# Regularity Criteria for a Coupled Navier-Stokes andQ-Tensor System

Authors: Jishan Fan; Tohru Ozawa
Journal: International Journal of Analysis
 (2013)

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## Abstract

We study a system describing the evolution of a nematic liquid crystal flow. The system couples a forced Navier-Stokes system describing the flow with a parabolic-type system describing the evolution of the nematic crystal director fields (Q-tensors). We prove some regularity criteria for the local strong solutions. However, we do not provide estimates on the rates of increase of high norms.


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## Body

## 1. Introduction

We consider the following coupled Navier-Stokes andQ-tensor system [1–4]:
(1)∂tQ+u·∇Q+QΩ-ΩQ=ΔQ-aQ+b(Q2-𝕀dtr(Q2))-cQtr(Q2),(2)∂tu+u·∇u+∇π=Δu-div(∇Q⊙∇Q)+div(QΔQ-ΔQQ),(3)divu=0,(4)(Q,u)(x,0)=(Q0,u0)inℝd(d=2,3).
Here the unknowns u,π, and Q denote the velocity field of the fluid, the pressure, and the order parameter, respectively. A Q-tensor is a symmetric and traceless d×d-matrix, Ω:=(1/2)(∇u-(∇u)T),a∈ℝ,b>0 and c>0 are physical constants, d is the space dimension, (∇Q⊙∇Q)ij:=tr(∂iQ∂jQ),trQ:=∑i=1dQii, and thus tr(Q2)=∑i,j=1dQij2.WhenQ≡0, (2) and (3) are the well-known Navier-Stokes system, for which Kozono et al. [5] and Kozono and Shimada [6] proved the well-known regularity criteria
(5)u∈L2/(1-s)(0,T;B˙∞,∞-s),0<s<1,(6)u∈L2(0,T;B˙∞,∞0),(7)ω:=curlu∈L1(0,T;B˙∞,∞0),
where B˙p,qs denotes the homogeneous Besov spaces [7].Very recently, Paicu and Zarnescu [8] proved the existence of global-in-time weak solutions in 3-dimensional space and of smooth solutions in 2-dimensional space. The aim of this paper is to study the regularity criteria.If one formally takesQ:=s+(n⊗n-(1/3)𝕀), with s+ a constant, then the equations reduce to the generally accepted equations of Leslie [9], which have been studied in [10–15]. We will prove the following.Theorem 1.
Letu0∈H1,Q0∈H2,divu0=0 in ℝd with d=2,3. Let (u,Q) be a unique strong solution in L∞(0,T1;H1)∩L2(0,T1;H2)×L∞(0,T1;H2)∩L2(0,T1;H3) with 0<T1<T<∞.(i)
Ifd=2,3 and u satisfies one of the conditions (5), (6), or (7) and ∇Q satisfies
(8)∇Q∈Lp(0,T;Lq)with2p+dq=1,d<q≤∞,
for some finite T<∞, then the solution (u,Q) can be extended beyond T>0.(ii)
Ifd=3 and u satisfies
(9)∇u∈Lp(0,T;Lq)with2p+3q=2,2<q≤3,
for some finite T<∞, then the solution (u,Q) can be extended beyond T>0.Remark 2.
By the well-known inequality∥curlu∥Lq≤∥∇u∥Lq≤C∥curlu∥Lq, the condition (9) can be replaced by
(10)ω:=curlu∈Lp(0,T;Lq)with2p+3q=2,32<q≤3.It has been proved in [8] that the system (1)–(4) has a Lyapunov functional:
(11)E(t):=∫‍12u2+12|∇Q|2+a2tr(Q2)-b3tr(Q3)+c4tr2(Q2)dx,
which satisfies
(12)ddtE=-∫‍|∇u|2dx-∫‍tr(ΔQ-aQ-b(Q2-𝕀dtr(Q2))-∫‍tr-cQtr(Q2)(Q2-𝕀dtr(Q2)))2dx≤0,
from which we easily obtain [8]
(13)u∈L∞(0,T;L2)∩L2(0,T;H1),Q∈L∞(0,T;H1)∩L2(0,T;H2).
When d=2, (13) give [8]
(14)u∈L4(0,T;L4)⊂L2/(1-1/2)(0,T;B˙∞,∞-1/2),∇Q∈L4(0,T;L4),
thus (5) and (8) hold true; this proves the existence of global-in-time strong solutions when d=2. In [8], this result was proven by complicated Littlewood-Paley theory, Bony’s paraproduct decomposition, and the logarithmic Sobolev inequality. The purpose of this paper is to make the argument in [8] much simpler. However, in [8], they obtained in addition the rate of increase of high norms.Our proof uses an energy method and relies on a simpleL∞ estimate of Q and the following cancellation property:Lemma 3 (see [8]).
LetQ~,Q:ℝd→ℝd×d be symmetric matrix-valued functions and let Ωαβ:=(1/2)(∂βuα-∂αuβ)∈ℝd×d be smooth and decaying and sufficiently fast at infinity (so that one can integrate by parts without boundary terms). Then
(15)∫tr((ΩQ~-Q~Ω)ΔQ)dx-∫∑α,β,γ‍∂β(Q~αγΔQγβ-ΔQαγQ~γβ)uαdx=0.


## 2. Proof of Theorem1

This section is devoted to the proof of Theorem1. Since it is easy to prove that there are T0>0 and a unique strong solution (u,Q) to the problem (1)–(4) in (0,T0], we only need to prove a priori estimates.First, we prove the following key estimate:(16)∥Q∥L∞(0,T;L∞)≤C.
To prove (16), we multiply (1) by 2pQtrp-1(Q2) and take the trace to obtain
(17)(∂t+u·∇)trp(Q2)=2pΔQαβQαβtrp-1(Q2)-2patrp(Q2)+2pbtr(Q3)trp-1(Q2)-2pctrp+1(Q2).Let us observe that forQ, a traceless, symmetric, 3×3 matrix, we have
(18)tr(Q3)≤3ϵ8tr2(Q2)+1ϵtr(Q2),∀ϵ>0.Integrating overℝd, integrating by parts, and using (3), (18), and the assumption c>0, we obtain
(19)ddt∫‍trp(Q2)dx≤-2p∫‍∇Qαβ∇Qαβtrp-1(Q2)dx-4p(p-1)∫‍∂γQαβQαβ∂γQδλQδλtrp-2(Q2)dx+cp∫‍trp(Q2)dx≤cp∫‍trp(Q2)dx,
which gives
(20)(∫‍trp(Q2)dx)1/(2p)≤eCT(∫‍trp(Q02)dx)1/(2p)
with C independent of p.Thanks to a simple lemma in [16, Page 102], we take p→∞ in (20), this proves (16).TakingΔ to (1), testing scalarly by ΔQ, and using (3); we find that
(21)12ddt∫‍|ΔQ|2dx+∫‍|∇ΔQ|2dx=∫‍(ΔΩQ-QΔΩ):ΔQdx+∫‍(ΩΔQ-ΔQΩ):ΔQdx+2∑i‍∫‍(∂iΩ∂iQ-∂iQ∂iΩ):ΔQdx-∫‍Δu·∇Q:ΔQdx+2∑i,j‍∫‍∂iuj∂j∂iQ:ΔQdx+∫‍Δ[-aQ+b(Q2-𝕀dtr(Q2))-cQtr(Q2)]:ΔQdx=:I1+I2+I3+I4+I5+I6.
Here “testing scalarly by ΔQ” means multiplying with respect to the Frobenius inner product of matrices, A:B=tr(AB) and integrating over ℝd.Testing (2) by -Δu and using (3), we infer that
(22)12ddt∫‍|∇u|2dx+∫‍|Δu|2dx=∫‍(u·∇)u·Δudx+∫‍div(∇Q⊙∇Q)Δudx-∫‍div(QΔQ-ΔQQ)Δudx=:J1+J2+J3.Summing (21) and (22) and using the cancellation I1+J3=0, due to Lemma 3, we have
(23)12ddt∫‍|ΔQ|2+|∇u|2dx+∫‍|∇ΔQ|2+|Δu|2dx=I2+I3+I4+I5+I6+J1+J2.(i) Let (8) hold trueUsing the integration by parts,I2 can be bounded as
(24)I2=∑i‍∫‍∂iQ∂i(ΩΔQ)-∂iΩ∂iQΔQ-Ω∂iQ∂iΔQdx≤C∥∇Q∥Lq∥Ω∥L2q/(q-2)∥∇ΔQ∥L2+C∥∇Q∥Lq∥∇Ω∥L2∥ΔQ∥L2q/(q-2)≤C∥∇Q∥Lq∥∇u∥L21-d/q∥Δu∥L2d/q∥∇ΔQ∥L2+C∥∇Q∥Lq∥Δu∥L2∥ΔQ∥L21-d/q∥∇ΔQ∥L2d/q≤C∥∇Q∥Lq(∥∇u∥L2+∥ΔQ∥L2)1-d/q×(∥Δu∥L2+∥∇ΔQ∥L2)1+d/q≤132∥Δu∥L22+132∥∇ΔQ∥L22+C∥∇Q∥Lqp×(∥∇u∥L22+∥ΔQ∥L22).
Here p and q satisfy the relation (8), and we have used the Gagliardo-Nirenberg inequality
(25)∥w∥L2q/(q-2)≤C∥w∥L21-d/q∥∇w∥L2d/q.Similarly, we get(26)I3,I4,I5,J2≤132∥Δu∥L22+132∥∇ΔQ∥L22+C∥∇Q∥Lqp(∥∇u∥L22+∥ΔQ∥L22).By using (16), I6 is simply bounded as
(27)I6≤C∥ΔQ∥L22.
Here we treat the term
(28)∫‍b∇Q∇QΔQdx≤C∥∇Q∥L42∥ΔQ∥L2≤C∥Q∥L∞∥ΔQ∥L22≤C∥ΔQ∥L22
by the Gagliardo-Nirenberg inequality
(29)∥∇Q∥L42≤C∥Q∥L∞∥ΔQ∥L2.Inserting the above estimates into (23), we derive
(30)12ddt∫‍|ΔQ|2+|∇u|2dx+12∫‍|∇ΔQ|2+|Δu|2dx≤J1+C∥∇Q∥Lqp(∥∇u∥L22+∥ΔQ∥L22)+C∥ΔQ∥L22.Now we estimateJ1 as follows.(1) Let (5) hold trueWe will use the following inequality [6]:
(31)∥u·∇u∥L2≤C∥u∥B˙∞,∞-s∥∇u∥H˙s,
and the Gagliardo-Nirenberg inequality
(32)∥∇u∥H˙s≤C∥∇u∥L21-s∥Δu∥L2s.Using (31) and (32), we bound J1 as follows:
(33)J1≤∥u·∇u∥L2∥Δu∥L2≤C∥u∥B˙∞,∞-s∥∇u∥L21-s∥Δu∥L2s≤116∥Δu∥L22+C∥u∥B˙∞,∞-s2/(1-s)∥∇u∥L22.
Substituting the above estimates into (30), we reach
(34)∥u∥L∞(0,T;H1)+∥u∥L2(0,T;H2)≤C,∥Q∥L∞(0,T;H2)+∥Q∥L2(0,T;H3)≤C.This completes the proof.(2) Let (6) hold trueUsing the following elegant inequality [17, 18]:
(35)∥∇u∥L42≤C∥u∥B˙∞,∞0∥Δu∥L2,
we bound J1 as follows:
(36)J1=∑i,j‍∫‍ui∂iu∂j2udx=-∑i,j‍∫‍∂jui∂iu∂judx≤C∥∇u∥L42∥∇u∥L2≤C∥u∥B˙∞,∞0∥∇u∥L2∥Δu∥L2≤116∥Δu∥L22+C∥u∥B˙∞,∞02∥∇u∥L22.Substituting the above estimates into (30), we have (34).This completes the proof.(3) Let (7) hold trueLet{ϕj}j∈ℤ be the Littlewood-Paley dyadic decomposition of unity that satisfies ϕ^∈C0∞(B2∖B1/2),ϕ^j(ξ)=ϕ^(2-jξ), and ∑j∈ℤϕ^j(ξ)=1 for any ξ≠0, where ϕ^ is the Fourier transform and Br is the ball with radius r centered at the origin.We decompose∂ju as follows:
(37)∂ju=∑k=-∞∞‍ϕk*∂ju=∑k<-N‍ϕk*∂ju+∑k=-NN‍ϕk*∂ju+∑k>N‍ϕk*∂ju,
where N is a positive integer to be chosen later. Plugging this decomposition into J1, we derive
(38)J1=-∑i,j‍∫‍∂jui∂iu(∑k<-N‍ϕk*∂ju+∑k=-NN‍ϕk*∂ju=-∑i,j‍∫‍∂jui∂iu+∑k>N‍ϕk*∂ju)dx=:J11+J12+J13.Recalling Bernstein’s inequality,(39)∥ϕk*f∥Lq≤C2dk(1/p-1/q)∥ϕk*f∥Lp,1≤p≤q≤∞,
with C being a positive constant independent of f and k, we apply Hölder’s inequality to deduce that
(40)J11≤∑i,j‍∥∂jui∥L2∥∂iu∥L2∑k<-N‍∥ϕk*∂ju∥L∞J11≤C∥∇u∥L22∑k<-N‍2(d/2)k∥ϕk*∇u∥L2J11≤C2-(d/2)N∥∇u∥L23,J12≤∑i,j‍∥∂jui∥L2∥∂iu∥L2∑k=-NN‍∥ϕk*∂ju∥L∞J12≤C∥∇u∥L22·N∥∇u∥B˙∞,∞0,J13≤∑i,j‍∥∂jui∥L3∥∂iu∥L3∑k>N‍∥ϕk*∂ju∥L3J13≤C∥∇u∥L32∑k>N‍2(d/6)k∥ϕk*∇u∥L2J13≤C∥∇u∥L32(∑k>N‍22(d/6-1)k)1/2J13×(∑k>N‍22k∥ϕk*∇u∥L22)1/2J13≤C∥∇u∥L322-(1-(d/6)N)∥Δu∥L2J13≤C2-(1-d/6)N∥∇u∥L22(1-d/6)∥Δu∥L21+d/3.Now we chooseN so that C2-(1-d/6)N∥∇u∥L22(1-d/6)≤1/16 and 2-(d/2)N∥∇u∥L2≤1/16 to conclude that
(41)J1≤C∥∇u∥L22+C∥∇u∥L22∥∇u∥B˙∞,∞0log(e+∥∇u∥L2+∥ΔQ∥L2)+116∥Δu∥L22+C.
Substituting the above estimates into (30), we arrive at (34).This completes the proof of part (i).(ii) Let (9) hold trueWe still have (23).I
2 is simply bounded as
(42)I2≤C∥Ω∥Lq∥ΔQ∥L2q/(q-1)2≤C∥∇u∥Lq∥ΔQ∥L22(1-3/2q)∥∇3Q∥L23/q≤132∥∇3Q∥L22+C∥∇u∥Lqp∥ΔQ∥L22.
Here we have used the Gagliardo-Nirenberg inequality
(43)∥w∥L2q/(q-1)≤C∥w∥L21-3/2q∥∇w∥L23/2q.I
3 is simply bounded as
(44)I3=-2∑i‍∫‍(Ω∂i(∂iQΔQ)-∂i2QΩΔQ-∂iQΩ∂iΔQ)dx≤C∥Ω∥Lq(∥ΔQ∥L2q/(q-1)2+∥∇Q∥L2q/(q-2)∥∇3Q∥L2)≤C∥∇u∥Lq∥ΔQ∥L22(1-3/2q)∥∇3Q∥L23/q≤132∥∇3Q∥L22+C∥∇u∥Lqp∥ΔQ∥L22.
Here we have used (43) and the Gagliardo-Nirenberg inequality
(45)∥w∥L2q/(q-2)≤C∥∇w∥L22-3/q∥Δw∥L23/q-1.
Similarly, I4,I5,J2, and J1 can be bounded as follows:
(46)I4+I5+J2≤C∥∇u∥Lq(∥ΔQ∥L2q/(q-1)2+∥∇Q∥L2q/(q-2)∥∇3Q∥L2)≤C∥∇u∥Lq∥ΔQ∥L22-3/q∥∇3Q∥L23/q≤132∥∇3Q∥L22+C∥∇u∥Lqp∥ΔQ∥L22,J1=-∑i,j‍∫‍∂iuj∂iu∂judx≤C∥∇u∥Lq∥∇u∥L2q/(q-1)2≤C∥∇u∥Lq∥∇u∥L22(1-3/2q)∥Δu∥L23/q≤132∥Δu∥L22+C∥∇u∥Lqp∥∇u∥L22.I
6 is bounded as above.Inserting the above estimates into (23), using the Gronwall inequality, we arrive at (34).This completes the proof.



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