Abstract

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 and one a priori estimate on the solution that is bounded for to three-dimensional incompressible Navier-Stokes equations in orthogonal curvilinear coordinate systems. These extent greatly the corresponding results on axisymmetric cylindrical flow.

1. 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, , denotes the velocity fields, is the scalar pressure, and is a given initial velocity with .

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 with . The uniqueness and global regularity of Leray-Hopf weak solutions is still one of the most challenging open problems in the mathematical fluid dynamics [1–6]. Many researchers are devoted to looking for certain sufficient conditions to ensure the smoothness of solutions, called the regularity criterion or Serrin-type criterion. Thanks to the pioneering work by [6–8], we have known that the weak solution 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 [9–12] 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 -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 [13–16]. However, it is still open for the global regularity with swirl ([17–20] 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 [21–26] 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 [29–31]. 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.

To state our main results in this paper, let us begin with some notations, see [32]. A point in is denoted by , where is general orthogonal curvilinear coordinates with a line element given by

Here, are orthogonal, of unit length and parallel to the coordinate lines with increasing; the nonnegative functions are the G.Lamé coefficients corresponding to the vectors , respectively. Throughout this paper, we assume are independent of , i.e., , and the measure of the set is zero in the sense of Lebesgue measure in .

In this paper, we consider the solution 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 on 3D incompressible Navier-Stokes equations in general orthogonal curvilinear coordinate systems.

Theorem 1 (the a priori estimate of ). Suppose that be a smooth solution of system (1) with the form (4) and the initial data (5) satisfying and for . Then, we have for any , and, moreover, if assume that the G.Lamé coefficient satisfies it holds, for any and any , that

Theorem 2 (the regularity criteria involving ). Let be a weak solution of system (1) with the form (4) and the initial data (5) satisfying and . Then, the solution is smooth in , if where .

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 notation introduced in Section 2, because is independent of . Thus, if is a radial function in , i.e., with then In this case, the assumption (7) is naturally true because the function 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 are be covered in Theorem 1 and Theorem 2, i.e., let 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 and G.Lamé coefficients are 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.

In this section, we will first derive the incompressible Navier-Stokes equations in orthogonal curvilinear coordinate systems given by Section 1.

We assume that , being one-to-one and onto mapping, transforms into with and where the domain is the bounded or unbounded domain of with the smooth boundary if is bounded, and the constants and satisfy .

Since , by the derivatives of the unit vectors and , 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 in orthogonal curvilinear coordinate systems:

Furthermore, for a vector field , we get the expressions of and in orthogonal curvilinear coordinate systems:

By the above expressions (17)–(19), then taking the inner product of equation (1)1with , , respectively, we can derive the Navier-Stokes equations in orthogonal curvilinear coordinate systems as follows: where and

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 as is equivalent to the following condition if the domain is bounded or of having partially bounded boundary.

By the expressions (4) and (20), using the vorticity , in orthogonal curvilinear coordinate systems, we have with the initial vorticity

Where

Moreover, with the help of (28), we can get the equation of from (21) as follows:

3. 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 in orthogonal curvilinear coordinate systems.

Lemma 6 (see [6–8]). Suppose that the initial data in (1), then any Leray-Hopf weak solution of 3D incompressible Navier-Stokes equations (1) is also a smooth solution in , if there holds that in which and 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 be a velocity field with its vorticity , then the inequality holds for any , where the constant depends only on .

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 with the form (4) be a field with zero divergence, then the estimate holds for any , where and the constant depends only on .

Proof. Since 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.

4. Proof of Theorem 1

In this section, we prove Theorem 1.

Proof of Theorem 1. Let , then putting into (21)3, we can obtain the equation for with the help of the following calculations Multiplying the both sides of equation (39) by and integrating over , we have Here, thanks to (15) and (25) and the incompressibility condition (23), by the fact of , we have For the and on the right of (41), by simple integration, one has Combining (42)–(44) with (41), we have Note that .
If , it has, immediately, by Gronwall’s inequality, If , 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 is immediate if we let in (48). Thus, we finish the proof of Theorem 1.

5. 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 , It is known that the vorticity equation for the vorticity for 3D incompressible Navier-Stokes equation is the following: Using the expressions (4), (17), and (26), we have Then, by multiplying on the both sides of equation (50) and integrating over , we get Just for the sake of clarity, we give the expressions as follows: Then, we estimate each integral in the right-hand 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 can be estimated similarly to as Similarly, On the other hand, by using Lemma 7, Hölder inequality, Sobolev imbedding inequality, and Young inequality, it yields, with the help of the relation (53), that The terms and can be estimated similarly to as Combining all above estimates about , from (52), we obtain Applying Gronwall’s inequality, it yields, for any Together with Lemma 7 and (49), it implies that, for any By Sobolev’s imbedding Theorem, we obtain that, for any Hence, with the help of Lemma 6, we finish the proof of Theorem 2.

Data Availability

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

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Authors’ Contributions

All authors contributed equally to writing of this paper. All authors read and approved the final manuscript.

Acknowledgments

This work was supported by the Natural Science Foundation of China (Grant Nos. 11831003, 11771031, and 11531010).