Weak solution to compressible hydrodynamic flow of liquid crystals in 1-D

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1 Weak solution to compressible hydrodynamic flow of liquid crystals in 1-D Shijin Ding Changyou Wang Huanyao Wen Abstract We consider the equation modeling the compressible hydrodynamic flow of liquid crystals in one dimension. As mentioned in [12], the weak solution was obtained with the initial density having a positive lower bound and H 1 - integrable. n this paper, we get a weak solution with initial density nonnegative and L γ -integrable. Key Words: Liquid crystal, compressible hydrodynamic flow, global weak solution. 1 ntroduction n this paper, we consider the one dimensional initial-boundary value problem: ρ t + (ρu) x =, (ρu) t + (ρu 2 ) x + P x = µu xx λ( n x 2 ) x, n t + un x = θ(n xx + n x 2 n), for (x, t) (, 1) (, + ), with the initial condition: (1.1) (ρ, ρu, n) t= = (ρ, m, n ) in [, 1], (1.2) School of Mathematical Sciences, South China Normal University, Guangzhou, 51631, China Corresponding author. cywang@ms.uky.edu. Department of Mathematics, University of Kentucky, Lexington, KY 456 1

2 where n : [, 1] S 2 and the boundary condition: (u, n x ) = (, ), t >, (1.3) where ρ denotes the density function, u denotes the velocity field, n denotes the optical axis vector of the liquid crystal that is a unit vector (i.e., n = 1), µ >, λ >, θ > are viscosity of the fluid, competition between kinetic and potential energy, and microscopic elastic relaxation time respectively. P = aρ γ, for some constants γ > 1 and a >, is the pressure function. The hydrodynamic flow of compressible (or incompressible) liquid crystals was first derived by Ericksen [1] and Leslie [2] in 196 s. However, its rigorous mathematical analysis was not taken place until 199 s, when Lin [3] and Lin-Liu [4, 5, 6] made some very important progress towards the existence of global weak solutions and partial regularity of the incompressible hydrodynamic flow equation of liquid crystals. When the Ossen-Frank energy configuration functional reduces to the Dirichlet energy functional, the hydrodynamic flow equation of liquid crystals in Ω R d can be written as follows (see Lin [3]): ρ t + div(ρu) =, (ρu) t + div(ρu u) + P = µ u λdiv( n n n 2 2 d ), n t + u n = θ( n + n 2 n), ( ) where u u = (u i u j ) 1 i,j d, and n n=(n xi n xj ) 1 i,j d. Observe that for d = 1, the system ( ) reduces to (1.1). When the density function ρ is a positive constant, then ( ) becomes the hydrodynamic flow equation of incompressible liquid crystals (i.e., div u = ). n a series of papers, Lin [3] and Lin-Liu [4, 5, 6] addressed the existence and partial regularity theory of suitable weak solution to the incompressible hydrodynamic flow of liquid crystals of variable length. More precisely, they considered the approximate equation of incompressible hydrodynamic flow of liquid crystals: (i.e., ρ = 1, and n 2 in ( ) 3 is replaced by (1 n 2 )n ), and proved [4], among many other results, the local existence of ɛ 2 classical solutions and the global existence of weak solutions in dimension two and 2

3 three. For any fixed ɛ >, they also showed the existence and uniqueness of global classical solution either in dimension two or dimension three when the fluid viscosity µ is sufficiently large; in [6], Lin and Liu extended the classical theorem by Caffarelli- Kohn-Nirenberg [7] on the Navier-Stokes equation that asserts the one dimensional parabolic Hausdorff measure of the singular set of any suitable weak solution is zero. See also [8, 9] for relevant results. For the incompressible case ρ = 1 and div u =, it remains to be an open problem that for ɛ whether a sequence of solutions (u ɛ, n ɛ ) to the approximate equation converges to a solution of the original equation ( ). t is also a very interesting question to ask whether there exists a global weak solution to the incompressible hydrodynamic flow equation ( ) similar to the Leray-Hopf type solutions in the context of Naiver-Stokes equation. We answer this question in [1] for d = 2. Moreover, for ρ, div u=, and d = 2 or 3, we also get an unique local strong solution to ( ) in [11]. Particularly, if initial density has a positive lower bound and d=2, we get an unique global strong solution with small initial data. When dealing with the compressible hydrodynamic flow equation ( ), in [12], we get the existence and uniqueness of global classical, strong solution and the existence of weak solution for < c 1 ρ c and the existence of strong solution for ρ when the dimension d = 1. As mentioned in the Remark 1.1 of [12], the global weak solution may be obtained with improved initial conditions. We answer this question in the paper. We remark that when the optical axis n is a constant unit vector, (1.1) reduces to the Navier-Stokes equation for compressible isentropic flow. Let s review some previous results about the weak solution to the compressible isentropic Navier-Stokes equation. n 1998, P.L. Lions in [13] got a weak solution with γ 9 5 for dimension d = 3. n 21, S. Jiang and P. Zhang in [14] obtained a weak solution to the Cauchy problem with spherically symmetric initial data for any γ > 1, and d=2 or 3. For general initial data, and d=3, E. Feireisl et al in [15] improved the condition of γ given by P.L. Lions in [13], i.e. γ > 3 2. Our ideas mainly come from [15]. While the proof in this paper is simpler, since we exploit the one-dimensional feature, and use integrals instead of commutators. Moreover, we only need γ > 1. 3

4 Since the constant a and µ, λ, θ in (1.1) don t play any role in the analysis, we assume henceforth that Notations: µ = λ = θ = a = 1. (1) = (, 1), = {, 1}, = (, T ) for T >. (2) f: f(x) = f(x) for x, and f(x) = for x R \. (3) η σ ( ) = 1 η( σ d σ ), where η is a standard mollifier. (4) C([, T ]; X ω): f C([, T ]; X ω) g X, f(t), g X X C([, T ]). Definition 1.1 For any T >, we call (ρ, u, n) a global weak solution of (1.1)-(1.3), if (1) ρ L (, T ; L γ ()), ρu 2 L (, T ; L 1 ()), ρ a.e. in, u L 2 (, T ; H 1 ()), n L (, T ; H 1 ()) L 2 (, T ; H 2 ()), n t L 2 (, T ; L 2 ()), n = 1 in, (ρ, ρu)(x, ) = (ρ (x), m (x)), weakly in L γ () L 2γ γ+1 (), n(x, ) = n (x) in, (n x (, t), n x (1, t)) = a.e. in (, T ). (2) (1.1) 1, (1.1) 2 are satisfied in D ( ), and (1.1) 3 holds a.e. in. (3) ( ρu2 2 + ργ γ 1 + n x 2 )(t) + (u 2 x + 2 n xx + n x 2 n 2 ) ( m2 + ργ 2ρ γ 1 + (n ) x 2 ), for a.e. t (, T ). (1.4) Our main result is as follows Theorem 1.1 f ρ, ρ L γ (), m ρ L 2 (), and n H 1 (, S 2 ), then there exists a global weak solution (ρ, u, n) : [, 1] [, + ) R + R S 2 to (1.1)-(1.3) such that for any T >, where E := ρ 2γ c(e, T ), denotes the total energy of the initial data. ( m2 2ρ + ργ γ 1 + (n ) x 2 ) 4

5 The rest of the paper is organized as follows. n section 2, we present some useful Lemmas which will be needed. n section 3, we derive some a priori estimates for the approximate solutions of (1.1)-(1.3), and prove the existence of weak solution. 2 Preliminaries Lemma 2.1 ([16]). Assume X E Y are Banach spaces and X E. Then the following embedding are compact: (i) { ϕ : ϕ L q (, T ; X), ϕ } t L1 (, T ; Y ) L q (, T ; E), if 1 q ; (ii) { ϕ : ϕ L (, T ; X), ϕ } t Lr (, T ; Y ) C([, T ]; E), if 1 < r. Lemma 2.2 ([15]). Let ρ L 2 (Ω (, T )), u L 2 (, T ; H 1 (Ω; Rd )) solve Then ρ t + div(ρu) =, in D (Ω (, T )). (2.1) t b(ρ) + div[b(ρ)u] + [b (ρ)ρ b(ρ)]divu =, in D (Ω (, T )), (2.2) for any b C 1 (R) such that b (z) for all z R large enough. Lemma 2.3 ([2]). Let ρ L 2 (R d ), u H 1 (R d ) Then η σ div(ρu) div(u(ρ η σ )) L 1 (R d ) C u H 1 (R d ) ρ L 2 (R d ), for some C > independent of σ, ρ, u. n addition, η σ div(ρu) div(u(ρ η σ )), in L 1 (R d ) as σ. Lemma 2.4 ([19]). Let Ω R d be a bounded Lipschitz domain. Let ρ L 2 (Ω (, T )), u L 2 (, T ; H 1 (Ω; Rd )) solve (2.1). Then ρ, u solve (2.1) in D (R d (, T )) provided they were extended to be zero outside Ω. Lemma 2.5 ([19]). Let O R M be compact and let X be a separable Banach space. Assume that v m : O X, m Z +, is a sequence of measurable functions such that esssup t O v m (t) X N, uniformly in m. 5

6 Moreover, let the family of functions v m, Φ : t v m (t), Φ, t O be equi-continuous for any Φ belonging to a dense subset in X. Then v m C(O; X ω) for m Z +, and there exists v C(O; X ω) such that after taking possible subsequences, v m v in C(O; X ω), as m. Lemma 2.6 ([19]). Let O R N be a measurable set and v m L 1 (O; R M ) for m Z + such that v m v weakly in L 1 (O; R M ). Let Φ : R M (, ] be a lower semi-continuous convex function such that Φ(v m ) L 1 (O) for any m, and Φ(v m ) Φ(v), weakly in L 1 (O). Then Φ(v) Φ(v), a.e. in O. 3 Existence of weak solution n this section, we mollify the initial data, give the initial density a positive lower bound, and get a sequence of classical approximate solutions. Then we derive some a priori estimates of the approximate solutions, and take limits. The main difficulties come from the convergence of the pressure. We overcome these difficulties by Lemmas By Sobolev s extension theorem in [18], there exists a ñ H 1 c (R) such that ñ = n in. We mollify the initial data as follows. ρ = η ρ +, u = 1 η ( m ), ρ ρ n = η ñ η ñ. 6

7 Then ρ >, ρ, u, n C 2+α () for < α < 1, satisfy as ρ ρ, in L γ (), ρ u m ρ, in L 2 (), ρ u m, in L 2γ γ+1 (), (3.1) n n in H 1 (). From [12], we get a sequence of global classical solutions (ρ, u, n ) such that (ρ ) t + (ρ u ) x =, ρ >, (ρ u ) t + (ρ u 2 ) x + (ρ γ ) x = (u ) xx ( (n ) x 2 ) x, (n ) t + u (n ) x = (n ) xx + (n ) x 2 n, n = 1, for (x, t) [, 1] (, + ), with the initial and boundary conditions: (3.1) (ρ, u, n ) t= = (ρ, u, n ) in [, 1], (u, x n ) = (, ). Lemma 3.1 ([12]) For any T > and t T, it holds = ( ρ u 2 2 ( ρ u ργ γ 1 + (n ) x 2 )(t) + t ( (u ) x (n ) xx + (n ) x 2 n 2 ) + ργ γ 1 + (n ) x 2 ), (3.2) and (n ) t 2 + (n ) xx 2 c(e, T ). (3.3) From (3.2), we have ρ γ L (, T ; L 1 ()). To take limits of ρ γ more regularity of ρ with respect to the space variable. Namely, as, we need Lemma 3.2 ρ 2γ c(e, T ). 7

8 Proof. Multiplying (3.1) 2 by x ργ x ργ, integrating the resulting equation over, and using integration by parts, we get ρ 2γ = ρ u ( T T x T ρ γ x ρ γ ) T T (ρ u 2 )(ργ ρ γ ) + ( (n ) x 2 (ρ γ ρ γ ) = V + V + V. = ρ u ( c sup ( t T c sup ( t T c(e ), x ρ γ x ρ γ ) T ρ u ρ γ ) ρ u 2 ρ u [ x ρ γ )2 + ρ γ ) + c sup ( t T (ρ γ ) t x (ρ γ ) t] T ρ (u ) x (ρ γ ρ γ ) ρ γ ) where we have used (3.2). To estimate, we multiply (3.1) 1 by γρ γ 1 and get (ρ γ ) t + (ρ γ u ) x + (γ 1)ρ γ (u ) x =. (3.4) Therefore, we have from (3.4) that = = T T x ρ u [(ρ γ u ) x + (γ 1)ρ γ (u ) x ] ρ γ+1 xρ u [(ρ γ u ) x + (γ 1)ρ γ (u ) x ] T u 2 + (γ 1) x ρ u ρ γ (u ) x T (γ 1) xρ u ρ γ (u ) x T u 2 + c ρ u ρ γ (u ) x ρ γ+1 ρ γ+1 T u 2 + c ρ γ+1 u 2 + c(e ) T ρ γ (u ) x ( ρ γ (u ) x. ρ + ρ u 2 ) 8

9 By Cauchy s inequality, Hölder s inequality, and (3.2), we have T ρ γ+1 u 2 + c(e ) ρ γ L 2γ () (u ) x L 2 () ρ γ+1 u ρ 2γ 4 + c(e ). T + V = V = 1 4 T T = ρ γ+1 (ρ u 2 )(ργ T u 2 + ρ γ+1 u 2 + c(e, T ). T (u ) x ρ γ (u ) x ρ γ (u ) x L 2 () ρ γ L 2γ () sup t T ρ 2γ + c(e, T ). T V = sup t T c(e, T ). T ρ γ ) + ( ρ γ )2 T ρ u 2 ρ γ + ( ρ γ )2 ρ γ (n ) x 2 (ρ γ ρ γ T (n ) x 2 ρ γ ) Putting these inequalities together, we have ρ 2γ = V + V + V 1 ρ 2γ 2 + c(e, T ). This completes the proof of the lemma. ( (u ) x 2 + 1) t follows from Lemma that there exists a subsequence of (ρ, u, n ), still denoted by (ρ, u, n ), such that as, for any T >, ρ ρ weak in L (, T ; L γ ()), and weakly in L 2γ ( ), (3.5) ρ γ ργ, weakly in L 2 ( ), (3.6) 9

10 u u, weakly in L 2 (, T ; H 1 ()), (3.7) n n, weak in L ( ), (3.8) (n ) x n x, weak in L (, T ; L 2 ), (3.9) ((n ) t, (n ) xx ) (n t, n xx ), weakly in L 2 ( ). (3.1) Since ρ L 2γ ( ), u L 2 (, T ; H 1 ()) L2 (, T ; L ()), we have ρ u L 2γ γ+1 (, T ; L 2γ ()). Therefore, t ρ = (ρ u ) x L 2γ γ+1 (, T ; H 1 ()). Since 2γ γ+1 > 1, ρ L (, T ; L γ ()), and L γ H 1 (), we have from Lemma 2.1 and Lemma 2.5 (3.7) and (3.12) imply Hence, ρ ρ, in C([, T ]; L γ ω), (3.11) ρ ρ in C([, T ]; H 1 ). (3.12) ρ u ρu, in D ( ). (3.13) ρ t + (ρu) x =, in D ( ). (3.14) Moreover, ρ u L (, T ; L 2 ) and ρ L (, T ; L 2γ ) give ρ u L (, T ; L 2γ γ+1 ). From (3.1) 2, we get (ρ u ) t = (ρ u 2 ) x (ρ γ ) x+(u ) xx ( (n ) x 2 ) x L 2 2γ 1, (, T ; W γ+1 ). By (3.13), Lemma 2.1 and Lemma 2.5, we obtain ρ u ρu, in C([, T ]; L 2γ γ+1 ω), (3.15) ρ u ρu, in C([, T ]; H 1 ), (also weak in L (, T ; L 2γ γ+1 )). (3.16) We have from (3.7) and (3.16) ρ u 2 ρu2, in D ( ), (also weakly in L 2 (, T ; L 2γ γ+1 )). (3.17) Similar to the above argument, we have from (3.8)-(3.1) and Lemma 2.1 n n, in C( ), (3.18) 1

11 n n, in L 2 (, T ; C 1 ([, 1])). (3.19) This, together with (3.6), (3.7), (3.9), (3.1), (3.13), and (3.17), implies (ρu) t + (ρu 2 ) x + (ρ γ ) x = u xx ( n x 2 ) x, in D ( ), (3.2) n t + un x = n xx + n x 2 n, in L 2 (, T ; L 2 ). (3.21) t follows from (3.1) 1, (3.1) 3, (3.11), and (3.15) that (ρ, ρu)(x, ) = (ρ (x), m (x)), weakly in L γ () L 2γ γ+1 (). By (3.1) 4 and (3.18), and the fact that n = 1, we have n(x, ) = n (x) in [, 1], and n = 1 in. (1.3) follows from (3.7) and (3.19). Since ρ > in, we have from (3.5) ρf = lim ρ f, for any nonnegative f C ( ). For f is arbitrary, we get ρ a.e. in. From (3.17), we have 1 ɛ t+ɛ t ρu 2 = 1 ɛ lim 1 ɛ t+ɛ t t+ɛ lim t ρ u 2 ρ u 2 (s), for t (, T ) and ɛ >. Let ɛ +, and use Lebesgue theorem, we get ρu 2 (t) lim ρ u 2 (t), for a.e. t. This, together with (3.1), (3.2), and the lower semi-continuity, implies the energy inequality (1.4). We still need to prove ρ γ = ρ γ. This requires the following lemmas. Lemma 3.3. t holds as [(u ) x ρ γ ]ρ (u x ρ γ )ρ, in D ( ). 11

12 Proof. For any ϕ C ((, T )), φ C ((, 1)), multiplying (3.1) 2 by ϕφ x ρ, integrating the resulting equation over, and using integration by parts, we have = = ϕ(t)φ(x)[(u ) x ρ γ ]ρ Q T x x ϕ (t)φ(x)ρ u ρ + ϕ(t)φ(x)ρ u ( ρ ) t + ϕ(t)φ(x)ρ 2 u2 Q T x x x + ϕ(t)φ (x)ρ u 2 ρ + ϕ(t)φ (x)ρ γ ρ ϕ(t)φ (x)(u ) x ρ x + ϕ(t)φ (x) (n ) x 2 ρ + ϕ(t)φ(x) (n ) x 2 ρ Q T x x x ϕ (t)φ(x)ρ u ρ + ϕ(t)φ (x)ρ u 2 ρ + ϕ(t)φ (x)ρ γ ρ x x ϕ(t)φ (x)(u ) x ρ + ϕ(t)φ (x) (n ) x 2 ρ + ϕ(t)φ(x) (n ) x 2 ρ, where we have used (3.1) 1. Since x ρ L (, T ; W 1,γ ), t ( x ρ ) = ρ u L (, T ; L 2γ γ+1 ), we have from Lemma 2.1 and (3.5) x ρ x ρ, in C( ), as. (3.22) This, combined with (3.5)-(3.7), (3.16), (3.17), and (3.19), gives lim ϕ(t)φ(x)[(u ) x ρ γ ]ρ Q T x = ϕ (t)φ(x)ρu ρ + ϕ(t)φ (x)ρu 2 Q T x ϕ(t)φ (x)u x ρ + ϕ(t)φ (x) n x 2 x x ρ + x ϕ(t)φ (x)ρ γ ρ ρ + ϕ(t)φ(x) n x 2 ρ. (3.23) To complete the proof, it suffices to show that the right side of (3.23) is equal to ϕ(t)φ(x)(u x ρ γ )ρ. The main difficulty is ρu / L 2 ( ). To overcome it, take 12

13 ϕφ x ρ σ as a test function of (3.2), where ρ σ = η σ ρ. Then ϕ(t)φ(x)(u x ρ γ ) ρ σ Q T x x = ϕ (t)φ(x)ρu ρ σ + ϕ(t)φ(x)ρu( ρ σ ) t + x ϕ(t)φ(x)ρu 2 ρ σ + ϕ(t)φ (x)ρu 2 ρ σ + x x ϕ(t)φ (x)ρ γ ρ σ ϕ(t)φ (x)u x ρ σ + x ϕ(t)φ (x) n x 2 ρ σ + ϕ(t)φ(x) n x 2 ρ σ. (3.24) Since ρ L 2 ( ), u L 2 (, T ; H 1 ()), we have from Lemma 2.4 ( ρ) t + ( ρû) x =, in D (R (, T )). (3.25) Denote r σ = ( ρ σ û) x ( ρû) x σ. t follows from Lemma 2.3 that r σ L 1 (R (, T )), and r σ, in L 1 (R (, T )), as σ. (3.26) Take η σ (x ) as a test function of (3.25), then ( ρ σ ) t + ( ρ σ û) x = r σ, a.e. in R (, T ). (3.27) ntegrating (3.27) over (, x), for < x 1, we have ( Therefore, x ρ σ ) t = ρ σ û + x ϕ(t)φ(x)ρu( ρ σ ) t + ϕ(t)φ(x)ρu 2 ρ σ Q T x = ϕ(t)φ(x)ρu ρ σ û + ϕ(t)φ(x)ρu r σ + ϕ(t)φ(x)ρu 2 ρ σ Q T x = ϕ(t)φ(x)ρu r σ, x r σ. 13

14 where we have used û = u in. This, together with (3.24), implies ϕ(t)φ(x)(u x ρ γ ) ρ σ Q T x x = ϕ (t)φ(x)ρu ρ σ + ϕ(t)φ(x)ρu r σ + x x ϕ(t)φ (x)ρu 2 ρ σ + ϕ(t)φ (x)ρ γ ρ σ x x ϕ(t)φ (x)u x ρ σ + ϕ(t)φ (x) n x 2 ρ σ + ϕ(t)φ(x) n x 2 ρ σ. (3.28) By the regularities of (ρ, u, n), (3.26), Lebesgue s Dominated convergence theorem, we get, after taking σ in (3.28), ϕ(t)φ(x)(u x ρ γ )ρ Q T x = ϕ (t)φ(x)ρu ρ + ϕ(t)φ (x)ρu 2 Q T x ϕ(t)φ (x)u x ρ + ϕ(t)φ (x) n x 2 x x ρ + x ϕ(t)φ (x)ρ γ ρ ρ + ϕ(t)φ(x) n x 2 ρ. (3.29) The conclusion follows from (3.23) and (3.29). Lemma. This completes the proof of the Lemma 3.4. t holds lim ρ (u ) x ρu x. Proof. Since ρ L 2γ ( ), u L 2 (, T ; H 1), we replace b in (2.2) by bl j, where j, l Z +, b l j C1 (R) and b l j (z) = (z + 1 l ) log(z + 1 l ) for z j, and bl j (z) = (j l ) log(j l ) for z j + 1. Since ρ L (, T ; L γ ), we have ρ < + a.e. in. This implies b l j (ρ) (ρ + 1 l ) log(ρ + 1 l ) a.e. in, as j. Let j in (2.2), the Lebesgue s Dominated convergence theorem implies t [(ρ+ 1 l ) log(ρ+ 1 l )]+[(ρ+ 1 l ) log(ρ+ 1 l )u] x +ρu x 1 l u x log(ρ+ 1 l ) =, in D ( ). (3.3) We obtain from ρ L 2γ ( ) that (ρ + 1 l ) log(ρ + 1 l ) L2 ( ). Similar to (3.25)- (3.27), we extend ρ, u in (3.3) to be zero outside, mollify (3.3), integrate the 14

15 resulting equation over, and take limits, we obtain ρu x = (ρ + 1 l ) log(ρ + 1 l ) (ρ + 1 l ) log(ρ + 1 l )(T ) + 1 u x log(ρ + 1 l l ) (3.31) Since (3.1) 1 is valid in classical sense, a direct calculation gives t [(ρ + 1 l ) log(ρ + 1 l )]+[(ρ + 1 l ) log(ρ + 1 l )u ] x +ρ (u ) x 1 l (u ) x log(ρ + 1 l ) =. (3.32) ntegrating (3.32) over, we have ρ (u ) x = (ρ + 1 l ) log(ρ + 1 l ) (ρ + 1 l ) log(ρ + 1 l )(T ) + 1 (u ) x log(ρ + 1 l l ) (ρ + 1 l ) log(ρ + 1 l ) (ρ + 1 l ) log(ρ + 1 l )(T ) + 1 l (u ) x L 2 ( ) ρ + 1 L 2 ( ) (ρ + 1 l ) log(ρ + 1 l ) (ρ + 1 l ) log(ρ + 1 l )(T ) + 1 l c(e, T ), (3.33) where we have used Hölder inequality, Lemma 3.1, and Lemma 3.2. Since ρ L (, T ; L γ ), we have (ρ + 1 l ) log(ρ + 1 l ) L (, T ; L τ ), (3.34) for some τ > 1. From (3.32), we get t [(ρ + 1 l ) log(ρ + 1 2γ 2γ 1, )] L γ+1 (, T ; W γ+1 ). (3.35) l (3.34), (3.35), and Lemma 2.5 give (ρ + 1 l ) log(ρ + 1 l ) (ρ + 1 l ) log(ρ + 1 l ), in C([, T ]; Lτ ω), as. This implies lim (ρ + 1 l ) log(ρ + 1 l )(T ) = (ρ + 1 l ) log(ρ + 1 )(T ). l 15

16 Since the function (z + 1 l ) log(z + 1 l ) is convex for z, Lemma 2.6 implies (ρ + 1 l ) log(ρ + 1 l ) (ρ + 1 l ) log(ρ + 1 l ), a.e. in. Therefore, lim (ρ + 1 l ) log(ρ + 1 l )(T ) (ρ + 1 l ) log(ρ + 1 )(T ). (3.36) l Take lim in (3.33), and use (3.36), we get ρ (u ) x (ρ + 1 l ) log(ρ + 1 l ) lim (ρ + 1 l ) log(ρ + 1 l )(T ) + 1 l c(e, T ) (ρ + 1 l ) log(ρ + 1 l ) (ρ + 1 l ) log(ρ + 1 l )(T ) + 1 l c(e, T ) = ρu x 1 u x log(ρ + 1 l l ) + 1 l c(e, T ) ρu x + 1 l u x L 2 ( ) ρ + 1 L 2 ( ) + 1 l c(e, T ). lim Since u x L 2 ( ), and ρ L 2 ( ), sending l yields The proof of the Lemma is complete. lim ρ (u ) x ρu x. Now we return to the proof of ρ γ = ρ γ. Assume ϕ m C (, T ), φ m C (, 1), ϕ m, φ m 1, and ϕ m, φ m 1, as m. For any ψ C ( ), denote v = ρ ɛψ for ɛ >, then = = = + (ρ γ v γ )(ρ v) Q T ϕ m φ m (ρ γ v γ )(ρ v) + (1 ϕ m φ m )(ρ γ v γ )(ρ v) Q T ϕ m φ m (ρ γ ρ ρ γ v v γ ρ + v γ+1 ) + (1 ϕ m φ m )(ρ γ v γ )(ρ v) Q T ϕ m φ m (ρ γ u x )ρ + (ϕ m φ m 1)ρu x + ρu x Q T ϕ m φ m ( ρ γ v v γ ρ + v γ+1 ) + (1 ϕ m φ m )(ρ γ v γ )(ρ v). Denote A m = (ϕ m φ m 1)ρu x + (1 ϕ m φ m )(ρ γ v γ )(ρ v). Together with 16

17 Lemma , (3.5), and (3.6), we have (ρ γ v γ )(ρ v) lim ϕ m φ m [ρ γ (u ) x ]ρ + lim ρ (u ) x + Q T lim ϕ m φ m ( ρ γ v vγ ρ + v γ+1 ) + A m Q T lim[ ϕ m φ m [ρ γ (u ) x ]ρ + ϕ m φ m ρ (u ) x Q T + ϕ m φ m ( ρ γ v vγ ρ + v γ+1 )] lim ρ 1 ϕ m φ m (u ) x + A m Q T ϕ m φ m (ρ γ vγ )(ρ v) lim ρ 1 ϕ m φ m (u ) x + A m. = lim By the monotonicity of z γ, we have Therefore, (ρ γ v γ )(ρ v) lim ϕ m φ m (ρ γ vγ )(ρ v). ρ 1 ϕ m φ m u x + A m lim 1 ϕ m φ m L 2γ γ 1 ( ) ρ L 2γ ( ) u x L 2 ( ) + A m c(e, T ) 1 ϕ m φ m L γ 1 2γ + A m, (3.37) ( ) where we have used Hölder inequality, Lemma 3.1, and Lemma 3.2. By the Lebesgue s Dominated Convergence Theorem, we have Let m in (3.37), we get 1 ϕ m φ m L γ 1 2γ, A m as m. ( ) Since v = ρ ɛψ, and ɛ >, we have (ρ γ v γ )(ρ v). [ρ γ (ρ ɛψ) γ ]ψ. (3.38) Sending ɛ yields (ρ γ ρ γ )ψ. 17

18 This clearly implies ρ γ = ρ γ, a.e. in. The proof of Theorem 1.1 is complete. Acknowledgment. The first author is supported by the National Natural Science Foundation of China (No.14715), by the National 973 Project of China (Grant No.26CB8592), by Guangdong Provincial Natural Science Foundation (Grant No.75795) and by University Special Research Foundation for Ph.D Program (Grant. No ). The second author is partially supported by NSF References [1] J. Ericksen, Hydrostatic Theory of Liquid Crystal, Arch. Rational Mech. Anal. 9 (1962), [2] F. Leslie, Some Constitute Equations for Anisotropic Fluids, Q. J. Mech. Appl. Math. 19 (1966), [3] F. H. Lin, Nonlinear theory of defects in nematic liquid crystal: phase transition and flow phenomena, Comm. Pure Appl. Math. 42, 1989, [4] F. H. Lin, C. Liu, Nonparabolic dissipative systems modeling the flow of liquid crystals, Comm. Pure Appl. Math. Vol. XLV, 1995, [5] F. H. Lin, C. Liu, Existence of solutions for the Ericksen-Leslie system, Arch. Rational Mech. Anal. 154(2) [6] F. H. Lin, C. Liu, Partial regularities of the nonlinear dissipative systems modeling the flow of liquid crystals, DCDS, 2(1996) [7] L. Caffarelli, R. Kohn, L. Nirenberg, Partial regularity of suitable weak solutions of the Navier-Stokes equations. Comm. Pure Appl. Math. 35 (1982), [8] C. Liu, N. J. Walkington, Mixed methods for the approximation of liquid crystal flows, Math. Modeling and Numer. Anal., 36(22),

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