An improved regularity criterion for the Navier–Stokes equations in terms of one directional derivative of the velocity field

In this paper, we establish a new multiplicative Sobolev inequality. As applications, we refine and extend the results in Kukavica and Ziane (J Math Phys 48:065203, 2007) and Cao (Discrete Contin Dyn Syst 26:1141–1151, 2010) simultaneously.


Introduction
The homogeneous incompressible fluid flow is governed by the following Navier-Stokes equations (NSE): ⎧ ⎨ ⎩ ∂ t u + (u · ∇)u − u + ∇π = 0, ∇ · u = 0, where u = (u 1 , u 2 , u 3 ) is the fluid velocity field, π is a scalar pressure, u 0 is the prescribed initial velocity field satisfying the compatibility condition ∇ · u 0 = 0, and The global existence of a weak solution to the evolutionary NSE (1) has been long established by Leray [19] and Hopf [9]; however, the issue of its regularity and uniqueness remains open up to now. Pioneered by Serrin [26], we began studying the regularity criterion for the NSE (1); that is, to find some sufficient condition to ensure the smoothness of the solution. The classical Prodi-Serrin conditions (see [8,24,26]) says that if u ∈ L p (0, T ; L q (R 3 )), then the solution is regular on (0, T ). This was be generalized by Beirão da Veiga [1] by considering the velocity gradient or vorticity, Notice that the case q ∈ 3 2 , 3 follows directly from (2) and the Sobolev inequality.
In view of the divergence-free condition ∇ ·u = 0, it is natural to ask whether or not we can reduce (2) and (3) to its partial components. One way is to consider regularity criteria involving only one velocity component, which were done in [3,11,16,20,34,36]. Another way is to study the possible components reduction of ∇u to ∇u 3 , see [16,23,27,35,36]; or to ∂ 3 u, see [2,17,21,22]. In [22], Penel-Pokorný showed that if then the solution is smooth. This was improved by Kukavica-Ziane [17] to be Notice that the range of q is not of full range 3 2 , ∞ . The reason is that in [17], the estimate of I 3 needs to be reconciled with the the estimate of K . Furthermore, this method was adjusted by Penel-Pokorný [21] to get an anisotropic criterion. For readers interested in this topic, please refer to [12][13][14][15]30] for recent progresses on regularity criteria of the MHD equations, which contains system (1) as a subsystem. Later on, Cao [2] employed multiplicative Sobolev inequalities and L q (7) to get the following extended regularity condition 1 Notice that the lower and upper bounds of q in (8) both are less than those in (5) respectively. Consequently, our best knowledge in this direction is the following sufficient condition Some of them was proved in [17], while other parts could only be seen [2].
In this paper, we shall further generalize (7), and improve (5) and (8) simultaneously. We will show that the condition Before stating the precise result, let us recall the weak formulation of (1), see [7,18,25,28] for instance.

Now, our main result reads
then the solution u is smooth in (0, T ] × R 3 . The proof of Theorem 2 will be given in Sect. 2. Before doing that, let us state our notations used throughout the paper, and prove a multiplicative Sobolev inequality. For simplicity of presentation, we do not distinguish between the spaces X and their N -dimensional vector analogs X N (e.g., N = 3 for u ∈ L 2 (R 3 ), N = 9 for ∇u ∈ L 2 (R 3 )); however, all vector-and tensor-valued functions are printed boldfaced. A constant C may change from line to line, depending only on the initial data or the norms that we have controlled. We denote by Generalizing (7) in [2], we have the following where {i, j, k} is a permutation of {1, 2, 3}.
Proof By Newton-Leibniz formula, we have Taking the sqrt of the multiplication of the above inequalities, we deduce Integrating in the x i variable and applying Hölder inequality, we obtain Successively, integrating in the x j and x k variables and applying Hölder inequality, we get Invoking Hölder inequality again, we find Dividing both sides by f , we finished the proof of Lemma 3. 38 Z. Zhang
By (11), we may find a Γ < Γ * such that where 0 <ε 1 will be determined later on. For convenience, we rewrite the NSE (1) as

H 1 estimate
Taking the inner product of (13) 1 with − u h and (1) 1 with −∂ 3 ∂ 3 u in L 2 (R 3 ) respectively, we obtain By [17, Lemma 2.2], To simplify I 2 , we take the divergence of (1) 1 to get and thus Finally, integrating by parts yields Gathering (15), (17) and (18) into (14), we deduce By Hölder and Gagliardo-Nirenberg inequalities, For J 2 , we first use Hölder inequality with to estimate as then invoke the interpolation and Gagliardo-Nirenberg inequalities to bound as and λ ≥ 3 2 will be specified later on. By Lemma 3, Plugging (20) and (23) into (19), we find Integrating in time and denoting by we deduce where Hölder inequality with is used.

Remark 4
Cao [2] took λ = 2 to deduce (8), which corresponds to the range of q in case λ = 2 in (40). In our paper, we treat all the possibilities to make the range of q as large as possible. The method involves a generalized multiplicative Sobolev inequality (see Lemma 3) and the general L 2λ estimate, but not just L 4 estimate. For some applications of general L 2λ estimates, we refer to [10,32], which improves [5].