# Two Logarithmically Improved Regularity Criteria for the 3D Nematic Liquid Crystal Flows

Authors: Ruirui Wang; Qiang Mu; Yulei Wang; Qiang Li
Journal: Journal of Mathematics
 (2022)

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## Abstract

In this paper, we study the regularity criterion for the local smooth solution of the 3D nematic liquid crystal flows. More precisely, it is proved the smooth solutionu,d can be extended beyond T provided that ∫0T∇huhB˙∞,∞0+∇dB˙∞,∞02/1+log1+∇uB˙∞,∞0+∇dB˙∞,∞0dt<∞ or ∫0T∇huhB˙∞,∞−r4/3−2r+∇dB˙∞,∞02/1+log1+∇uB˙∞,∞0+∇dB˙∞,∞0dt<∞,0≤r≤1.


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## Body

## 1. Introduction

Liquid crystals is a state of the matter which has both properties of the liquid and the solid crystal. And as a kind of liquid crystals, the nematic liquid crystal can flow like fluids and has very nice properties. Ericksen et al. during 1960s (see [1, 2]) established the hydrodynamic theory for describing the nematic liquid crystal flows. Owing to the complexity of original Ericksen-Leslie equations and for further research, Lin [3] simplify the original Ericksen-Leslie equations, which still retains most of the essential features of original equations. In this paper, we investigate the following simplified version for nematic liquid crystal flows in 3-dimensions(1)∂tu+u⋅∇u−μΔu+∇p=−λ∇⋅∇d⊙∇d,∂td+u⋅∇d=γΔd+∇d2d,∇⋅u=0,d=1,ux,0=u0x,dx,0=d0x,here u=ux,t∈ℝ3 denotes the velocity field, d=dx,t∈S2 (the unit sphere in ℝ3) the macroscopic average of molecular orientation field and p=px,t represents the scalar pressure, ∇⋅u=0 is the incompressible condition. And μ, λ, γ are positive constants, which shall be assumed to be all equal to 1 in consideration of their concrete values playing no role in our arguments. The notation ∇d⊙∇d represents the 3×3 matrix whose the i,jth component is given by(2)∑k=13∂idk∂jdki,j≤3.It is well-known that the system (1) has a unique local smooth solution (see [4]). More precisely, if initial data u0∈Hsℝn,ℝn with ∇⋅u0=0 and d0∈Hs+1ℝn,S2 for s≥n, then(3)u∈C0,T;Hsℝn∩C10,T;Hs−1ℝn,d∈C0,T;Hs+1ℝn,S2∩C10,T;Hsℝn,S2.However, the global existence of solutions is an difficult problem. Hence much efforts have been paid to study the regularity criteria to extend local solutions. For the regularity criteria readers may refer to [5–13] and references therein.On one hand, the above system (1) reduces to the incompressible Navier-Stokes equations when the orientation field d equals a constant. It is well-known that Navier-Stokes equations has an unique smooth solution (see [14]) provided that the solution u satisfies(4)∫0TωL∞dt<∞,where ω=∇×u. Later, Kozono and Taniuchi [15], Kozono et al. [16] generalized the criterion (4) to(5)∫0TωBMOdt<∞,∫0TωB˙∞,∞0dt<∞,respectively, where BMO is the space of Bounded Mean Oscillation and B˙∞,∞0 represents the homogeneous Besov space. And based on (5), Fan et al. [17], Guo and Gala [18] respectively improve (5) by the following conditions(6)∫0T∇uB˙∞,∞01+lne+∇uB˙∞,∞0dt<∞,(7)∫0T∇uB˙∞,∞−121+lne+∇uB˙∞,∞0dt<∞.It is obvious that the logarithmic improvement is here, in time only, and that can be seen as a natural Gronwall type extension of the Prodi-Serrin conditions. On the other hand, when the velocity fieldu=0, the system (1) becomes to the heat flow of harmonic maps onto a sphere. And Wang [19] established a blow up criterion, which implies the unique smooth solution d∈C∞ℝn;0,T is global if(8)∫0T∇dLndt<∞.Inspired by the conditions (4) and (8), Huang and Wang [4] established a BKM type blow-up criterion for the system (1). That is, if T is the maximal time, 0<T<∞, then(9)∫0TωL∞+∇dL∞2dt=∞.Naturally, similar to (6) and (7), Liu and Zhao [20] extend (9) to the Logarithmically improved regularity criterion. Namely, the local smooth solution u,d can continuously past any time T>0 if the following holds(10)∫0TωB˙∞,∞0+∇dB˙∞,∞021+lne+∇uB˙∞,∞0+∇dB˙∞,∞0dt<∞,or(11)∫0T∇uB˙∞,∞−12+∇dB˙∞,∞02lne+∇uB˙∞,∞0+∇dB˙∞,∞0dt<∞.In view of it is difficult to reduce the condition ond, we are mainly concerned with reducing the condition on u. Inspired by the references above, we will use the components of ∇u to replace the condition (11). Our main results are stated as follows:Theorem 1.
Assumeu,dis a local smooth solution to the system (1) on the time interval0,Tfor some0<T<∞. And let initial datumu0∈H3ℝ3with∇⋅u0=0,d0∈H4ℝ3,S2. Ifu,dsatisfies(12)∫0T∇huhB˙∞,∞0+∇dB˙∞,∞021+log1+∇uB˙∞,∞0+∇dB˙∞,∞0dt<∞,then u,d can be extended beyond T smoothly, where ∇h=∂1,∂2,0,uh=u1,u2,0. That is to say, if the solution blows up at T, then(13)∫0T∇huhB˙∞,∞0+∇dB˙∞,∞021+log1+∇uB˙∞,∞0+∇dB˙∞,∞0dt=∞.Remark 1.
(1)
In view of the fact that the normsωB˙∞,∞0 and∇uB˙∞,∞0are approximate, it is obvious that the condition(10) is weaker than the condition (8) and (9) in some sense. And it can be seen that if the condition (10) reduces to(14)∫0T∇huhB˙∞,∞0+∇dB˙∞,∞02dt<∞,then the conclusion of Theorem (1) still remains valid, which is also an improved result compared to the regularity criterion (9).(2)
Noting that the norm∇uhB˙∞,∞−1 is equivalent to uhB˙∞,∞0, combining (10)–(12), the condition (12) can be replaced by the following condition:(15)∫0TuhB˙∞,∞02+∇dB˙∞,∞021+log1+∇uB˙∞,∞0+∇dB˙∞,∞0dt<∞.Remark 2.
It is well-known thatL∞ℝ3⊂BMOℝ3⊂B˙∞,∞0ℝ3. Thus the conclusion of Theorem 1still remains true if the condition (12) is substituted by(16)∫0T∇huhBMO+∇dBMO21+log1+∇uBMO+∇dBMOdt<∞.Theorem 2.
Assumeu,dis a local smooth solution to the system (1) on the time interval0,Tfor some0<T<∞. And let initial datumu0∈H3ℝ3with∇⋅u0=0,d0∈H4ℝ3,S2. Ifu,dsatisfies(17)∫0T∇huhB˙∞,∞−r4/3−2r+∇dB˙∞,∞021+log1+∇uB˙∞,∞0+∇dB˙∞,∞0dt<∞,here 0≤r≤1, then u,d can be extended beyond T smoothly. That is to say, if the solution blows up at T, then(18)∫0T∇huhB˙∞,∞−r4/3−2r+∇dB˙∞,∞021+log1+∇uB˙∞,∞0+∇dB˙∞,∞0dt<∞.Remark 3.
Owing toB˙∞,∞−rcontaining the caseB˙∞,∞0asr=0, the condition (17) is an improvement in some sense compared to the condition (12). However,4/3−2r=4/3whenr=0, hence the condition (12) is better than the condition (17) in the end point.


## 2. Preliminaries

In this section, we collect some useful analytic tools which play an important part in our proof.Lemma 1(Page 82 in [21]).
Let1<q<p<∞andαbe a positive real number. Then there exists a constant C such that(19)fLp≤CfB˙∞,∞−α1−θfB˙q,qβθ,withβ=αpq−1,θ=qp.In particular, whenβ=1, q=2 and p=4, we have α=1 and(20)fL4≤CfB˙∞,∞−11/2∇fL21/2.Lemma 2.
(Product and Commutator estimate[22, 23]). Lets>0,1<p<∞, and1/p=1/p1+1/p2=1/p3+1/p4withp2,p3∈1,+∞andp1,p4∈1,+∞. Then,(21)ΛsfgLp≤CgLp1ΛsfLp2+ΛsgLp3fLp4,(22)Λs,f⋅∇gLp≤C∇fLp1ΛsgLp2+ΛsfLp3∇gLp4,where Λs,fg=Λsfg−fΛsg.Lemma 3 (see [24], Lemma 2).
Let∇f∈B˙∞,∞−1,f∈Hsfor alls>3/2. Then there exists a positive constant C such that(23)fL∞≤C1+∇fB˙∞,∞−1log+1+fHs1/2,where Hs denotes the standard Sobolev space and(24)log+x=logx,x>e,1,0≤x≤e.


## 3. Proof of Main Results

Proof of Theorem 1.
In this section, we shall first show the proof of Theorem1. Since the existence of local smooth solutions is obvious owing to the initial value condition for u,d in Theorem 1, we only need to show the priori estimate for the local smooth solution. And by the condition (12), we will give the following priori estimate(25)limt⟶T−sup∇ΔuL22+Δ2dL22<∞,which is enough to guarantee the smooth solution (u, d) pasts time T smoothly.
Firstly, we will show theL1 estimate of u and ∇d together because the terms ∇⋅∇d⊙∇d and u⋅∇d can be cancelled when integrating. Applying u to the equation 11 and integrating over ℝ3 yields(26)12ddtuL22+∇uL22=−∫ℝ3∇d⋅Δd⋅udx,where the following equalities have been used(27)∫ℝ3u⋅∇u⋅udx=0,∫ℝ3∇p⋅udx=0.
Then, multiplying equation12 by −Δd and integrating over ℝ3 one has(28)12ddt∇dL22+ΔdL22=∫ℝ3u⋅∇d⋅Δd−|∇d|2dΔddx.
By adding the above equalities and using the factsd=1,Δd2=0⇒∇d2=−d⋅Δd, we have(29)12ddtuL22+∇dL22+∇uL22+ΔdL22≤ΔdL22.
Integrating above inequality (29) in time yields(30)sup0<t<TutL22+∇dtL22+∫0T∇utL22dt≤Cu0L22+∇d0L22.
Besides, it is sufficient to show the boundness fordL∞. Multiplying equation 12 by pdp−2d for p>2 and integrating both sides on ℝ3, one has(31)ddtdLpp+2∇dp/2L22+pdLp+2p+2=pdLpp,where we use the equality(32)∫ℝ3u⋅∇d⋅pdp−2ddx=−∫ℝ3∇⋅udpdx=0.
Hence the (31) implies(33)ddtdLp≤dLp.
Applying the Gronwall inequality and lettingp⟶∞, it can be deduced from above that(34)dL∞≤eTd0L∞<∞.
Now we shall show theH1 estimate of u and ∇d. Similarly going on the above process, multiplying the equation 11 by −Δu and integrating over ℝ3, then taking Δ on the equation 12, multiplying with Δd and integrating over ℝ3, and combining that two equations, one obtains that(35)12ddt∇uL22+ΔdL22+ΔuL22+∇ΔdL22=∫ℝ3u⋅∇u⋅Δudx+∫ℝ3∇⋅∇d⊙∇d⋅Δudx−∫ℝ3Δu⋅∇d⋅Δddx+∫ℝ3Δ∇d2d⋅Δddx≔I1+I2+I3+I4.
In the following, we will estimate the termsIii=1,2,3,4. For I1, making use of the incompressibility condition and integration by parts several times, one can be concluded that(36)I1=∫ℝ3u⋅∇u⋅Δudx=∫ℝ3∑i,j,k=13ui∂iuj∂k∂kujdx,=−∫ℝ3∑i,j,k=13∂kui∂iuj∂kuj+∑i,j,k=1312ui∂i∂kuj2dx=−∫ℝ3∑i,k=12∑j=13∂kui∂iuj∂kuj+∑i=12∑j=13∂3ui∂iuj∂3uj,+∑k=12∑j=13∂ku3∂3uj∂kuj+∑j=13∂3u3∂3uj∂3ujdx=−∫ℝ3∑i,k=12∑j=13∂kui∂iuj∂kuj+∑i,j=12∂3ui∂iuj∂3uj+∑i=12∂3ui∂iu3∂3u3+∑j,k=12∂ku3∂3uj∂kuj+∑k=12∂ku3∂3u3∂ku3+∑j=13∂3u3∂3uj∂3ujdx.
Noting that∂3u3=−∂1u1+∂2u2, and using the Lemma 3, I1 can be estimated as follows(37)I1≤C∫ℝ3∇huh∇u2dx≤C∇huhL∞∇uL22≤C1+∇∇huhB˙∞,∞−1log1/21+∇ΔuL2∇uL22≤C1+∇huhB˙∞,∞0log1/21+∇ΔuL2∇uL22≤C∇huhB˙∞,∞01+log1+∇uB˙∞,∞0+∇dB˙∞,∞0log1/21+∇ΔuL2×1+log1+∇uB˙∞,∞0+∇dB˙∞,∞0∇uL22+C∇uL22≤C∇huhB˙∞,∞01+log1+∇uB˙∞,∞0+∇dB˙∞,∞0log1/21+∇ΔuL2+Δ2dL2×1+log1+∇uB˙∞,∞0+∇dB˙∞,∞0∇uL22+C∇uL22≤C1+∇huhB˙∞,∞01+log1+∇uB˙∞,∞0+∇dB˙∞,∞0log1+∇ΔuL2+Δ2dL2∇uL22,where we have used the following inequality(38)1+log1+∇uB˙∞,∞0+∇dB˙∞,∞0≤Clog1+∇uL∞+∇dL∞≤Clog1+∇uL21/6∇ΔuL25/6+‖∇d‖L21/2‖Δ2d‖L21/2≤Clog1+∇ΔuL22+Δ2dL22.
In view ofI2 and I3 containing terms that could be cancelled, adding I2 and I3 together and by the incompressibility condition ∇⋅u=0, it follows that(39)I2+I3=∫ℝ3∑i,j,k=13∂i∂jdk∂jdk+∂idk∂j∂jdkΔui−Δui∂idkΔdk+2∇ui∂i∇dkΔdk+ui∂iΔdkΔdk,dx=∫ℝ3∑i,j,k=13−2∇ui∂i∇dkΔdkdx≤C∫ℝ3∇u∇∇dΔddx.
Hence, it can be deduced fromHo.. lder inequality, Young inequality and the inequality (20) that(40)I2+I3≤C∇uL2ΔdL42≤C∇uL2∇dB˙∞,∞0∇Δd|L2≤C∇dB˙∞,∞02∇uL22+14∇ΔdL22≤C∇dB˙∞,∞021+log1+∇uB˙∞,∞0+∇dB˙∞,∞01+log1+∇uB˙∞,∞0+∇dB˙∞,∞0∇uL22+14∇ΔdL22≤C∇dB˙∞,∞021+log1+∇uB˙∞,∞0+∇dB˙∞,∞0log1+∇ΔuL2+Δ2dL2∇uL22+14∇ΔdL22,where the fact that the norms ΔdB˙∞,∞−1 and ∇dB˙∞,∞0 are equivalent has been used. For I4, by the product estimate (21) and inequality (20), we obtain(41)I4=∫ℝ3Δ∇d2d⋅Δddx≤Δ∇d2dL43ΔdL4≤C∇ΔdL2∇dL4dL∞+ΔdL4∇dL42ΔdL4≤CΔdL42∇dL42+18∇ΔdL22≤C∇dB˙∞,∞0∇ΔdL2dL∞ΔdL2+18∇ΔdL22≤C∇dB˙∞,∞02ΔdL22+14∇ΔdL22≤C∇dB˙∞,∞021+log1+∇uB˙∞,∞0+∇dB˙∞,∞0log1+∇ΔuL2+Δ2dL2ΔdL22+14∇ΔdL22,here the following Gagliardo-Nirenberg inequality has been used(42)∇dL4≤CdL∞1/2ΔdL21/2.
Combining (35), (37), (39), and (41) one has(43)ddt∇uL22+ΔdL22+ΔuL22+∇ΔdL22≤C1+∇huhB˙∞,∞0+∇dB˙∞,∞021+log1+∇uB˙∞,∞0+∇dB˙∞,∞0log1+∇ΔuL2+Δ2dL2×∇uL22+ΔdL22.
Noting (12), one may conclude that for any small constant ϵ>0, there exists T0<T such that(44)∫T0T∇huhB˙∞,∞0+∇dB˙∞,∞021+log1+∇uB˙∞,∞0+∇dB˙∞,∞0dt<ϵ.
For anyT0≤t<T, we set(45)Mt=supT0≤s≤t∇ΔusL22+Δ2dsL22.
Using (44) and (45), and applying Gronwall inequality to (43) in the interval T0,t gives(46)∇utL22+ΔdtL22+∫T0tΔusL22+∇ΔdsL22ds≤∇uT0L22+ΔdT0L22×expC∫T0t1+∇huhB˙∞,∞0+∇dB˙∞,∞021+log1+∇uB˙∞,∞0+∇dB˙∞,∞0log1+∇ΔuL22+Δ2dL22ds≤∇uT0L22+ΔdT0L22expCT−T0+Cϵlog1+Mt≤C0CTexpCϵlog1+Mt≤C0CT1+MtCϵ,where the letter C0 means a constant depending on ∇uT0L22+ΔdT0L22, CT depends on the maximum value of time T, and C is a generic constant which may be different from line to line.
At last, the boundedness of the norm∇ΔuL2 and Δ2dL2 are needed so as to guarantee the validness of inequality (25) and (46). Employing ∇Δ and Δ2 to the equations 11 and 12 respectively, and taking the L2 inner product with ∇Δu,Δ2d, we see that(47)12ddt∇ΔuL22+Δ2dL22+Δ2uL22+∇Δ2dL22=−∫ℝ3∇Δu⋅∇u⋅∇Δudx−∫ℝ3∇Δ∇d⋅Δd⋅∇Δudx−∫ℝ3Δ2u⋅∇d⋅Δ2ddx−∫ℝ3Δ2∇d2d⋅Δ2dx≔J1+J2+J3+J4.
ForJ1, applying ∇⋅u=0, Ho.. lder inequality, the commutator estimate (22) and Young inequality, we have(48)J1=−∫ℝ3∇Δ,u⋅∇u⋅∇Δudx≤∇Δ,u⋅∇uL43∇ΔuL4≤C∇uL2∇ΔuL4+∇ΔuL4∇uL2∇ΔuL4≤C∇uL2∇ΔuL42≤C∇uL2∇uL21/6‖Δ2u‖L211/6≤C∇uL214+16Δ2uL22,here we have used the following Gagliardo-Nirenberg inequality:(49)∇ΔuL4≤C∇uL21/6Δ2uL25/6.
ForJ2, by the above inequalities used for J1 and product estimate (21), we have(50)J2=∫ℝ3Δ∇d⋅Δd⋅Δ2udx≤Δ∇dΔdL2Δ2uL2≤∇dL4Δ2dL4+∇ΔdL4ΔdL4Δ2uL2≤C∇dL42Δ2dL42+C∇ΔdL42ΔdL42+16∇ΔuL22≤CdL∞ΔdL2ΔdL21/6∇Δ2dL211/6+CΔdL25/6∇Δ2dL27/6ΔdL23/2∇Δ2dL21/2+16∇ΔuL22≤CΔdL27/6∇Δ2dL211/6+CΔdL27/3Δ2dL25/3+16Δ2uL22≤CΔdL214+16∇Δ2dL22+16∇ΔuL22,here we have used the following Gagliardo-Nirenberg inequalities:(51)∇dL4≤CdL∞1/2ΔdL21/2,ΔdL4≤CΔdL23/4∇Δ2dL21/4,∇ΔdL4≤CΔdL25/12∇Δ2dL27/12,Δ2dL4≤CΔdL21/12∇Δ2dL211/12.
ForJ3, similar as (48) and by the above Gagliardo-Nirenberg inequalities, one may conclude(52)J3=−∫ℝ3Δ2,u⋅∇d⋅Δ2ddx≤Δ2,u⋅∇dL43Δ2dL4≤∇uL2Δ2dL4+∇dL4Δ2uL2Δ2dL4≤C∇uL2Δ2dL42+C∇dL42Δ2dL42+16Δ2uL22≤C∇uL2Δ2dL21/6∇Δ2dL211/6+CdL∞ΔdL2ΔdL21/6∇Δ2dL211/6+16Δ2uL22≤C∇uL212ΔdL22+CΔdL214+16∇Δ2dL22+16Δ2uL22.
ForJ4, by the product estimate (21) and the fact ∇d2=−d⋅Δd, we infer that(53)J4=−∫ℝ3Δ2∇d2d⋅Δ2ddx=∫ℝ3∇Δ∇d2d⋅∇Δ2ddx≤∇Δ∇d2dL2∇Δ2dL2≤C∇Δ∇d2dL2+∇d2∇ΔdL2∇Δ2dL2≤C∇dL4Δ2dL4dL∞+ΔdL4∇ΔdL4dL∞∇Δ2dL2≤CΔdL21/2ΔdL21/12∇Δ2dL211/12+ΔdL23/4∇Δ2dL21/4ΔdL25/12∇Δ2dL27/12∇Δ2dL2≤CΔdL27/12∇Δ2dL211/12+ΔdL27/6∇Δ2dL25/6∇Δ2dL2≤CΔdL214+16∇Δ2dL22.
Inserting the above estimates (48), (50), (52), (53) to (47), and combining (46) yields(54)ddt1+∇ΔuL22+Δ2dL22+Δ2uL22+∇Δ2dL22≤C∇uL214+ΔdL214≤CC0CT1+Mt7Cϵ.
Integrating the above inequality with respect to time fromT0 to t, T0≤t<T, it follows that(55)1+∇ΔutL22+Δ2dtL22+∫T0tΔ2uL22+∇Δ2dL22dτ≤1+∇ΔuT0L22+Δ2dT0L22+∫T0tCC0CT1+Mτ7Cϵdτ=1+∇ΔuT0L22+Δ2dT0L22+∫T0tCC0CT1+Mτdτ,here we choose ϵ=1/7C. The above inequality and equality (45) imply that(56)1+Mt+∫T0t∇ΔuL22+Δ2dL22dτ≤1+∇ΔuT0L22+Δ2dT0L22+∫T0tCC0CT1+Mτdτ.
Therefore, employing Gronwall’s inequality leads to(57)1+Mt+∫T0t∇ΔuL22+Δ2dL22dτ≤1+∇ΔuT0L22+Δ2dT0L22expCC0CTT−T0,which indicates the truth of equality (25). Thus the Proof of Theorem 1 is completed.Proof of Theorem 2.
For the proof of Theorem2, we only need reestimate I1 again. By the Lemma 1, we have(58)I1≤C∫ℝ3∇huh∇u2dx≤C‖∇huh‖L4‖∇u‖L8/32≤C∇huhB˙∞,∞−r1/2∇huhH˙r1/2∇uL25/4ΔuL23/4≤C∇huhB˙∞,∞−r1/2∇huhL21−r/2ΔuL2r/2∇uL25/4ΔuL23/4≤C∇huhB˙∞,∞−r1/2∇uL27−2r/4ΔuL23+2r/4≤C∇huhB˙∞,∞−r4/5−2r∇uL24/5−2r∇uL22+14ΔuL22≤C∇huhB˙∞,∞−r4/3−2r+∇uL22∇uL22+14ΔuL22≤C∇uL22+∇huhB˙∞,∞−r4/3−2r1+log1+∇uB˙∞,∞0+∇dB˙∞,∞0log1+∇ΔuL2+Δ2dL2×∇uL22+14ΔuL22.
Going the same process to (39)–(57), the desired result will be obtained. Thus the Proof of Theorem 2 is completed.



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