Regularity Criterion for the 3D Nematic Liquid Crystal Flows
Jishan Fan1and Tohru Ozawa2
Academic Editor: G. A. Seregin, Y. Liu, A. Carpio
Received17 Jan 2012
Accepted14 Feb 2012
Published11 Apr 2012
Abstract
We study the hydrodynamic theory of liquid crystals. We prove a logarithmically improved regularity criterion for two simplified Ericksen-Leslie systems.
1. Introduction
The hydrodynamic theory of liquid crystals was established by Ericksen and Leslie [1β4]. However, since the equations are too complicated, we consider the first simplified Ericksen-Leslie system:
which include the velocity vector , the scalar pressure being and the direction vector . with , a positive constant. , and hence .
Lin-Liu [5] proved that the system (1.1)β(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 denotes the homogeneous Besov space.
The first aim of this paper is to prove a new regularity criterion as follows.
Theorem 1.1. Let with in . Let be a smooth solution to the problem (1.1)β(1.4) on . If satisfies
for some with , then the solution can be extended beyond .
When the penalization parameter , (1.1)β(1.4) reduce to
When , then (1.9) is the well-known harmonic heat flow equation onto a sphere.
Fan-Gao-Guo [8] proved the following blow-up criteria:
We will prove the folowing theorem
Theorem 1.2. Let with in . Let be a smooth solution to the problem (1.7)β(1.10) on . If the following condition is satisfied:
for some with , then the solution can be extended beyond .
This section is devoted to the proof of Theorem 1.1. Since it is well-known that there are and a unique smooth solution to the problem (1.1)β(1.4) in , we only need to show a priori estimates.
Here we used the Gagliardo-Nirenberg inequality
Inserting the above estimates into (2.10), we derive
Due to (1.6), one concludes that for any small constant , there exists such that
For any , we set
Applying Gronwallβs inequality to (2.16) in the interval , one has
Now, we derive a bound on defined by (2.18). To this end, we will use the following commutator and product estimates due to Kato-Ponce [10]:
with and .
Applying to (1.1), testing by , and using (1.3), (2.20), (2.21) and (2.19), we obtain
Here we have used the following Gagliardo-Nirenberg inequalities:
Taking to (1.2), testing by , and using (1.3), (2.20), (2.23), and (2.6), we have
Summing up (2.22) and (2.24) and taking small enough, we arrive at
In this section, we will prove Theorem 1.2. Since it is easy to prove that there are and a unique smooth solution to the problem (1.7)β(1.10) in , we only need to prove a priori estimates.
First, as in the previous section, we still have (2.1).
Similarly to (2.10), we have
Here , and are the same as that in (2.10) and can be bounded as in the previous section. The corresponding last term is written and bounded as
Here we have used the following inequality [11, 12]:
and the Gagliardo-Nirenberg inequality
Substituting the above estimates into (3.3), we obtain
Due to (1.12), one concludes that for any small constant , there exists such that
Applying Gronwallβs inequality to (3.7) in the interval , one has (2.19).
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