Regularity Criterion for the 3 D Nematic Liquid Crystal Flows

We study the hydrodynamic theory of liquid crystals. We prove a logarithmically improved regularity criterion for two simplified Ericksen-Leslie systems.


Introduction
The hydrodynamic theory of liquid crystals was established by Ericksen and Leslie 1-4 .However, since the equations are too complicated, we consider the first simplified Ericksen-Leslie system: which include the velocity vector u : u 1 , u 2 , u 3 t , the scalar pressure π being and the direction vector d : Lin-Liu 5 proved that the system 1.1 -1.4 has a unique smooth solution globally in 2 space dimensions and locally in 3 dimensions.They also proved the global existence of weak solutions.However, the regularity of solutions to the system is still open.Fan-Guo 6 and Fan-Ozawa 7 showed the following regularity criteria: where Ḃ0 ∞,∞ denotes the homogeneous Besov space.
The first aim of this paper is to prove a new regularity criterion as follows.
for some s with 0 < s < 1, then the solution u, d can be extended beyond T > 0.
When the penalization parameter η → 0, 1.1 -1.4 reduce to When u 0, then 1.9 is the well-known harmonic heat flow equation onto a sphere.Fan-Gao-Guo 8 proved the following blow-up criteria:

1.11
We will prove the folowing theorem: Let u, d be a smooth solution to the problem 1.7 -1.10 on 0, T .If the following condition is satisfied: for some s with 0 < s < 1, then the solution u, d can be extended beyond T > 0.

Proof of Theorem 1.1
This section is devoted to the proof of Theorem 1.1.Since it is well-known that there are T 0 > 0 and a unique smooth solution u, d to the problem 1.1 -1.4 in 0, T 0 , we only need to show a priori estimates.Testing 1.1 by u and using 1.3 , we see that Testing 1.2 by Δd − f d and using 1.3 , we find that d dt Summing up 2.1 and 2.2 , we infer that

2.3
Testing 1.2 by d and using 1.3 , we deduce that 1 2 Next, we prove the following estimate: Without loss of generality, we assume that 1 ≤ d 0 L ∞ .Multiplying 1.2 by d, we get L ∞ ≤ 0. Then 2.6 follows immediately from φ ≤ 0 by the maximum principle.
Testing 1.1 by −Δu and using 1.3 , we see that

2.9
Summing up 2.8 and 2.9 , we get

2.10
By using 2.6 , I 4 is simply bounded as

2.11
By using the inequalities 9 2.12 ISRN Mathematical Analysis 5 I 1 can be bounded as follows:

2.13
We bound I 2 and I 3 as follows:

2.14
Here we used the Gagliardo-Nirenberg inequality

2.15
Inserting the above estimates into 2.10 , we derive

2.16
Due to 1.6 , one concludes that for any small constant > 0, there exists T * < T such that

2.22
Here we have used the following Gagliardo-Nirenberg inequalities:

2.24
Summing up 2.22 and 2.24 and taking small enough, we arrive at

2.25
This completes the proof.

Proof of Theorem 1.2
In this section, we will prove Theorem 1.2.Since it is easy to prove that there are T 0 > 0 and a unique smooth solution u, p, d to the problem 1.7 -1.10 in 0, T 0 , we only need to prove a priori estimates.First, as in the previous section, we still have 2.1 .Testing 1.9 by −Δd, using dΔd −|∇d|

3.3
Here I 1 , I 2 , and I 3 are the same as that in 2.10 and can be bounded as in the previous section.The corresponding last term I 5 is written and bounded as

3.4
Here we have used the following inequality 11, 12 : and the Gagliardo-Nirenberg inequality Substituting the above estimates into 3.3 , we obtain Δd 2 L 2 .

3.7
Due to 1.12 , one concludes that for any small constant > 0, there exists T * < T such that 3.9 J 1 is bounded as that in 2.24 ; 3.10 then J 2 can be bounded as that in 2.24 .Combining 2.22 and 3.9 and taking small enough, we conclude that u L ∞ 0,T ;H 3 u L 2 0,T ;H 4 ≤ C, ∇d L ∞ 0,T ;H 3 ∇d L 2 0,T ;H 4 ≤ C.

3.11
This completes the proof.