Global weak solutions to the active hydrodynamics of liquid crystals

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Global weak solutions to the active hydrodynamics of liquid crystals Wei Lian a,b , Rongfang Zhang b,∗ a College of Automation, Harbin Engineering University, Harbin 15000, China b Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA

Received 12 May 2019; accepted 20 October 2019

Abstract We consider the incompressible flow of the active hydrodynamics of liquid crystals with inhomogeneous density in the Beris-Edwards hydrodynamics framework. The Landau-de Gennes Q-tensor order parameter is used to describe the liquid crystalline ordering. Faedo-Galerkin’s method is adopted to construct the solutions for the initial-boundary value problem. Two levels of approximations are used and the weak convergence is obtained through compactness estimates by new techniques due to the active terms. The existence of global weak solutions in dimension two and three is established in a bounded domain. © 2019 Elsevier Inc. All rights reserved. MSC: 35Q35; 76D05; 76A15 Keywords: Active hydrodynamics; Liquid crystals; Incompressible Navier-Stokes equations; Weak solutions; Compactness

1. Introduction We are concerned with the active hydrodynamics, describing fluids with active constituent particles that have collective motion and are constantly maintained out of equilibrium by internal energy sources, rather than by the external force applied at the boundary of the system. Active hydrodynamics has wide applications in biophysics and so on [3,5,9,10,23–25,34,44,45,47,49, * Corresponding author.

E-mail addresses: [email protected] (W. Lian), [email protected] (R. Zhang). https://doi.org/10.1016/j.jde.2019.10.020 0022-0396/© 2019 Elsevier Inc. All rights reserved.

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55] and has attracted lots of attention recently. Active nematic systems are distinguished from their well-studied passive counterparts since the constituent particles are active; that is, it is the energy consumed and dissipated by the active particles that drives the system out of equilibrium. We refer the readers to [6,15,18,21,22,24,35–38,46,48,50,51,55] for more discussions on the novel effects observed in active systems. In this paper, we use one of the most comprehensive descriptions of the nematic liquid crystals, the Landau-de Gennes Q-tensor description, which describes the nematic state by the Q-tensor order parameter, a symmetric traceless d × d matrix (d = 2 or 3). In particular, we consider the following hydrodynamic equations of inhomogeneous incompressible flows of active nematic liquid crystals [19,20,23] in a bounded domain O ⊆ Rd : ⎧ ⎪ ⎪ ⎪ ⎨

ρt + ∇ · (ρu) = 0, (ρu)t + ∇ · (ρu ⊗ u) + ∇P − μu = ∇ · τ + ∇ · σ, ⎪ ∂t Q + (u · ∇)Q + Q − Q − λ|Q|D = H [Q], ⎪ ⎪ ⎩ ∇ · u = 0,

(1.1)

where ρ is the density of the fluid, u ∈ Rd represents the flow velocity, Q is the nematic tensor order parameter, P stands for the pressure, μ > 0 denotes the viscosity coefficient, −1 > 0 is the rotational viscosity, λ ∈ R is the nematic alignment parameter, D = 12 (∇u + ∇u ) and  = 1  2 (∇u −∇u ) are the symmetric and antisymmetric part of the strain tensor with (∇u)αβ = ∂β uα . Hereafter, we use the Einstein summation convention, i.e., we sum over the repeated indices. Moreover, the molecular tensor  k tr(Q2 )  H = H [Q] = KQ − (c − c∗ )Q + b Q2 − Id − cQ tr(Q2 ), 2 d describes the relaxation dynamics of the nematic phase and can be derived from the Landau-de Gennes free energy, i.e., Hαβ = −δF/δQαβ , where 

b c K k F= (c − c∗ )tr(Q2 ) − tr(Q3 ) + |tr(Q2 )|2 + |∇Q|2 dA, 4 3 4 2 with K the elastic constant for the one-constant elastic energy density, c the concentration of active units and c∗ the critical concentration for the isotropic-nematic transition, and k > 0 and b ∈ R the material-dependent constants. We note that the analysis of this paper holds for all real b, although b is usually taken to be positive in the literature. In what follows, we set K = k = 1 for simplicity of notation. The stress tensor σ = (σ ij ) is ij

ij

σ ij = σr + σa , with ij

σr = −λ|Q|H ij [Q] + Qik H kj [Q] − H ik [Q]Qkj , ij

ij

σa = σ∗ c2 Qij , ij

where σr is the elastic stress tensor due to the nematic elasticity and σa is the active contribution which describes contractile or extensile stresses exerted by the active particles in the direction

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of the director field (σ∗ > 0 for the contractile case and σ∗ < 0 for the extensile case). The symmetric additional stress tensor is denoted by: τ ij = −∂j Qkl ∂i Qkl = −(∇Q  ∇Q)ij . In the rest of this paper, we consider the case where c > 0 is a constant. We set, 1 a = (c − c∗ ), 2

κ = σ∗ c 2 .

Then system (1.1) becomes: ρt + ∇ · (ρu) = 0,

(1.2)

(ρu)t + ∇ · (ρu ⊗ u) + ∇P = μu − ∇ · (∇Q  ∇Q) − λ∇ · (|Q|H [Q]) + ∇ · (QQ − QQ) + κ∇ · Q,

(1.3)

∂t Q + (u · ∇)Q + Q − Q − λ|Q|D = H [Q],

(1.4)

∇ · u = 0,

(1.5)

with  tr(Q2 )  H = H [Q] = Q − aQ + b Q2 − Id − cQ tr(Q2 ), d the constants c > 0, > 0, μ > 0, a, b, λ, κ ∈ R, and (x, t) ∈ O × R+ . The system is subject to the following initial conditions: ρ|t=0 = ρ(x) ¯ ∈ L∞ (O), ρu|t=0 = m(x) ¯ ∈ L2 (O),

ρ¯ ≥ 0,

(1.6)

m ¯ = 0 where ρ¯ = 0,

|m| ¯ 2 ρ¯

∈ L1 (O),

¯ Q|t=0 = Q(x) ∈ H 1 (O), and Q¯ ∈ S0d a.e. in O,

(1.7) (1.8)

and the following boundary conditions: u(x, t) = 0,

¯ Q(x, t) = Q(x),

for

(x, t) ∈ ∂O × (0, ∞),

(1.9)

 where S0d := Q ∈ Md×d : Qij = Qij , tr(Q) = 0, i, j = 1, · · · , d is the space of Q-tensors in d-dimension. There have been a lot of studies on liquid crystals in the Q-tensor framework, especially for the passive case. The existence of global weak solutions to the incompressible Q-tensor system in dimension two and three, and the regular solution in dimension two were studied by Paicu-Zarnescu [42,43]. The existence and regularity of weak solutions on the d–dimensional torus over a certain singular potential was established by Wilkinson [56]. For the nematic liquid crystal system with arbitrary physically relevant initial data in case of a singular bulk potential

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proposed in Ball-Majumdar [1], Feireisl-Rocca-Schimperna-Zarnescu [14] derived the globalin-time weak solutions. For the coupled compressible Navier-Stokes and Q-tensor system, the existence and long-time dynamics of globally defined weak solutions was obtained by Wang-XuYu [52]. For the active liquid crystals, Chen-Majumdar-Wang-Zhang [7] studied the existence of global weak solutions to the incompressible active liquid crystals for d = 2, 3, and the regularity and weak-strong uniqueness of the solutions for d = 2. In [8] the compressible flow of active hydrodynamics was studied. See [4] and the references therein for more results and discussions, and [1,2,10–12,16,17,26,28–30,32,33,40,41,53,54] for more discussions on the nematic liquid crystal models. In this paper, we consider the inhomogeneous version of the active hydrodynamic model studied in [7], in particular, the fluid flow is governed by the inhomogeneous incompressible Navier-Stokes equations, the motion of the order parameter Q is represented by a parabolic type equation, and both with the extra nonlinear coupling terms as forcing terms. We shall establish the existence of global weak solutions in dimension two and three for this inhomogeneous coupled system (1.2)−(1.5) with the initial-boundary conditions (1.6)-(1.9). In our system, the highly nonlinear terms cause more difficulties mathematically. First, by using the general energy method, we obtain a priori estimates for our system, based on some crucial cancellations. Those cancellations turn out to be very important in the proof of the existence of weak solutions in dimension two and three. Due to the appearance of the active term κQαβ , we can only obtain an energy inequality, instead of the perfect Lyapunov functional for the smooth solutions to the system. Here we mention that the symmetry and traceless properties of the Q-tensor play a key role in the validity of the cancellations (see Lemma A.1). Also the property (2.3) of the Q-tensor is very important in order to derive the H 1 -estimate for the Q-tensor in Proposition 2.1, since the bulk potential in the Landau-de Gennes energy density (the terms independent of ∇Q) is not always positive and we need to redefine a positive energy E M (t) of the system. The Friedrichs’ scheme was used to construct the solutions in [7] in the whole space. In this paper we adopt the Faedo-Galerkin’s method as in [8] to construct the solutions for the initial-boundary value problem in a bounded domain. Two levels of approximations will be used and the weak convergence will be obtained through compactness estimates. One level of the approximations is related to lifting the density above zero to avoid the vacuum. The other is the approximation from the finite dimensional space to the infinite one. In order to obtain the necessary compactness results, the force term in the inhomogeneous Navier-Stokes equations is required to be in Hx−1 as in Lions 1 2 2 [31]. However, we can only obtain L∞ t Hx ∩ Lt Hx regularity for Q from the Q-tensor equation. By using the structure of the system, we are able to overcome this difficulty from the similar cancellations we mentioned in the a priori estimates. Finally with all the compactness estimates we can pass to the limit in the approximate solutions to establish the existence of global weak solutions to the system (1.2)−(1.5) with the initial-boundary conditions (1.6)-(1.9). The rest of the paper is organized as follows. In Section 2, we obtain the a priori estimates, give the definition of the weak solutions, and state our main result. In Section 3, we prove the existence of solutions to the approximation problems by using the Faedo-Galerkin’s method. In Section 4, we pass to the limit to recover the solutions to the original system by deducing the estimates and the compactness results on the approximation solutions. In the appendix, we provide some preliminaries that we use extensively in this paper.

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2. The a priori estimates In this section, we derive the a priori estimates for the system (1.2)−(1.5). We use H k (O) to denote the Sobolev space for k ≥ 1 integer with the standard norm · H k , and denote by H −k (O) the dual spaces of H0k (O). For the inner product (·, ·) in L2 (O), if a, b are vectors, then  (a, b) =

a(x) · b(x)dx, O

and if A, B are matrices, then  (A, B) =

 A : Bdx =

O

tr(AB)dx. O

For Q-tensors, we define the following Sobolev space: H 1 (O, S0d ) :=

⎧ ⎨ ⎩

⎫ ⎬

 Q : O → S0d : O

|∇Q(x)|2 + |Q(x)|2 dx < ∞ , ⎭

where |Q|2 := tr(Q2 ) and |∇Q|2 = ∂k Qij ∂k Qij . We also denote |Q|2 = Qij Qij . Moreover, we introduce the following function spaces: V1 = {v ∈ L2 (O) : div u = 0, in D }, V2 = {v ∈ H01 (O) : div u = 0}. For the sake of convenience, we denote the Landau-de Gennes free energy for the nematic liquid crystals (cf. [17]) by   F(Q) := O

 a b c 1 |∇Q|2 + |Q|2 − tr(Q3 ) + |Q|4 dx. 2 2 3 4

(2.1)

Moreover, the energy of the system (1.2)-(1.5) can be represented as  E(t) := F(Q) + O

1 ρ|u|2 dx, 2

(2.2)

by adding the kinetic energy to the Landau-de Gennes free energy. Since the bulk potential in the Landau-de Gennes energy density is not always positive, we can redefine the energy of the system in the following way. By Remark A.1 in [7], we know ε 1 tr(Q3 ) ≤ |tr(Q2 )|2 + tr(Q2 ) 4 ε

for any ε > 0, and d × d matrix Q.

Then there exists a sufficiently large constant M = M(a, b, c) > 0, such that

(2.3)

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a b c (M + )|Q|2 − tr(Q3 ) + |Q|4 2 3 4   a b 1 c 3ε 2 2 2 2 ≥ (M + )|Q| − tr (Q ) + tr(Q ) + |Q|4 2 3 8 ε 4 a c b b = (M + − )|Q|2 + ( − ε)|Q|4 2 3ε 4 8 M c ≥ |Q|2 + |Q|4 ≥ 0. 2 8

(2.4)

Then we define the positive energy of the system by   E (t) := M

O

 1 a b c 1 2 2 2 3 4 ρ|u| + |∇Q| + ( + M)|Q| − tr(Q ) + |Q| dx. 2 2 2 3 4

(2.5)

Proposition 2.1. Let (ρ, u, Q) be a smooth solution of the problem (1.2)-(1.5), with smooth ¯ ¯ initial data (ρ(x), ¯ m(x), ¯ Q(x)). If (ρ(x), ¯ m(x), ¯ Q(x)) ∈ L∞ × L2 × H 1 , the following energy inequality holds for any T > 0, d M 1 E (t) + dt 2

t  (μ|∇u|2 + |Q|2 + c2 |Q|6 )dxds 0 O

 2

|m| ¯ Ct ¯ 4 + |∇ Q| ¯ 2 dx, ¯ 2 + |Q| + |Q| ≤ Ce ρ¯

(2.6) for a.e. t ∈ [0, T ].

O

Moreover, we have 0 ≤ ρ(x, t) ≤ ρ ¯ L∞ (O) , for any t ∈ (0, T ), √ 2 ρu L∞ (0,T ;L2 (O)) + μ ∇u 2L2 (0,T ;L2 (O)) ≤ C,

(2.7)

Q L∞ (0,T ;H 1 (O)∩L4 (O))∩L2 (0,T ;H 2 (O))∩L6 (0,T ;L6 (O)) ≤ C.

(2.9)

(2.8)

Hereafter C is a constant that depends on the material coefficients a, b, c, κ, μ, λ, and the initial data. Proof. First, since ρ satisfies the transport-type equations, we obtain (2.7). By using the density equation (1.2) and the boundary equation (1.9), we know 1 d ((ρu)t , u) + (ρu ⊗ u, ∇u) = 2 dt

 ρ|u|2 dx.

(2.10)

O

Then we take the summation of the equation (1.3) multiplied by u and the equation (1.4) multiplied by −H + 2MQ (M is a sufficiently large constant as in (2.4)), take the trace, and integrate by parts over O to get

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d M E (t) + μ ∇u 2L2 + Q 2L2 + (a 2 + 2Ma) Q 2L2 + c2 Q 6L6 + 2(a + M) ∇Q 2L2 dt  tr(Q2 ) 2 + 2(a + M)c Q 4L4 + b2 tr Q2 − Id dx d O



 tr(Q2 )

2 2 I d + cQ|Q| − (Q − Q, Q) = (u · ∇Q, Q) − u · ∇Q, aQ − b Q − d   tr(Q2 )

2 2 I d + cQ|Q| − λ (|Q|D, H ) + Q − Q, aQ − b Q − d − (∇ · (∇Q  ∇Q), u) + λ (|Q|H, ∇u) + (∇ · (QQ − QQ), u) − κ (Q, ∇u)







− 2b Q, Q2 + 2bc Qtr(Q2 ), Q2 + 2(a + M)b Q, Q2 + 2c Q, Qtr(Q2 ) + 2λM (|Q|D, Q) =

14 

Ii .

i=1

We now derive the estimates for the terms Ii , 1 ≤ i ≤ 14, one by one. First, I2 = 0 (by div u = 0) and I3 + I8 = 0 (by Lemma A.1). In the following, we shall show that I1 + I6 = 0, I5 + I7 = 0, I4 = 0, I13 ≤ 0, and the estimates for all other terms. A direct computation shows that I1 + I6 = (u · ∇Q, Q) − (∇ · (∇Q  ∇Q), u)   = ui ∂i Qj k Qj k dx − ∂i ∂j Qkl ∂j Qkl ui + ∂i Qkl ∂j ∂j Qkl ui dx O

O



∂i ∂j Qkl ∂j Qkl ui dx =

=− O

1 2

 |∇Q|2 divudx = 0. O

By the fact that Q is symmetric and  is skew-symmetric we have,    tr(Q2 )  I4 = Q − Q, aQ − b Q2 − Id + cQ|Q|2 d    tr(Q2 )  = − Q + Q, aQ − b Q2 − Id + cQ|Q|2 d    tr(Q2 )  + 2 Q, aQ − b Q2 − Id + cQ|Q|2 d = 0, I5 + I7 = λ(|Q|H, ∇u) − λ(|Q|D, H ) = λ(|Q|H, ∇u) − λ(|Q|H, D) = λ(|Q|H, ∇u − D) = λ(|Q|H, ) = 0. Moreover, by Young’s inequality, we get

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μ ∇u 2L2 + C Q 2L2 , 4 I10 = −2b (Q, Q2 ) ≤ Q 2L2 + C Q 4L4 , 2 μ I14 = 2λM(|Q|D, Q) ≤ ∇u 2L2 + C Q 4L4 . 4 I9 = −κ(Q, ∇u) ≤

From (2.3), we have the following estimates for I11 , I12 , of course, by choosing an appropriate ε > 0, 

I11 = 2bc Qtr(Q2 ), Q2 = 2bc tr(Q)3 |Q|2 dx   ≤ 2|b|c O

=

c2 2

O



1 3ε |Q|4 + |Q|2 |Q|2 dx 8 ε

Q 6L6 + C Q 4L4 ,



I12 = 2(a + M)b Q, Q2 = 2(a + M)b tr(Q)3 dx ≤ C( Q 2L2 + Q 4L4 ). O

Finally, 

I13 = 2c Q, Qtr(Q2 ) = 2c ∂kk Qij Qij tr(Q2 )dx O





∂k Qij ∂k Qij tr(Q2 )dx − 2c

= −2c O





|∇Q|2 |Q|2 dx − c

= −2c O

∂k Qij Qij ∂k tr(Q2 )dx O

|∇tr(Q2 )|2 dx ≤ 0. O

With all the above estimates, we have d M μ c2 E (t) + ∇u 2L2 + Q 2L2 + Q 6L6 dt 2 2 2 ≤ C( ∇Q 2L2

+ Q 2L2

(2.11)

+ Q 4L4 ),

where C = C(a, b, c, κ, μ, λ, , M). Then the desired estimates (2.6), (2.8) and (2.9) follow from (2.11) and the Grönwall’s inequality. 2 Next, let us introduce the definition of the global weak solutions. Definition 2.1. (ρ, u, Q) is a global weak solution to the system (1.2)-(1.5) with the initial and boundary conditions (1.6)-(1.9), if for any T > 0, the following conditions are satisfied:

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⎧ ∞ p ⎪ ⎨ρ ≥ 0, ρ ∈ L ((0, T ) × O), ρ ∈ C([0, T ]; L (O)), 1 ≤ p < ∞, u ∈ L2 (0, T ; V2 (O)), and ρ|u|2 ∈ L∞ (0, T ; L1 (O)), ⎪ ⎩ Q ∈ L∞ (0, T ; H 1 (O)) ∩ L2 (0, T ; H 2 (O)), and Q ∈ S0d a.e. in [0, T ] × O;

9

(2.12)

moreover, for any ζ ∈ C 1 ([0, T ] × O) with ζ (T , ·) = 0, T  −

 (ρ∂t ζ + ρu · ∇x ζ )dxdt =

0 O

ρζ ¯ (0, x)dx;

(2.13)

O

for any ψ ∈ C 1 ([0, T ] × O) with divx ψ = 0 and ψ(T , ·) = 0, T 

 (−ρu · ∂t ψ − (ρu ⊗ u) : ∇x ψ + μ∇x u : ∇x ψ)dxdt −

0 O

T

m(x) ¯ · ψ(0, x)dx O



(2.14)

(∇Q  ∇Q + λ|Q|H [Q] − QQ + QQ − κQ) : ∇x ψdxdt;

= 0 O

for any ϕ ∈ C 1 ([0, T ] × O) with ϕ ∈ S0d and ϕ(T , ·) = 0, T 



 Q : ∂t ϕ + Q : ϕ + Q : (u · ∇x ϕ) − (Q − Q − λ|Q|D) : ϕ dxdt

0 Rd

=

T 

 tr(Q2 )  Id + cQtr(Q2 ) : ϕdxdt aQ − b Q2 − d

(2.15)

0 Rd



−

¯ Q(x) : ϕ(0, x)dx;

Rd

and finally, the following energy inequality holds: d M 1 E (t) + dt 2

t  (μ|∇u|2 + |Q|2 + c2 |Q|6 )dxds 0 O

 2

|m| ¯ Ct ¯ 4 + |∇ Q| ¯ 2 dx, ¯ 2 + |Q| + |Q| ≤ Ce ρ¯

(2.16) for a.e. t ∈ [0, T ].

O

Our main result states as follows: Theorem 2.1. The problem (1.2)−(1.5) admits a weak solution (ρ, u, Q) in the sense of Definition 2.1 under the assumptions (1.6)-(1.9) on the initial and boundary conditions.

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We will prove Theorem 2.1 using the Faedo-Galerkin approximation for constructing the solution. 3. The approximation scheme In this section, we present the approximation system, and prove the existence of the approximate solutions. 3.1. The approximation scheme Let {ψn } ∈ C0∞ (O) be an orthonormal basis of V1 . Now, we define a sequence of finite dimensional spaces Xn = span{ψ1 , ψ2 , · · · , ψn },

n = 1, 2, · · · ,

(3.1)

and let Yn = C([0, T ]; Xn ). Because the presence of vacuum is allowed, we will choose our approximate initial density to be bounded away from zero as follows. We set  ρ, ¯ in O, ρ˜ = 1, Rd \ O, and let (ρ) ¯ ε = (ρ˜ ∗ ηε )|O , where η is the standard mollifier in Rd . Then the initial density for the approximation system is ρ|t=0 = ρ ε = (ρ) ¯ ε + ε.

(3.2)

There exists some constant C independent of ε such that ε ≤ ρ ε ≤ C, and ρ ε ∈ C ∞ , ρ ε → ρ¯ in Lp (O) for all 1 ≤ p < ∞, as ε → 0. For the initial data of the velocity u, we first recall the Hodge-de Rham type decomposition in Lions [31]. Lemma 3.1. Let N ≥ 2, ρ ∈ L∞ (Rd ) such that 0 <  ≤ ρ a.e. on Rd for some  ∈ (0, ∞). Then there exists two bounded operators Pρ and Qρ on L2 (Rd ) such that for all m ∈ L2 (Rd ), (mp , mq ) = (Pρ m, Qρ m) is the unique solution in L2 (Rd ) of m = mp + mq ,

1

(−)− 2 div(ρ −1 mp ) = 0,

1

(−)− 2 rot(mq ) = 0.

Furthermore, if ρn ∈ L∞ (Rd ),  ≤ ρn ≤  a.e. on Rd for some 0 <  ≤  < ∞ and ρn converges a.e. to ρ, then (Pρn mn , Qρn mn ) converges weakly in L2 (Rd ) to (Pρ m, Qρ m) whenever mn converges weakly to m.

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We set   ( ρ) ¯ ε = ( ρ˜ ∗ ηε )|O , and define 1  mε = m ¯ ρ¯ − 2 ( ρ) ¯ ε.

Obviously, mε → m ¯

in L2 (O),

1

1

and mε (ρ ε )− 2 → m ¯ ρ¯ − 2

in L2 (O).

By Lemma 3.1, we have mε = ρ ε u0,ε + Qρ ε mε , where u0,ε ∈ L2 (O) and div u0,ε = 0, and Qρ ε mε ∈ L2 (O), ∇ × Qρ ε mε = 0 in D . Now we impose the initial data for the velocity as u|t=0 = u0,n = Pn u0,ε ,

(3.3)

with Pn the orthogonal projection in V1 onto Xn . Lastly, we impose the initial data for the Q-tensor as ¯ Q|t=0 = Q.

(3.4)

With the initial data defined above, our approximate solution can be stated as: Definition 3.1. (ρn , un , Qn ) is an approximate solution if the following equations are satisfied in the weak sense of Definition 2.1, (ρn )t + ∇ · (ρn un ) = 0,

(3.5)

(ρn un )t + ∇ · (ρn un ⊗ un ) + ∇pn − μun = −∇ · (∇Qn  ∇Qn ) − λ∇ · (|Qn |H [Qn ]) + ∇ · (Qn Qn − Qn Qn ) + κ∇ · Qn ,

(3.6)

(Qn )t + (un · ∇)Qn + Qn n − n Qn − λ|Qn |Dn = H [Qn ],

(3.7)

∇ · un = 0,

(3.8)

with the initial conditions (3.2)-(3.4), boundary conditions (1.9), and the test function space in (2.14) replaced by the restriction on Xn . Moreover, ρn ∈ C([0, T ] × O),

un ∈ Yn ,

Qn ∈ L∞ (0, T ; H 1 (O)) ∩ L2 (0, T ; H 2 (O)).

Under the above definition, we have the following existence result of approximate solutions.

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Theorem 3.1. For any T > 0, the problem (3.5)-(3.8) admits a global weak solution (ρn , un , Qn ) with the initial conditions (3.2)-(3.4) and the boundary conditions (1.9). We remark that the approximate solution (ρn , un , Qn ) depends also on ε. 3.2. The Neumann problem for the density and the Q-tensor In order to prove Theorem 3.1, let us first introduce the following existence results for the density and Q-tensor equations, which are classical and may be found in [13]. ¯ Rd )) with div u = 0, there exists a mapping S = Lemma 3.2. For each u ∈ C([0, T ]; C0∞ (O, S[u], ¯ Rd )) → C([0, T ]; C ∞ (O)), ¯ S : C([0, T ]; C ∞ (O, satisfying the following properties: (1) ρ = S[u] is the unique classical solution of the initial-value problem  ρt + div (uρ) = 0, ¯ ρ|t=0 = ρ¯ ∈ C ∞ (O),

(3.9)

(2) ρ ∈ C 1 ([0, T ]; C ∞ (O)), and ε ≤ S[u](t, x) ≤ C, for all t ∈ [0, T ]; (3) S[u1 ] − S[u2 ] C([0,T ];L2 (O)) ≤ T C(T ) u1 − u2 C([0,T ];C 1 (O)) , 0

(3.10)

for any u1 , u2 in the set ¯ Rd )), div v = 0, and v NN := {v|v ∈ C([0, T ]; C0∞ (O, C([0,T ];C 1 (O¯ )) ≤ N }, 0

(3.11)

for some suitable constant N > 0. Proof. Since div u = 0, (3.9) is just a transport equation. We integrate along the characteristic line, i.e., solve the following problem: dX = u(X, s), ds

X(t; (x, t)) = x,

for (x, t) ∈ O × [0, T ].

(3.12)

By the standard theory of the transport equation, and the fact that u ∈ C([0, T ]; C0∞ (O)), there exists a unique smooth solution X to (3.12) and the solution satisfies ρ(t, x) = ρ ε (X(0; (x, t))),

for all t ∈ [0, T ], x ∈ O.

(3.13)

It is easy to see that ε ≤ ρ ≤ C. Since ρ ε is smooth, we know that ρ is bounded in C([0, T ]; C ∞ (O)) and it is unique.

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13

In order to show the continuity of S, let ρ1 , ρ2 represent the solutions to (3.9) corresponding to u1 , u2 ∈ NN . And we denote by ρ˜ = ρ1 − ρ2 , it satisfies, 

ρ˜t + u2 · ∇ ρ˜ + (u1 − u2 ) · ∇ρ1 = 0, ρ| ˜ t=0 = 0.

Multiplying ρ˜ on the both sides of the above equation, and integrating by parts, we have d ρ ˜ 2L2 = −2 dt

 (u1 − u2 ) · ∇ρ1 ρdx ˜ ≤ C u1 − u2 2L2 + C ρ ˜ 2L2 . O

Then by the Grönwall’s inequality, we get t ρ(t, ˜ ·) 2L2

≤ Ce

(u1 − u2 )(s, ·) 2L2 ds,

Ct 0

2

which implies (3.10).

¯ Rd )) with u|∂ O = 0, and Q(x) ¯ Lemma 3.3. For each u ∈ C([0, T ]; C02 (O, satisfied (1.8), there exists a unique solution to the following initial-boundary value problem, 

∂t Q + (u · ∇)Q + Q − Q − λ|Q|D = H, ¯ Q|t=0 = Q|∂ O = Q(x),

(3.14)

with Q ∈ L∞ ([0, T ]; H 1 (O)) ∩ L2 ([0, T ]; H 2 (O)). Moreover, the above mapping u → Q[u] is continuous from NN to L∞ ([0, T ]; H 1 (O)) ∩ L2 ([0, T ]; H 2 (O)), and Q[u] ∈ S0d a.e. in [0, T ] × O. Proof. The existence of the solution Q can be achieved by the standard parabolic theory [27]. Next, we show that Q belongs to L∞ (0, T ; H 1 (O)) ∩ L2 (0, T ; H 2 (O)). Assume u ∈ NN . First, similar to Proposition 2.1, we multiply the first equation in (3.14) by −H [Q] + 2MQ (M is a sufficient large constant as in (2.4)), take the trace, integrate by parts over O, and then obtain, d M E (t) + Q 2L2 + a 2 Q 2L2 + c2 Q 6L6 + b2 dt

 O

tr(Q2 ) 2 tr Q2 − Id dx d

   tr(Q2 )  Id + cQ|Q|2 − (Q − Q, Q) = (u · ∇Q, Q) − u · ∇Q, aQ − b Q2 − d     2 tr(Q ) 2 2 Id + cQ|Q| − λ(|Q|D, H [Q]) + 2a (Q, Q) + Q − Q, aQ − b Q − d







− 2b Q, Q2 + 2bc Qtr(Q2 ), Q2 + 2(a + M)b Q, Q2 + 2c Q, Qtr(Q2 )

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14

− 2M ∇Q 2L2 − 2aM Q 2L2 + 2λM(|Q|D, Q) − 2(a + M)c Q 4L4 ≤

c2 Q 2L2 + Q 6L6 + C(N, M)( ∇Q 2L2 + Q 2L2 + Q 4L4 ). 2 2

Then by Grönwall’s inequality, we have Q L∞ (0,T ;H 1 (O)) + Q L2 (0,T ;H 2 (O)) ≤ C, ¯ H 1 (O ) . where C > 0 depends on M, N, a, b, c, λ, , T , Q ˜ For the uniqueness, we denote by Q = Q1 − Q2 , with Q1 and Q2 two solutions to the problem (3.14). Then it holds, ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

˜ − Q ˜ + Q ˜ − λ(|Q1 | − |Q2 |)D − Q˜ + a Q ˜ ˜ t + u · ∇Q Q         ˜ 1 +Q2 ) tr Q(Q 2 ˜ ˜ ˜ Id − cQtrQ1 − cQ2 tr Q(Q1 + Q2 ) , = b Q(Q1 + Q2 ) − d

(3.15)

˜ ∂ O = 0. ˜ t=0 = Q| Q|

˜ take the trace, and integrate by parts over O to obtain, Multiply equation (3.15) by Q, 1 d ˜ 2 ˜ 22 Q L2 + ∇ Q L 2 dt





˜ 2 2 − (u · ∇ Q, ˜ Q) ˜ + Q ˜ − Q, ˜ Q ˜ + λ (|Q1 | − |Q2 |)D, Q˜ = −a Q L      ˜ 1 + Q2 ) Id    tr Q(Q 2 ˜ ˜ ˜ ˜ − cQtrQ1 − cQ2 tr Q(Q1 + Q2 ) , Q + b Q(Q1 + Q2 ) − d

˜ 2 2 + N ∇ Q ˜ L2 Q ˜ L2 + C Q ˜ L2 Q ˜ L6 ( Q1 + Q2 L3 + Q1 2 6 + Q2 2 6 ) ≤ C Q L L L ≤

˜ 2 2, ˜ 2 2 + C Q ∇ Q L L 2

where C depends on a, b, c, λ, , N, Q1 L∞ (0,T ;H 1 (O)) , Q2 L∞ (0,T ;H 1 (O)) ; and in the last step, we used Sobolev embedding inequality, Poincaré inequality and Young’s inequality. Then, by applying Grönwall’s inequality to the above inequality, we obtain the desired uniqueness result. In the following, we shall show that the map u → Q[u] is continuous. Let {un } be a bounded sequence in NN , with lim un − u C(0,T ;C 2 (O¯ )) = 0,

n→∞

0

(3.16)

¯ for some u ∈ C(0, T ; C02 (O)). Set Qn = Qn [un ], Q = Q[u], and Q˜ n = Qn − Q. Taking the ˜ n , taking difference of the equations of Qn and Q, multiplying the resulting equation by −Q the trace, integrating by parts over O, we obtain,

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15

1 d ˜ n 2 2 ˜ n 2 2 + Q ∇ Q L L 2 dt



˜ n + (un − u) · ∇Q, Q ˜ n − n Q ˜ n + (n − )Q, Q˜ n = un · ∇ Q



˜ n − λ |Qn |(Dn − D) + (|Qn | − |Q|)D, Q˜ n ˜ n n + Q(n − ), Q + Q  

˜ n (Qn + Q)) tr( Q ˜ n , Q ˜ n − b Q ˜n ˜ n (Qn + Q) − + a Q Id , Q d

  ˜ n + Qtr Q ˜ n (Qn + Q) , Q˜ n + c |Qn |2 Q =

7 

Ii .

i=1

In the following, we shall estimate the terms on the right hand side of the above equation one by one, by using the Sobolev embedding inequality, Poincaré inequality and Young’s inequality:

˜ n + (un − u) · ∇Q, Q ˜n I1 = un · ∇ Q

˜ n L2 + un − u L∞ ∇Q L2 Q ˜ n L2 ≤ un L∞ ∇ Q ˜ n 2 2 + C un − u 2L∞ . ˜ n 2 2 + C ∇ Q Q L L 12

˜ n + (n − )Q, Q˜ n I2 = −  n Q ≤

˜ n L2 Q ˜ n L2 + ∇un − ∇u L∞ Q L2 Q ˜ n L2 ≤ n L∞ Q ˜ n 2 2 + C ∇un − ∇u 2L∞ ˜ n 2 2 + C Q Q L L 12 ˜ n 2 2 + C ∇un − ∇u 2L∞ . ˜ n 2 2 + C ∇ Q ≤ Q L L 12

˜ n n + Q(n − ), Q ˜n I3 = Q ≤

˜ n L2 + Q L2 n −  L∞ Q ˜ n L2 ˜ n L2 n L∞ Q ≤ Q ˜ n 2 2 + C ∇un − ∇u 2L∞ ˜ n 2 2 + C Q Q L L 12 ˜ n 2 2 + C ∇un − ∇u 2L∞ . ˜ n 2 2 + C ∇ Q ≤ Q L L 12

I4 = −λ |Qn |(Dn − D) + (|Qn | − |Q|)D, Q˜ n

˜ n L2 ∇u L∞ Q ˜ n L2 ≤ Qn L2 ∇un − ∇u L∞ + Q ≤

≤

˜ n 2 2 + C ∇un − ∇u 2L∞ ˜ n 2 2 + C Q Q L L 12

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16

˜ n 2 2 + C ∇un − ∇u 2L∞ . ˜ n 2 2 + C ∇ Q Q L L 12

˜ n , Q˜ n = −a ∇ Q ˜ n 2 2 . I5 = a Q L   ˜ n (Qn + Q)) tr( Q ˜ n (Qn + Q) + I6 = −b Q I3 , Q˜ n 3 ≤

˜ n L3 Qn + Q L6 Q ˜ n L2 ≤ C Q ˜ n 2 2 . ˜ n 2 2 + C ∇ Q Q L L 12

˜ n + Qtr(Q˜ n (Qn + Q)), Q˜ n I7 = c |Qn |2 Q ≤

˜ n L6 Q ˜ n L2 ≤ C ( Qn 2L6 + Qn L6 Q L6 + Q 2L6 ) Q ≤

˜ n 2 2 . ˜ n 2 2 + C ∇ Q Q L L 12

Combining all the above estimates, we get, 1 d ˜ n 2 2 . ˜ n 2 2 + Q ˜ n 2 2 ≤ C un − u 2 2 + C ∇ Q ∇ Q L L L C0 (O ) 2 dt 2 Then by Grönwall’s inequality, we have ˜ n (t, ·) 2 2 ∇ Q L

t +

˜ n (s, ·) 2 2 ds ≤ CT eCt un − u 2 Q . L C([0,T ];C 2 (O ))

(3.17)

0

0

As n → ∞, we conclude that ˜ n L2 (0,T ;H 2 (O)) ) = 0. lim ( Q˜ n L∞ (0,T ;H 1 (O)) + Q

n→∞

(3.18)

Finally, we shall show Q ∈ S0d , that is, QT = Q, and trQ = 0 a.e. in [0, T ] × O. It is obvious to see that if Q is a solution to the problem (3.14), so is QT . Then by the uniqueness of the solution we proved earlier, we know that QT = Q. The only thing left is to show that trQ = 0. Let us take the trace on both sides of the first equation in (3.14) to get ∂t (trQ) + u · ∇trQ = (trQ − atrQ − ctrQtr(Q2 )),

(3.19)

with trQ|t=0 = trQ|∂ O = 0. Here we use the fact that QT = Q, T = − and trD = div u = 0. Then we multiply the equation (3.19) by trQ, and integrate by parts over O to obtain d trQ 2L2 + ∇trQ 2L2 = −a trQ 2L2 − c dt ≤ −a trQ 2L2

+ C Q 2L6 trQ L6 trQ L2

 |trQ|2 tr(Q2 )dx O

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17

≤ −a trQ 2L2 + C ∇trQ L2 trQ L2 ≤

∇trQ 2L2 + C trQ 2L2 . 2

Applying the Grönwall’s inequality again, we complete the proof. 2 3.3. Existence of the solution to the approximation scheme In this subsection, we shall finish the proof of Theorem 3.1. Proof of Theorem 3.1. Let u¯ n ∈ Yn , we first consider the following two problems: 

(ρn )t + ∇ · (u¯ n ρn ) = 0, ρn |t=0 = ρ ε ,  ¯n− ¯ n Qn − λ|Qn |D¯n = H [Qn ], (Qn )t + (u¯ n · ∇)Qn + Qn  ¯ Qn |t=0 = Qn |∂ O = Q,

(3.20)

(3.21)

¯ n = (∇ u¯ n − ∇ u¯  ¯ with  ¯ n + ∇ u¯  n )/2 and Dn = (∇ u n )/2. By Lemma 3.2, we have a unique classical solution ρn = S[u¯ n ] to the initial-value problem (3.20), and from Lemma 3.3 the problem (3.21) has a unique solution Qn = Q[u¯ n ]. Then we consider the Navier-Stokes equations and look for the solution un ∈ Yn to the following variational approximation problem: 

t  ρ(t, x)un (t, x) · ψ(x)dx +

O

t  div (ρun ⊗ ∇un ) · ψdxds + μ

0 O

t

∇un : ∇ψdxds 0 O



=

 (∇Q  ∇Q + λ|Q|H [Q] − QQ + QQ − κQ) : ∇ψdxds +

0 O

ρ ε u0,n · ψ(x)dx, O

(3.22) for any t ∈ [0, T ], and ψ ∈ Xn . Next, we introduce a family of operators as in [13], M[ρ] : Xn → Xn∗ ,

 M[ρ]v(w) =

ρv · wdx,

(3.23)

O

for any v, w ∈ Xn . As stated in [13], we know that this operator is invertible provided ρ is strictly positive on O, and the functional ρ → M−1 [ρ]

(3.24)

maps from Nη = {ρ ∈ L1 (O)| infx∈O ρ ≥ η > 0} into L(Xn∗ , Xn ) with the following property,

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18

M−1 [ρ1 ] − M−1 [ρ2 ] L(Xn∗ ,Xn ) ≤ C(n, η) ρ1 − ρ2 L1 (O) .

(3.25)

From the above together with Theorems 3.2 and 3.3, where we take ρn = S[un ], and Qn = Q[un ], we can rewrite the variational problem (3.22) as: ⎛ un (t) = M−1 [ρn ] ⎝q ∗ +

t

⎞ N [ρn (s), un (s), Qn (s)]ds ⎠ ,

(3.26)

ρ ε u0,n · ψdx,

(3.27)

0

with (q ∗ , ψ) =

O

 (N [ρn , un , Qn ], ψ) = − 

 

div(ρn un ⊗ un ) · ψdx − μ O

∇un : ∇ψdx O

(3.28)

(∇Qn  ∇Qn + λ|Qn |H [Qn ] − Qn Qn + Qn Qn − κQn ) : ∇ψdx,

+ O

for any t ∈ [0, T ], and ψ ∈ Xn . Therefore, combining (3.10), (3.17) and (3.25), we achieve a local solution (ρn , un , Qn ) to the problem (3.5), (3.7), (3.22), with initial data (3.2)-(3.4) and boundary data (1.9) on a short time interval [0, Tn ], Tn ≤ T , by using the standard fixed point theorem on C([0, T ]; Xn ). In order to extend the existence time Tn to T for any n = 1, 2, · · · , we need to prove that un stays bounded in Xn for the whole interval [0, Tn ]. Hence, in the following, we will prove an energy inequality in the same way as Proposition 2.1. Taking the sum of (3.6) multiplied by un and (3.7) multiplied by −H [Qn ] + MQn (M is a sufficiently large constant as in (2.4)), taking the trace, and integrating by parts over O, we get d M E (t) + μ ∇un 2L2 + Qn 2L2 + (a 2 + 2aM) Qn 2L2 + c2 Qn 6L6 dt n  tr(Q2n ) 2 2 4 2 + 2M ( ∇Qn L2 + c Qn L4 ) + b tr Q2n − dx d O



= −κ (Qn , ∇un ) + 2a (Qn , Qn ) − 2ac Qn 4L4 − 2b Qn , Q2n





+ 2bc Qn tr(Q2n ), Q2n + 2(a + M)b Qn , Q2n + 2c Qn , Qn tr(Q2n ) + 2λM(|Qn |Dn , Qn )) ≤

c2 μ ∇un 2L2 + Qn 2L2 + Qn 6L6 + C( Qn 2L2 + Qn 4L4 ), 2 2 2

which gives, d M μ c2 En (t) + ∇un 2L2 + Qn 2L2 + Qn 6L6 ≤ C( Qn 2L2 + Qn 4L4 ), dt 2 2 2

(3.29)

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19

where C is independent of n and ε, and   EnM (t) = O

 1 a b c 1 2 2 2 3 4 ρn |un | + |∇Qn | + ( + M)|Qn | − tr(Qn ) + |Qn | dx. 2 2 2 3 4

(3.30)

Then from Grönwall’s inequality, the above inequality yields, for a.e. t ∈ [0, Tn ],  O

c (ρn |un |2 + |∇Qn |2 + M|Qn |2 + |Qn |4 )dx + 4 

≤ CeCt

t  (μ|∇un |2 + |Qn |2 + c2 |Qn |6 )dxds 0 O

¯ 2 + |Q| ¯ 4 + |∇ Q| ¯ 2 )dx (ρ ε |u0,n |2 + |Q|

O



≤ CeCt

¯ 2 + |Q| ¯ 4 + |∇ Q| ¯ 2 )dx ≤ C, (ρ ε |u0,ε |2 + |Q|

O

(3.31)

where C is independent of n. Then we obtain the following uniform estimates: Tn ∇un 2L2 dt ≤ C,

μ 0

 ρn |un |2 dx ≤ C.

sup

t∈[0,Tn ]

(3.32)

(3.33)

O

Since Xn is a finite dimensional space, we can deduce from Lemma 3.2 that there exists a constant ¯ a, b, c, O), such that C = C(n, ρ, ¯ m, ¯ Q, 0 < ε ≤ ρn (t, x) ≤ C,

∀ t ∈ (0, Tn ), x ∈ O,

(3.34)

which, combined with (3.33) and the fact that L∞ and L2 norms are equivalent on Xn , yields sup ( un (t, ·) L∞ (O) + ∇un (t, ·) L∞ (O) ) ≤ C(n, Eδ (0), M, O).

t∈[0,Tn ]

(3.35)

Then we can extend the existence time interval [0, Tn ) of the solution (un , ρn , Qn ) to [0, T ], for any T ∈ [0, ∞). 2 4. Convergence of the approximate problem In this section, we shall complete the proof of the main Theorem 2.1, by taking the limit of the approximation system as n → ∞ and ε → 0.

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20

4.1. The existence of the first level approximate solutions as n → ∞ First we keep ε as a fixed constant, then ρn is bounded away from zero uniformly in n. Thus from the energy inequality (3.31), we obtain the following estimates of the solution (ρn , un , Qn ) which are independent of n, 0 < ε ≤ ρn (t, x) ≤ C, ∀t ∈ (0, T ), x ∈ O, √ ρ n un L∞ (0,T ;L2 (O)) ≤ C,

(4.1)

un L∞ (0,T ;L2 (O))∩L2 (0,T ;H 1 (O)) ≤ C,

(4.3)

Qn L∞ (0,T ;H 1 (O)∩L4 (O))∩L2 (0,T ;H 2 (O))∩L6 (0,T ;L6 (O)) ≤ C.

(4.4)

(4.2)

0

The above estimates give the following convergence results for any fixed ε > 0 as n → ∞: un  u in L2 (0, T ; H01 (O)),

(4.5)

Qn  Q in L2 (0, T ; H 2 (O)), √ √ ρn un ∗ ρu in L∞ (0, T ; L2 (O)).

(4.6) (4.7)

However the above convergence is not strong enough to pass every term in the system to a limit. We need to prove more compactness results as follows. First, we apply Theorem A.1 to obtain some compactness for ρn and un . The only condition we need to verify is that for any ψ ∈ D with div ψ = 0, the following holds |(∂t (ρn un ), ψ)| ≤ C ψ L2 (0,T ;W 1,3 (O)) .

(4.8)

To this end, we will show that the other terms aside from ∂t (ρn un ) in equation (3.6) belong to 3 W −1, 2 (O). In fact, by using the Sobolev embedding inequality and the Poincaré inequality, we have

ρn un ⊗ un

3

L2t Lx2

⎛ T ⎞ 12 ⎛ T ⎞ 12   √ √ ≤ C ⎝ ρn un 2L2 un 2L6 dt ⎠ ≤ C ⎝ ρn un 2L2 un H 1 dt ⎠ 0

0

√ ≤ C ρn un L∞ 2 u L2 H 1 , t Lx t

λ|Qn |H [Qn ]

3 L2t Lx2

x

= λ|Qn |(Qn − aQn + b[Q2n −

tr(Q2n ) Id ] − cQn |Qn |2 ) 3 d L2t Lx2

⎛ T ⎞ 12  ≤ C ⎝ Qn 2 6 Qn 2 2 + Qn 4 3 + Qn 2 3 Qn 4 6 + Qn 8 6 dt ⎠ L

L

L

L

L

0 2 3 ≤ C( Qn L∞ 1 + Qn ∞ 1 + Qn ∞ 1 ) Qn L2 H 2 , L H L H t Hx t

x

t

t

x

∇Qn  ∇Qn + Qn Qn − Qn Qn + κQn

3

L2t Lx2

x

L

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21

⎛ T ⎞ 12  1 ≤ C ⎝ ∇Qn 2 2 ∇Qn 2 6 + Qn 2 6 Qn 2 2 + Qn 2 2 |O| 3 dt ⎠ L

L

L

L

L

0

≤ C(1 + Qn L∞ 1 ) Qn L2 H 2 , t Hx t

x

3

which imply that the first order derivatives of all the above terms are in L2 (0, T ; W −1, 2 (O)), and so is ∂t (ρn un ). Then by Theorem A.1, we have ρn → ρ in C([0, T ]; Lq (O)), √ ρn un → ρu in Lp (0, T ; Lr (O)),

(4.10)

ρn un → ρu in Lp (0, T ; Lr (O)),

(4.11)

(4.9)

2dp for any 1 ≤ q < ∞, 2 < p < ∞, and 1 ≤ r < dp−4 . Moreover, similarly to the above statement, from the estimates (4.3) and (4.4) along with the 3 fact that ∂t Qn satisfies (3.7), we know that ∂t Qn ∈ L2 (0, T ; L 2 (O)), and

(un · ∇)Qn

3

L2t Lx2

⎛ T ⎞ 12  ≤ C ⎝ un 2L6 ∇Qn 2L2 dt ⎠ ≤ C un L2 (H 1 )x ∇Qn L∞ 2, t Lx t

⎛ λ|Qn |Dn

H [Qn ] ⎛ ≤C⎝

3

L2t Lx2

3 L2t Lx2

T

0

0

≤C⎝

T

⎞ 12 Qn 2L6 ∇un 2L2 dt ⎠ ≤ C Qn L∞ 1 ∇un L2 , t Hx t,x

0

= Qn − aQn + b[Q2n −

tr(Q2n ) Id ] − cQn |Qn |2 3 d L2t Lx2

⎞ 12

Qn 2L2 |O| 3 + Qn 2L2 |O| 3 + Qn 4L3 + Qn 4L6 Qn 2L3 dt ⎠ 1

1

x

x

x

x

x

0



2 ≤ C Qn L2 + Qn L2 + Qn L2 H 2 ( Qn L∞ 1 + Qn ∞ 1 ) . H L H t x t,x

t,x

t

x

t

x

Then by the Aubin-Lions Lemma (Lemma A.2), we have Qn  Q in L2 (0, T ; H 2 (O)),

Qn → Q in L2 (0, T ; H 1 (O)).

(4.12)

Obviously, (3.3) implies the strong convergence of u0,n to u0,ε in L2 ((0, T ) × O). Thus we can pass to limit as n goes to infinity in system (3.5)-(3.7) in D ((0, T ) × O), i.e., (ρ, u, Q) is a weak solution to this system. Regarding the limit of the energy, from the above convergence results, we know that

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E M (t) ≤ lim inf EnM (t), n→∞

t

 (μ|∇u|2 + |Q|2 + c2 |Q|6 )dxds

0 O

t  (μ|∇un |2 + |Qn |2 + c2 |Qn |6 )dxds.

≤ lim inf n→∞

0 O

For the initial data in the energy inequality, by the definition of u0,n we have 

 ρ ε |u0,n |2 dx →

O

ρ ε |u0,ε |2 dx. O

In summary, we have obtained the following results: Proposition 4.1. For any T > 0, there is a solution (ρε , uε , Qε ) which satisfies the system (1.2)-(1.5) in the weak sense of Definition 2.1, with the initial data ρ|t=0 = ρ ε ,

u|t=0 = u0,ε ,

¯ Q|t=0 = Q,

(4.13)

and the boundary condition (1.9). In addition, the solution satisfies the following energy inequality: 1 E (t) + 2

T  (μ|∇u|2 + |Q|2 + c2 |Q|6 )dxdt

M

 ≤ Ce

0 O

(4.14)

(ρ ε |u

CT

0,ε |

2

¯ + |Q| ¯ + |∇ Q| ¯ )dx. + |Q| 2

4

2

O

4.2. Pass to the limit as ε → 0 The difficulty here is the degeneracy of the lower bound of ρ, but we can resolve this by using Theorem A.1. First let us consider the initial kinetic energy. Since we have ∇ × Qρ ε mε = 0 in D , we know 

 ρ ε |u

O



= O



= O

0,ε |

2

dx = O

1 ε |m − Qρ ε mε |2 dx ρε

1 (|mε |2 + |Qρ ε mε |2 − 2mε Qρ ε mε )dx ρε 1 (|mε |2 + |Qρ ε mε |2 − 2(ρ ε u0,ε + Qρ ε mε )Qρ ε mε )dx ρε

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 = O

23

1 (|mε |2 − |Qρ ε mε |2 )dx. ρε

For each ε, the solution (ρε , uε , Qε ) satisfies the following energy inequality from (4.14), T  2EεM (t) + 

μ|∇uε |2 + |Qε |2 + c2 |Qε |6 dxdt 0 O

≤ CeCT O

¯ 2 + |Q| ¯ 4 + |∇ Q| ¯ 2 )dx (ρ ε |u0,ε |2 + |Q|

 

≤ CeCT O

 |mε |2 ¯ 2 + |Q| ¯ 4 + |∇ Q| ¯ 2 dx, + | Q| ρε

with   EεM (t) := O

 1 a b c 1 2 2 2 3 4 ρε |uε | + |∇Qε | + ( + M)|Qε | − tr(Qε ) + |Qε | dx. 2 2 2 3 4

1

1

Since mε (ρ ε )− 2 → m ¯ ρ¯ − 2 in L2 (O) as ε → 0, we have the following uniform bounds with respect to ε, √ ρ ε uε L∞ (0,T ;L2 (O)) ≤ C,

(4.15)

uε L2 (0,T ;H 1 (O)) ≤ C,

(4.16)

Qε L∞ (0,T ;H 1 (O)∩L4 (O))∩L2 (0,T ;H 2 (O))∩L6 (0,T ;L6 (O)) ≤ C.

(4.17)

0

Then we obtain uε  u in L2 (0, T ; H01 (O)),

(4.18)

Qε  Q in L2 (0, T ; H 2 (O)), √ √ ρε uε ∗ ρu in L∞ (0, T ; L2 (O)).

(4.19) (4.20)

In order to pass to the limit as ε → 0 in the approximate system, the above convergence results are not strong enough. Then similar to the Subsection 4.1, we need to get some compactness results. By the same way as (4.8), we also can verify the condition |(∂t (ρε uε ), ψ)| ≤ C ψ L2 (0,T ;W 1,3 (O)) . Then Theorem A.1 yields

(4.21)

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ρε → ρ in C([0, T ]; Lq (O)), √ ρε uε → ρu in Lp (0, T ; Lr (O)), ρε uε → ρu in Lp (0, T ; Lr (O)), 2dp for any 1 ≤ q < ∞, 2 < p < ∞, and 1 ≤ r < dp−4 . Moreover, similarly to (4.12), we also have

Qε  Q in L2 (0, T ; H 2 (O)),

Qε → Q in L2 (0, T ; H 1 (O)).

(4.22)

The above boundedness and convergence results imply that div (ρε uε ⊗ uε ) → div(ρu ⊗ u) in D , T  ∇ · (∇Qε  ∇Qε + λ|Qε |H [Qε ] − Qε Qε + Qε Qε − κQε ) : ∇x ψdxdt 0 O

T  ∇ · (∇Q  ∇Q + λ|Q|H [Q] − QQ + QQ − κQ) : ∇x ψdxdt,

→ 0 O

T  (uε · ∇x Qε + Qε ε − ε Qε − λ|Qε |Dε ) : ϕdxdt 0 O

T  (u · ∇x Q + Q − Q − λ|Q|D) : ϕdxdt,

→ 0 O

T



T  H [Qε ] : ϕdxdt →

0 O

H [Q] : ϕdxdt, 0 O

for any ψ ∈ D with div ψ = 0, and ϕ ∈ D with ϕ ∈ S0d . For the initial data in the Navier-Stokes equations, by the decomposition and the convergence of mε , we have    ρ ε u0,ε · ψdx = mε · ψdx → m ¯ · ψdx, (4.23) O

O

O

for any ψ ∈ O with div ψ = 0. Then we can pass to the limit to obtain the weak solution (ρ, u, Q) of the problem (1.2)-(1.5) with the initial condition (1.6)-(1.8) and the boundary condition (1.9). Acknowledgments R. Zhang’s research was supported in part by the National Science Foundation under grants DMS-1312800, DMS-1613375. The results of this paper are part of the thesis [57]. We would like to thank Professor Dehua Wang for advising and valuable discussions.

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25

Appendix A. Preliminaries In this appendix, we recall some basic lemmas and theorem used in the previous sections. Lemma A.1. Let Q and Q be two d × d symmetric matrices, and  = 12 (∇u − ∇u ) be the vorticity with (∇u)αβ = ∂β uα . Then, 

   Q − Q , Q − ∇ · (Q Q − QQ ), u = 0.

Proof. See Lemma A.1 in [7].

2

Lemma A.2. (Aubin-Lions Lemma) Let X0 , X and X1 be three Banach spaces with X0 ⊆ X ⊆ X1 , with X0 compactly embedded in X, and X is continuously embedded in X1 . For 1 ≤ p, q ≤ +∞. Let W = {u ∈ Lp (0, T ; X0 )|u˙ ∈ Lq (0, T ; X1 )}. Then, (1) If p < +∞, then the embedding of W into Lp (0, T ; X) is compact. (2) If p = +∞ and q > 1, then the embedding of W into C(0, T ; X) is compact. Lemma A.3. (Gagliardo-Nirenberg interpolation inequality [39]) Let 1 ≤ q, r ≤ ∞, and the function u : O → R defined on a bounded Lipschitz domain O ⊆ Rd . For 0 ≤ j < m, the following inequalities hold s D j u Lp ≤ C1 D m u aLr u 1−a Lq + C2 u L ,

where 1 j 1 m 1 = + a( − ) + (1 − a) , p n r d q j ≤ a ≤ 1, m and s > 0 is arbitrary, C1 and C2 depend on O, m and d. Theorem A.1. (Theorem 2.4 in [31]) Assume 0 ≤ ρn ≤ C div un = 0

a.e. on O × (0, T ), ∂ρn + div (ρn un ) = 0, ∂t

ρ0,n → ρ¯

in L1 (O),

a.e. on O × (0, T ),

un L2 (0,T ;H 1 (O)) ≤ C, in D (Rd × (0, T )),

un  u in L2 (0, T ; H01 (O)),

(A.1)

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where ρ¯ satisfies 0 ≤ ρ¯ ≤ C a.e., and C denotes various positive constants independent of n. Then (1) ρn converges in C([0, T ]; Lp (O)) for all 1 ≤ p < ∞ to the unique solution ρ, bounded on O × (0, T ), of 

+ div (ρu) = 0, in D (Rd × (0, T )), ρ ∈ C([0, T ]; L1 (O)), ρ|t=0 = ρ¯ a.e. in O. ∂ρ ∂t

(A.2)

(2) If we assume in addition that ρn |un |2 is bounded in L∞ (0, T ; L1 (O)) and that we have for some q ∈ (1, ∞), m ≥ 1, |(

∂ (ρn un ), ψ)| ≤ C ψ Lq (0,T ;W m,q (O)) ∂t

(A.3)

√ for all ψ ∈ Lq (0, T ; W m,q (O)), such that, div ψ = 0 on Rd × (0, T ). Then, ρn un con√ 2dp , and un converges to u verges to ρu in Lp (0, T ; Lr (O)) for 2 < p < ∞, 1 ≤ r < dp−4 dθ

in Lθ (0, T ; L d−2 (O)) for 1 ≤ θ < 2 on the set {ρ > 0} (if d = 2, arbitrary r in [0, ∞)).

dθ d−2

is replaced by an

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