# A Regularity Criterion for Compressible Nematic Liquid Crystal Flows

Authors: Jishan Fan; Tohru Ozawa
Journal: ISRN Mathematical Analysis
 (2013)

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## Abstract

We prove a blow-up criterion for local strong solutions to a simplified hydrodynamic flow modeling the compressible, nematic liquid crystal materials in a bounded domain.


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## Body

## 1. Introduction

LetΩ⊂ℝ3 be a bounded domain with smooth boundary ∂Ω. We consider the following simplified version of Ericksen-Leslie system modeling the hydrodynamic flow of compressible nematic liquid crystals:
(1)∂tρ+div(ρu)=0,(2)∂t(ρu)+div(ρu⊗u)+∇p(ρ)-μΔu-(λ+μ)∇divu=-Δd·∇d,(3)∂td+u·∇d=Δd+|∇d|2d,|d|=1inΩ×(0,∞),(4)u=0,d=d0(x)on∂Ω×(0,∞),(5)(ρ,u,d)(x,0)=(ρ0,u0,d0)(x),|d0|=1,x∈Ω⊂ℝ3.
Here ρ is the density of the fluid, u is the fluid velocity, d represents the macroscopic average of the nematic liquid crystal orientation field, and p(ρ):=aργ is the pressure with positive constants a>0 and γ>1. Two real constants μ and λ are the shear viscosity and the bulk viscosity coefficients of the fluid, respectively, which are assumed to satisfy the following physical condition:
(6)μ>0,3λ+2μ≥0.
Equations (1) and (2) are the well-known compressible Navier-Stokes system with the external force -Δd·∇d. Equation (3) is the well-known heat flow of harmonic map when u=0.Recently, Huang et al. [1] prove the following local-in-time well-posedness.Proposition 1.
Letρ0∈W1,q for some q∈(3,6] and ρ0≥0 in Ω, u0∈H2, d0∈H3 and |d0|=1 in Ω. If, in addition, the compatibility condition
(7)-μΔu0-(λ+μ)∇divu0-∇p(ρ0)-Δd0·∇d0=ρ0gforsomeg∈L2(Ω)
holds, then there exist T0>0 and a unique strong solution (ρ,u,d) to the problem (1)–(5).Based on the above proposition, Huang et al. [2] prove the regularity criterion
(8)∫0T‍(∥𝒟(u)∥L∞+∥∇d∥L∞2)dt<∞
to the problem (1)–(3), (5) with the boundary condition
(9)u=0=∂d∂νon∂Ω×(0,∞)
or
(10)u·ν=curlu×ν=∂d∂ν=0on∂Ω×(0,∞).
Here,
(11)𝒟(u):=12(∇u+t∇u),
where ν is the unit outward normal vector to ∂Ω.WhenΩ=ℝ3, Huang and Wang [3] show the following regularity criterion:
(12)∥ρ∥L∞(0,T;L∞)+∥u∥Ls1(0,T;Lr1)+∥∇d∥Ls2(0,T;Lr2)<∞,
with ri and si satisfying
(13)2si+3ri=1,3<ri≤∞,i=1,2.When the term|∇d|2d in (3) is replaced by d-|d|2d, the problem (1)–(5) has been studied by L. M. Liu and X. G. Liu [4]; they proved the following regularity criterion:
(14)∫0T‍(∥∇u∥L24+∥∇u∥L∞)dt<∞.The aim of this paper is to study the regularity criterion of local strong solutions to the problem (1)-(5). We will proveTheorem 2.
Let the assumptions in Proposition1 hold true. If (12) holds true with 0<T<∞, then the solution (ρ,u,d) can be extended beyond T>0.Remark 3.
Theorem2 is also true for the boundary condition (9). But it is an open problem to prove (12) when the homogeneous Dirichlet boundary condition u=0 is replaced by
(15)u·ν=0,curlu×ν=0on∂Ω×(0,∞).


## 2. Proof of Theorem2

Since(ρ,u,d) is the local strong solution, we only need to prove a priori estimates.First, testing (2) and (3) by u and Δd+|∇d|2d, respectively, and adding the resulting equations together, we see that
(16)ddt∫‍(12ρ|u|2+12|∇d|2+aργγ-1)dx+∫‍(μ|∇u|2+(λ+μ)(divu)2+|Δd+|∇d|2d|2)dx=0,
which gives
(17)∫(ρ|u|2+|∇d|2)dx+∫0T‍∫‍(|∇u|2+|Δd+|∇d|2d|2)dxdt≤C.We decompose the velocityu into two parts: u=v+w, where v(t)∈H01(Ω)∩H2(Ω) satisfies
(18)μΔv+(λ+μ)∇divv=∇p(ρ),
and thus w(t)∈H01(Ω)∩H2(Ω) satisfies
(19)μΔw+(λ+μ)∇divw=ρu˙+Δd·∇d,
where we used u˙:=∂tu+u·∇u to denote the material derivative of u. Then, together with the standard W1,p theory and H2 theory for elliptic systems, we obtain
(20)∥∇v∥L6≤C∥p(ρ)∥L6,∥∇w∥L6+∥∇2w∥L2≤C∥ρu˙∥L2+C∥Δd∇d∥L2.Testing (3) by Δ∂td and using (4), (20), (3), and the identity 0=Δ(d∂td)=dΔ∂td+∂tdΔd+2∇d∂td, we derive
(21)12ddt∫‍|Δd|2dx+∫‍|∇∂td|2dx=∫‍(u·∇d-|∇d|2d)Δ∂tddx=-∫‍∇(u·∇d)∇∂tddx-∫‍|∇d|2(dΔ∂td)dx=-∫‍(u∇2d+∇u·∇d)∇∂tddx+∫‍|∇d|2(∂tdΔd+2∇d∇∂td)dx=-∫‍(u·∇2d+∇u·∇d)∇∂tddx+∫‍|∇d|2(Δd+|∇d|2d-u·∇d)Δddx+2∫‍|∇d|2∇d∇∂tddx≤∥u∥Lr1∥∇2d∥L2r1/(r1-2)∥∇∂td∥L2+∥∇d∥Lr2∥∇u∥L2r2/(r2-2)∥∇∂td∥L2+C∥∇d∥Lr22∥Δd∥L2r2/(r2-2)+C∥u∥Lr12∥Δd∥L2r1/(r1-2)2+ϵ∥∇∂td∥L22≤Cϵ∥∇∂td∥L22+C∥u∥Lr1s1∥∇2d∥L22+C∥∇d∥Lr2s2∥∇u∥L22+ϵ∥∇u∥L62+C∥∇d∥Lr2s2∥Δd∥L22+ϵ∥d∥H32+C≤Cϵ∥∇∂td∥L22+Cϵ∥d∥H32+Cϵ∥ρu˙∥L22+C∥∇d∥Lr2s2(∥Δd∥L22+∥∇u∥L22)+C∥u∥Lr1s1∥∇2d∥L22+C
for any 0<ϵ<1, where we have used the Hölder inequality
(22)∥∇u∥L2r1/(r1-2)≤C∥∇u∥L21-(3/r1)∥∇u∥L63/r1
and the Gagliardo-Nirenberg inequality
(23)∥∇2d∥L2r2/(r2-2)≤C∥∇2d∥L21-(3/r2)∥d∥H33/r2,∥∇u∥L6≤∥∇v∥L6+∥∇w∥L6≤C+∥∇w∥L6.By theH3 theory of the elliptic equations, it follows from (3) that
(24)∥d∥H3≤C(1+∥∇Δd∥L2)≤C(1+∥∇(∂td+u·∇d-|∇d|2d)∥L2)≤C(1+∥∇∂td∥L2+∥u∥Lr1∥∇2d∥L2r1/(r1-2)+∥∇d∥Lr2∥∇u∥L2r2/(r2-2)+∥∇d∥Lr2∥∇2d∥L2r2/(r2-2))≤C(1+∥∇∂td∥L2+ϵ∥d∥H3+∥u∥Lr1s1/2∥∇2d∥L2+ϵ∥∇u∥L6+∥∇d∥Lr2s2/2∥∇u∥L2+∥∇d∥Lr2s2/2∥Δd∥L2),
which yields
(25)∥d∥H3≤C(1+∥∇∂td∥L2+∥u∥Lr1s1/2∥∇2d∥L2+∥∇d∥Lr2s2/2∥∇u∥L2+∥∇d∥Lr2s2/2∥∇2d∥L2).Testing (2) by ∂tu and setting M(d):=∇d⊙∇d-(1/2)|∇d|2𝕀3, we find that
(26)12ddt∫‍(μ|∇u|2+(λ+μ)(divu)2)dx+∫‍ρ|u˙|2dx-ddt∫‍pdivudx-ddt∫‍M(d):∇udx=∫‍ρu˙·(u·∇u)dx-∫‍∂tpdivudx-∫‍∂tM(d):∇udx≤∥ρu˙∥L2∥u∥Lr1∥∇u∥L2r1/(r1-2)+C∥∇d∥Lr2∥∇u∥L2r2/(r2-2)∥∇∂td∥L2-∫‍ptdivudx.Now we deal with the last term.First, (1) implies that
(27)∂tp+div(pu)+(γ-1)pdivu=0.Using (27) and (20), we have
(28)-∫‍∂tpdivudx=-∫‍∂tpdivvdx-∫‍∂tpdivwdx=∫‍∇∂tpvdx+∫‍div(pu)divwdx+(γ-1)∫‍pdivudivwdx=-ddt∫‍(μ2|∇v|2+λ+μ2(divv)2)dx-∫‍pu∇divwdx+(γ-1)∫‍pdivudivwdx≤-ddt∫‍(μ2|∇u|2+λ+μ2(divv)2)dx+C∥ρu∥L2∥∇divw∥L2+C∥divu∥L2∥divw∥L2.Inserting (28) into (26) and using (20), we have
(29)12ddt∫‍(μ|∇u|2+(λ+μ)(divu)2)dx+ddt∫‍(μ2|∇v|2+λ+μ2(divv)2)dx-ddt∫‍pdivudx-ddt∫‍M(d):∇udx+∫‍ρ|u˙|2dx≤∥ρu˙∥L2∥u∥Lr1∥∇u∥L2r1/(r1-2)+C∥∇d∥Lr2∥∇u∥L2r2/(r2-2)∥∇∂td∥L2+C∥∇divw∥L2+C∥divu∥L22+C≤ϵ∫‍ρ|u˙|2dx+C∥u∥Lr1s1∥∇u∥L22+ϵ∥∇u∥L62+C∥∇d∥Lr2s2∥∇u∥L22+ϵ∥∇∂td∥L22+C∥∇d∥Lr2s2∥Δd∥L22+C∥divu∥L22+C≤Cϵ∫‍ρ|u˙|2dx+Cϵ∥∇∂td∥L22+C∥u∥Lr1s1∥∇u∥L22+C∥∇d∥Lr2s2(∥∇u∥L22+∥Δd∥L22)+C.Combining (21), (25), and (29), taking ϵ small enough, and using the Gronwall inequality, we conclude that
(30)∥u∥L∞(0,T;H1)+∥d∥L∞(0,T;H2)+∥d∥L2(0,T;H3)+∥ρu˙∥L2(0,T;L2)≤C.Now by the same calculations as those in [3, 5], we prove that
(31)∥ρ∥L∞(0,T;W1,q)+∥∂tρ∥L∞(0,T;Lq)≤C,∥ρ∂tu∥L∞(0,T;L2)+∥∂tu∥L2(0,T;H1)≤C,∥u∥L∞(0,T;H2)+∥u∥L2(0,T;W2,q)≤C,∥d∥L∞(0,T;H3)≤C,∥∂td∥L∞(0,T;H1)≤C.This completes the proof.



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*Source: 271324-2013-10-31.xml*