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Global regularity criterion for the 3D Navier-Stokes equations involving one entry of the velocity gradient tensor

In this paper we provide a sufficient condition, in terms of only one of the nine entries of the gradient tensor, i.e., the Jacobian matrix of the velocity vector field, for the global regularity of strong solutions to the three-dimensional Navier-Stokes equations in the whole space, as well as for the case of periodic boundary conditions.


Introduction
The three-dimensional Navier-Stokes equations (NSE) of viscous incompressible fluid read: u(x 1 , x 2 , x 3 , 0) = u 0 (x 1 , x 2 , x 3 ), where u = (u 1 , u 2 , u 3 ), the velocity field, and p, the pressure, are the unknowns, and ν > 0, the viscosity, is given. We set ∇ h = (∂ x1 , ∂ x2 ) to be the horizontal gradient operator and ∆ h = ∂ 2 x1 + ∂ 2 x2 the horizontal Laplacian, while ∇ and ∆ are the usual gradient and the Laplacian operators, respectively. In this paper we consider finite energy solutions of the system (1)- (3) in the whole space R 3 , that decay at infinity. However, we remark that one can apply our proof, nearly line by line, to establish same result for the three-dimensional Navier-Stokes equations in a periodic domain.
The question of global regularity for the 3D Navier-Stokes equations is a major challenging problem in applied analysis. Over the years there has been an intensive work by many authors attacking this problem (see, e.g., [6], [7], [9], [19], [21], [22], [24], [25], [32], [35], [36], [37] and references therein). It is well-known that the 2D Navier-Stokes equations have a unique weak and strong solutions which exist globally in time (cf., for example, [7], [19], [32], [35], [36]). In the 3D case, the weak solutions are known to exist globally in time. But, the uniqueness, regularity, and continuous dependence on initial data for weak solutions are still open problems. Furthermore, strong solutions in the 3D case are known to exist for a short interval of time whose length depends on the physical data of the initial-boundary value problem. Moreover, this strong solution is known to be unique and depend continuously on the initial data (cf., for example, [7], [19], [32], [35]).
We denote by L q and H m the usual L q −Lebesgue and Sobolev spaces, respectively (cf. [1]), and by We set V = {φ : the three-dimensional 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 in L 2 under L 2 −topology, and in H 1 under H 1 −topology, respectively. Let u 0 ∈ H, we say u is a Leray-Hopf weak solution to the system (1)-(3) on the interval [0, T ] with initial value u 0 if u satisfies the following three conditoins: (2) the weak formulation of the NSE: for every test function φ ∈ C ∞ ([0, T ], V), and for almost every t, t 0 ∈ [0, T ]; (3) the energy inequality: for every t and almost every t 0 . Moreover, if u 0 ∈ V , a weak solution is called strong solution of (1)- In this case, one also has energy equality in (6) instead of inequality, and the equality holds for every t 0 .
In this paper, we provide sufficient conditions, in terms of only one of the nine components of the gradient of velocity field, i.e., the velocity Jacobian matrix, that guarantee the global regularity of the 3D NSE. Specifically, if u 0 ∈ V , and if for some T > 0 and some k, j, with 1 ≤ k, j ≤ 3, we have or where u = (u 1 , u 2 , u 3 ) is a weak solution with the initial datum u 0 on [0, T ], then u is a strong solution of the 3D Navier-Stokes equations which exists on the interval [0, T ]. Moreover, u is the only weak and strong solution on the interval [0, T ] with the initial datum u 0 . In particular, if (7) or (8) holds for all T > 0, then there is a unique global (in time) strong solution for the 3D NSE with the initial datum u 0 .
For convenience, we recall the following version of the three-dimensional Sobolev and Ladyzhenskaya inequalities in the whole space R 3 (see, e.g., [1], [7], [12], [20]). There exists a positive constant C r such that for every ψ ∈ H 1 (R 3 ) and every r ∈ [2,6]. Observe that in case of periodic boundary conditions, one would have instead of (9) the following inequality for every ψ ∈ H 1 (Ω) and every r ∈ [2,6]. Here, Ω is the periodic box [0, L] 3 . We remark that one can apply inequality (10) instead of inequality (9), and the methods presented in this paper to establish the same results in the case of periodic boundary conditions. The details of the proof in the periodic case are omitted.

The Main Result
In this section we will prove our main result, which states that the strong solution to system (1)-(3) exists on the interval [0, T ] provided the assumption (7) or (8) holds. Theorem 1. Let u 0 ∈ V, and let u = (u 1 , u 2 , u 3 ) be a Leray-Hopf weak solution to 3D NSE, system (1)-(3), with the initial value u 0 . Let T > 0, and suppose that, for some k, j, with 1 ≤ k, j ≤ 3, u satisfies the condition (7) or (8), namely, for some M > 0. Then u is a strong solution of 3D NSE, system (1) Before we prove the main Theorem 1, we show the following two lemmas.
Proof. Observe, first, that it is enough to prove the inequality for functions φ, f, g ∈ C ∞ 0 (R 3 ) and then passing to the limit using a density argument.
Proof. Here, again, it is enough to prove the inequality for functions φ, f, g ∈ C ∞ 0 (R 3 ).
Proof of the Theorem 1. Without loss of generality, we will assume that j = 3 and k = 1 in (11) and (12), namely, or, It is well-known that there exists a global in time Leray-Hopf weak solution to the 3D NSE, the system (1)-(3), in whole space R 3 if u 0 ∈ H (see, e.g., [7], [21], [23], [25], [32], [36]). It is also well-known that there exists a unique strong solution for a short time interval if u 0 ∈ V . In addition, this strong solution is the only weak solution, with the initial datum u 0 , on the maximal interval of existence of the strong solution.
Suppose that u is the strong solution with the initial value u 0 ∈ V such that u ∈ C([0, T * ), V )∩L 2 ([0, T * ), H 2 ), 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 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 condition (15) or (16) is valid. As a result the interval [0, T * ) can not be a maximal interval of existence, and consequently T * ≥ T . Which will conclude our proof.
From now on we focus on the strong solution, u, on its maximal interval of existence [0, T * ), where we assume that T * < T. As we have observed earlier the strong solution u will also be the only weak solution on the interval [0, T * ). Therefore, by the energy inequality (6), for Leray-Hopf weak solutions, we have (see, e.g., [7], [32], [35] or [36] for details) for all t ≥ 0, where Next, let us show that the H 1 norm of the strong solution u is bounded on interval [0, T * ).
2.1. ∇ h u 2 estimates. First we obtain some estimates of the horizontal gradient. Taking the inner product of the equation (1) with −∆ h u in L 2 , we obtain By integration by parts few times, and using the incompressibility condition (2), we get Next, we will estimate the right-hand side of the above inequality using either Lemma 2 or Lemma 3. Each will be used for dealing with either one of the conditions (15) or (16). On the one hand by applying (13), with φ = |u 3 |, f = |∇u|, g = |∇ h ∇u|, and r = 3α−2 α , we get On the other hand by applying (14), with φ = |u 3 |, f = |∇u|, g = |∇ h ∇u|, we obtain In case we use (19) we obtain Alteratively, if we use (20), we obtain Therefore, integrating (21) and using Hölder inequality and applying (17) we get for all t ∈ [0, T * ). Alteratively, integrating (22) and using Hölder inequality and applying (17) we get a different estimate for all t ∈ [0, T * ).
2.2. ∇u 2 estimates. Taking the inner product of the equation (1) with −∆u in L 2 , we obtain Applying the Cauchy-Schwarz inequality and (9), with r = 4, we obtain Now, we are ready to complete our proof for the case when condition (15) holds. By (19) and Young's inequality, we have As a result of the above and (25), we get Integrating the above inequality and using Hölder inequality we obtain Thanks to (17) and (23), we get By Young's and Hölder inequalities, we obtain ∇u(t) Thanks again to (17), we get Therefore, in case (11) holds, we apply Gronwall inequality to obtain for all t ∈ [0, T * ). Therefore, if the condition (15) holds the H 1 norm of the solution u is bounded, and this completes our proof in this case. Next, we complete the proof when u 3 satisfies (16). Thanks to (25), (20) and ( Integrating the above inequality and using Hölder inequality we obtain ∇u(t) Thanks to (17) and (24), we get ∇u(t) Thanks again to (17), we get ∇u(t) for all t ∈ [0, T * ). Therefore, 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.