Weak solution to the steady compressible flow of nematic liquid crystals

Accepted Manuscript Weak solution to the steady compressible flow of nematic liquid crystals

Zhong Tan, Qiuju Xu

PII: DOI: Reference:

S0022-247X(16)30696-5 http://dx.doi.org/10.1016/j.jmaa.2016.11.013 YJMAA 20866

To appear in:

Journal of Mathematical Analysis and Applications

Received date:

22 March 2016

Please cite this article in press as: Z. Tan, Q. Xu, Weak solution to the steady compressible flow of nematic liquid crystals, J. Math. Anal. Appl. (2017), http://dx.doi.org/10.1016/j.jmaa.2016.11.013

This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

WEAK SOLUTION TO THE STEADY COMPRESSIBLE FLOW OF NEMATIC LIQUID CRYSTALS ZHONG TAN AND QIUJU XU Abstract. The three-dimensional equations for the steady compressible flow of nematic liquid crystals are considered in a bounded domain for the adiabatic exponent γ > 1. We establish the existence of a weak solution by three level approximation and adapting the weak convergence method.

Keywords 3D steady compressible nematic liquid crystal flow, weak solution, approximate problem, weighted estimates.

1. Introduction In the paper, we study a simplified system for the steady compressible nematic liquid crystal flow in a bounded domain Ω ⊂ R3 with C 2 boundary: ⎧ ⎪ ⎪ div(ρu) = 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎨div(ρu ⊗ u) + ∇p(ρ) = μΔu + μ ˜∇divu    1 1 2 2 2 ⎪ ⎪ I3 + ρf, −div ∇d  ∇d − + (1 − |d| ) |∇d| ⎪ ⎪ 2 42 ⎪ ⎪ ⎪ ⎩u · ∇d = Δd + 1 (1 − |d|2 )d. 2

(1.1)

Here ρ ≥ 0, u = (u1 , u2 , u3 ) ∈ R3 , d = (d1 , d2 , d3 ) ∈ R3 denote the density, the velocity and the direction field of the averaged macroscopic molecular orientations respectively, the pressure p for isentropic flows is given by p(ρ) = ργ with γ > 1 being the specific heat ratio, and f = (f1 , f2 , f3 ) is a given external force.  > 0 is a constant, the viscosity coefficients μ and μ ˜ satisfy the usual physical conditions μ > 0, μ ˜ = μ + λ ≥ 0. The symbol ⊗ denotes the Kronecker tensor product, I3 is the 3 × 3 identity matrix, and ∇d  ∇d denotes the 3 × 3 matrix whose (i, j)-th entry is given by ∂xi · ∂xj for 1 ≤ i, j ≤ 3. Indeed, ∇d  ∇d = (∇d)T ∇d, where (∇d)T denotes the transpose of the 3 × 3 matrix ∇d. In addition, the total mass of the fluid is prescribed

 ρ = M > 0. Ω

2010 Mathematics Subject Classification. 76A15; 76N10; 35Q35. Corresponding author: Qiuju Xu, [email protected]. Supported by the National Natural Science Foundation of China (No. 11271305, 11531010). 1

2

ZHONG TAN AND QIUJU XU

System (1.1) is a simplified version of the celebrated Ginzburg-Landau approximation model for the steady flows of nematic liquid crystals, the director d has variable degree of orientations. When the incompressible flows of nematic liquid crystals is considered, Lin-Liu [18] have initiated the mathematical analysis, the weak solutions and classical solutions with large viscosity μ were obtained. In [19], they went on the partial regularity of the suitable weak solutions, analogous to the classical theorem of Caffarelli-Kohn-Nirenberg [2] for the incompressible Navier-Stokes equation. The existence and uniqueness of the global strong solution have been proved by Hu-Wang [9] with small initial data. For the density-dependent incompressible case, Jiang-Tan [13] proved the global existence of weak solutions under the condition that the initial density belongs to Lγ (Ω) for γ > 32 , which weakened the initial density ρ0 ∈ L2 in Liu-Zhang [26]. We note that the compressible flow of liquid crystals is much more complicate and difficult to explore the mathematical analysis. The global existence of weak solutions has been obtained by Wang-Yu [32] and Liu-Qin [25] in dimension three. In addition, Yang-Dou-Ju [33] established weak-strong uniqueness property in class of finite energy weak solutions for two different compressible liquid crystal systems by the method of relative entropy. However, when the steady compressible flow is discussed, there are very few results concerning weak solutions. It should be mentioned the simplified Ericksen-Leslie system modeling the flow of nematic liq1 (1−|d|2 )2 vanishes and the Ginzburg42 |∇d|2 d. Due to the significant mathematical

uid crystals, namely, the Ginzburg-Landau penalization Landau penalty function

1 (1−|d|2 )d 2

is replaced by

difficulties induced by the supercritical nonlinearity |∇d|2 d and the nonlinear constraint |d| = 1, the simplified Ericksen-Leslie system becomes more complicate and is highly challenging. Some progress has also been made on the analysis of the simplified Ericksen-Leslie system for the incompressible fluid, we can refer to [7, 20, 23] in the direction of weak solutions, to [31] for strong solutions, to [11, 8] in the direction of blow up criterion. See also [4, 5, 15, 22, 30] for some related discussions. For the compressible case, the global existence of weak solutions was proved in recent works [12, 21], in [3, 10] the global existence and uniqueness of strong solutions were gained, Ma [27] obtained the classical solutions. For more results, the readers can refer to [24] that has introduced some recent developments of analysis for hydrodynamic flow of nematic liquid crystal flows and references therein. To our knowledge, there are no works on the simplified Ericksen-Leslie system for the steady compressible flow of nematic liquid crystals. The aim of this paper is to solve the existence of weak solutions to the problem (1.1) along with the Dirichlet boundary condition u = 0,

d=g

on ∂Ω.

(1.2)

Before defining a weak solution to (1.1)–(1.2), we explain the notations and conventions used throughout this paper. Notations: We shall write Lp (Ω) for the Lebesgue spaces with norm · Lp , W k,p (Ω) for the Sobolev spaces with norm · W k,p . In particular, H k (Ω) ≡ W k,2 (Ω) with norm · H k . We also

WEAK SOLUTION TO THE STEADY COMPRESSIBLE FLOW OF NEMATIC LIQUID CRYSTALS

3

use D(Ω) to denote C0∞ (Ω), and D (Ω) to denote the sense of distributions. The integration Ω · dx is written simply as Ω ·. Moreover, let us introduce the Bogovskii operator B (see [1])

  p B = (B1 , B2 , B3 ) : w ∈ L (Ω) w = 0 → W01,p (Ω) Ω

such that v = B[w] solves the problem divv = w

a.e. in Ω,

v = 0 on ∂Ω,

which is a bounded linear operator satisfying B[w] W 1,p ≤ C(p) w Lp 0

for any 1 < p < ∞.

(1.3)

Now let us define the renormalized bounded energy weak solution to the problem (1.1)–(1.2) in the following sense: • ρ ≥ 0, ρ ∈ Lγ (Ω), u ∈ H01 (Ω), d ∈ H 2 (Ω), • Equations (1.1) hold in the sense of

D (Ω);



Ωρ

= M > 0;

• Equation (1.1)1 is satisfied in the sense of renormalized solutions, it means that

  div b(ρ)u + b (ρ)ρ − b(ρ) divu = 0 holds in the sense of D (Ω) for any b ∈ C 1 (R+ ) such that b (z) = 0 for z ∈ R+ large enough. • The energy inequality    2   1  2 2 2  μ|∇u| + μ ˜|divu| + Δd + 2 (1 − |d| )d ≤ ρf u.  Ω Ω Our existence result of renormalized bounded energy weak solutions to the Dirichlet problem (1.1)–(1.2) is the following. Theorem 1.1. Let Ω ⊂ R3 be a bounded domain with C 2 boundary, and the external force f ∈ L∞ (Ω) and g ∈ H 3 (Ω). Then for any γ > 1, the Dirichlet problem (1.1)–(1.2) has a renormalized bounded energy weak solution such that   1   u H 1 + Δd + 2 (1 − |d|2 )d 2 + ρ Lγq + ρ|u|2 Ls ≤ C  L

(1.4)

for some q > 1, s > 1, and the constant C depending only on μ, μ ˜, γ, f, M, Ω. To prove Theorem 1.1, we first consider the three level approximation of the problem (1.1), i.e., artificial pressure, relaxed continuity equation, relaxed continuity equation with dissipation. In the frame of Lions [16] and Novotn´ y et al. [28] for steady Navier-Stokes equations, we obtain the weak solution of the problem (1.1) with artificial pressure. Secondly, based on weighted estimates for the energy density and the pressure, we shall derive the bounded estimates independent of artificial pressure which ensure the vanishing of artificial pressure and the strong compactness of the density. Then we finally establish existence of weak solution by vanishing artificial pressure and adapting the weak convergence method.

4

ZHONG TAN AND QIUJU XU

The rest of the paper is arranged as follows. We are going to establish weak solution of the problem (1.1) with artificial pressure in Section 2. In Section 3, we will do some uniform estimates for the problem (1.1) with artificial pressure. In Section 4 and 5, we shall derive the uniform estimates independent of artificial pressure. The proof of Theorem 1.1 is given in the last section. 2. The three level approximate problem The task of this section is to construct the approximate solutions to the problem (1.1)–(1.2). First, we introduce an artificial pressure term pδ (ρ) := ργ + δρ4 . Here we choose ρ4 just for technical reason, and in fact we can take ρc for any c ≥ max{γ, 3} instead of ρ4 . In what follows, we work with the three level approximate problem with positive parameters α, ε, δ: ⎧ ⎪ ⎪ ⎪α(ρ − h) + div(ρu) − εΔρ = 0, ⎪ ⎪ ⎪ 1 1 ⎪ ⎨α(ρ + h)u − μΔu − μ ˜∇divu + div(ρu ⊗ u) + ρu · ∇u + ∇pδ 2  2    1 1 ⎪ 2 2 2 ⎪ = −div ∇d  ∇d − + (1 − |d| ) |∇d| I3 + ρf, ⎪ ⎪ 2 42 ⎪ ⎪ ⎪ ⎩u · ∇d = Δd + 1 (1 − |d|2 )d, 2

(2.1)

subject to the boundary conditions ∂n ρ = 0, where we take h =

M |Ω|

u = 0,

d=g

on ∂Ω,

(2.2)

for simplicity, n denotes the outward normal to ∂Ω and g is the same as

Theorem 1.1. For the orientation field equation in the approximate problem (2.1), we have the following regularity results on the orientation field d. Lemma 2.1. Assume d satisfies the orientation field equation with the boundary condition ⎧ ⎨ u · ∇d = Δd + 1 (1 − |d|2 )d in Ω, 2 (2.3) ⎩ d=g on ∂Ω. If g ∈ H 3 (Ω), u ∈ L∞ (Ω), then d ∈ L∞ (Ω). Further, if Δd + d ∈ H 2 (Ω) and ∇d ∈ L4 (Ω).

1 (1 2

− |d|2 )d ∈ L2 (Ω), then

Proof. It is obvious to see that d ∈ L∞ (Ω) when |d| < 1, then we only deal with the case |d| ≥ 1. Taking the scalar product of (2.3)1 with d and utilizing the fact that Δ|d|2 = 2|∇d|2 + 2Δd · d, we have −Δ|d|2 + 2u · ∇|d|2 = −2|∇d|2 +

2 (1 − |d|2 )d2 ≤ 0. 2

WEAK SOLUTION TO THE STEADY COMPRESSIBLE FLOW OF NEMATIC LIQUID CRYSTALS

5

Due to the maximum principle for elliptic system, one obtains d ∈ L∞ (Ω) and d L∞ ≤ g L∞ . Moreover, by elliptic estimates, we get d ∈ H 2 (Ω). Notice that the Galiardo-Nireberg inequality, for some constant C > 0, 1

1

∇d L4 ≤ C Δd L2 2 d L2 ∞ + C d L∞ , which implies ∇d ∈ L4 (Ω).  Next, we employ the Leray-Schauder fixed point theorem (see Section 1.4.11.8 in [28]) to obtain a weak solution to (2.1)–(2.2). More precisely, we have Lemma 2.2. For ε > 0, there exists a weak solution (ρ, u, d) to the problem (2.1)–(2.2) satisfying the following properties: (i) ρ ∈ W 2,2 (Ω),

u ∈ H01 (Ω),

(ii)

d ∈ H 2 (Ω),



Ωρ

= M > 0,

ρ ≥ 0;

u H 1 ≤ C(δ, Ω, f, g, h),

(2.4)

ρ L8 ≤ C(δ, Ω, f, g, h),

(2.5)

ε ∇ρ 2L2 ≤ C(δ, Ω, f, g, h),   1   Δd + 2 (1 − |d|2 )d 2 ≤ C(δ, Ω, f, g, h).  L Here C is independent of ε and α particularly.

(2.6) (2.7)

Proof. We will following the standard process of steady compressible Navier-Stokes equations with some modifications. Here we only give the outline of the proof. See [16] or [28] for the detail. Let 1 < p < ∞, v ∈ W01,∞ (Ω) such that v W 1,∞ ≤ K for some K > 0. For t ∈ [0, 1], we denote by ξ → ρt , Tt :      1,p 1,p W (Ω) ∩ ξ = M → W (Ω) ∩ ρt = M Ω

the solution of the problem 

Ω

−εΔρ = −tdiv(ξv) + αt(h − ξ) in Ω, ∂n ρ = 0

(2.8)

on ∂Ω.

We use (2.8)1 with ξ = ρ subsequently test it by ρ and apply conveniently a bootstrapping argument to conclude that ρ W 1,p ≤ C(α, ε, p, Ω)(1 + K) h Lp ≤ C(α, ε, p, Ω, K, h), where C is a positive constant independent of t. By the theory of Neumann problem for the elliptic equations and the compact imbedding W 2,p (Ω) →→ W 1,p (Ω), the operator Tt is a

6

ZHONG TAN AND QIUJU XU

compact mapping for every t ∈ [0, 1]. Thus by the Leray-Schauder fixed point theorem, there exists at least one ρt satisfy Tt (ρt ) = ρt , t ∈ [0, 1]. Putting ρ = ρ1 = T1 (ρ), we can construct the operator T T : W

1,∞

v → ρ = T1 (ρ), (Ω) → W

1,p

(Ω) ∩



 ρ=M Ω

such that ρ = T (v) solves (2.1)1 . This proves the existence of solution to the mass equation (2.1)1 . Next, we turn to prove the existence of solutions to the momentum equation (2.1)2 and the direction field equation (2.1)3 . For this purpose, we consider the following problem ⎧ ⎪ ˜∇divu = Ft (ρ, v, m) in Ω, ⎪ ⎨ −μΔu − μ −Δd = Gt (v, m) ⎪ ⎪ ⎩ (u, d) = (0, g)

where

in Ω,

(2.9)

on ∂Ω,

 1 1 Ft = − t α(h + ρ)v + div(ρv ⊗ v) + ρv · ∇v + ∇(ργ + δρ4 ) 2 2     1 1 2 |∇b| + 2 (1 − |m|2 )2 I3 − ρf , + div ∇m  ∇m − 2 4   1 Gt = − t v · ∇m − 2 (1 − |m|2 )m . 

Taking v = u, m = d, then testing (2.9)1 by u and (2.9)2 by Δd and 12 (1 − |d|2 )d, one obtains    γ 2 2 αt h|u| + αt ρ|u| + α t (ρ − h)(ργ−1 − hγ−1 ) γ−1 Ω Ω Ω     4 3 3 2 2 2 ˜ |divu|2 + αδt (ρ − h)(ρ − h )dx + 4δεt ρ |∇ρ| + μ |∇u| dx + μ 3 Ω Ω Ω Ω    t 2 + t Δd∇du + 2 (1 − |d| )d∇du ≤ t ρf u,  Ω Ω Ω    t − |Δd|2 = −t u∇dΔd + 2 (1 − |d|2 )dΔd,  Ω Ω Ω     2 t 1 1  2 2 Δd(1 − |d| )d = − 2 u∇d(1 − |d| )d + t  2 (1 − |d|2 )d . − 2  Ω  Ω Ω  Adding up the above equalities implies that √      2 t   2 2 2 2 |∇(ρ )| + μ |∇u| + μ ˜ |divu| + εδ Δd + 2 (1 − |d|2 )d ≤ C(α, Ω, f, h). (2.10)  Ω Ω Ω Ω It is obviously observed that d = 0 on the boundary ∂Ω, so we set ω = d − g such that ω = 0 on the boundary ∂Ω. Then (2.9)2 takes the form ˜ t (v, m) ˜ −Δω = G Here

in Ω.

  1 1 ˜ t = −t v · ∇m ˜ + v · ∇g − 2 (1 − |m|2 )m ˜ − 2 (1 − |m|2 )g − Δg . G  

(2.11)

WEAK SOLUTION TO THE STEADY COMPRESSIBLE FLOW OF NEMATIC LIQUID CRYSTALS

7

Taking m ˜ = ω, m = d, v = u, and testing (2.11) by ω, we have from Lemma 2.1 that  |∇ω|2 ≤ C(Ω, g)( u 2H 1 + 1). Ω

This further yields ∇d 2L2 ≤ C(Ω, g)( u 2H 1 + 1). On the other hand, we apply the regularity to the elliptic system −εΔρ = div(−ρu + αB(h − ρ)) to get ρ W 2,2 ≤ C(α, δ, ε, Ω, f, h). By use of Sobolev imbedding inequalities, one obtains Ft (ρ, u) L4 ≤ C(α, δ, ε, Ω, f, g, h),

˜ t (u, ω) L4 ≤ C(α, δ, ε, Ω, f, g, h). G

This in turn gives u W 2,4 + ω W 2,4 ≤ C(α, δ, ε, Ω, f, g, h). Further, it holds that u W 1,∞ + ω W 1,∞ ≤ C(α, δ, ε, Ω, f, g, h). Besides, by the classical regularity theory of the elliptic system and the compact imbedding W 2,p (Ω)

→→ W 1,∞ (Ω) for p > 3, the operator St St :

(v, m) ˜ → (ut , ωt ),

W01,∞ (Ω) × W01,∞ (Ω) → W01,∞ (Ω) × W01,∞ (Ω) as the solution to the problem (2.9)1 and (2.11) is compact for every t ∈ [0, 1]. Therefore there exists a fixed point (u, ω) = S1 (u, ω) and complete it with ρ = T (u). Finally, we use the Bogovskii operator B(ρ4 −

1 4 |Ω| ρ )

as the test function to test (2.1)2 , we

obtain ρ L8 ≤ C(δ, Ω, f, g, h). This in turn implies the bounds (2.4) and (2.6)–(2.7) from the estimate (2.10).  Now we employ the estimates obtained in Lemma 2.2 to pass to the limit for α → 0, ε → 0 in (2.1). By virtue of the method of weak convergence, we get the existence of weak solutions to the following system: ⎧ ⎪ ⎪ div(ρu) = 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎨−μΔu − μ ˜∇divu + div(ρu ⊗ u) + ∇pδ (ρ) − μΔu − μ ˜∇divu    1 1 2 2 2 ⎪ ⎪ = −div ∇d  ∇d − + (1 − |d| ) |∇d| I3 + ρf, ⎪ ⎪ 2 42 ⎪ ⎪ ⎪ 1 ⎩u · ∇d = Δd + (1 − |d|2 )d, 2 complemented by the boundary condition (1.2).

(2.12)

8

ZHONG TAN AND QIUJU XU

As a consequence, we have Proposition 2.1. Let f, g be assumed as in Theorem 1.1, there exists a renormalized weak solution (ρ, u, d) with the following properties:  8 1 2 (i) ρ ∈ L (Ω), u ∈ H0 (Ω), d ∈ H (Ω), ρ = M > 0, Ω D (Ω),

ρ ≥ 0;

i.e., for ϕ ∈ D(Ω), (ii) The system (2.12) holds in the sense of  ρu∇ϕ = 0,   Ω

(2.13)

Ω

ρu ⊗ u : ∇ϕ + pδ (ρ)divϕ − μ∇u : ∇ϕ − μ ˜divudivϕ

(2.14)    1 2 2 − div ∇d  ∇d − |∇d| + 2 (1 − |d| ) I3 ϕ + ρf ϕ = 0, 2 4    1 (2.15) u∇d · ϕ − Δd · ϕ − 2 (1 − |d|2 )d · ϕ = 0.  Ω In particular, one easily verifies by a density argument and (i) that (2.13) and (2.15) hold with 

1

2

any test function ϕ ∈ H 1 (Ω), (2.14) holds with any test function ϕ ∈ H01 (Ω); (iii) The mass equation (2.12)1 holds in the sense of renormalized; (iv) Moreover, it holds that    2   1  2 2 2  μ|∇u| + μ ˜|divu| + Δd + 2 (1 − |d| )d = ρf u.  Ω Ω

(2.16)

Proof. The properties (i), (ii), (iii) can be proved by the similar argument in [28]. Then it remains to prove (iv). From the properties (iii) with ψ(ρ) =  pδ (ρ)divu = 0.

1 γ γ−1 ρ

+ 3δ ρ4 , we find

Ω

This in turn yields, from (2.14) with ϕ = u,    1 ρu ⊗ u : ∇u − μ|∇u|2 − μ ˜|divu|2 − Δd∇d · u − 2 (1 − |d|2 )d∇d · u + ρf u = 0.  Ω 1 (1 2

− |d|2 )d in (2.15) respectively leads to    1 u∇dΔd − |Δd|2 − 2 (1 − |d|2 )dΔd = 0,  Ω   1 2  1 1  2 2 2  u∇d · (1 − |d| )d − Δd · (1 − |d| )d − (1 − |d| )d  2  = 0. 2 2  Ω  Summing up (2.17)–(2.19) implies    2  1   ˜|divu|2 − Δd + 2 (1 − |d|2 )d + ρf u = 0. ρu ⊗ u : ∇u − μ|∇u|2 − μ  Ω

Taking ϕ = Δd and ϕ =

(2.17)

(2.18) (2.19)

(2.20)

3

Since |u|2 ∈ W 1, 2 (Ω), there exists a sequence ξn ∈ C 1 (Ω) such that ∇ξn → ∇(|u|2 )

3

in L 2 (Ω).

It is observed that ρu ∈ L3 (Ω), which, together with (2.13) and (2.21), implies that   ρu∇ξn = 0 → ρu∇(|u|2 ). Ω

Ω

(2.21)

WEAK SOLUTION TO THE STEADY COMPRESSIBLE FLOW OF NEMATIC LIQUID CRYSTALS

Then one has



1 ρu ⊗ u : ∇u = 2 Ω



ρu∇(|u|2 ) = 0.

9

(2.22)

Ω



Therefore, (2.16) follows from (2.20) and (2.22) immediately. 3. Uniform estimates for the approximate problem (2.12)

The section is devoted to formal derivation of the uniform estimates for the approximate problem (2.12). The estimates will play a crucial role. First of all, we introduce the four quantities, for every γ > 1,   1 1 3(1 − 8r2 ) β(s − 1) r= 1− , β= , s = 1 + 2r2 , q = 1 + . 8 γ 2(3 − 8r2 ) β + (1 − β)s

(3.1)

It follows from a simple calculation that 1 0<β< , 2

1 0
1
33 . 32

(3.2)

Moreover, for every c > 0, we denote by Ω2c and Ωc the annuluses Ω2c = {x ∈ R3 : dist(x, ∂Ω) < c},

Ωc = Ω2c ∩ Ω.

And the signed distance function dx (x) is defined by  ¯ dist(x, ∂Ω), for x ∈ Ω, dx (x) = −dist(x, ∂Ω), for x ∈ R3 \Ω. According to the Chap. 14.6 in [6], since ∂Ω ∈ C 2 (Ω), there exists a > 0, depending on Ω, such that ¯ 2a ), dx (x) ∈ C 2 (Ω

¯ 2a . |∇dx (x)| = 1 in Ω

Thus we could give the following definition. Definition 3.1. We define a function φ : Ω2a ∪ Ω → R satisfying the two properties: 1. φ ∈ C 2 (Ω2a ∪ Ω), φ(x) = dx (x) in Ω2a ; 2. There is t > 0 such that φ > t in Ω\Ω2a . Remark 3.1. Obviously, φ is positive in Ω. Due to the Whitney extension theorem, there exists such a function φ. 3.1. Pressure estimates. In this subsection, we are in a position to derive the uniform estimates for the pressure function. Lemma 3.1. Let (ρ, u, d) be the solution of the approximate problem (2.12) obtained by Proposition 2.1, then it holds that      −β 2 −β pδ (ρ)φ + ρ|u · ∇φ| φ ≤C pδ (ρ)φ1−β + ρ|u|2 φ1−β Ω

Ω

Ω



+ C 1 + u H 1

   (3.3) 1  2  + Δd + 2 (1 − |d| )d 2 ∇d L2 ,  L

10

ZHONG TAN AND QIUJU XU

   1   pδ (ρ)φ1−β Ls ≤C 1 + u H 1 + Δd + 2 (1 − |d|2 )d 2 ∇d L2  L    2 1−β −β  s +C pδ (ρ)φ . + C ρ|u| φ L 

(3.4)

Ω

Here the constant C depends only on μ, μ ˜, γ, f, M, Ω, but not on δ and , and β is defined by (3.1). Proof. For 0 < β < 12 , setting the vector field ϕ(x) = φ1−β (x)∇φ(x),

x ∈ Ω,

we have ∇ϕ = (1 − β)φ−β ∇φ ⊗ ∇φ + φ1−β ∇2 φ, divϕ = (1 − β)φ−β |∇φ|2 + φ1−β Δφ. With the help of the properties of φ in Definition 3.1, we arrive at |φ−β ∇φ ⊗ ∇φ| ≤ Cφ−β ,

|φ1−β ∇2 φ| ≤ C.

Hence it is seen that ϕ ∈ L∞ (Ω) ∩ W01,p (Ω) for p ∈ [1, β1 ). Then substituting ϕ into (2.14) yields that  (ρu ⊗ u : ∇ϕ + pδ divϕ) Ω    1    1 |∇d|2 + 2 (1 − |d|2 )2 I3 ϕ − ρf ϕ = μ∇u : ∇ϕ + μ ˜divudivϕ + div ∇d  ∇d − 2 4 Ω (3.5)     1  1 2 2 2 2 Δd + 2 (1 − |d| )d ∇d + C ≤C (|∇u| + |divu| ) + C ρ  Ω Ω Ω     1   ≤C 1 + u H 1 + Δd + 2 (1 − |d|2 )d 2 ∇d L2 .  L On the other hand, we have 1 ρu ⊗ u : ∇ϕ = (1 − β)φ−β ρ|u · ∇φ|2 + φ1−β ρu ⊗ u : ∇2 φ ≥ φ−β ρ|u · ∇φ|2 − Cφ1−β ρ|u|2 (3.6) 2 and 1 pδ divϕ = (1 − β)pδ φ−β |∇φ|2 + pδ φ1−β Δφ ≥ pδ φ−β − Cpδ φ1−β . 2 Hence combining (3.5)–(3.7) gives (3.3).

(3.7)

Next, we turn to prove (3.4). For any function 

w(x) ∈ w ∈ L

s s−1

 (Ω) |

 w=0 , Ω

it follows from (1.3) that B[w]

s 1, s−1

W0

s 1, s−1

Notice that the embedding W0

≤ C w

¯ since (Ω) →→ C(Ω)

B[w] L∞ + φ−1 B[w]

s

L s−1

s

L s−1 s s−1

.

(3.8)

> 3, we deduce that

≤ w

s

L s−1

.

(3.9)

WEAK SOLUTION TO THE STEADY COMPRESSIBLE FLOW OF NEMATIC LIQUID CRYSTALS

11

Here we have used the simple fact φ−1

s

L s−1

≤ φ−1 Lp ≤ C,

1 ≤ p < 3.

Define the vector field ϕ(x) = φ1−β (x)B[w](x),

x ∈ Ω,

then   ∇ϕ = (1 − β)φ−β ∇φ ⊗ B[w] + φ1−β ∇B[w] = φ1−β (1 − β)φ−1 ∇φ ⊗ B[w] + ∇B[w] . It follows from (3.8)–(3.9) that   (1 − β)φ−1 ∇φ ⊗ B[w] + ∇B[w] s L s−1   −1 s ∇φ L∞ + ∇B[w] ≤ (1 − β)φ B[w] s−1 L

s L s−1

≤ C w

(3.10) s L s−1

.

Further, we have ϕ L∞ + ϕ

s

L s−1

≤ C w

s

L s−1

.

(3.11)

Substitute ϕ into (2.14) to find that   pδ (ρ)divϕ = (μ∇u∇ϕ + μ ˜divudivϕ − ρu ⊗ u : ∇ϕ) Ω Ω     1 Δd∇d + 2 (1 − |d|2 )d∇d ϕ − ρf ϕ +  Ω Ω   s ≤ C u H 1 ∇ϕ L2 + ρ|u|2 φ1−β Ls (1 − β)φ−1 ∇φ ⊗ B[w] + ∇B[w] s−1 L    1   + C Δd + 2 (1 − |d|2 )d 2 ∇d L2 ϕ L∞ + C f L∞ ϕ L∞ ρ  L Ω     1   s , ≤ C 1 + u H 1 + ρ|u|2 φ1−β Ls + Δd + 2 (1 − |d|2 )d 2 ∇d L2 w s−1 L  L (3.12) where we have used (3.10) and (3.11) in the last inequality of (3.12). In addition, one has  Ω

 pδ divϕ =

Ω

pδ φ1−β w + (1 − β)

 Ω

pδ φ−β ∇φ · B[w].

(3.13)

It follows from (3.9) that  Ω

pδ φ−β ∇φ · B[w] ≤ C w

 s L s−1

Ω

pδ φ−β .

(3.14)

Combining (3.12)–(3.14) implies        1   s . pδ φ1−β w ≤ C 1 + u H 1 + Δd + 2 (1 − |d|2 )d 2 ∇d L2 + C ρ|u|2 φ1−β Ls w s−1 L  L Ω This in turn gives (3.4).



12

ZHONG TAN AND QIUJU XU

3.2. Quantities A and B. Define  A= ρ|u|2(2−r) φ2β , Ω



ργ φ−β ,

B=

(3.15)

Ω

we shall derive a series of estimates which can controlled by the quantities A and B with some exponents. Lemma 3.2. Let (ρ, u, d) be the solution of the approximate problem (2.12) obtained by Proposition 2.1, then it holds that   1 1 1   u H 1 + Δd + 2 (1 − |d|2 )d 2 ≤ CA 4(2−r) B 2(2γ−1)(2−r) ,  L   r 1 ρ|u|2 φ1−β  s ≤ CA 2−r B 2(2γ−1)(2−r) . L

(3.16) (3.17)

Here the constant C depends only on μ, μ ˜, γ, f, M, Ω, but not on δ and , and β, r are defined by (3.1), A, B are given by (3.15). Proof. It follows from (2.16) that u 2H 1

  2 1  2  + Δd + 2 (1 − |d| )d 2 ≤ C |ρu|.  L Ω

(3.18)

Notice that  θ 1  θ 2  θ1 − θ2 2 ρ1−θ1 −γθ2 , |ρu| = ρ|u|2(2−r) φ2β ργ φ−β φ−2β where θ1 =

1 2(2−r) ,

0 < θ2 < 1. It is assumed that θ1 + θ2 + θ1 −

θ2 2

(3.19)

+ 1 − θ1 − γθ2 = 1, we can

compute from (3.1) that θ1 1 = . γ − 1/2 (2γ − 1)(2 − r) We apply H¨older inequality to (3.19) to infer that θ1   θ 2  θ1 − θ  1−θ1 −γθ2   2 |ρu| ≤ ρ|u|2(2−r) φ2β ργ φ−β φ−2β ρ θ2 =

Ω

Ω

≤ CA

1 2(2−r)

Ω

B

1 (2γ−1)(2−r)

Ω

Ω

.

Using this inequality together with (3.18), we get (3.16) immediately. Now we prove (3.17) by applying the same argument. Similar to (3.19), we have θ 3  θ4  θ 5  (ρ|u|2 φ1−β )s = ρ|u|2(2−r) φ2β ργ φ−β φ−2β ρs−θ3 −γθ4 φs(1−β)−2βθ3 +βθ4 +2βθ5 , (3.20) where s sr , 0 < θ4 < 1, 0 < θ5 < 1, θ4 = . 2−r (2γ − 1)(2 − r) We can compute that, if θ3 + θ4 + θ5 + s − θ3 − γθ4 = 1, θ3 =

θ5 = 1 − s +

sr(γ − 1) . (2γ − 1)(2 − r)

Thus from (3.1), we find that s(1 − β) − 2βθ3 + βθ4 + 2βθ5 = s(1 − β) + β(2 − 3s) =

10r2 (1 + 8r2 ) > 0. 3 − 8r2

WEAK SOLUTION TO THE STEADY COMPRESSIBLE FLOW OF NEMATIC LIQUID CRYSTALS

13

By virtue of (3.20), H¨older inequality leads to  θ 3   θ 4  θ5  s−θ3 −γθ4   2 1−β s 2(2−r) 2β γ −β −2β (ρ|u| φ ) ≤C ρ|u| φ ρ φ φ ρ Ω

Ω

≤ CA

Ω

s (2−r)

B

sr (2γ−1)(2−r)

Ω

Ω

, 

which implies (3.17).

Lemma 3.3. Let (ρ, u, d) be the solution of the approximate problem (2.12) obtained by Proposition 2.1, then it holds that        2 pδ φ−β + ρ|u · ∇φ|2 φ−β + pδ φ1−β Ls ≤ C 1 + A 3 , Ω

    2 1   u H 1 + Δd + 2 (1 − |d|2 )d 2 ≤ C 1 + A 45 (7−8r) ,  L     1+4r 2 ρ|u|2 φ1−β  s ≤ C 1 + A 3 − 3(1+8r)(2−r) . L

(3.21) (3.22) (3.23)

Here the constant C depends only on μ, μ ˜, γ, f, M, Ω, but not on δ and , and β, r are defined by (3.1), A is given by (3.15). Proof. By virtue of (3.2)–(3.3), one obtains      pδ φ−β + ρ|u · ∇φ|2 φ−β + pδ φ1−β Ls Ω        1   ≤C 1 + u H 1 + Δd + 2 (1 − |d|2 )d 2 ∇d L2 + ρ|u|2 φ1−β Ls + pδ φ−β .  L Ω

(3.24)

By the interpolation, we have  1 1−β  p 4 φ 4  δ

L4

 1 1−β θ  1 1−β 1−θ ≤ pδ4 φ 4 L1 pδ4 φ 4 L4s

for a certain 0 < θ < 1 satisfying 14 = θ + 1−θ 4s . Due to the following inequality  1 1−β θ  1 p 4 φ 4  1 ≤ C p 4 θ 1 ≤ C ρ θ 1 L δ δ L L and H¨ older inequality, we arrive at 1−θ 1−θ  1 pδ φ1−β  4 1 ≤ C ρ θ 1 pδ φ1−β s4 ≤ C pδ φ1−β s4 . L L L L

Inserting the above inequality into (3.24) and using Young inequality, we deduce from (3.16)– (3.17) that



 Ω

   pδ φ−β + ρ|u · ∇φ|2 φ−β + pδ φ1−β Ls

      1   ≤C 1 + u H 1 + Δd + 2 (1 − |d|2 )d 2 ∇d L2 + ρ|u|2 φ1−β Ls  L   1 1 3 3 r 1 ≤C 1 + A 4(2−r) B 2(2γ−1)(2−r) + A 8(2−r) B 4(2γ−1)(2−r) + A 2−r B (2γ−1)(2−r)   3 3 r 1 ≤C 1 + A 8(2−r) B 4(2γ−1)(2−r) + A 2−r B (2γ−1)(2−r) . Here we have used the fact ∇d 2L2 ≤ C(1 + u H 1 ).

(3.25)

14

ZHONG TAN AND QIUJU XU

Next, we employ Young inequality to bound the last two terms in (3.25). For every  > 0, 3

3

A 8(2−r) B 4(2γ−1)(2−r) ≤ C()Aκ1 + B, where 3 4(2γ − 1)(2 − r)/3 3 · = 8(2 − r) 4(2γ − 1)(2 − r)/3 − 1 2 3 1 + 8r 3 = · < · 2 4(1 + 8r)(2 − r) − 3(1 − 8r) 2

κ1 =

Here we have used the relation γ =

1 1−8r

2γ − 1 4(2γ − 1)(2 − r) − 3 4 2 = . 9 3

·

with 0 < r < 18 .

Similarly, we have r

1

A 2−r B (2γ−1)(2−r) ≤ C()Aκ2 + B, where 1 (2γ − 1)(2 − r)/r 2γ − 1 · = 2 − r (2γ − 1)(2 − r)/r − 1 (2γ − 1)(2 − r) − r 1 r 1 1 r = + · < + . 2 2 1 + 7r 2 2

κ2 =

Thus we get 3

3

2

A 8(2−r) B 4(2γ−1)(2−r) ≤ C()A 3 + B, r

1

1

r

A 2−r B (2γ−1)(2−r) ≤ C()A 2 + 2 + B. These, together with (3.25), imply        2 pδ φ−β + ρ|u · ∇φ|2 φ−β + pδ φ1−β Ls ≤ CB + C() 1 + A 3 . Ω

Since observe that

 B=

γ −β

ρ φ Ω

 ≤C

Ω

pδ φ−β ,

(3.26)

(3.27)

from (3.26), there holds       2 −β ρ|u · ∇φ|2 φ−β + pδ φ1−β Ls ≤ C() 1 + A 3 . (1 − C) pδ φ + Ω

Ω

It is clear that we can choose sufficiently small  > 0 such that (3.21) holds. Next, to prove (3.22), it is easy to see from (3.27) that   2 B ≤ C 1 + A3 . Then (3.16) yields 1     1 2 1 2(2γ−1)(2−r)   u H 1 + Δd + 2 (1 − |d|2 )d 2 ≤ CA 4(2−r) 1 + A 3  L

≤ C(1 + Aκ3 ), where κ3 =

1 1 1 6γ + 1 7 − 8r 2 + = = < (7 − 8r). 4(2 − r) 3 (2γ − 1)(2 − r) 12(2γ − 1)(2 − r) 12(1 + 8r)(2 − r) 45

Here we estimate κ3 by the same way as the estimate of κ1 . Hence (3.22) was proved.

WEAK SOLUTION TO THE STEADY COMPRESSIBLE FLOW OF NEMATIC LIQUID CRYSTALS

15

Finally, it remains to estimate (3.23). By virtue of (3.17) and (3.27), one obtains r     1 ρ|u|2 φ1−β  s ≤ CA 2−r 1 + A 23 (2γ−1)(2−r) ≤ C (1 + Aκ4 ) . L Here κ4 =

1 2 r 2 1 + 4r + = − . 2 − r 3 (2γ − 1)(2 − r) 3 3(1 + 8r)(2 − r) 

Hence we deduce (3.23). 4. uniform boundness for u, d and pδ

At this stage, we are ready to establish the uniform boundness for u, d and pδ . More precisely, we have Proposition 4.1. Let (ρ, u, d) be the solution of the approximate problem (2.12) obtained by Proposition 2.1, then we have   1   u H 1 + Δd + 2 (1 − |d|2 )d 2 + pδ Lq ≤ C.  L

(4.1)

Here the constant C depends only on μ, μ ˜, γ, f, M, Ω, but not on δ and , and q is defined by (3.1). 4.1. Weighted estimates for the energy density and the pressure. In this subsection, we will estimate two weighted estimates for the energy density and the pressure in terms of A which will be used in the proof of the uniform bounded result. Lemma 4.1. Let (ρ, u, d) be the solution of the approximate problem (2.12) obtained by Proposition 2.1, then for every τ ∈ (0, 1) and x0 ∈ Ω, it holds that 3      φ 2 −β 2 2 3 ρ|u| + pδ ≤ C 1 + A . |x − x0 |τ Ω

(4.2)

Here the constant C depends only on μ, μ ˜, γ, f, M, Ω and τ , but not on δ and , and β is defined by (3.1), A is given by (3.15). Proof. For any τ ∈ (0, 1) and x0 ∈ Ω, let the vector field ϕ(x) be defined by 3

φ 2 −β ϕ(x) = (x − x0 ). |x − x0 |τ It is easily seen that  τ 3 − 2β φ 2 −β  I − (x − x ) ⊗ (x − x ) + ) ⊗ ∇φ . (x − x ∇ϕ = 3 0 0 0 |x − x0 |τ |x − x0 |2 2φ 3

Since 0 < β < 12 , we find that |∇ϕ| ≤

C . |x − x0 |τ

Further, we have ϕ ∈ W 1,p for all p ∈ [1, τ3 ). In particular, ϕ H 1 ≤ C,

ϕ L∞ (Ω) ≤ C.

16

ZHONG TAN AND QIUJU XU

Then we substitute ϕ into the integral identity (2.14) to infer that  (ρu ⊗ u : ∇ϕ + pδ divϕ) Ω    1    1 μ∇u : ∇ϕ + μ ˜divudivϕ + div ∇d  ∇d − = |∇d|2 + 2 (1 − |d|2 )2 I3 ϕ − ρf ϕ 2 4 Ω (4.3)      1  2  ≤C u H 1 ϕ H 1 + Δd + 2 (1 − |d| )d 2 ∇d L2 ϕ L∞ + f L∞ ϕ L∞ ρ  L Ω     1   ≤C 1 + u H 1 + Δd + 2 (1 − |d|2 )d 2 ∇d L2 .  L Meanwhile, we have  2 3 − 2β  φ 2 −β  2 τ ρ |u| − ) + ) · u)(∇φ · u) u · (x − x ((x − x 0 0 |x − x0 |τ |x − x0 |τ 2φ 3

ρu ⊗ u : ∇ϕ =

3

≥ (1 − τ )

1 φ 2 −β ρ|u|2 − Cφ 2 −β ρ|u||∇φ · u| τ |x − x0 |

≥ (1 − τ )

φ 2 −β ρ|u|2 − Cφ1−β ρ|u|2 − Cφ−β ρ(∇φ · u)2 , |x − x0 |τ

3

(4.4) and 3

divϕ = (3 − τ )

1

3 φ 2 −β φ 2 −β + ( (x − x0 )∇φ − β) |x − x0 |τ 2 |x − x0 |τ

(4.5)

3

φ 2 −β ≥ (3 − τ ) − Cφ−β . |x − x0 |τ In view of (4.3)–(4.5), one arrives at 

 Ω

2

ρ|u| + pδ



3     φ 2 −β 1  2  ≤ C 1 + u (1 − |d| )d ∇d 1 + Δd + 2  H L |x − x0 |τ 2 L2     1−β 2 −β 2 φ ρ|u| + φ ρ(∇φ · u) + pδ φ−β . +C

Ω

Ω

Ω

Recalling (3.21)–(3.23) and Lemma 2.1, we conclude that 

 Ω

2

ρ|u| + pδ



  1+4r 2 2 3 φ 2 −β − 3(1+8r)(2−r) (7−8r) 3 3 + A 45 ≤ C 1 + A + A |x − x0 |τ   2 ≤ C 1 + A3 . 3

 Lemma 4.2. Let (ρ, u, d) be the solution of the approximate problem (2.12) obtained by Proposition 2.1, then for every x0 ∈ Ω, it holds that    2 φ2β ρ|u|2(1−r) ≤ C 1 + A3 . |x − x0 | Ω

(4.6)

Here the constant C depends only on μ, μ ˜, γ, f, M, Ω, but not on δ and , and β, r are defined by (3.1), A is given by (3.15).

WEAK SOLUTION TO THE STEADY COMPRESSIBLE FLOW OF NEMATIC LIQUID CRYSTALS

Proof. To begin, choose τ=

17

1 − 16r2 ∈ (0, 1). 1 − 8r2

We note that

    3 r 3 − β + (1 − r) −β γ 2 2 by the relations (3.1). It is observed that  r  1−r  r− r 3 3 γ −β −β 2β γ 2 2 1 2(1−r) φ γ φ 2 φ . = ρ ρ|u| ρ|u| τ τ 2 |x − x0 | |x − x0 | |x − x0 | |x − x0 | 2β =

Then applying Young inequality, we infer that 3

ρ|u|

2(1−r)

3

−β 2 φ2β φ 2 −β 1 2 φ + ρ|u| + , ≤ Cpδ τ τ |x − x0 | |x − x0 | |x − x0 | |x − x0 |2



which, together with (4.2), implies (4.6). 4.2. Proof of Proposition 4.1. We first give the following auxiliary lemma. Lemma 4.3. Let Ω be a bounded domain with C 2 boundary. If Ψ(x) ∈ L2 (Ω) satisfies  Ψ(x) Ψ(x) ≥ 0, ≤ E for all x0 ∈ Ω. Ω |x − x0 | Then there esxits C > 0, depending only on Ω, such that  |u|2 Ψ ≤ CE u 2H 1 for all u ∈ H01 (Ω). 0

Ω



Proof. This can be found in [29] (Lemma 4). It follows from (4.6), (3.22) and Lemma 4.3 that   φ2β  u 2H 1 A= ρ|u|2(2−r) φ2β ≤ C sup ρ|u|2(1−r) |x − x | 0 Ω Ω    2 2 (7−8r) 1 + A3 ≤ C 1 + A 45   44 16 ≤ C 1 + A 45 − 45 r . Since 0 < r < 18 , we have by Young inequality A ≤ C.

(4.7)

It is clear that the estimate (4.1) for u and d follows from (3.22) and (4.7). It remains to bound pδ . From the definition of q, observe that s· β   (1−β)s  β+(1−β)s q 1−β −β β+(1−β)s pδ φ . pδ = pδ φ Applying Young inequality to (4.8), we get from (3.21) and (4.7) that      s  2 s q 1−β pδ ≤ C +C pδ φ−β ≤ C 1 + A 3 ≤ C. pδ φ Ω

Ω

Ω

Thus this completes the proof of Proposition 4.1.

(4.8)

18

ZHONG TAN AND QIUJU XU

5. Uniform boundedness for the kinetic energy density In this section, we shall establish the uniform boundedness for the kinetic energy density with the help of weighted estimates for the energy density and the pressure. Thus we have Proposition 5.1. Let (ρ, u, d) be the solution of the approximate problem (2.12) obtained by Proposition 2.1, then we have   ρ|u|2 

Ls

≤ C.

(5.1)

Here the constant C depends only on μ, μ ˜, γ, f, M, Ω, but not on δ and , and s is defined by (3.1). 5.1. Weighted estimates for the energy density and the pressure. This subsection is devoted to deriving the weighted pressure estimate. Recalling the definition in the beginning of Section 3, we introduce the vector field Λ(x), for any τ ∈ (0, 1) and x0 ∈ Ω2a ,



 φ(x) − φ(x0 ) φ(x) + φ(x0 ) Λ(x) = ∇φ, + − (x, x0 )τ + (x, x0 )τ

where ± (x, x0 ) = |φ(x) ± φ(x0 )| + |x − x0 |2 . Now we shall state the following lemmas which constitute the basic properties of Λ(x). Lemma 5.1. For every x, x0 ∈ Ω2a , there is a constant C, depending only on τ and Ω, such that  1 1 + +1 , |Λ(x)| ≤ C, |∇Λ(x)| ≤ C − (x, x0 )τ + (x, x0 )τ   1 1−τ 1 ∂Λi (x) |u · ∇φ|2 − C|u|2 , ui uj ≥ + ∂xj 2 − (x, x0 )τ + (x, x0 )τ   1 1 1−τ − C. + divΛ(x) ≥ 2 − (x, x0 )τ + (x, x0 )τ 

(5.2)

(5.3) (5.4)

In particular, for every x0 ∈ Ωa , we have ∇Λ(x) L2 (Ωa ) ≤ C.

(5.5) 

Proof. The proof is in the Appendix of [29]. Next, we will derive the weighted pressure estimates in the following three lemmas.

Lemma 5.2. Let (ρ, u, d) be the solution of the approximate problem (2.12) obtained by Proposition 2.1 and Ω a2 = {x ∈ R3 : dist(x, ∂Ω) < a2 } ∩ Ω. Assume that η ∈ C ∞ (Ω) satisfies η ≥ 0 in Ω

and

η = 0 in Ω \ Ω a2 .

WEAK SOLUTION TO THE STEADY COMPRESSIBLE FLOW OF NEMATIC LIQUID CRYSTALS

19

Then for every τ ∈ (0, 1) and x0 ∈ Ω, it holds that      ηpδ (ρ)(x) 1  2  ≤C 1 + u + Δd + (1 − |d| )d 1   2 ∇d L2 + ρ|u|2 L1 + pδ L1 . (5.6) H τ 2  L Ω |x − x0 | Here the constant C depends only on μ, μ ˜, γ, f, M, Ω and η, τ , but not on δ and . Proof. We first deal with the case of x0 ∈ Ωa . By virtue of the estimate (5.2) and (5.5) in Lemma 5.1, one has that ηΛ ∈ H01 (Ω) and ηΛ H 1 ≤ C. Taking ϕ = ηΛ in (2.14) implies  (ηρu ⊗ u : ∇Λ + ηpδ divΛ) Ω    1   1 μ∇u : ∇(ηΛ) + μ ˜divudiv(ηΛ) + div ∇d  ∇d − |∇d|2 + 2 (1 − |d|2 )2 I3 ηΛ = 2 4 Ω (5.7)  − ρf ηΛ − ρ(u · ∇η)(u · Λ) − pδ ∇η · Λ     1   ≤C 1 + u H 1 + Δd + 2 (1 − |d|2 )d 2 ∇d L2 + ρ|u|2 L1 + pδ L1  L for all x0 ∈ Ωa . Moreover, (5.3) leads to   ∂Λi ηρu ⊗ u : ∇Λ = ηρ ui uj ∂xj Ω Ω     1 1 2 ρ|u · ∇φ| − C ≥C + ρ|u|2 . τ τ  (x, x )  (x, x ) − 0 + 0 Ω Ω It follows from (5.4) that    ηpδ divΛ ≥ C ηpδ Ω

Ω

1 1 + − (x, x0 )τ + (x, x0 )τ



(5.8)

 −C

Ω

pδ .

Then in view of (5.7)–(5.9), we arrive at    1 1 ηpδ + − (x, x0 )τ + (x, x0 )τ Ω     1   ≤C 1 + u H 1 + Δd + 2 (1 − |d|2 )d 2 ∇d L2 + ρ|u|2 L1 + pδ L1  L for all x0 ∈ Ωa . We may note that for x, x0 ∈ Ωa ,

(5.9)

(5.10)

|φ(x) − φ(x0 )| ≤ |x − x0 |. Then one has − (x, x0 ) ≤ C|x − x0 |, where C > 0 is dependent only on Ω. Thus we find from (5.10) that, for all x0 ∈ Ωa ,      ηpδ 1  2  2 ≤ C 1 + u (1 − |d| )d ∇d 1 + Δd + 2 + ρ|u| L1 + pδ L1 . (5.11)  H L τ 2 L2 Ω |x − x0 | Now let us consider the other case of x0 ∈ Ω\Ωa . Since 2|x − x0 | ≥ C for all x ∈ suppη and x0 ∈ Ω\Ωa , it holds that

 Ω

ηpδ ≤ C pδ L1 |x − x0 |τ

for all x0 ∈ Ω\Ωa .

Hence (5.6) follows from (5.11)–(5.12) immediately.

(5.12) 

20

ZHONG TAN AND QIUJU XU

Lemma 5.3. Let (ρ, u, d) be the solution of the approximate problem (2.12) obtained by Proposition 2.1, and the function ηˆ ∈ C0∞ (Ω), ηˆ ≥ 0 in Ω. Then for every x0 ∈ Ω, it holds that      ηˆpδ (ρ)(x) 1   ≤C 1 + u H 1 + Δd + 2 (1 − |d|2 )d 2 ∇d L2 + ρ|u|2 L1 + pδ L1 . (5.13)  L Ω |x − x0 | Here the constant C depends only on μ, μ ˜, γ, f, M, Ω and ηˆ, but not on δ and . Proof. Let us define the vector field ζ(x) by x − x0 |x − x0 |

ζ(x) = for fixed x0 ∈ Ω. It is easy to see that ∇ζ(x) =

x − x0 1 I3 − (x − x0 ) ⊗ |x − x0 | |x − x0 |3

implies ∇ζ(x) ≤

C . |x − x0 |

η ζ H 1 ≤ C. Then we have ηˆζ ∈ H01 (Ω) and ˆ Next, (2.14) with ϕ replaced by ηˆζ yields  (ˆ η ρu ⊗ u : ∇ζ + ηˆpδ divζ) Ω    1   1 μ∇u : ∇(ˆ η ζ) + μ ˜divudiv(ˆ η ζ) + div ∇d  ∇d − |∇d|2 + 2 (1 − |d|2 )2 I3 ηˆζ = 2 4 Ω (5.14)  − ρf ηˆζ − ρ(u · ∇ˆ η )(u · ζ) − pδ ∇ˆ η·ζ     1   ≤C 1 + u H 1 + Δd + 2 (1 − |d|2 )d 2 ∇d L2 + ρ|u|2 L1 + pδ L1 .  L Note that u ⊗ u : ∇ζ =

1 (u · u − (u · ζ)(u · ζ)) ≥ 0, |x − x0 |

and divζ =

2 . |x − x0 |

These together with (5.14) imply (5.13).



Lemma 5.4. Let (ρ, u, d) be the solution of the approximate problem (2.12) obtained by Proposition 2.1. Then for every τ ∈ (0, 1) and x0 ∈ Ω, it holds that  τ   pδ (ρ)(x) ≤C 1 + ρ|u|2 L1 . Ω |x − x0 | Here the constant C depends only on μ, μ ˜, γ, f, M, Ω and τ , but not on δ and . Proof. Choose a nonnegative function η ∈ C ∞ (Ω) such that  η = 1 in a neighborhood of ∂Ω, η = 0 in Ω\Ω a2 .

(5.15)

WEAK SOLUTION TO THE STEADY COMPRESSIBLE FLOW OF NEMATIC LIQUID CRYSTALS

21

Then we have 1 − η ∈ C0∞ (Ω). It follows from (5.6) and (5.13) that    pδ ηpδ (1 − η)pδ ≤ + C τ τ |x − x | |x − x | 0 0 Ω Ω Ω |x − x0 |    (5.16)  1  2  2 ≤C 1 + u H 1 + Δd + 2 (1 − |d| )d 2 ∇d L2 + ρ|u| L1 + pδ L1 .  L We apply Young inequality to obtain τ  1−τ  pτδ 1 Cpδ C pδ ≤ + . = |x − x0 | |x − x0 |τ |x − x0 |1+τ |x − x0 |τ |x − x0 |1+τ This in turn implies 

pτδ ≤C |x − x0 |

Ω

 Ω

pδ + C. |x − x0 |τ

(5.17)

Sum up (5.16)–(5.17) to give      pτδ 1   ≤C 1 + u H 1 + Δd + 2 (1 − |d|2 )d 2 ∇d L2 + ρ|u|2 L1 + pδ L1 .  L Ω |x − x0 |

(5.18) 

Hence we deduce (5.15) from (5.18) and (4.1). 5.2. Proof of Proposition 5.1. Since 0 < r < 18 , (3.1) gives 2s − (3 − s) = 2(1 − 8r)(1 + 2r2 ) − 2 + 2r2 = 2r(3r − 16r2 − 8) < 0. γ

2s ∈ (0, 1). It follows from Lemma 4.3, Lemma 5.4 and Proposition 4.1 that We can set τ = γ(3−s)       pτδ  pτδ |u|2 ≤ C sup u 2H 1 ≤ C 1 + ρ|u|2 L1 u 2H 1 ≤ C 1 + ρ|u|2 L1 . Ω Ω |x − x0 | (5.19)

Notice that 2s

ρ 3−s = ργτ ≤ Cpτδ , we find from (5.19) that  Ω

  2s ρ 3−s |u|2 ≤ C 1 + ρ|u|2 L1 .

(5.20)

Applying H¨ older inequality and embedding H 1 (Ω) → L6 (Ω), we infer from (4.1) and (5.20) that



s

Ω

ρ |u|

2s

≤

 ρ

2s 3−s

Ω

≤C

 ρ Ω

|u|

2s 3−s

2

 3−s  

|u|2

2

 3−s

Ω

|u|6

 s−1 2

2

(5.21)

  3−s 2 ≤ C 1 + ρ|u|2 L1 . We apply H¨older inequality again to get   1 1 s s ρ|u|2 L1 + 1 ≤ C ρs |u|2s + 1 ≤ C ρs |u|2s + 1 . Ω

Ω

(5.22)

22

ZHONG TAN AND QIUJU XU

The estimates (5.21) and (5.22) imply    3−s 2s ρs |u|2s ≤ C ρs |u|2s + 1 , Ω

Ω

which yields (5.1) by using the fact 0 <

3−s 2s

< 1.

6. Proof of Theorem 1.1 In this section, we will take the limit as δ → 0 for the approximate problem (2.12) to show the existence of weak solution to (1.1) for any γ > 1, which is based on the uniform bounded results Proposition 4.1–5.1 and the weak convergence argument. Proof of Theorem 1.1. Assume that (ρδ , uδ , dδ ) are weak solutions to the problem (2.12). By Proposition 4.1 and Proposition 5.1, we have the following limits: uδ  u

weakly in H01 (Ω),

strongly in Lp (Ω), 1 ≤ p < 6,

uδ → u

ρδ  ρ ργδ  ργ

weakly in Lγq (Ω),

(6.1)

weakly in Lq (Ω).

It follows from Lemma 2.1 that dδ  d

weakly in H 2 (Ω),

dδ → d

strongly in H 1 (Ω),

∇dδ  ∇d

weakly in L4 (Ω),

(6.2)

1 1 (1 − |dδ |2 )dδ  Δd + 2 (1 − |d|2 )d weakly in L2 (Ω). 2  √ √ In view of (4.1) and (5.1), ρδ is bounded in L2γq (Ω) and ρδ uδ is bounded in L2s (Ω). It is √ observed that q < s by the relations (3.1), then ρδ uδ is bounded in L2q (Ω). Thus ρδ uδ is Δdδ +

2γq

bounded in L 1+γ (Ω). Obviously we see that

2γq 1+γ

> 1. So it holds that 2γq

ρδ uδ  ρu weakly in L 1+γ (Ω), ρδ uδ ⊗ uδ  ρu ⊗ u weakly in Ls (Ω). By the interpolation, we arrive at   3q 4· 3q 4(1− 3q ) 1− 3q  1− 3q ρ4δ ≤ δ ρδ L4q4q−1 ρδ L1 4q−1 ≤ Cδ 4q−1 δ ρδ 4L4q 4q−1 ≤ Cδ 4q−1 → 0 δ

(6.3)

(6.4)

Ω

as δ → 0. Consequently, letting δ → 0 in (2.12) and making use of (6.1)–(6.4), the limit of (ρδ , uδ , dδ ) satisfies the following system: ⎧ ⎪ ⎪ div(ρu) = 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎨−μΔu − μ ˜∇divu + div(ρu ⊗ u) + ∇ργ    1 1 2 2 2 ⎪ ⎪ = −div ∇d  ∇d − + (1 − |d| ) |∇d| I3 + ρf, ⎪ 2 ⎪ 2 4 ⎪ ⎪ ⎪ ⎩u · ∇d = Δd + 1 (1 − |d|2 )d. 2

(6.5)

WEAK SOLUTION TO THE STEADY COMPRESSIBLE FLOW OF NEMATIC LIQUID CRYSTALS

23

Therefore, in order to show that (ρ, u, d) is a weak solution of (1.1), we only need to prove ργ = ρ γ

a.e. on Ω.

(6.6)

(6.6) can be fulfilled by following [14, 28]. The basic idea is to establish the strong convergence of the density ρδ to ρ in L1 (Ω) by exploiting smooth property of effective viscous flux and the oscillation of defect measures in the same manner as the arguments for the 3D steady Navier-Stokes equtions. We will not give the details here.

References [1] M.E. Bogovskii, Solution of the first boundary value problem for the equation of continuity of an incompressible medium, Soviet Math. Dokl., 20 (1979), 1094–1098. [2] L. Caffarelli, R. Kohn, L. Nirenberg, Partial regularity of suitable weak solutions of Navier-Stokes equations, Comm. Pure Appl. Math., 35 (1982), 771–831. [3] S.J. Ding, J.Y. Lin, C.Y. Wang, H.Y. Wen, Compressible hydrodynamic flow of liquid crystals in 1-D, Discrete Contin. Dynam. Systems, 32 (2012), 539–563. [4] Y. Du, K.Y. Wang, Space-time regularity of the Koch and Tataru solutions to the liquid crystal equations, SIAM J. Math. Anal., 45 (2013), 3838–3853. [5] Y. Du, K.Y. Wang, Regularity of the solutions to the liquid crystal equations with small rough data, J. Differential Equations, 256 (2014), 65–81. [6] D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Heidelberg, New-York, 1983. [7] M.C. Hong, Global existence of solutions of the simplified Ericksen-Leslie system in dimension two, Calc. Var. Partial Differential Equations, 40 (2011), 15–36. [8] Y.H. Hao, X.G. Liu, The existence and blow-up criterion of liquid crystals system in critical Besov space, Commun. Pure Appl. Anal., 13 (2014), 225–236. [9] X.P. Hu, D.H. Wang, Global solution to the three-dimensional incompressible flow of liquid crystals, Comm. Pure Appl. Math., 296 (2010), 861–880. [10] X.P. Hu, G.C. Wu, Global existence and optimal decay rates for three-dimensional compressible viscoelastic flows, SIAM J. Math. Anal., 45 (2013), 2815–2833. [11] T. Huang, C.Y. Wang, Blow up criterion for nematic liquid crystal flows, Comm. Partial Differential Equations, 37 (2012), 875–884. [12] F. Jiang, S. Jiang, D.H. Wang, On multi-dimensional compressible flows of nematic liquid crystals with large initial energy in a bounded domain, J. Funct. Anal., 265 (2013), 3369–3397. [13] F. Jiang, Z. Tan, Global weak solution to the flow of liquid crystals system, Math. Methods Appl. Sci., 32 (2009), 2243–2266. [14] S. Jiang, C. Zhou, Existence of weak solutions to the three-dimensional steady compressible Navier-Stokes equations, Ann. Inst. H. Poincar´e Anal. Non Lineaire, 28 (2011), 485–498. [15] J.K. Li, Global strong and weak solutions to nematic liquid crystal flow in two dimensions, Nonlinear Anal., 99 (2014), 80–94. [16] P.L. Lions, Mathematical Topics in Fluid Mechanics II, Compressible Models, Clarendon Press, Oxford, 1998. [17] F.H. Lin, Nonliner theory of defects in nematic liquid crystal: phase transition and flow phenomena, Comm. Pure Appl. Math., 42 (1989), 789–814.

24

ZHONG TAN AND QIUJU XU

[18] F.H. Lin, C. Liu, Nonparabolic dissipative systems modeling the flow of liquid crystals, Comm. Pure Appl. Math., 48 (1995), 501–537. [19] F.H. Lin, C. Liu, Partial regularity of the dynamic system modeling the flow of liquid crystals, Discrete Contin. Dynam. Systems, 2 (1998), 1–22. [20] F.H. Lin, J.Y. Lin, C.Y. Wang, Liquid crystal flows in two dimensions, Arch. Ration. Mech. Anal., 197 (2010), 297–336. [21] J.Y. Lin, B.S. Lai, C.Y. Wang, Global finite energy weak solutions to the compressible nematic liquid crystal flow in dimension three, SIAM J. Math. Anal., 47 (2015), 2952–2983. [22] L. Li, Q. Liu, X. Zhong, Global strong solution to the two-dimensional density-dependent nematic liquid crystal flows with vacuum, arXiv:1508.00235. [23] F.H. Lin, C.Y. Wang, Global existence of weak solutions of the nematic liquid crystal flow in dimension three, Comm. Pure Appl. Math., 69 (2016), 1532–1571. [24] F.H. Lin, C.Y. Wang, Recent development of analysis for hydrodynamic flow of nematic liquid crystals, arXiv:1408.4138. [25] X.G. Liu, J. Qing, Globally weak solutions to the flow of compressible liquid crystals system, Discrete Contin. Dyn. Syst., 33 (2013), 757–788. [26] X.G. Liu, Z.Y. Zhang, Existence of the flow of liquid crystals system, Chinese Ann. Math. Ser. A, 30 (2009), 1–20. [27] S.X. Ma, Classical solutions for the compressible liquid crystal flows with nonnegative initial densities, J. Math. Anal. Appl, 397 (2013), 595–618. [28] A. Novotn´ y, I. Straˇskraba, Introduction to the Mathematical Theory of Compressible Flow, Oxford Univ. Press, Oxford, 2004. [29] P.I. Plotnikov, W. Weigant, Steady 3D viscous compressible flows with adiabatic exponent γ ∈ (1, ∞), J. Math. Pures Appl., (9)104 (2015), 58–82. [30] C.Y. Wang, Well-posedness for the heat flow of harmonic maps and the liquid crystal flow with rough initial data, Arch. Ration. Mech. Anal., 200 (2011), 1–19. [31] H.Y. Wen, S.J. Ding, Solution of incompressible hydrodynamic flow of liquid crystals, Nonlinear Anal. Real World Appl., 12 (2011), 1510–1531. [32] D.H. Wang, C. Yu, Global weak solution and large-time behavior for the compressible flow of liquid crystals, Arch. Ration. Mech. Anal., 204 (2012), 881–915. [33] Y.F. Yang, C.S. Dou, Q.C. Ju, Weak-strong uniqueness property for the compressible flow of liquid crystals, J. Differential Equations, 255 (2013), 1233–1253. School of Mathematical Sciences and Fujian Provincial Key Laboratory on Mathematical Modeling and Scientific Computing, Xiamen University, Xiamen, 361005, China. E-mail address, Z. Tan: [email protected] School of Mathematical Sciences, Xiamen University, Xiamen, 361005, China. E-mail address, Q. J. Xu: [email protected]