Regularity Criteria for a Coupled Navier-Stokes and Q-Tensor System

where ?̇?s p,q 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.

Our proof uses an energy method and relies on a simple  ∞ estimate of  and the following cancellation property: Lemma 3 (see [8]).Let Q,  : R  → R × be symmetric matrix-valued functions and let

Proof of Theorem 1
This section is devoted to the proof of Theorem 1.Since it is easy to prove that there are  0 > 0 and a unique strong solution (, ) to the problem ( 1)-( 4) in (0,  0 ], we only need to prove a priori estimates.
First, we prove the following key estimate: To prove ( 16), we multiply (1) by 2 tr −1 ( 2 ) and take the trace to obtain Let us observe that for , a traceless, symmetric, 3 × 3 matrix, we have Integrating over R  , integrating by parts, and using (3), (18), and the assumption  > 0, we obtain which gives with  independent of .
Taking Δ to (1), testing scalarly by Δ, and using (3); we find that Here "testing scalarly by Δ" means multiplying with respect to the Frobenius inner product of matrices,  :  = tr() and integrating over R  .Testing (2) by −Δ and using (3), we infer that Summing ( 21) and ( 22) and using the cancellation  1 + 3 = 0, due to Lemma 3, we have (i) Let (8) hold true.Using the integration by parts,  2 can be bounded as Here  and  satisfy the relation ( 8), and we have used the Gagliardo-Nirenberg inequality Similarly, we get By using ( 16),  6 is simply bounded as Here we treat the term by the Gagliardo-Nirenberg inequality Inserting the above estimates into (23), we derive Now we estimate  1 as follows.
2 is simply bounded as Here we have used the Gagliardo-Nirenberg inequality 3 is simply bounded as Inserting the above estimates into (23), using the Gronwall inequality, we arrive at (34).
This completes the proof.