Regularity criterion for 3D Navier-Stokes Equations in Besov spaces

Several regularity criterions of Leray-Hopf weak solutions $u$ to the 3D Navier-Stokes equations are obtained. The results show that a weak solution $u$ becomes regular if the gradient of velocity component $\nabla_{h}{u}$ (or $ \nabla{u_3}$) satisfies the additional conditions in the class of $L^{q}(0,T; \dot{B}_{p,r}^{s}(\mathbb{R}^{3}))$, where $\nabla_{h}=(\partial_{x_{1}},\partial_{x_{2}})$ is the horizontal gradient operator. Besides, we also consider the anisotropic regularity criterion for the weak solution of Navier-Stokes equations in $\mathbb{R}^3$. Finally, we also get a further regularity criterion, when give the sufficient condition on $\partial_3u_3$.


Introduction
In the present paper, we address sufficient conditions for the regularity of weak solutions of the Cauchy problem for the Navier-Stokes equations in R 3 × (0, T ): where u = (u 1 , u 2 , u 3 ) : R 3 × (0, T ) → R 3 is the velocity field, p : R 3 × (0, T ) → R 3 is a scalar pressure, and u 0 is the initial velocity field, ν > 0 is the viscosity. We set ∇ h = (∂ x 1 , ∂ x 2 ) as the horizontal gradient operator and ∆ h = ∂ 2 x 1 + ∂ 2 x 2 as the horizontal Laplacian, and ∆ and ∇ are the usual Laplacian and the gradient operators, respectively. Here we use the classical notations and for sake of simplicity, we denote ∂ x i by ∂ i .
It is well known that the weak solution of the Navier-Stokes equations (1.1) is unique and regular in two dimensions. However, in three dimensions, the regularity problem of weak solutions of Navier-Stokes equations is an outstanding open problem in mathematical fluid mechanics. Strong solutions are known to exist for a short interval of time whose length depends on the initial data. Moreover, this strong solution is known to be unique and to depend continuously on the initial data (see, for example, [22], [24]). Let us recall the definition of Leray-Hopf weak solution. We set V = {φ : the 3D vector valued C ∞ 0 functions and ∇ · φ = 0}, which will form the space of test functions. Let H and V be the closure spaces of V under L 2 -topology, and under H 1 -topology, respectively.
Researchers are interested in the classical problem of finding sufficient conditions for weak solutions of (1.1) such that the weak solutions become regular, and the first result is usually referred as Prodi-Serrin conditions (see [20] and [21]), which states that if a weak solution u is in the class of then the weak solution becomes regular. The full regularity of weak solutions can also be proved under alternative assumptions on the gradient of the velocity ∇u. Specifically (see [3]), if then u is regular. In the marginal case r = ∞, H. Kozono and Y. Taniuchi [10] proved the regularity of weak solutions under the condition where BM O is the space of bounded mean oscillation defined by Recently, the study of the regularity of weak solution involving Besov space becomes popular. For example, by establishing the logarithmic Sobolev inequality in Besov spaces, H. Kozono, T. Ogawa and Y. Taniuchi [11] refined the above two conditions to Here and thereafter,Ḃ s p,q stands for the homogeneous Besov space, see Section 2 for the definition. On the other hand, Chen and Zhang in [5] proved regularity criterion by imposing only the two-component vorticity field. More precisely, they proved the regularity of weak solutions in the class of ω ∈ L q (0, T ;Ḃ 0 r,σ ) withω = (ω 1 , ω 2 , 0), In [12], H. Kozono and Y. Yatsu showed that if the Leray-Hopf weak solution u of (1.1) satisfies then u is regular. More generally, B. Yuan and B. Zhang in [25] prove the weak solution u became regular if ω satisfies ) for 0 < α < 1. As to the endpoint case, S. Gala in [8] showed that if , then the solution u was regular.
We point out that H. Kozono, T. Ogawa and Y. Taniuchi in [11] (Theorem 3.5) also got the full regularity of weak solutions under alternative assumptions on the velocity u. More precisely, if u satisfies then the solution u is regular. More generally, A. Cheskidov and R. Shvydkoy in [6] proved that the solution becames smooth if for some r ∈ (2, ∞), where B s p,q stands for the nonhomogeneous Besov space (for detail see [1]). Motivated by the mentioned above, in this article, we consider assumptions on the gradient of velocity ∇u or the gradient of velocity component ∇ h u (or ∇u 3 ).
Our main results can be stated in the following: Then u is regular.
By the definition of Besov space and Bernstein inequality (see (2.2) in section 2), we have (1.12) (ii) u satisfies the following condition Then u is regular.  (1.14) or, for any (small) positive real number ε, satisfies Then u is regular. In framework of the Lebesgue spaces, the regularity criterion problem has been in-deep study with the conditions in terms of one component ∇u 3 ( for example, see [26], [19]) or one directional derivative ∂ 3 u ( for example, see [18], [13]). Because of the embedding with 2 ≤ p < r, q ≤ ∞, a natural idea is to extend the regularity criterion results to the framework of Besov spaces. To our knowledge, there are some results in term of the whole velocity or the vorticity. Motivated by the literature [18], [13], we want to consider additional condition only on ∂ 1 u 3 , ∂ 2 u 3 , ∂ 3 u 3 instead of ∇u in the more general spaces. Our result reads as: Theorem 1.6. Let u 0 and u be as in Theorem 1.1. Suppose that the additional conditions of u are satisfied and Then u is regular.
Theorem 1.7. Let u 0 and u be as in Theorem 1.1. Suppose that u the additional condition Then u is regular.
Remark 1.8. By the embedding (1.16), we know that the condition (1.17) is corresponding to the endpoint case of the Prodi-Serrin conditions in the class of L q (0, T ; L p (R 3 )) with p = 3 and q = 2, which is consistent with (1.2). While the condition (1.18) is in consistent with (1.2), however, we give a same range of q with 2 < q < ∞ when 0 < s < 2 5 . Furthermore, if we provide the sufficient condition, only in terms of ∂ 3 u 3 , we shall have a more strict condition, which is shown in Theorem 1.7. One can see the corresponding q varies from 3 to ∞.
For the convenience, we recall the following version of the three-dimensional Sobolev and Ladyzhenskaya inequalities in the whole space R 3 (see, for example, [7], [14]). There exists a positive constant C such that for every u ∈ H 1 (R 3 ) and every r ∈ [2,6], where C is a constant depending only on r. Taking ∇div on both sides of (1.1) for smooth (u; p), one can obtain therefore, the Calderon-Zygmund inequality in R 3 (see [23]) holds, where C is a positive constant depending only on q. And there is another estimates for the pressure Recall also that if divu = 0 then the vorticity ω =curlu= ∇ × u has the following estimates (see [18]): can be reduced to (1.25) On the other hand, note that divω = 0, applying (1.23), we have (1.26) Therefore, by (1.23) and (1.25), we have

Preliminaries
We begin this section with some notations and Lemmas, which is useful for us to prove the main results. In order to define Besov spaces, we first introduce the Littlewood-Paley decomposition theory. Let S(R 3 ) be the Schwartz class of rapidly decreasing function, given f ∈ S(R 3 ), its Fourier transformation Ff =f is defined bŷ More generally, the Fourier transform of any f ∈ S ′ (R 3 ), the space of tempered distributions, is given by for any g ∈ S(R 3 ). The Fourier transform is a bounded linear bijection from S ′ to S ′ whose inverse is also bounded. We fix the notation Its dual is given by In other words, two distributions in S ′ h are identified as the same if their difference is a polynomial. Let us choose two nonnegative radial functions Let h = F −1 ϕ andh = F −1 χ, and then we define the homogeneous dyadic blocks∆ j and the homogeneous low-frequency cut-off operatorṠ j as follows: Informally,∆ j is a frequency projection to the annulus {|ξ| ∼ 2 j }, whileṠ j is a frequency projection to the ball {|ξ| 2 j }. And one can easily verify that∆ j∆k f = 0 if |j − k| ≥ 2. Especially for any f ∈ L 2 (R 3 ), we have the Littlewood-Paley decomposition: with the norm It is of interest to note that the homogeneous Besov spaceḂ s 2,2 (R 3 ) is equivalent to the homogeneous Sobolev spaceḢ s (R 3 ). The following Bernstein inequalities will be used in the next section.
Lemma 2.1. (see [1]) Let B be a ball and C an annulus. A constant C exists such that for any nonnegative integer k, and couple (p,q) Lemma 2.2. (see [1]) Let 1 ≤ q < p < ∞ and α be a positive real number. A constant C exists such that In particular, for β = 1, q = 2 and p = 4, we get α = 1 and and further, if we give the suitable values to parameters β, q, p, α, we get other inequalities, for example [1]) A constant C exists which satisfies the following properties. If s 1 and s 2 are real numbers such that s 1 < s 2 and θ ∈ (0, 1)), for any (p, r) ∈ [1, ∞] 2 and any f ∈ B s 1 p,r Ḃ s 2 p,r , then we have (2.6)

Proof of Main Results
In this section, under the assumptions of the Theorem 1.1, Theorem 1.3 or Theorem 1.6 in Section 1 respectively, we prove our main results. First of all, we note that, by the energy inequality, for Leray-Hopf weak solutions, we have (see, for example, [22], [24] for detail) for all 0 < t < T, where K 1 = u 0 2 L 2 . It is well known that there exists a unique strong solution u local in time if u 0 ∈ V . In addition, this strong solution u ∈ C([0, T * ); V ) ∩ L 2 (0, T * ; H 2 (R 3 )) is the only weak solution with the initial datum u 0 , where (0, T * ) is the maximal interval of existence of the unique strong solution. If T * ≥ T, then there is nothing to prove. If, on the other hand, T * < T, then our strategy is to show that the H 1 norm of this strong solution is bounded uniformly in time over the interval (0, T * ), provided additional conditions in Theorem 1.1, Theorem 1.3 or Theorem 1.6 in Section 1 are valid. As a result the interval (0, T * ) can not be a maximal interval of existence, and consequently T * ≥ T, which concludes our proof.
In order to prove the H 1 norm of the strong solution u is bounded on interval (0, T * ), combing with the energy equality (3.1), it is sufficient to prove where the constant C depends on T , K 1 . Proof of Theorem 1.1 Taking the curl on (1.1), we obtain We taking the inner product of above inequality with ω in L 2 (R 3 ), and by using of the Hölder's and Young's inequalities, as well as (1.23), (1.27) and (2.4), we obtain Absorbing the first term in right hand side and integrating the above inequality, we obtain Therefore, by Gronwall's inequality, one has By using of Gronwall's inequality and condition (1.10), we have Therefore, by (1.23) and (1.27), we get the H 1 norm of the strong solution u is bounded on the maximal interval of existence (0, T * ). This completes the proof of Theorem 1.1.
Proof of Theorem 1.3 Firstly, we deal with (i). Taking the inner product of the equation (1.1) with −∆ h u in L 2 (R 3 ), By integrating by parts a few times and using the incompressibility condition, we obtain Applying Hölder's and Young's inequalities, as well as (2.4), we have (3.6) Absorbing the first term in right hand side and integrating the above inequality, we obtain Next, we also use −∆u as test function, and get 1 2 The calculation has been shown in [26], for the convenience of readers, we list it below. By integrating by parts a few times and using the incompressibility condition, we get L 1 (t), L 2 (t), L 3 (t) as follows Therefore, by (1.20) and Hölder's inequalities, for every i (i = 1, 2, 3) we have and hance we have 1 2 Integrating (3.9), applying Hölder's inequality and combing (3.7) and (3.8), we obtain (3.10) By using of the Hölder's and Young's inequalities, it follows that Thanks again to the energy inequality, we get (3.12) Therefore, by using of Gronwall's inequality, we finally obtain by condition (1.12), we get the H 1 norm of the strong solution u is bounded on the maximal interval of existence (0, T * ). This completes the proof of (i). Now we prove (ii). Taking the inner product of the equation (1.1) with −∆ h u in L 2 (R 3 ), we have (see [4] for detail) By using of the Littlewood-Paley decomposition, we decompose u 3 as follows: where σ is a real number determined later, and [·] denotes the integer part of σ. Therefore, (3.13) becomes 1 2 In what following, we estimate I 1 (t) and I 2 (t). For I 1 (t), by using of the Hölder's and Young's inequalities, as well as Lemma 2.1, we have the last inequality, we use the fact that [σ] ≤ σ. As to I 2 (t), we take the same strategy to I 1 (t), by the definition of norm of the Besov space, for any 0 < ε < 1, we have Now, we choose σ such that Integrating (3.18), combing above two inequalities and the energy inequality (3.1) we have Integrating (3.9), applying Hölder's inequality and combing (3.1) and (3.19), we obtain (3.20) By using of the Hölder's and Young's inequalities, it follows that Thanks again to the energy inequality, we get If we set s = 1 − ε, then we have with 0 < s < 1.
Therefore, by using of Gronwall's inequality, we finally obtain by condition (1.13), we get the H 1 norm of the strong solution u is bounded on the maximal interval of existence (0, T * ). This completes the proof of Theorem 1.3.
Proof of Theorem 1.7 Firstly, we begin with (3.13), and by (1.20), we have where p and q satisfy Therefore, integrating above inequality and using Hölder's inequality, it follows that (3.37) For u 3 L q , we have the same estimate to (3.31), in which the parameters satisfy (3.32) and (3.38) Next, in view of (3.9), integrating (3.9), applying Hölder's and Young's inequalities and combing (3.37), we obtain  We finally get (3.41) In above inequality, we note that β and q satisfy the additional condition 4q β(3q − 10) < 1.
By using of Gronwall's inequality, we obtain