# Continuation Criterion for the 2D Liquid Crystal Flows

Authors: Jishan Fan; Tohru Ozawa
Journal: ISRN Mathematical Analysis
 (2012)

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## Abstract

We consider the 2D liquid crystal systems, which consists of Navier-Stokes system coupled with wave maps or biharmonic wave maps, respectively. By logarithmic Sobolev inequalities, we obtain a blow-up criterion∇d,∂td∈L1(0,T;B˙∞,∞0(ℝ2)) for the case with wave maps, and we prove the existence of a global-in-time strong
solutions for the case with biharmonic wave maps.


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## Body

## 1. Introduction

First, we consider the following simplified liquid crystal flows in two space dimensions [1]:(1.1)∂tu+u⋅∇u+∇π-Δu=∑k∂tdk⋅∇dk,(1.2)div⁡u=0,(1.3)∂t2d+u⋅∇d-Δd=d(|∇d|2-|∂td|2),|d|=1,(1.4)(u,d,∂td)(x,0)=(u0,d0,d1)(x),x∈R2,|d0|=1,d0⋅d1=0,
where u is the velocity, π is the pressure, and d represents the macroscopic average of the liquid crystal orientation field with values in the unit circle.The first two equations (1.1) and (1.2) are the well-known Navier-Stokes system with the Lorentz force ∑k∂tdk·∇dk. The last equation (1.3) is the well-known wave maps when u=0.It is a simple matter to show that the system (1.1)–(1.4) has a unique local-in-time smooth solution when u0,∇d0,d1∈H1+s(ℝ2) with s>0,divu0=0,|d0|=1,d0·d1=0 in ℝ2. The aim of this paper is to study the regularity criterion of smooth solutions to the problem (1.1)–(1.4). We will prove the following.Theorem 1.1.
Letu0,∇d0,d1∈H1+s(ℝ2) with s>0,divu0=0,|d0|=1,d0·d1=0 in ℝ2 and let (u,d) be a smooth solution of (1.1)–(1.4) on some interval [0,T] with 0<T<∞. Assume that
(1.5)∇d,∂td∈L1(0,T;Ḃ∞,∞0(R2)).
Then the solution (u,d) can be extended beyond T>0.Ḃ∞,∞0 is the homogeneous Besov space. We have L∞⊂BMO⊂Ḃ∞,∞0; see Triebel [2].In the proof of Theorem1.1, we will use the logarithmic Sobolev inequalities [3–6]:(1.6)‖u‖L∞≤C‖u‖H1log⁡1/2(e+‖u‖H1+s),(1.7)‖∇d‖L∞≤C(1+‖∇d‖Ḃ∞,∞0log⁡(e+‖∇d‖H1+s)),(1.8)‖∂td‖L∞≤C(1+‖∂td‖Ḃ∞,∞0log⁡(e+‖∂td‖H1+s)),
for s>0, and the Gagliardo-Nirenberg inequalities:(1.9)‖w‖L4≤C‖w‖L2α‖Λ1+sw‖L21-α,‖Λsw‖L4≤C‖w‖L21-α‖Λ1+sw‖L2α,
with Λ∶=(-Δ)1/2,α∶=1-(1/2)·(1/(1+s)), and s>0, and the product estimate due to Kato-Ponce [7]:(1.10)‖Λs(fg)‖Lp≤C(‖f‖Lp1‖Λsg‖Lq1+‖g‖Lp2‖Λsf‖Lq2),
with s>0 and 1/p=1/p1+1/q1=1/p2+1/q2.Motivated by the problem (1.1)–(1.4), we consider the following liquid crystal flows:(1.11)∂tu+u⋅∇u+∇π-Δu=∑k∂tdk⋅∇dk,(1.12)div⁡u=0,(1.13)∂t2d+u⋅∇d+(-Δ)2d=-λd,|d|=1,(1.14)λ∶=|∂td|2+|Δd|2+Δ|∇d|2+2∑k∂kd⋅Δ∂kd,(1.15)(u,d,∂td)(x,0)=(u0,d0,d1)(x,0),x∈R2,|d0|=1.
The last two equations (1.13) and (1.14) are the biharmonic wave maps. It is also a simple matter to show that the problem (1.11)–(1.15) has at least one local-in-time strong solution. The aim of this paper is to prove the global-in-time regularity. We obtain the following.Theorem 1.2.
Letu0∈H2,(∇d0,d1)∈H3×H2 with divu0=0,|d0|=1,d0·d1=0 in ℝ2. Then there exists at least a global-in-time smooth solution:
(1.16)(u,∇d,∂td)∈L∞(0,T;H2)×L∞(0,T;H3)×L∞(0,T;H2)
for any T>0.Remark 1.3.
We are unable to prove the uniqueness of strong solutions in Theorem1.2.


## 2. Proof of Theorem1.1

We only need to prove a priori estimates.Testing (1.1) by u, using (1.2), we see that(2.1)12ddt∫u2dx+∫|∇u|2dx=∫(u⋅∇)d⋅∂tddx.Testing (1.3) by ∂td, using |d|=1 and d·∂td=0, we find that(2.2)12ddt∫|∇d|2+|∂td|2dx=-∫(u⋅∇)d⋅∂tddx.Summing up (2.1) and (2.2), we get(2.3)12ddt∫u2+|∇d|2+|∂td|2dx+∫|∇u|2dx=0,
from which we get(2.4)∫u2+|∇d|2+|∂td|2dx+∫0T∫|∇u|2dxdt≤C.ApplyingΛ1+s to (1.1), testing by Λ1+su, using (1.2) and (1.10), we derive(2.5)12ddt∫|Λ1+su|2dx+∫|Λ2+su|2dx=-∫Λ1+sdiv⁡(u⊗u)⋅Λ1+sudx+∫Λ1+s(∂td⋅∇d)⋅Λ1+sudx≤C‖u‖L∞‖Λ2+su‖L2‖Λ1+su‖L2+C(‖∂td‖L∞‖Λ2+sd‖L2+‖∇d‖L∞‖Λ1+s∂td‖L2)‖Λ1+su‖L2≤12‖Λ2+su‖L22+C‖u‖L∞2‖Λ1+su‖L22+C‖(∂td,∇d)‖L∞(y2+‖Λ1+su‖L22),
where(2.6)y2∶=‖Λ1+s∂td‖L22+‖Λ2+sd‖L22.TakingΛ1+s to (1.3), testing by Λ1+s∂td, we have(2.7)12ddty2=∫Λ1+s[d(|∇d|2-|∂td|2)]⋅Λ1+s∂tddx-∫Λ1+s(u⋅∇d)⋅Λ1+s∂tddx=:I1+I2.By using (1.10), (2.4), and (1.9), I1 can be bounded as follows:(2.8)I1≤C[‖d‖L∞‖Λ1+s(|∇d|2-|∂td|2)‖L2+‖Λ1+sd‖L4(‖∇d‖L∞‖∇d‖L4+‖∂td‖L∞‖∂td‖L4)]‖Λ1+s∂td‖L2≤C[(‖∇d‖L∞‖Λ2+sd‖L2+‖∂td‖L∞‖Λ1+s∂td‖L2)+cyα(‖∇d‖L∞y1-α+‖∂td‖L∞y1-α)]‖Λ1+s∂td‖L2≤C‖(∂td,∇d)‖L∞y2.By using (1.10), I2 can be bounded as(2.9)I2≤C(‖u‖L∞‖Λ2+sd‖L2+‖∇d‖L∞‖Λ1+su‖L2)‖Λ1+s∂td‖L2≤C‖u‖L∞y2+C‖∇d‖L∞(y2+‖Λ1+su‖L22).Combining (2.5), (2.7), (2.8), and (2.9) and using (1.6), (1.7), (1.8), and the Gronwall lemma, we arrive at(2.10)‖u‖L∞(0,T;H1+s)+‖u‖L2(0,T;H2+s)≤C,‖(∇d,∂td)‖L∞(0,T;H1+s)≤C.This completes the proof.


## 3. Proof of Theorem1.2

For simplicity, we only present a priori estimates.First, we still have (2.1).Testing (1.13) by ∂td, using d·∂td=0, we have(3.1)12ddt∫|Δd|2+|∂td|2dx=-∫(u⋅∇)d⋅∂tddx.Summing up (2.1) and (3.1), we get(3.2)∫u2+|Δd|2+|∂td|2dx+∫0T∫|∇u|2dxdt≤C,12ddt∫|∇d|2dx=-∫Δd⋅∂tddx≤‖Δd‖L2‖∂td‖L2≤C,which yields(3.3)∫|∇d|2dx≤C.ApplyingΔ to (1.11), testing by Δu, using (1.2) and (1.10), we deduce that(3.4)12ddt∫|Δu|2dx+∫|∇Δu|2dx=-∫Δdiv⁡(u⊗u)⋅Δudx+∫Δ(∂td,∇d)⋅Δudx≤C‖u‖L∞‖∇Δu‖L2‖Δu‖L2+C(‖∂td‖L∞‖∇Δd‖L2+‖∇d‖L∞‖Δ∂td‖L2)‖Δu‖L2≤12‖∇Δu‖L22+C‖u‖L∞2‖Δu‖L22+C(‖∂td‖L21/2‖Δ∂td‖L21/2‖Δd‖L21/2‖Δ2d‖L21/2+‖∇d‖L∞‖Δ∂td‖L2)‖Δu‖L2≤12‖∇Δu‖L22+C‖u‖L∞2‖Δu‖L22+Cy2+C‖Δu‖L22+C‖∇d‖L∞(y2+‖Δu‖L22),
where(3.5)y2∶=‖Δ∂td‖L22+‖Δ2d‖L22.ApplyingΔ to (1.13), we have(3.6)Δ∂t2d+Δ3d=-(λΔd+2∇λ⋅∇d+dΔλ)-Δ(u⋅∇d).Since(3.7)0=Δ(d∂td)=dΔ∂td+∂tdΔd+2∑k∂kd∂k∂td,
we easily see that(3.8)-dΔ∂td=∂tdΔd+∂t|∇d|2.Testing (3.6) by Δ∂td, using (3.8), we obtain(3.9)12ddty2=-∫λΔd⋅Δ∂td+∇λ[2∇d⋅Δ∂td+∇(∂tdΔd+2∇d⋅∇∂td)]dx-∫Δ(u⋅∇d)⋅Δ∂tddx=:J1+J2.By the same calculations as those in [8], we have(3.10)J1≤C(1+‖∇d‖L∞)y2≤C(1+‖∇d‖H1log⁡(e+y))y2.By using (1.10), J2 can be bounded as(3.11)J2≤C(‖Δu‖L2‖∇d‖L∞+‖u‖L∞‖∇Δd‖L2)‖Δ∂td‖L2≤C‖∇d‖L∞(y2+‖Δu‖L22)+C‖u‖L21/2‖Δu‖L21/2‖Δd‖L21/2‖Δ2d‖L21/2‖Δ∂td‖L2≤C‖∇d‖L∞(y2+‖Δu‖L22)+Cy2+C‖Δu‖L22.Combining (3.4), (3.9), (3.10), and (3.11) and using (1.6) and the Gronwall lemma, we conclude that(3.12)‖u‖L∞(0,T;H2)+‖u‖L2(0,T;H3)≤C,‖∇d‖L∞(0,T;H3)+‖∂td‖L∞(0,T;H2)≤C.
This completes the proof.



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*Source: 248473-2012-04-08.xml*