On the interior regularity criteria of the 3-D Navier-Stokes equations involving two velocity components

We present some interior regularity criteria of the 3-D Navier-Stokes equations involving two components of the velocity. These results in particular imply that if the solution is singular at one point, then at least two components of the velocity have to blow up at the same point.


Introduction
In this paper, we study the incompressible Navier-Stokes equations (N S) ∂ t u − ∆u + u · ∇u + ∇π = 0, divu = 0, where u(x, t), π(x, t) denote the velocity and the pressure of the fluid respectively. In a seminal paper [11], Leray proved the global existence of weak solution with finite energy. In two spatial dimensions, Leray weak solution is unique and regular. In three spatial dimensions, the regularity and uniqueness of weak solution is an outstanding open problem in the mathematical fluid mechanics. It was known that if the weak solution u of (1) satisfies so called Ladyzhenskaya-Prodi-Serrin(LPS) type condition u ∈ L q (0, T ; L p (R 3 )) with 2 q then it is regular in R 3 ×(0, T ), see [16,6,17,5], where the regularity in the class L ∞ (0, T ; L 3 (R 3 )) was proved by Escauriaza, Seregin andŠverák [5].
Concerning the partial regularity of weak solution, it was started by Scheffer [15], and later Caffarelli, Kohn and Nirenberg [1] showed that one dimensional Hausdorff measure of the possible singular set is zero. The proof relies on the following small energy regularity result: there exists some ε 0 > 0 so that if u is a suitable weak solution of the Navier-Stokes equations and satisfies sup R>0 1 R Q R (z) |∇u| 2 dxdt ≤ ε 0 , then u is regular at the point z ( i.e., u is bounded in a Q r (z) for some r > 0). Here and in what follows z = (x, t), Q R (z) = (−R 2 + t, t) × B R (x) and B R (x) is a ball of radius r centered at x. One could check [12,10,18,7,19] for the simplified proof and improvements.
Recently, there are many interesting works devoted to the LPS type criterions involving the partial components of the velocity, see [2,3,4,9,14] and references therein. The authors [20] considered the interior regularity criteria involving the partial components of the velocity. Let G(u, p, q; r) r x (Qr) . It was proved in [20] that if (u, π) is a suitable weak solution of (1) in Q 1 and satisfies sup 0<r<1 G(u 3 , p, q; r) < M for some M > 0, (2) and lim sup r→0 G(u h , p, q; r) = 0, where u h = (u 1 , u 2 ) and 1 ≤ 3 p + 2 q < 2, 1 < q ≤ ∞, then (0, 0) is a regular point. The goal of this paper is to get rid of the extra condition (2). Making full use of the structure of nonlinear term and divu = 0, we obtain the following interior regularity criteria involving two components of the velocity. Theorem 1.1 Let (u, π) be a suitable weak solution of (1) in R 3 × (−1, 0). If u satisfies one of the following three conditions: for some r 0 ∈ (0, 1), then u is regular at (0, 0).
The range of (p, q) can be extended if we impose a similar condition on the velocity in a cylinder domain. The proof relies on a new pressure decomposition formula.

Suitable weak solution and ε-regularity criterion
Let us first introduce the definition of suitable weak solution.
Definition 2.1 Let Ω ⊂ R 3 and T > 0. We say that (u, π) is a suitable weak solution of (1) in 2. the (NS) equation is satisfied in the sense of distribution; 3. the local energy inequality: for any nonnegative φ ∈ C ∞ c (R 3 × R) vanishing in a neighborhood of the parabolic boundary of Ω T , Let (u, π) be a solution of (1) and introduce the following scaling u λ (x, t) = λu(λx, λ 2 t), π λ (x, t) = λ 2 π(λx, λ 2 t), for any λ > 0, then the family (u λ , π λ ) is also a solution of (1). Let us introduce some invariant quantities under the scaling (3): We also introduce is the average of f in the ball B r (x 0 ). These scaling invariant quantities will play an important role in the interior regularity theory. For the simplicity, we denote Q r (0) by Q r and B r (0) by B r , and we will use the following notations: A(u, r, (0, 0)) = A(u, r), E(u, r, (0, 0)) = E(u, r).
Here and in what follows, we define a solution u to be regular at z 0 = (x 0 , t 0 ) if u ∈ L ∞ (Q r (z 0 )) for some r > 0. We recall the following ε-regularity result.
Proposition 2.2 [7] Let (u, π) be a suitable weak solution of (1) in Q 1 (z 0 ) and w = ∇ × u. There exists ε 1 > 0 such that if one of the following two conditions holds, then u is regular at z 0 .
3 Proof of Theorem 1.1 Throughout this section, we assume that (u, π) is a suitable weak solution of (1) in
Lemma 3.1 It holds that for any r ∈ (0, 1), Here C is a constant independent of r.
Proof. By scaling invariance, it suffices to consider the case of r = 1. By Hölder inequality and Sobolev interpolation inequality ( for example, see [1]), we get This gives the first inequality. The proof of the second inequality is similar.
In the following, we derive the local energy inequality. We denote Lemma 3.2 Let 0 < 4r < ρ < r 0 and 1 ≤ p, q ≤ ∞. Then we have where the constant C is independent of r, ρ, and π 1 , π 3 and ∂ 3 π 4 is given by Proof. Let ζ be a cutoff function, which vanishes outside of Q ρ and equals 1 in Q ρ 2 , and satisfies Define the backward heat kernel as Let φ = Γζ. Due to the local energy inequality and noting that (∂ t + △)Γ = 0, we obtain sup t Bρ It is easy to verify the following facts: where By Hölder inequality and Lemma 3.1, we have and using the facts that ∇ · u = 0 and 3 2p ′ + 2 2q ′ = 2, we get by integrating by parts and Hölder inequality that This gives that The main trouble comes from the term including the pressure. Let where We get by Hölder inequality that To deal with II 2 , recall that the pressure π satisfies We get by using ∇ · u = 0 that Consequently, we obtain Noting that 3 hence, Now the lemma follows by summing up the estimates of I, II 1 and II 2 .
The following lemma is devoted to the estimates of the pressure.
Proof. Let ζ be a cut-off function,which equals 1 in Q ρ 2 and vanishes outside of Q ρ . We decompose π 1 intoπ 1 +π 2 with By Calderon-Zygmund inequality, we have Then we get by Lemma 3.1 that The first equality of the lemma is proved. The proof of the second inequality is almost the same. Let us turn to the proof of the third inequality. Recall that π satisfies We decompose ∇ h π intoπ 1 +π 2 with By Calderon-Zygmund inequality, we have Noting that 3 p ′ + 2 q ′ = 4, we infer from Lemma 3.1 that The third inequality is proved. The proof of the fourth inequality is similar.
Proof. As in the proof of Lemma 3.1, we have from which and Calderon-Zygmund inequality, it follows that The lemma follows by taking suitable (s, l) and Hölder inequality. Now we are in position to prove Case 1 in Theorem 1.1. Given any ε > 0, there exists ρ ∈ (0, r 0 ) so that G(u h , p, q; ρ) ≤ ε.
Take r so that 0 < 8r < ρ < r 0 . It follows from Lemma 3.2 that where δ > 0 will be determined later. Let Then it follows from Lemma 3.3 that Take r = θρ with 0 < θ < 1 8 . The above inequality yields that We first choose θ small enough, then choose δ small, and finally choose ε small enough so that On the other hand, Lemma 3.3 and Lemma 3.4 imply that F (r 0 ) ≤ C with C depending on r 0 and u L ∞ (−1,0;L 2 (R 3 ))∩L 2 (−1,0;H 1 (R 3 )) . Then a standard iteration argument ensures that there exists r 1 > 0 such that F (r) ≤ ε 1 for any 0 < r < r 1 < r 0 , which implies Case 1 of Theorem 1.1 by Proposition 2.2.

Proof of Case 2 and Case 3
Let us claim that Case 2 and Case 3 in Theorem 1.1 can be deduced from the following theorem.
In what follows, we assume that 3 p + 2 q = 2, 1 < q < ∞. We denote by (p ′ , q ′ ) the conjugate index of (p, q). To prove Theorem 3.5, we need the following local energy inequality. Lemma 3.6 Let 0 < 4r < ρ < r 0 . It holds that where the constant C is independent of r, ρ.
Proof. Since the proof is very similar to Lemma 3.2, we only present a sketch. Using the same test function φ in the proof of Lemma 3.2, we have sup t Bρ where By Hölder inequality and ∇ · u = 0, we have and noting that ∂ 3 |u| 2 ≤ |∇u h ||u|, we get which along with Lemma 3.1 imply that where We have by Hölder inequality and Lemma 3.1 that The lemma follows by summing up the estimates of I, II 1 and II 2 .
The proof of the following lemma is similar to Lemma 3.3. So, we omit the details.
Lemma 3.7 It holds that for any 0 < 8r < ρ < r 0 , where the constant C is independent of r, ρ.
Take r > 0 so that 0 < 8r < ρ < r 0 . It follows from Lemma 3.6 that where δ > 0 will be determined later. Let Then it follows from Lemma 3.7 that Take r = θρ with 0 < θ < 1 8 . The above inequality yields that We first choose θ small enough, then choose δ small, finally choose ε small enough so that On the other hand, it is easy to see that with C depending on r 0 and u L ∞ (−1,0;L 2 (R 3 ))∩L 2 (−1,0;H 1 (R 3 )) . Then a standard iteration argument ensures that there exists r 1 > 0 such that F (r) ≤ ε 1 for all 0 < r < r 1 < r 0 .
which implies Theorem 3.5 by Proposition 2.2.

Proof of Theorem 1.2
Throughout this section, we assume that (u, π) be a suitable weak solution of (1) in R 3 × (−1, 0). Let us first introduce some notations.
r (x 0 ) is a ball of radius r centered at the horizontal part of x 0 . For the simplicity, we denote Q * r (0) by Q * r and B * r (0) by B * r . As in Section 2, we will still use the notations like A(u, r), E(u, r), G(f, p, q; r), H(f, p, q; r), G(f, p, q; r), H(π, p, q; r) etc. The differences are that here the integral domain is replaced by Q * r or B * r , and the mean value in G, H is taken only on B 2 r . We denote by (p ′ , q ′ ) the conjugate index of (p, q). Lemma 4.1 Let 0 < 4r < ρ < r 0 and 1 ≤ p, q ≤ ∞. We have where C is a constant independent of r, ρ.
Proof. Let ζ be a cutoff function, which vanishes outside of Q * ρ and equals 1 in Q * ρ 2 , and satisfies Define the backward heat kernel as Taking the test function φ = Γζ in the local energy inequality, and noting It is easy to verify that from which and Hölder inequality, it follows that This completes the proof of the lemma.
In the sequel, we assume that (p, q) satisfies Lemma 4.2 For any 0 < r < r 0 , we have where C is a constant independent of r.
Proof. Recall a well-known Sobolev's interpolation inequality (for example, see [1]): where 2 ≤ ℓ ≤ 6 and a = 3 4 (ℓ − 2). Applying (6) with ℓ = 2p ′ (Note that 2p ′ ≤ 6 since p ≥ 3 2 ) and a suitable localization, we get then the lemma follows by noting that aq ′ p ′ = 1 and − 2a p ′ + 2 q ′ = − 3 p − 2 q + 2 = 0. In the following, we will introduce a new pressure decomposition formula in a cylinder domain based on the following properties of harmonic function, which is new even for harmonic function to our knowledge. Lemma 4.3 Let f be a harmonic function in a cubic D 1 ⊂ R 3 . Let Then it holds that Proof. For |h| ≤ 1 5 , let and g(x) = f (x) − P 3 f (x 1 , x 2 ). It is easy to see that Since f is a harmonic function in D 1 , we have The gradient estimate of harmonic function yields that This proves that for any |h| ≤ 1 5 , The second inequality of (7) implies by Mean value theorem that given x ∈ B 1/2 , there exists h = h(x) with |h| ≤ 1 5 so that which along with (7) gives the first inequality of the lemma. The proof of the second inequality of the lemma is similar. LetH (π, p ′ , q ′ ; r) = r Lemma 4.4 For any 0 < 8r < ρ < r 0 , it holds that H(π, p ′ , q ′ ; r) ≤ C ρ r G (u, 2p ′ , 2q ′ ; ρ) 2 + C r ρ 2 p ′H (π, p ′ , q ′ ; ρ), where C is a constant independent of r, ρ.
which implies Theorem 1.2 by Proposition 2.2.