The Regularity Criteria and the A Priori Estimate on the 3D Incompressible Navier-Stokes Equations in Orthogonal Curvilinear Coordinate Systems

The paper considers the regularity problem on three-dimensional incompressible Navier-Stokes equations in general orthogonal curvilinear coordinate systems. We establish one regularity criteria of the weak solutions involving only in a vorticity component ω3 and one a priori estimate on the solution that kH3ukL∞ð0,T ;LpðR3ÞÞ is bounded for 1 ≤ p ≤∞ to three-dimensional incompressible Navier-Stokes equations in orthogonal curvilinear coordinate systems. These extent greatly the corresponding results on axisymmetric cylindrical flow.


Introduction
In this paper, we investigate the regularity problem on the following three-dimensional (3D) incompressible Navier-Stokes equations in general orthogonal curvilinear coordinate systems. Here, x = ðx 1 , x 2 , x 3 Þ ∈ ℝ 3 , u = ðu 1 , u 2 , u 3 Þ ðt, xÞ denotes the velocity fields, P = P ðt, xÞ is the scalar pressure, and u 0 is a given initial velocity with div u 0 = 0. The existence of global weak solutions to (1) is known since the famous work of Leray [1] (see also Hopf [2] for the bounded domain case) for initial data u 0 ∈ L 2 ðℝ 3 Þ with div u 0 = 0. The uniqueness and global regularity of Leray-Hopf weak solutions is still one of the most challenging open problems in the mathematical fluid dynamics [1][2][3][4][5][6]. Many researchers are devoted to looking for certain sufficient con-ditions to ensure the smoothness of solutions, called the regularity criterion or Serrin-type criterion. Thanks to the pioneering work by [6][7][8], we have known that the weak solution u will be smooth as long as Afterwards, there are many progresses on the regularity criteria involving only one component of the velocity fields, one can refer to [9][10][11][12] for details.
An interesting problem is to study the globally stabilizing effects of the geometry structures of the domain or/and solutions on the evolution of solution in time to the 3D incompressible Navier-Stokes equations. For example, the axisymmetric flow makes the 3D flow close to 2D flow, that is, all velocity components (radial, angular (or swirl) and x 3 -component) and the pressure are independent of the angular variable in the cylindrical coordinates. It is well known that the 3D incompressible axisymmetric Navier-Stokes equations without swirl have the unique global smooth solution [13][14][15][16]. However, it is still open for the global regularity with swirl ( [17][18][19][20] and therein). These results indicate that the swirl of the fluid plays a crucial role in the issue of global regularity. Subsequently, to understand this problem better, many efforts have been devoted to looking for suitable regularity criteria, see [21][22][23][24][25][26] for details. The paper is motivated by the studies on the axisymmetric flow (see [5,13,14,16]) and the helical flow (see [27] and references therein) of the 3D incompressible flows and on the absence of simple hyperbolic blow-up for the 3D incompressible Euler and quasigeostrophic equations [28], we investigate the regularity criteria of the weak solutions to the 3D incompressible Navier-Stokes equations in general orthogonal curvilinear coordinate systems. Recently, global well-posedness results on the smooth solution for 3D incompressible Navier-Stokes equations in spherical coordinates are obtained in [29][30][31]. The main purpose of this paper is to establish the a priori estimate and the regularity criteria for the 3D incompressible Navier-Stokes equations in general orthogonal curvilinear coordinate systems.
In this paper, we consider the solution ðu, PÞ to the 3D incompressible Navier-Stokes (1) of having the form with the initial data Our main results are the a priori estimates and the regularity criteria involving only in a vorticity component ω 3 on 3D incompressible Navier-Stokes equations in general orthogonal curvilinear coordinate systems.

Remark 3.
The assumption (7) comes from the geometry on the harmonic mapping in some sense. It is easy to see that in (7), based on the notationΔ introduced in Section 2, because H 3 is independent of ξ 3 . Thus, if H 3 is a radial function in ℝ 2 , i.e., then In this case, the assumption (7) is naturally true because the function ln H 3 is the harmonic one.

Remark 4.
The assumption (7) of Theorem 1 is satisfied in the cases of cylindrical coordinates. More precisely, we have the known results on 3D problem in the case of cylindrical coordinate system in ℝ 3 are be covered in Theorem 1 and Theorem 2, i.e., let Journal of Function Spaces we consider an axisymmetric solution of the Navier-Stokes equations of the form (4) in the cylindrical coordinate system, where the mapping is taken as x = xðξ 1 , ξ 2 , ξ 3 Þ = ðξ 1 cos ξ 3 , ξ 2 , ξ 1 sin ξ 3 Þ, and G.Lamé coefficients are H 1 = H 2 = 1, H 3 = ξ 1 , satisfying the assumption in Theorem 1. Then, Theorem 1 is equivalent to Proposition 1 in [9], Theorem 2 is equivalent to Theorem 1.3 in [21].
Remark 5. This difference from the case of curvilinear cylindrical coordinates may be imply that one should care about the advantage or overcome the difficulty brought by the choice of curvilinear coordinates, including nonorthogonal curvilinear coordinates, which will be discussed in the future.
The remaining of this paper is organized as follows. In Section 2, we will derive the Navier-Stokes equations in orthogonal curvilinear coordinate systems. In Section 3, we introduce some basic lemmas and one estimate used for the proof of main theorems. In Section 4 and Section 5, we prove Theorem 1 and Theorem 2 separately.

Navier-Stokes Equations in Orthogonal Curvilinear Coordinate Systems
In this section, we will first derive the incompressible Navier-Stokes equations in orthogonal curvilinear coordinate systems ξ 1 , ξ 2 , ξ 3 , given by Section 1. We assume that x = xðξÞ, being one-to-one and onto mapping, where the domain D is the bounded or unbounded domain of ℝ 2 with the smooth boundary ∂D if D is bounded, and the constants α and β satisfy −∞ < α ≤ β < ∞.
Since H i = H i ðξ 1 , ξ 2 Þði = 1, 2, 3Þ, by the derivatives of the unit vectors e ξ 1 , e ξ 2 and e ξ 3 , we have Using the definition of gradient, we get the expression of the gradient operator ∇ in orthogonal curvilinear coordinate systems: we also obtain the expression of Laplacian operator Δ in orthogonal curvilinear coordinate systems: Furthermore, for a vector field V = Vðξ 1 , ξ 2 , ξ 3 Þ = V 1 e ξ 1 + V 2 e ξ 2 + V 3 e ξ 3 , we get the expressions of div V and rotV in orthogonal curvilinear coordinate systems: By the above expressions (17)- (19), then taking the inner product of equation (1) 1 with e ξ 1 , e ξ 2 , e ξ 3 , respectively, we can derive the Navier-Stokes equations in orthogonal curvilinear coordinate systems as follows: 3 Journal of Function Spaces where ðξ 1 , ξ 2 Þ ∈ D ⊂ ℝ 2 , t > 0 and ∇ ξ 1 ,ξ 2 = e ξ 1 1 The incompressible constraint is It is clear that equations (21) and (23) completely determine the evolution of 3D Navier-Stokes equations in orthogonal curvilinear coordinate systems, respectively, once the initial value and/or boundary conditions are given.
We take initial condition for the system (21) as follows: Moreover, the boundary condition u → 0 as | x | → ∞,t ≥ 0 is equivalent to the following condition if the domain D is bounded or of having partially bounded boundary. By the expressions (4) and (20), using the vorticity ω = ∇× u, in orthogonal curvilinear coordinate systems, we have with the initial vorticity Where Moreover, with the help of (28), we can get the equation of ω 3 from (21) as follows:

Some Useful Estimates
To study the main estimates of Theorem 1 and Theorem 2, we need to introduce two basic lemmas and one estimate relates to ω 3 in orthogonal curvilinear coordinate systems.
Lemma 6 (see [6][7][8]). Suppose that the initial data u 0 ∈ H 2 ðℝ 3 Þ in (1), then any Leray-Hopf weak solution u of 3D incompressible Navier-Stokes equations (1) is also a smooth solution in ð0, T × ℝ 3 , if there holds that in which p and q satisfy the conditions And we would like to recall the well-known relation between the velocity and vorticity of 3D flow. Lemma 7 (see [33]). Let u ∈ W 1,p ðℝ 3 Þ be a velocity field with its vorticity ω = ∇× u, then the inequality holds for any p ∈ ð1,∞Þ, where the constant C p depends only on p.
As one kind of fluid with the special geometry structure of form (4), the incompressible 3D flow also has one particular property, which is shown as follows.
Proposition 8. Suppose that u ∈ W 1,p ðℝ 3 Þ with the form (4) be a field with zero divergence, then the estimate holds for any p ∈ ð1,∞Þ, wherẽ and the constantC p depends only on p.

Journal of Function Spaces
Proof. Since div u = 0 and we have Thus, we obtain On the other hand, by the expressions (20), (34), and (28), we get Consequently, by Lemma 7 and using (36) and (37), one has This finishes the proof of the proposition.

Proof of Theorem 1
In this section, we prove Theorem 1.
Proof of Theorem 1. Let Fðt, ξ 1 , we can obtain the equation for F with the help of the following calculations Multiplying the both sides of equation (39) by jFj p−2 F and integrating over ℝ 3 , we have Here, thanks to (15) and (25) and the incompressibility condition (23), by the fact of dx = H 1 H 2 H 3 dξ 1 ξ 2 ξ 3 , we have 5 Journal of Function Spaces For the I 1 and I 2 on the right of (41), by simple integration, one has If p ∈ ½1, 2Þ ∪ ð2,∞, from (45) and (7), the right hand of (45) can be estimated by and, then, by Gronwall's inequality again, one also has The case p = ∞ is immediate if we let p → ∞ in (48). Thus, we finish the proof of Theorem 1.

Proof of Theorem 2
In this section, we prove Theorem 2.
Proof of Theorem 2. From (1:1) 1 , we have the basic energy inequality, for any T > 0, sup 0≤t≤T u k k 2 Then, we estimate each integral J i ði = 1,⋯,9Þ in the righthand side of (52) by using the relation (53) and the relation (54), respectively.
On the one hand, by applying Proposition 8, Hölder inequality, Sobolev's imbedding inequality, and Young inequality, it follows, with the help of the relation (54), that The term J 2 can be estimated similarly to J 1 as

Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.