Regularity criterion for 3D nematic liquid crystal flows in terms of finite frequency parts in B ̇∞,∞−1$\dot{B}_{\infty,\infty }^{-1}$

In this paper, we establish the regularity criterion for the weak solution of nematic liquid crystal flows in three dimensions when the L∞(0, T ; Ḃ–1 ∞,∞)-norm of a suitable low frequency part of (u,∇d) is bounded by a scaling invariant constant and the initial data (u0,∇d0). Our result refines the corresponding one in (Liu and Zhao in J. Math. Anal. Appl. 407:557-566, 2013) and that in (Ri in Nonlinear Anal. TMA 190:111619, 2020).

In 2008, Fan and Guo [4] showed that, if u satisfies one of the following conditions: u ∈ L s 0, T;Ṁ p,q R 3 with 2 s then (u, d) is extended beyond t = T. Later Liu, Zhao and Cui [12] obtained the regularity criterion to the system (1.1) under the assumption that ∂ 3 u ∈ L β (0, T; L α ) with 2 β + 3 α ≤ 1, α > 3. Recently, Wei, Li and Yao [16] proved that, if the weak solution (u, d) satisfies then (u, b) can be extended beyond t = T. Liu and Zhao [13] proved that the solution (u, d) to (1.1) is smooth up to time T provided that When d = 0, the system (1.1) becomes an incompressible Navier-Stokes equation. There is a large literature on the regularity criteria on the Navier-Stokes equation; see [1,6,7,15].
By traditional turbulence theory, viscous incompressible flows develop in such a way that energy is transferred from large scales to neighboring smaller scales. Hence, it is important to study regularity for the Navier-Stokes equation based on various wave-number band parts of weak solutions is important since it reveals in a way the relationship between regularity of weak solutions and turbulent flows. Cheskidov and Shvydkoy [2] proved that a Leray-Hopf weak solution u to the Navier-Stokes equation is regular in where u k is high frequency part of u with Fourier models |ξ | ≥ k. Kim, Kwak and Yoo [5] proved that, if sufficiently high frequency parts of a weak solution to the Navier-Stokes equation on a torus belong to Serrin's class, then the weak solution is regular. Very recently, Ri [14] proved that a Leray-Hopf weak solution u to 3D Navier-Stokes equations is regular if the L ∞ (0, T; B -1 ∞,∞ (R 3 ))-norm of a suitable low frequency part of u is bounded by a scaling invariant constant depending on the kinematic viscosity ν and initial value u 0 . Motivated by [2,5,13] and [14], we will investigate the regularity criteria for the weak solution (u, d) to the liquid crystal fluid flows (1.1) in the critical function space L ∞ (0, T;Ḃ -1 ∞,∞ (R 3 )) based on low and medium frequency parts, respectively. Before stating our result, we shall present some symbols and notations. Let Here andû denotes Fourier transform of u. Our result is stated as follows.
The rest of this paper is organized as follows. Some useful facts are presented in Sect. 2. The proof of Theorem 1.2 is given in Sect. 3.

Preliminaries and some basic facts
In order to define Besov spaces, we first introduce the Littlewood-Paley decomposition theory. Let S(R n ) be the Schwartz class of rapidly decreasing functions.
For given f ∈ S(R n ), its Fourier transform F(f ) =f and its inverse Fourier transform F -1 (f ) =f are given bŷ respectively. Let us choose two nonnegative radial functions χ, ϕ ∈ S(R n ) satisfying supp χ ⊂ B = {ξ ∈ R n : |ξ | ≤ 4 3 } and supp ϕ ⊂ C = {ξ ∈ R n : 3 4 ≤ |ξ | ≤ 8 3 } such that j∈Z ϕ 2 -j ξ = 1, for any ξ ∈ R n \{0} and For j ∈ Z, the homogeneous Littlewood-Paley projection operators S j and˙ j are defined byṠ j is a frequency projection to the annulus {|ξ | ∼ 2 j }, andṠ j is a frequency projection to the ball with the norm On the other hand, we recall some facts that can be found in [14]. If u ∈ L 2 (R 3 ), then it follows from the definition of u k and u k that Moreover, for 0 ≤ r < s, by Plancherel's theorem, with some c > 0. Moreover, it can be easily seen that

Proof of Theorem 1.2
For convenience, we assume μ = λ = 1 throughout the proof of Theorem 1.2.
Proof Assume that a weak solution (u, d) of (1.1) is regular in (0, T), but not in (0, T]. Then Notice that, for all smooth solutions to system (1.1), one has the following basic energy law (see [10]): Thus, We can see from [13] that, if there exists a positive constant ε 0 > 0 such that then the solution (u, d) is smooth up to time T. Now we multiply the first equation of (1.1) withu k and integrate over R 3 to get by (2.2) Applying ∇ to the second equation of (1.1) and making an L 2 inner product with respect to ∇ d k , we can verify Adding (3.3) and (3.4) gives rise to (3.5) Next we estimate I 1 -I 5 , respectively. From [14], we have Then (3.7) Note that d k = d k 2 +d k 2 ,k and the Fourier transform of ( d k ·∇)d k is supported in {|ξ | ≤ 2k}, thus we deduce , u k,2k Similarly, which along with (3.9) implies With the help of Hölder's inequality, (2.4), Gagaliardo-Nirenberg's inequality, Sobolev's embedding and Young's inequality, one has (3.12) By the definition of theḂ -1 ∞,∞ -norm, we have From the Hölder inequality, (3.13) and the Young inequality, we can conclude that (3.14) Similarly, (3.15) Combining (3.7), (3.11), (3.12), (3.14) and (3.15), one arrives at (3.16) To estimate I 3 , we make the following decomposition: Then Applying the same method to the bound (3.12) gives rise to (3.18) Similarly to (3.8), we have (3.20) Similarly, (3.23) We now address the term I 4 . We decompose I 4 into the following form: we can get (3.26) The Hölder inequality, the Young inequality and (3.13) imply (3.27) Similarly, (3.28) Arguing as (3.8), we have , ∇ d k,2k := I 4141 + I 4142 .