A New Regularity Criterion for the Navier–stokes Equations in Terms of the Two Components of the Velocity

This paper establishes a new regularity criterion for the Navier–Stokes equations in terms of two velocity components. We show that if the two velocity components u = (u 1 , u 2 , 0) satisfy T 0 ˜ u(s) 2 ˙ B 0 ∞,∞ ds < ∞, then the solution can be smoothly extended after t = T. This gives an answer to an open problem in [B.


Introduction
Consider the Navier-Stokes equations for the viscous incompressible fluid flow in the whole space R 3 : where u = u(x, t) is the velocity field, p = p(x, t) is the scalar pressure and u 0 (x) with div u 0 = 0 in the sense of distribution is the initial velocity field.For simplicity, we assume that the external force has a scalar potential and is included in the pressure gradient.
It is well-known that Navier-Stokes equations are an important mathematical model in fluid dynamics (see [12,22]).The question of global regularity for smooth solutions in the 3D case remains generally open.Therefore, it is interesting to consider the regularity criterion for solutions under some additional growth conditions.The research on this direction started from Serrin in the 1960's and attracted more and more attention over the last few decades (see e.g.[1,6,10,26,31,32] and the references therein).
Another approach is to consider the regularity condition in terms of the two velocity components u = (u 1 , u 2 , 0).In [4] (see also [2]), Bae and Choe proved that if then the solution is smooth on (0, T).Later on, Zhang et al. [33] extended the condition (1.2) into BMO space in the marginal case when q = ∞, i.e.
Based on the Littlewood-Paley decomposition of equations (1.1), Dong and Zhang [7] extended the regularity criterion by means of horizontal derivatives of the two velocity components ∇ h ũ = (∂ 1 ũ, ∂ 2 ũ, 0) in the homogeneous Besov space Ḃ0 ∞,∞ : It is still an open question, asked by Dong and Zhang [7, p. 2417], whether (1.4) can be replaced by the following condition The purpose of this paper is to give an answer to this problem posed in [7] and we claim that we obtain the regularity for weak solutions if the two components are in the sharp critical space Before giving the main result, we recall the following definition of Leray-Hopf weak solution.
Definition 1.1.Let u 0 ∈ L 2 (R 3 ) and ∇ • u 0 = 0.A measurable vector field u(x, t) is called a Leary-Hopf weak solution to the Navier-Stokes equations (1.1) on (0, T), if u satisfies the following properties: (iv) u satisfy the energy inequality Next, we recall the definition of the space that we are going to use (see e.g.[3,28]).
Definition 1.2.Let ϕ j j∈Z be the Littlewood-Paley dyadic decomposition of unity that sat- where B R is the ball in R 3 centered at the origin with radius R > 0. The homogeneous Besov spaces Ḃs p,q (R 3 ) are defined to be where S is the space of tempered distributions and P is the space of polynomials.
This definition ensures the following homogeneous property Throughout this paper, C will denote a generic positive constant which can vary from line to line.Our main result is the following.
then the solution can be extended after t = T.In other words, if the solution blows up at t = T, then Duality relation (H 1 ) * = BMO and the fact that ω.∇ω ∈ H 1 due to [5] play an important role of our apriori estimate, where H 1 denotes the Hardy space.For the proof of our result, we recall the following logarithmic Sobolev inequality in Besov spaces due to Kozono-Ogawa-Taniuchi [20, Theorem 2.1].

Lemma 1.4. Let s > 3
2 .There exists a constant C such that the estimate holds for all f ∈ H s R 3 .
In order to prove our main result, we need the following lemma.
Lemma 1.5.Assume that u = (u 1 , u 2 , u 3 ) is a smooth and divergence-free (∇.u = 0) vector field, let u = (u 1 , u 2 , 0).Then we have, for the generic constant C Proof.Due to the divergence-free condition ∇.u = ∑ 3 i=1 ∂ i u i = 0 and integration by parts, observe first that where we have used the fact We now estimate RHS.When i = 1, 2 or j = 1, 2, by integrating by parts, In the case i = j = 3, by using the fact −∂ 3 u 3 = ∂ 1 u 1 + ∂ 2 u 2 and integrating by parts, we have Combining the two inequalities with (1.8) yields The proof of Lemma 1.5 is complete.

Proof of Theorem 1.3
Now we are in a position to prove our main result.
Proof.We take L 2 -inner products on (1.1) with u, integrate by parts and use incompressibility to obtain 1 2 This identity allows us to get Next, multiplying the first equation in (1.1) by ∆u, after integration by parts and taking the divergence free property into account, we have by Lemma 1.5 We recall the following property of Hardy spaces and BMOs for any ∇.f = 0 and ∇ × g = 0.According to above inequality (2.2) and Young's inequality, we estimate Due to a fact that Ḃ0 ∞,2 ⊂ BMO, inserting the above estimates into (2.1),we derive 1 2 (2.4) For any T 0 < t ≤ T, we set z(t) := sup (2.5) By the Gronwall inequality on (2.4) for the interval [T 0 , t], one has ln(e + z(s)))ds .
Hence, we obtain from the above estimate ∇u(t) ≤ C 0 (e + z(t)) 2C , provided that for any small constant > 0, there exists T 0 < T such that here C 0 means a constant depending on T 0 .Now we do the estimate for z(t) defined by (2.5).Taking the operation Λ 3 = (−∆) 3 2 on both sides of (1.1), then multiplying them by Λ 3 u, after integrating over R 3 , we have 1 2 Noting that ∇ • u = 0 and integrating by parts, we write (2.7) as 1 2 In what follows, we will use the following inequality due to Kenig, Ponce and Vega [19]: for α > 1, and 1 p = 1 p 1 + 1 q 1 = 1 p 2 + 1 q 2 .Hence Π can be estimated as where we used (2.9) with α = 3, p = 3 2 , p 1 = q 1 = p 2 = q 2 = 3, and the following Gagliardo-Nirenberg inequalities L 2 , and  Gronwall's inequality implies the boundedness of H 3 -norm of u provided that < 1 13C , which can be achieved by the absolute continuous property of integral (1.5).This completes the proof of Theorem 1.3.