L u, d : u η R

7 THE ERICKSEN-LESLIE SYSTEM IN R 7 + d u η d d : u η t d, d η..7 R Denote the first term in the right h side of.7 as I. In order to estimate I, we need to use.8 3 to make crucial cancelations among terms of I as follows. while λ N Ω d = λ Ni Ω ij dj = λ A ik dk d i + λ d T A d d i Ω ij dj = λ A ik dk d i Ω ij dj = λ A dω d Ω d, d,.8 λ N A d = λ Ni A ij dj = λ λ A ij dj λ λ d i + d di + λ λ d T A d d i A ij dj = λ λ A d λ λ d T Ad + λ λ A d, d + λ λ d T A d d..9 Since A : d d = d T A d, we have, by substituting.8.9 into I, that I = µ λ A : λ d d + µ 5 + µ 6 + λ A λ d..0 R Substituting.0 into.7, we finally obtain d u + d η + µ u + dt R R λ d + d d η = η µ λ A : R λ d d + µ 5 + µ 6 + λ A d λ + u + P u η + d d σ L u, d : u η R + d u η d d : u η t d, d η + µ u u, η.. R Here we have used the fact u = 0 the following identity: A η = u η u u, η.. R R R If η, then. the condition.5 imply d u + d + µ u + dt R R λ d + d d = µ λ A : λ d d + µ 5 + µ 6 + λ A λ d 0..3 R Integrating.3 over 0 t t T yields.. This completes the proof of lemma.. We also need the following local energy inequality in the proofs of our main theorems. Lemma.. For 0 < T +, assume the conditions.6,.7,.5 hold. If u L t L x L t HxR 0, T, R, d L 0, T, He 0 R, S L 0, T, He 0 R, S, P L R 0, T is a suitable weak solution of the Ericksen-Leslie system.8. Then for any 0 t < t T η C0 R, it holds t η u + d t + η µ u + R t R λ d + d d t η u + d t + C u + u + u d + d + d + P u η R t R + u + u d + d + d d η..

8 8 J. HUANG, F. LIN, AND C. WANG Proof. It suffices to estimate the last two terms in the right h of.. Denote these two terms by II III. To do so, first observe that by.8 3 it holds that hence t d C u d + u + d + d, σ L u, d C A + N C u + t d + u d C u + u d + d + d. With these estimates, we can show that II C u + P + d + σ L u, d u η R C u + P + d + u d + u + d u η, R III C d u + t d d + u u η R C d u + u u + u + u d + d + d d η. R Putting these estimates of II III into. yields.. This completes the proof. 3. ɛ-regularity of the Ericksen-Leslie system.8 in R In this section, we will establish the regularity of suitable weak solutions of the Ericksen-Leslie system.8 in R, under a smallness condition. The crucial step is the following decay lemma under the smallness condition. Lemma 3.. Assume that the conditions.6,.7,.5 hold. There exist ɛ 0 > 0 θ 0 0, such that for 0 < T < + a bounded domain O R, if u L t L x L t HxO 0, T, R, d L 0, T, H O, S L 0, T, H O, S, P L O 0, T is a suitable weak solution of the Ericksen-Leslie system.8 in O 0, T, which satisfies, for some z 0 = x 0, t 0 O 0, T 0 < r 0 min{dx 0, O, t 0 }, then Here we denote Φu, d, P, z 0, r ɛ 0, Φu, d, P, z 0, θ 0 r Φu, d, P, z 0, r. 3. Φu, d, P, z 0, r := u + u + d + d + P. P rz 0 P rz 0 P rz 0 P rz 0 P rz 0 Proof. First observe that since.8 is invariant under translations dilations, we have that u r x, t = rux 0 + rx, t 0 + r t, d r x, t = dx 0 + rx, t 0 + r t, P r x, t = r P x 0 + rx, t 0 + r t, is a suitable weak solution of.8 in P 0. Thus it suffices to prove the lemma for z 0 = 0, 0 r =. We argue it by contradiction. Suppose that the lemma were false. Then there would exist ɛ i 0 a sequence of suitable weak solutions u i, d i, P i L t L x L t H xp, R L t H x L t H xp, S L P of.8 such that but, for any θ 0,, it holds Φ u i, d i, P i, 0, 0, = ɛ i 0, 3. Φ u i, d i, P i, 0, 0, θ > ɛ i. 3.3

9 THE ERICKSEN-LESLIE SYSTEM IN R 9 Now we define a blow-up sequence: ũ i = u i, di = d i d i, ɛ i ɛ i Pi = P i, ɛ i 3. where d i = d i is the average of d i over P. P P It is easy to see that ũ i, d i, P i satisfies t ũ i + ɛ i ũ i ũ i + P i = ɛ i d i d i + ɛ i σ L u i, d i, ũ i = 0, t di + ɛ i ũ i d i ɛ Ω i d i + λ i A i d i = i λ ɛ λ d i + ɛ i d i d i + λ λ ɛ i d T i Ai d i d i. 3.5 Moreover, we have Φ ũ i, d i, P i, 0, 0, =, Φ ũ i, d i, P i, 0, 0, θ >. 3.6 It follows from the equation that L t di P 3 C ũ i d i L P + ũ i L P + d i L P After taking possible subsequences, we may assume that there exists ũ, d, P L t L x L t H xp, R L t H x L t H xp, R 3 L P C. 3.7 a point d 0 S such that It is easy to check that ũ i ũ in L t L xp L t HxP, P i P in L t L xp, d i d in L t Wx, P L t HxP, d i d in H P 3, d i d 0 a.e. P 3. while ɛ i ɛ i Ω i = ũi ũ i T Ω := ũ ũ T in L P 3, ɛ i A i = ũi + ũ i T à := ũ + ũ T in L P 3, ɛ i d T i A i di d 0T à d 0, ɛ i A i di à d 0 in L P 3, i t d i + u i d i Ω i di Ñ := d t Ω d 0 in L P 3, N i = ɛ ɛ i σ L u i, d i := µ d i d i : ɛ i A i d i d i + µ ɛ i N i d i + µ 3 di ɛ i N i + µ ɛ i A i +µ 5 ɛ i A i d i d i + µ 6 di ɛ i A i d i σ L ũ, d 0 := µ d 0 d 0 : à d 0 d 0 + µ Ñ d 0 + µ 3 d0 Ñ + µ à +µ 5 à d 0 d 0 + µ 6 d0 à d 0 in L P 3. Since d i =, an elementary argument from the differential geometry implies that dx, t T d0 S a.e. x, t P

10 0 J. HUANG, F. LIN, AND C. WANG Therefore ũ, d, P satisfies 3.8 the following linear system in P 3 : t ũ + P = σ L ũ, d 0, ũ = 0, t d Ω d0 + λ λ à d 0 = λ d + λ λ d0t à d 0 d 0. By the lower semicontinuity, we have 3.9 Φ ũ, d, P, 0, 0,. 3.0 By the regularity lemma 3. below, we know that ũ, d, P is smooth in P there exists 0 < θ 0 < such that Φ ũ, d, P, 0, 0, θ 0 Cθ 0 <. 3. In order to reach the desired contradiction, we need to apply the local energy inequality. for ũ i, d i, P i. First, observe that the equation 3.5 can be written as t ũ i + P i µ ũ i = g i := ɛ i ũ i ũ i + d i d i + ɛ i σ L u i, d i µ A i. 3. It follows from 3.6 that g i L, 0, H B g i L,0,H B ũ i L P + d i L P + ũ i L P + di L P Φ ũ i, d i, P i, 0, 0, C. Hence by the stard estimate on Stokes system cf. 3 we have that t ũ i L 3, 0, H B 3 L t ũ i Φ ũ i, d i, P L i, 0, 0, + g i C ,0,H B 3,0,H B It follows from 3.6, 3.7, 3.3 that we can apply the Aubin-Lions compactness lemma cf. 3 to conclude that, after taking possible subsequences, ũ i ũ, d i d in L P By Fubini s theorem, for any θ 0, there exists τ 0 θ, θ such that ũ i ũ + d i d x, τ 0 dx Cθ ũ i ũ + d i d Cθ o. B θ Here o denotes the constant such that we have B θ B θ P θ lim o = 0. Since i + ũ + d x, τ 0 dx Cθ, ũ i + d i x, τ 0 dx C θ + θ o. 3.5 Since u i, d i, P i satisfies the local energy inequality., we see that by rescalings ũ i, d i, P i satisfies the following local energy inequality: for any τ 0 t 0 any η C0 B, t η ũ i + d i t + η µ ũ i + R τ 0 R λ d i + ɛ i d d i η ũ i + d i τ 0 R t +C ɛ i ũ i + ũ i + ɛ i ũ i d i + ɛ i d i + di + P i ũ i η τ 0 R

11 THE ERICKSEN-LESLIE SYSTEM IN R + ũ i + ɛ i ũ i d i + ɛ i d i + di d i η. 3.6 By the weak strong convergence properties for ũ i, d i, P i listed as above, we have that, as i +, t E i := ɛ i ũ i + ũ i + ɛ i ũ i d i + ɛ i d i + di + P i ũ i η τ 0 R + ũ i + ɛ i ũ i d i + ɛ i d i + di d i η t E := ũ + d + P ũ η + ũ + d d η. 3.7 τ 0 R Now we choose η C 0 B such that 0 η, η in B τ 0, η 0 outside B τ0, η Cτ 0 Cθ. Then we have that for θ 0,, E θ ũ + d + P ũ + ũ + d d so that P θ d L P θ + ũ L P θ + P L P θ d L P θ + ũ L P θ Φ ũ, d, P, 0, 0, θ Cθ E i Cθ + o. 3.8 Substituting into 3.6 yields sup ũ i + d i t + µ ũ i + θ t 0 B θ λ d i + ɛ i d d i C θ + θ o. 3.9 P θ Since d i P θ d i + ɛ i d d i + ɛ i P θ d i P θ d i + ɛ i d d i + Cɛ i, P θ by the H -estimate di d i + θ P θ d i P θ d i + θ + θ o, P θ we obtain sup θ t 0 P θ ũ i + d i t + B θ P θ ũ i + di C θ + ɛ i + θ o. 3.0 Recall Ladyzhenskaya s inequality in R cf. 5: f f r f + f, f H B r. 3. B r B r B r Applying 3. to ũ i d i integrating over t-variable using 3.0, we obtain ũ i + d i ũ i + d i θ ũ i + d i + ũ i + di B θ P θ sup t θ,0 P θ C θ + ɛ i + θ o. 3. To estimate P i, let s take divergence of the equation 3.5 to get P i = ɛ i ũ i ũ i + d i d i + ɛ i σ L u i, d i in B θ. 3.3 Let φ C 0 R such that 0 φ, φ in B θ, φ 0 outside B θ, φ θ. 8

12 J. HUANG, F. LIN, AND C. WANG Define Q i by Q i x, t = ygx y : φ y ɛ i ũ i ũ i + d i d i ɛ i σ L u i, d i y, t dy, R where G is the fundamental solution of the Laplace equation on R. Then we have Q i = φ ɛ i ũ i ũ i + d i d i ɛ i σ L u i, d i in R. By Calderon-Zygmund s L -theory we have Q i L R ɛ i ũ i L B θ + d i L B θ + ɛ i σ L u i, d i L B θ ɛ i ũ i L B θ + d i L B θ + ũ i L B θ + ɛ i N i L B θ ũ i L B θ + d i L B θ + ũ i L B θ + di L B θ. 3. Integrating 3. over t θ, 0, using , we obtain 0 Q i ũ i + d i + ũ i + di C θ + ɛ i + θ o. 3.5 θ R P θ Set R i = P i Q i in P θ. Then we have R i t = 0 in B θ, t θ, 0, so that by the stard estimate of harmonic functions 3.5 we have R i θ R i Cθ P i + Q i C θ + θ + ɛ i + θ o. 3.6 P θ P θ P θ Putting 3.5 together with 3.6 yields P i C θ + θ + ɛ i + θ o. 3.7 P θ Combining all these estimates 3.0, 3., 3.7, we obtain Φ ũ i, d i, P i, 0, 0, θ C θ + ɛ i + θ o, 3.8 provided that we first choose sufficiently small θ then choose sufficiently large i. This gives the desired contradiction. The proof is complete. The following lemma plays an important role in the blow-up process, which may have its own interest. Lemma 3.. Assume.6,.7,.5 hold. For any point d 0 S, if ũ L t L x L t Hx P 3, R, d L t Hx L t Hx P 3, T d 0 S, P L P 3 solves the linear system 3.9 satisfies the condition 3.0, then ũ, d, P C P satisfies the following estimate: Φ ũ, d, P, 0, 0, θ Cθ, θ 0,. 3.9 Proof. To simplify the notations, we write u, d, P for ũ, d, P in the proof below. The argument is based on the higher order local energy inequality argument. Taking x i of the linear system 3.9 yields t u xi + P xi = σ L u, d 0 xi, u xi = 0, 3.30 t d xi Ω xi d0 + λ λ A xi d0 = λ d x i + λ λ dt 0 A xi d0 d 0.

13 THE ERICKSEN-LESLIE SYSTEM IN R 3 For any η C0 B, multiplying the equation 3.30 by u xi η the equation by d xi η integrating the resulting equations over B, we obtain 3 d u η = 3.3 dt B P u η + u : η + σ L u, d 0 x i : u xi η + η σ L u, d 0 x i : u xi, B B B d d η + d η dt B λ B = t d xi d xi η Ω xi d0 λ A xi d0, B B λ d xi + λ λ d T 0 A xi d0 d 0, d xi η. = t d xi d xi η Ω xi d0 λ A xi d0, B λ d xi η, 3.3 B where we have used in the last step the fact d T d0 S in order to deduce that d 0, d xi = 0 a.e. in B. Similar to the calculations in the proof of lemma., we have η σ L u, d 0 : u x xi i B = η µ d 0 d 0 : A xi d 0 d 0 + µ Ñ xi d 0 + µ 3 d0 Ñx i + µ A xi B +µ 5 A xi d 0 d 0 + µ 6 d0 A xi d 0 : A xi + Ω xi = µ d T 0 A xi d0 + µ A xi λ Ñ xi A xi d 0 + λ Ñ xi Ω xi d 0 B η By the equation , we have +µ 5 + µ 6 A xi d 0 + λ A xi d 0 Ω xi d λ Ñ xi Ω xi d 0 = λ A xi d 0 Ω xi d 0 Ω xi d0, d xi, λ Ñ xi A xi d 0 = λ A xi λ d 0 λ λ d T 0 A xi d0 + λ A xi d0, λ d xi. Substituting these identities into 3.33, we obtain = η σ L u, d 0 x i : u xi B µ λ λ d T 0 A xi d0 + µ A xi + λ A xi d0 Ω xi d0, λ d xi B η +µ 5 + µ 6 + λ A xi λ d Putting 3.3 into 3.3 adding the resulting 3.3 with 3.3, we have, by.5, d u + d η + µ u + dt B B λ 3 d η = η µ λ B λ d T 0 A xi d0 + µ 5 + µ 6 + λ A xi λ d 0 P u η + u : η σ L u, d 0 : u x xi η i B B 3 Strictly speaking, we first need to take finite quotient D i h of the system 3.9 then multiply the first equation the third equation of the resulting equations by Dh i ux, tη = ux+he i,t ux,t D h h i dη respectively for h > 0 i =,, with e =, 0 e = 0,. Then the desired estimate follows from the estimates on the finite quotients by sending h to zero.

14 J. HUANG, F. LIN, AND C. WANG + B B µ u η : u t d xi d xi η λ d d : η P u η + u : η σ L u, d 0 x i : u xi η + B B µ u η : u t d xi d xi η + λ 3 d d xi + d d xi η xi := R + R + R 3. Now we estimate each term of the right h side of 3.35 as follows. R P u η η + u η + η B µ u η + C P + u η + η, 6 B B R u + 3 d u η η B µ u η + 3 d η + C u η, 6 B λ B B R 3 u u η η + u + 3 d d η η B µ u η + 3 d η + C u + d η. 6 B λ B B Putting these estimates into 3.35, we obtain d dt u + d η + B C B B µ u + λ 3 d η 3.35 P + u η + η + d η, 3.36 By Fubini s theorem, there exists t, 0 such that u + d η t 8 u + d η. B P Integrating 3.36 over t t, 0 yields that sup u + d 0 η t + u + 3 d η t 0 B B C P + u η + η + d η + C u + d η P P For the pressure P, taking divergence of the equation 3.30 yields that for any t 0, P xi = σ L u, d 0 in B x i Similar to the pressure estimates obtained in the proof of lemma 3., we have P σ L u, d 0 xi + P u + 3 d + P P 38 P P 56 Let η C0B be a cut-off function of B 3, i.e. η in B 3, η 0 outside B 3, 0 η, 8 8 η + η 6. Then, by combining 3.37 with 3.39, we obtain sup t 0 B u + d t + P u + 3 d + P

15 THE ERICKSEN-LESLIE SYSTEM IN R 5 C P + u + d. 3.0 P 3 It turns out that the above energy method can be extended to any high order. Here we only give a sketch of the proof. In fact, if we denote α = k x as the k-th order derivative for any multiple index α = α α, α k = α = α + α, take α of the system 3.9, then we obtain t α u + α P = α σ L u, d 0, α u = 0, t α d α Ω d 0 + λ λ α A d 0 = λ α d + λ λ dt 0 α A d 3. 0 d 0. Multiplying 3. by α uη 3. 3 by α dη integrating the resulting equations over B, repeating the above calculations cancelations, we would obtain d k u + k+ d η µ + dt B B k+ u + λ k+ d η C k P + k u η + η + k+ d η. 3. For P, since we have P B α P = α σ L u, d 0 in B 3, 3.3 k P k P + k+ u + k+ d. 3. P 38 Following the same lines of proof as above, we can choose suitable time slice t, 0 such that k u + k+ d η t 8 k u + k+ d η. B P By choosing suitable test functions similar to the above ones, we can reach that for any k, it holds sup k u + k+ d t + k+ u + k+ d + k P t 0 B P C k P + k u + k+ d. 3.5 P 3 It is clear that with suitable adjusting of the radius, we see that implies that sup k u + k+ d t + k+ u + k+ d + k P t 0 B P C P + u + d 3.6 P 3 holds for all k. Now we can apply the regularity theory for both the linear Stokes equations c.f. 3 the linear heat equation cf. 5 to conclude that u, d C P. Furthermore, apply the elliptic estimate for the pressure equation 3.38, we see that P C P first we have k P C 0 P, then note that l tp also satisfies a similar elliptic equation, so that k tp l C 0 P. Therefore u, d, P C P the desired estimate 3.9 holds. The proof of lemma 3. is complete. In order to show the smoothness of solutions to.8 under the condition 3., we need to iterate the decay inequality 3. establish higher integrability of u, d by applying the techniques of Morrey space estimates for Riesz potentials, similar to that by Hineman-Wang.

16 6 J. HUANG, F. LIN, AND C. WANG Lemma 3.3. Assume that the conditions.6,.7,.5 hold. For any 0 < T + a bounded domain O R, there exists ɛ 0 > 0 such that u, P, d L t L x L t HxO 0, T L O 0, T L t HxO 0, T, S is a suitable weak solution of.8, satisfies, for z 0 = x 0, t 0 O 0, T P r0 z 0 O 0, T, Φ u, d, P, z 0, r 0 ɛ 0, 3.7 then u, d L q loc P r 0 z 0 for any < q < +. Moreover, it holds u, d L q P r 0 z 0 Cq, r 0 ɛ Proof. Set r = r0. Then it is easy to see that 3.7 also holds for u, P, d with z 0, r 0 replaced by z, r for any z P r 0 z 0. Applying lemma 3. for u, P, d on P r z, we conclude that there exists θ 0 0, such that for any 0 < r r, it holds that Iterating this inequality k-times, k, yields Φ u, d, P, z, θ 0 r Φ u, d, P, z, r Φ u, d, P, z, θ k 0r k Φ u, d, P, z, r. It is well known that this implies that there exists α 0, such that for any 0 < τ < r r, it holds Φ u, d, P, z, τ τ r α Φ u, d, P, z, r 3.9 holds for any z P r 0 z 0 0 < r r0. Now we proceed with the Riesz potential estimates of u, d between Morrey spaces as follows. First, let s recall the notion of Morrey spaces on R R, equipped with the parabolic metric δ: { δ x, t, y, s = max x y, } t s, x, t, y, s R R. For any open set U R +, p < +, 0 λ, define the Morrey Space M p,λ U by { } M p,λ U := v L p loc U : v p M p,λ U sup r λ v p < z U,r>0 P rz U It follows from 3.9 that for some α 0,, u, d M, α P r 0 z 0, u, d, P M, α P z 0 z Write the equation.8 3 as t d d = f, with f := λ By 3.5, we see that u d + Ω d λ A λ d + λ d d + λ λ d T A dd. 3.5 f M, α P r 0 z 0. As in 3, let η C 0 R + be a cut-off function of P r 0 z 0 : 0 η, η in P r 0 z 0, t η + η Cr 0. Set w = η d. Then we have where d z0, r 0 estimate t w λ w = F, F := η f + t η λ η d d z0, r 0 λ η d, 3.53 is the average of d over P r 0 z 0. It is easy to check that F M, α R + satisfies the F C M, α R + Φ u, d, P, z 0, r 0 + f M, α P r 0 z 0 Cɛ Let Γx, t denote the fundamental solution of the heat operator t λ on R. Then by the Duhamel formula for 3.53 the estimate see also 3 lemma 3.: Γ x, t δ 3, x, t 0, 0, x, t, 0, 0

17 THE ERICKSEN-LESLIE SYSTEM IN R 7 we have wx, t t 0 Γx y, t s F y, s C R R 3 F y, s δ 3 x, t, y, s := CI F x, t, 3.55 where I β is the Riesz potential of order β on R 3 β 0,, defined by gy, s I β g = δx, t, y, s β, g Lp R R 3 Applying the Riesz potential estimates see 3 Theorem 3., we conclude that w M α α, α R 3 w α M α, α R 3 F Cɛ M, α R 3 α Choosing α using lim α α = +, we can conclude that for any < q <, w Lq P r0 z 0 w Cq, r 0ɛ L q P r0 z 0 Since d w solves t d w λ d w = 0 in P r 0 z 0, it follows from the stard estimate on the heat equation that for any < q < +, d L q P r 0 z 0 d Lq P r 0 z 0 Cq, r 0ɛ Now we proceed with the estimation of u. Let v : R 0, + R solve the Stokes equation: t v µ v + Q = η d d + u u + η σ L u, d µ A in R 0,, v = 0 in R 0,, v, 0 = 0 in R By using the Oseen kernel see Leray 6, an estimate for v, similar to 3.55, can be given by t Xy, s vx, t C δx, t, y, s 3 CI X x, t, x, t R 0, +, R where X = η d d + u u + η σ L u, d µ A. As above, we can check that X M, α R 3 X C u + d M, α R 3 M, α P r 0 z + 0 u + d M, α P r 0 z 0 Cɛ 0. Hence, by 3 Theorem 3., we have that v M α α, α R 3, v α CX Cɛ M M, α R 3 α, α R 3 By sending α, 3.6 implies that for any < q < +, v L q P r0 z 0 v Cq, r 0ɛ Lq P r0 z 0 Since u v satisfies the linear homogeneous Stokes equation in P r 0 z 0 : t u v µ u v + P Q = 0, u v = 0 in P r 0 z 0. It is well-known that u v L P r 0 z 0. Therefore we conclude that for any < q < +, u L q P r 0 z 0, u Lq P r 0 z 0 The estimate 3.8 follows from This completes the proof. Cq, r 0 ɛ

18 8 J. HUANG, F. LIN, AND C. WANG Now we utilize the integrability estimate 3.8 of u, d to prove the smoothness of u, d. The argument is based on local inequalities of higher order energy of u, d. Lemma 3.. Assume that the conditions.6,.7,.5 hold. For any 0 < T + a bounded domain O R, there exists ɛ 0 > 0 such that u, P, d L t L x L t H xo 0, T L O 0, T L t H x L t H O 0, T, S is a suitable weak solution of.8, satisfies, for z 0 = x 0, t 0 O 0, T P r0 z 0 O 0, T, Φ u, d, P, z 0, r 0 ɛ 0, 3.65 then l u, l+ d L t L x L t Hx P + z l+ 0 for any l 0, the following estimate holds r 0 l u + l+ d sup t 0 + l+ r 0 t t 0 + B + l+ x 0 r 0 P + l+ z 0 r 0 l+ u + l+ d + l P Clɛ Proof. For simplicity, assume z 0 = 0, 0 r 0 =. We will prove 3.66 by an induction on l 0. i l = 0: 3.66 follows from the local energy inequality., similar to that given by lemma 3.. ii l : Suppose that 3.66 holds for l k. We want to show 3.66 also holds for l = k. From the hypothesis of induction, we have that for all 0 l k, sup l u + l+ d + l+ t 0 + P + l+ B + l+ l+ u + l+ d + l P Clɛ Hence by the Ladyzhenskaya inequality 3. we have l u + l+ d Clɛ 0, 0 l k By lemma 3.3, we also have P + l+ u L q P 3 + d Lq P 3 Cqɛ 0, < q < Take k-th order spatial derivative k of the equation.8, we have 5 Let η C 0 B such that t k u + k u u + k P = k d d + k σ L u, d η, η in B + k+, η = 0 outside B + k, η + η k+. Multiplying 3.70 by k uη integrating over B, we obtain 6 d dt B k u η = k u u : k uη + k P k u η B B + k d d : k uη k σ L u, d : k uη B B := I + I + I 3 + I. 3.7 We estimate I, I, I 3 as follows. Applying Hölder s inequality the following interpolation inequality: f η f η f η + f η, f H R, 3.7 B B B B In fact, we only need to use u, d L 8 in the proof for the cases k =,. 5 Strictly speaking we need to take the finite quotient D i h k of.8 then taking limit as h tends to zero. 6 Strictly speaking, we need to multiply the equation by D i h k uη.

19 THE ERICKSEN-LESLIE SYSTEM IN R 9 we have k I u k u + j u k j u k+ u η + k u η η B j= k δ + C k u η k+ u η + C j u B B j=0 +C k u + k u k u η 3.73 B where δ > 0 is a small constant to be chosen later. For I I 3, we have I k P k+ u η η + k u η B δ k+ u η + C k P + k u. 3.7 B k I 3 d k+ d + j+ d k+ j d k+ u η + k u η B j= k δ + C k+ d η k+ u + k+ d η + C u + j d B B j= +C k u + k+ d + k+ d k+ d η B For I, we need to proceed as follows. Set σ L k u, d = µ d d : k A d d + µ k N d + µ3 d k N + µ k A + µ 5 k A d d + µ 6 d k A d, ωk L u, d := k σ L u, d σk L u, d. Then we have k σ L u, d : k uη = σk L u, d : k+ uη B B σk L u, d : k u η B ωk L u, d : k+ uη + k u η B := J + J + J To estimate J, J, J 3, we take k of the equation.8 3 to get 7 k N + λ k A λ d = k d + k d d + λ k λ λ d T A dd Since d =, we have that d = d, d. Denote by # the multi-linear map with constant coefficients. It is well known see that for any l 0, l d#d L d L l d L l d L, l d#d#d L d L l d#d L + d#d L l d L l d L. Therefore by we have that for any 0 l k, d#d, d#d#d L t Hx l L t Hx l+ L P + l+. d#d + d#d#d L L t 7 Strictly speaking, we need to take D i h k of the equation. Clɛ Hl x L t Hl+ x P + l+

20 0 J. HUANG, F. LIN, AND C. WANG The estimate 3.78 also holds for d#d#d#d. Applying the equation 3.77 we have J k+ u + k N k u η η B k k+ u + k+ d + j u k j+ d k u η η + + B k l=0 k l=0 j= l+ d k l d#d k u η η k l d#d#d l+ u k u η η := J + J + J 3. Direct calculations imply J δ + C k u + k+ d η k+ u + k+ d η B B k + C k u + k+ d k u + k+ d η + C B J J 3 B l=0 i= i d + i u, k k d d + k+ d d + l+ d k l d#d k u η η C k u η k+ u η + C k u k u η B B B k +C d 8 + k d + k+ d + i d, B l= i= k k u d + k+ l d#d#d l u k u η η C k u + k+ d η B k+ u + k+ d η B k + C d 8 + k u + k+ d + C j u + j+ d + C j=0 k u + k+ d k u + k+ d η. B By the definition of ω L k u, d the equation.8 3, we have ω L k u, d k l= k l u k+ l d#d + k+ l d#d#d + l N k l d k l u k+ l d#d + k+ l d#d#d + d k+ d + l= k + l Ad + l d#d#d + l A#d#d#d k l d l=0 l=0 k l d k u + k+ d d + d#d + d#d#d k + l u + l d + l d#d + l d#d#d l= l=

21 THE ERICKSEN-LESLIE SYSTEM IN R Hence we can estimate J 3 ωk L u, d k+ u η + k u η η B δ + C k u + k+ d η k+ u + k+ d η + C B B k +C l u + l d + l d#d + l d#d#d l= +C k u + k+ d k u + k+ d k u + k+ d η. B 3.80 The most difficult term to hle is J, since the integrs involve terms consisting of the highest order factors k+ u k+ d. Here we need to apply.8 to cancel some of those terms employ the condition.5 to argue that other terms are non-positive. For this, we proceed as follows similar to lemma 3.. J = µ d d : k A d d + µ k N d + µ3 d k N B +µ k A + µ 5 k A d d + µ 6 d k A d : k Aη + k Ωη = µ d T k A d η + µ k A η + µ 5 + µ 6 k A d η B Applying the equation 3.77, we obtain +λ k N k Ω dη λ k N k A dη + λ k A d k Ω dη. 3.8 λ k N k Ω d + λ k A d k Ω d = k Ω d, k d k d d, k Ω d +λ k d T A d d, k Ω d λ k A d k A d, k Ω d, 3.8 λ k N k A d = λ k d, k A λ d λ λ d T k Ad + λ k A λ d Putting into 3.8 yields J + λ λ k A d k A d, k A d + λ λ d T k A d k d T A d d, k A d + λ k d d, k A λ d = µ λ B λ d T k A d + µ k A + µ 5 + µ 6 + λ k A λ d + λ k A λ d k Ω d, k d η λ k d T A d d k d d λ k A d k A d, k Ω d + λ k d d, k A B λ d + λ k A λ d k A d, k A d + λ λ d T k A d d k d T A d d, k A d η. 3.8 Multiplying the equation 3.77 by k dη integrating over B, we have 8 d dt B k+ d η + k d η = λ B B + B λ k A λ d k Ω d, k d η k u d + k Ω d k Ω d + λ λ k A d k A d + λ k d d, k d η 8 Strictly speaking, we need to multiply the equation by D i h k dη.

22 J. HUANG, F. LIN, AND C. WANG t k d k d η λ k B λ d T A dd, k d η B Adding , using.5, we obtain d k u + k+ d η + µ k A + dt B B λ k+ d η I + I + I 3 + J + J 3 + k u d + k Ω d k Ω d + λ k A B λ d k A d, k d η t k d k d η + k d d λ k B B λ λ d T A dd, k d η λ k d T A d d k d d λ k A d k A d, k Ω d + λ k d d, k A B λ d + λ k A λ d k A d, k A d λ k λ d T A d d d T k A d d, k A d η := I + I + I 3 + J + J 3 + K + K + K 3 + K The terms K, K, K 3, K can be estimated as follows. k K d k u + u k+ d + l u + l+ d k+ d η B l= δ + C k u + k+ d η k k+ u + k+ d η + C l u + l+ d B B l=0 +C k u + k+ d k u + k+ d η, 3.87 B For K, by the equation 3.77 the fact d = d, d we have so that t k d k u d + k Ω d + k A d + k+ d + k d T A dd + k d, d d, K k+ d + k+ u + k+ d u + d + k u d + d#d#d B k + l u + l d + l d#d + l d#d#d k+ d η η l= δ + C k u + k+ d η k+ u + k+ d η B B k + l u + l d + l d#d + l d#d#d 3.88 l= + C k u + k+ d + C For K, we first estimate the terms inside the integr. Since it follows k d T A d d, k Ω d = 0, k u + k+ d k u + k+ d η. B k d T A d d, k Ω d = k d T A d d k d T A d d, k Ω d k k u d + l+ u k l d#d#d k+ u l=0 k k u d + l u + l+ d#d#d k+ u. l=

23 THE ERICKSEN-LESLIE SYSTEM IN R 3 We also have k d d, Ω d = k d d k d d, k Ω d k k d d + l d k l d k+ u k d d + l=0 k l d + l d#d k+ u, l= k A d k A d, k Ω d k k u d + l u + l+ d k+ u, l= k d d, k A d k+ d d + k d d + k l d + l d#d k+ u, k A d k A d, k A d k k u d + l u + l+ d k+ u, l= l= k d T A d d d T k A d d, k A d k k u d#d#d + l u + l+ d#d#d k+ u. Putting all these estimates together, we would have K k u + k+ d d + k d d + k u d#d#d k+ u η B k + l u + l+ d + l+ d#d + l+ d#d#d k+ u η B l= δ + C k u + k+ d η k+ u + k+ d η B B k +C l u + l+ d + l+ d#d + l+ d#d#d l=0 +C l= k u + k+ d k u + k+ d η + C B d To estimate K 3, we first estimate both terms inside the integr. k d d, k d k k d d + k+ d d + l d + l d#d k+ d. Since d =, it follows d, d = d Therefore we have k d T A dd, k d l= k d, k d = k d j d, k j d. = d T k A d d, k d + k j A k j d#dd, k d + k j d T A d k j d, k d j=0 j=0 j=0

24 J. HUANG, F. LIN, AND C. WANG = d T k A d k k d + j d, k j d + k j A k j d#dd, k d + k j=0 j d T A d k j d, k d j=0 k+ d d + k d d k+ u + k u d#d + d#d#d k+ d + k l d k+ u + k l u + l+ d#d + l+ d#d#d k+ d. l= Substituting these two estimates into K 3, we would obtain l= K 3 k+ d d + k d d + k u d#d + d#d#d k+ u + k+ d η B k + l u + l d + l+ d#d + l+ d#d#d k+ u + k+ d η B l= δ + k u + k+ d η k+ u + k+ d η B B k +C d + d 8 + l u + l+ d + l+ d#d + l+ d#d#d +C l= k u + k+ d k u + k+ d η. B 3.90 Finally, by substituting all these estimates on I i s, J i s, K i s, L i s into 3.86 we obtain d k u + k+ d η + µ k+ u + dt B B λ k+ d η δ + C k u + k+ d η k+ u + k+ d η B B +C k u + k+ d k u + k+ d η B k +C u 8 + d 8 + l u + l+ d + l+ d#d + l+ d#d#d +C l=0 j=0 k l u + l+ d + l P. 3.9 l= By Fubini s theorem, we can choose t, 9 such that k u + k+ d η t 8 k u + k+ d η 8ɛ 0. B P For a large constant C > 8 to be chosen later, define T t, 0 by { T = sup k u + k+ d η s < C ɛ 0, t s t t t 0 B Claim. If ɛ 0 > 0 is sufficiently small, then T = 0. First, by continuity we know that T > t. Suppose that T < 0. Then we have k u + k+ d η s < C ɛ 0 B t s < T ; k u + k+ d η T = C ɛ 0. B 3.9 }.

25 THE ERICKSEN-LESLIE SYSTEM IN R 5 By choosing sufficiently small ɛ 0 > 0 δ > 0, we may assume δ + C k u + k+ d η t δ + CC ɛ 0 { } B min µ,, t t T. λ Set t φt := k u + k+ d s ds, t t T. t Then by integrating 3.9 over t t, T applying Gronwall s inequality, we have k u + k+ d η T + T µ k+ u + B t B λ k+ d η e CφT B k k u + k+ d η t + C l u + l+ d + l P + C P 3 u 8 + d 8 k + C e Cɛ0 8ɛ 0 + Cɛ 0, l=0 P + l+ l= P + l+ l u + l+ d + l+ d#d + l+ d#d#d 3.93 where we have used both 3.67, 3.68, 3.69 in the last step. It is easy to see that we can choose C > C + 8e Cɛ0 so that e Cɛ0 8ɛ 0 + Cɛ 0 < C ɛ 0. Hence 3.93 implies k u + k+ d η T < C ɛ 0, B which contradicts the definition of T. Thus the claim holds true. Since k P satisfies k P = k u u + d d σ L u, d, 3.9 the elliptic theory 3.93 with T = 0 then yield k P sup P + k+ + k+ t 0 + P + k+ B + k+ This yields that the conclusion holds for l = k. Thus the proof is complete. k u + k+ d P + k+ u + k+ d Cɛ Completion of Proof of Theorem.3: It is readily seen that by the Sobolev embedding theorem, lemma 3. implies that k u, k+ d L P 3r 0 z 0 for any k. This, combined with the theory of linear Stokes equation heat equation, would imply the smoothness of u, d in P r 0 z 0.. Existence of global weak solutions of Ericksen-Leslie s system.8 In this section, by utilizing both the local energy inequality. for suitable weak solutions of.8 the regularity Theorem.3 for suitable weak solutions to.8, we will establish the existence of global weak solutions to.8. that enjoy the regularity described as in Theorem.. The argument is similar to 3 Section 5. First, we recall the following version of Ladyzhenskaya s inequality see 30 lemma 3. for the proof. Lemma.. There exists C 0 > 0 such that for any T > 0, if u L t L x L t HxR 0, T, then for 0 < R +, it holds u C 0 sup u, t u + R u.. R 0,T x,t R 0,T B R x R 0,T R 0,T Similar to 3 lemma 5., we can estimate the life span of smooth solutions to.8 in term of Sobolev profiles of smooth initial data.

26 6 J. HUANG, F. LIN, AND C. WANG Lemma.. Assume.6,.7,.5 hold. There exist ɛ > 0 θ > 0 depending on u 0, d 0 such that if u 0, d 0 C R, R S k 0 H k R, R He k+ 0 R, S satisfies u 0 + d 0 ɛ. sup x R B R0 x for some R 0 > 0. Then there exist T 0 θ R0 a unique solution u, d C R 0, T 0, R S of.8. in R, satisfying sup x,t R 0,T 0 u + d t ɛ..3 B R0 x Proof. By the theorem of Wang-Zhang-Zhang 33 on the local existence of smooth solutions, there exist T 0 > 0 a unique smooth solution u, d C R 0, T 0, R S to.8.. Let 0 < t 0 T 0 be the maximal time such that Hence we have sup x R B R0 x u + d t ɛ, 0 t t 0.. sup u + d t 0 = ɛ..5 x R B R0 x Without loss of generality, we assume t 0 R0. Set Et = u + d t, E 0 = u 0 + d 0. R R Then by lemma. we have that for any 0 < t t 0, Et + µ u + d + d d E 0..6 λ R 0,t By lemma., we have that for all 0 t t 0, d C 0 where R 0,t E R 0 t = sup x,s R 0,t C 0 ER 0 t R 0,t B R0 x d s d + R 0,t R0 d + te 0 R 0 R 0,t sup u s, ER 0 t = sup d s, E R0 t = x,s R 0,t B R0 x x,s R 0,t B R0 x By., we have E R0 t ɛ, 0 t t 0 so that d C 0 ɛ Hence we obtain R 0,t 0 d = R 0,t 0 C 0 ɛ R 0,t 0 Choosing 0 < ɛ, we would have C 0 d = R 0,t 0 R 0,t 0 R 0,t 0 d ER i 0 t. i=.7 d + t 0E 0 R0..8 d + d d + d λ E 0 + d C0 t 0 ɛ + R0 + λ E 0. R 0,t 0 d R 0,t 0 d λ + C 0ɛ t 0 R0 E 0 C E 0..9

27 THE ERICKSEN-LESLIE SYSTEM IN R 7 This, combined with.8, also yields d C ɛ E 0..0 R 0,t 0 We can also estimate u C 0 ER 0 t 0 u + R 0,t 0 R 0,t 0 R0 u R 0,t 0 C 0 ER E0 0 t 0 + t 0E 0 µ R0 C ɛ E 0.. Now we need to estimate E R0 t. Before we do it, we need to recall the following global L -estimate of P : t0 t0 t0 P u + d + σ L u, d u + d + u + d.. 0 R 0 R 0 R For any x R, let η C 0 B R0 x be a cut-off function of B R0 x: 0 η, η on B R0 x, η 0 outside B R0 x, η R 0. Then, by applying lemma. with this η the estimates.6,.0,.9,.,., we have t0 sup u + d + c 0 u + d + d d E R0 0 0 t t 0 R 0,t 0 B R0 x 0 B R0 x u 3 + u u + u d + u d + u P + d d + u d + d u η R 0 u + d L R 0,t 0 t0 C E R0 0. Thus, by taking supremum over x R we obtain ɛ = sup x R, 0 t t 0 This implies u + d + P L R 0,t0 + u + d L R 0,t 0 E R0 0 + C B R0 x t0 R 0 u + d t E 0 ɛ + C 0 t0 R 0 t 0 ɛ C0 E R0 = θ R0, with θ ɛ 0 C0 E. 0 Set T 0 = t 0. Then we have that T 0 θ R0.3 holds. This completes the proof..3 E 0.. Before proving Theorem., we need the following density property of Sobolev maps see 8 for the proof. Lemma.3. For n = any given map f H e 0 R, S, there exist {f k } C R, S l Hl e 0 R, S such that lim k f k f H R = 0. Proof of Theorem.: Since u 0 H, there exists u k 0 C 0 R, R, with u k 0 = 0, such that lim k uk 0 u 0 L R = 0. Since d 0 H e 0 R, S, lemma.3 implies that there exist {d k 0} C R, S l Hl e 0 R, S such that lim k dk 0 d 0 H R = 0.

28 8 J. HUANG, F. LIN, AND C. WANG By the absolute continuity of u 0 + d 0, there exists R 0 > 0 such that sup u 0 + d 0 ɛ x R B R0 x,.5 where ɛ > 0 is given by lemma.. By the strong convergence of u k 0, d k 0 to u 0, d 0 in L R, we have that sup u k 0 + d k 0 ɛ, k >>..6 x R B R0 x For simplicity, we assume.6 holds for all k. By lemma., there exist θ 0 = θ 0 ɛ, E 0 0, T k 0 θ 0 R 0 such that there exist solutions u k, d k C R 0, T k 0, R S to.8. with the initial condition: that satisfies u k, d k t=0 = u k 0, d k 0,.7 sup u k + d k t ɛ, k..8 x,t R 0,T0 k B R0 x By lemma., we have that for all k, sup u k + d k t + µ u k + 0 t T0 k R R 0,T0 k λ d k u k 0 + d k 0 CE 0..9 R Combining.8.9 with lemma., we conclude that u k + d k Cɛ E 0,.0 R 0,T0 k R 0,T k 0 t d k + d k + P k CE 0,. T k 0 sup u k + d k + P k Cɛ.. x R 0 B R0 x Furthermore,.8 implies that for any φ J, t u k, φ = u k φ + u k u k + d k d k σ L u k, d k : φ, R R where, denotes the pair between H R H R, we conclude that t u k L 0, T0 k, H R t u k CE L 0,T0 k,h R 0..3 It follows from.0. that Φ u k, d k, P k, x, t, R 0 Cɛ, x R, R 0 t T k 0. Hence, by Theorem.3, we conclude that for any δ > 0, u k, d k Cl R δ,t k 0 C l, ɛ, E 0, δ, l.. After passing to a subsequence, we may assume that there exist T 0 θ 0 R0, u L t L x L t HxR 0, T 0, R, d L t He 0 L t He 0 R 0, T 0, S, P L R 0, T 0 such that u k u in L t H xr 0, T 0, R, d k d in L t H e 0 R 0, T 0, S, P k P in L R 0, T 0. It follows from..3 that we can apply Aubin-Lions lemma to conclude that u k u, d k d in L locr 0, T 0.

29 THE ERICKSEN-LESLIE SYSTEM IN R 9 By., we may assume that for any 0 < δ < T 0, 0 < R < + l, lim u k, d k u, d Cl B R δ,t = 0. 0 k It is clear that u C R 0, T 0, R L t L x L t HxR 0, T 0, R, d C R 0, T 0, S L t He 0 L t He 0 R 0, T 0, S solves.8 in R 0, T 0. It follows from.3. that we can assume u, dt u 0, d 0 in L R, as t 0. In particular, by the lower semicontinuity we have that On the other h,.9 implies E0 lim inf Et. t 0 E0 lim sup Et. t 0 This implies that u, dt u 0, d 0 in L R hence u, d satisfies the initial condition.. Let T T 0, + be the first finite singular time of u, d, i.e., Then we must have u, d C R 0, T, R S but u, d / C R 0, T, R S. lim sup t T max x R B R x u + d t ɛ, R > 0..5 Now we look for an extension of this weak solution beyond T. To do it, we define the new initial data at t = T. Claim. u, d C 0 0, T, L R. This follows easily from.3.. By Claim, we can define ut, dt = lim t T ut, dt in L R. By lemma. we have that d L 0, T, L R so that dt dt in L R. In particular, ut H dt He 0 R, S. Use ut, dt as an initial data to obtain a continuation of u, d beyond T as a weak solution of.8., we will show that this procedure will cease in finite steps afterwards we will have constructed a global weak solution. In fact, at any such singular time there is at least a loss of energy amount of ɛ. By.5, there exist t i T x 0 R such that lim sup u + d t i ɛ for all R > 0. t i T This implies u + d T = lim R R 0 lim lim inf R 0 t i T B R x 0 R \B R x 0 u + d t i lim sup R t i T u + d T lim lim inf u + d t i R 0 t i T R \B R x 0 u + d t i E 0 ɛ. B R x 0 Hence the number of finite singular times must be bounded by L = E0 ɛ. If 0 < T L < + is the largest finite singular time, then we can use ut L, dt L as the initial data to construct a weak solution u, d to.8. in R T L, +. Thus i of Theorem. is established. It is also clear that iii of Theorem. holds for the solution constructed. Now, we perform a blow-up analysis at each finite singular time. It follows from.5 that there exist 0 < t 0 < T, t m T, r m 0 such that ɛ = sup x R,t 0 t t m B rm x u + d t..6