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.
1. Introduction
We consider the following coupled Navier-Stokes and Q-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.
When Q≡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 takes Q:=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.
Let u0∈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<∞.
If d=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.
If d=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 simple L∞ estimate of Q and the following cancellation property:
Lemma 3 (see [8]).
Let Q~,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 Theorem 1
This section is devoted to the proof of Theorem 1. 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 for Q, a traceless, symmetric, 3×3 matrix, we have
(18)tr(Q3)≤3ϵ8tr2(Q2)+1ϵtr(Q2),∀ϵ>0.
Integrating over ℝd, 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).
Taking Δ 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 true
Using 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 estimate J1 as follows.
(1) Let (5) hold true
We 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 true
Using 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 true
Let {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 choose N 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 true
We still have (23).
I2 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.
I3 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.
I6 is bounded as above.
Inserting the above estimates into (23), using the Gronwall inequality, we arrive at (34).
This completes the proof.
Acknowledgments
This paper is supported by NSFC (no. 11171154). The authors are indebted to the referee for nice suggestions which improved the paper.
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