On the 3D Incompressible Boussinesq Equations in a Class of Variant Spherical Coordinates

Abstract

This paper investigates the global stabilizing effects of the geometry of the domain at which the flow locates and of the geometric structure of the solution to the incompressible flows by studying the three-dimensional (3D) incompressible, viscosity, and diffusivity Boussinesq system in spherical coordinates. We establish the global existence and uniqueness of the smooth solution to the Cauchy problem for a full 3D incompressible Boussinesq system in a class of variant spherical coordinates for a class of smooth large initial data. We also construct one class of nonempty bounded domains in the three-dimensional space , in which the initial boundary value problem for the full 3D Boussinesq system in a class of variant spherical coordinates with a class of large smooth initial data with swirl has a unique global strong or smooth solution with exponential decay rate in time.

1. Introduction and Main Results

In this paper, we consider the Cauchy problem for the three-dimensional(3D) incompressible Boussinesq () equations and the initial boundary value problem for the 3D incompressible Boussinesq () equations in the bounded domain respectively. Here, ; the unknowns denote the fluid velocity vector field; is the scalar pressure and is the scalar density; are viscosity and thermal diffusivity, respectively; is the unit outer normal vector of bounded domain ; is the unit vector in the vertical direction; and and are the given initial velocity and initial density, respectively, with . It should be noted that, if , (1) comes back to the classical 3D incompressible Navier-Stokes equations.

It is well known that the 3D incompressible Navier-Stokes equations have at least one global weak solution with the finite energy [1, 2]. However, the issue of the regularity and uniqueness for the global weak solution is still a challenging open problem in the field of mathematical fluid dynamics [3–8].

Recently, motivated by the studies on the axisymmetric flow (see [6–11] and the references therein), the helical flow (see [12] and the references therein), and the 3D incompressible Euler and the SQG (surface quasigeostrophic) equations [13–15], we investigate further the global dynamical stabilizing effects of the geometry of the domain at which the flow locates and of the geometry structure of the solution to the 3D incompressible Navier-Stokes equations. As an example, we study the 3D incompressible Navier-Stokes and Euler equations in the spherical coordinate system, see S. Wang and Y.X. Wang [16], where the existence and uniqueness of the global strong solution of the 3D incompressible Navier-Stokes and Euler equations in the spherical coordinates are obtained for a class of large smooth initial data with swirl or without swirl.

As stated in the beginning, the present paper is focused on the Boussinesq system, which plays an important role in the atmospheric and oceanographic sciences [11, 17–20]. Considering the 2D standard Boussinesq equations with the viscosity or diffusive coefficient, Hou and Li [21] and Chae [22] obtain the global well-posedness results similar to the 2D incompressible Navier-Stokes equations [6]. On global regularity on the smooth solution for the 2D Boussinesq system, see also, e.g., [23–25] and the references therein. On the other hand, comparing with the magnitude of research conducted on the Boussinesq equations on Euclidean domains, the qualitative behaviour of the model on Riemannian manifolds has been investigated relatively little, see [26], in which the convergence of the average of weak solutions of the 3D equations to a 2D problem is proved by Saito, and see [27], in which the nondegenerate and partially degenerate Boussinesq equations on a closed surface are studied by Li et al..

The global well-posedness for a 3D axisymmetric Boussinesq system without swirl and with partial viscosity or thermal diffusivity in the system of cylindrical coordinates is obtained by Abidi et al. in [28], Hmidi and Keraani in [29], and Hmidi et al. [30, 31], respectively. For the general 3D Boussinesq system, there exist some results on the local well-posedness problem, partial regularity, or the global regularity with respect to small initial data; see [32–37], etc.

In this paper, we further investigate the global stabilizing effects of the geometry of the domain and the solution to the three-dimensional incompressible flows by studying the 3D incompressible axisymmetric Boussinesq system in the system of a class of variant spherical coordinates.

Let the matrix be a real orthogonal matrix, i.e., , where is an identity matrix and is a transpose of the matrix . For the given and the constant , introduce a class of variant spherical coordinates defined as

Because the matrix is an orthogonal one, we have where

Note that, for variant spherical coordinates , the coordinate is spherical symmetric in , but the coordinate and coordinate are not axisymmetric with respect to the Cartesian coordinates except that . Denote

Also, denote the special bounded domain described by variant spherical coordinates by where are given fixed positive constants. Here, we give an explicit example for the domain by taking

Now, we consider the 3D incompressible Boussinesq equations (1) and (2) with the form with

When the matrix is an orthogonal matrix, the gradient operator and Laplacian have the expression respectively.

Then, one can derive the evolution equations for for 3D incompressible Boussinesq equations as follows: where and

Note that equations (15) completely determine the evolution of the 3D Boussinesq equations in a class of variant spherical coordinates once the initial conditions and/or the boundary value conditions are given. Also, the 3D incompressible Boussinesq system in a class of variant spherical coordinates is completely different from the one in cylindrical coordinates because of the complexity of the last equation in system (15) and of Laplace operator given by (16).

We take the initial condition for system (15) as follows:

Moreover, the boundary condition is equivalent to the following condition:

It is easy to know, by direct computation, that the vorticity can be expressed as with the initial vorticity where

It is clear that

In addition, we can obtain the equation of from (15) as

We now state our main results as follows:

Theorem 1 (the case of 3D incompressible Boussinesq equations in without swirl in the sense of spherical coordinates). Assume that and . Let be given by (13) with . Let . If with and then the Cauchy problems (15) and (17) have a unique global strong solution with satisfying , given by (12). Moreover, assume that is smooth with , and furthermore, with some compatibility conditions for the initial data with respect to and , then the Cauchy problem (1) to the 3D incompressible Boussinesq equations has a unique global smooth solution in time.

Theorem 2 (the exponential decay rate in time and the global strong solution of 3D incompressible Boussinesq equations in the special bounded domain of with swirl in the sense of spherical coordinates). Assume that and . Let in (2), given by (9). Let be given by (13) with . If with and , then the initial-boundary value problems (15), (17), and (18) to the incompressible Boussinesq equation (2) have a unique global strong solution satisfying , , given by (12), and the exponential decay rate in time for some constants and , independent of . Moreover, any Leray-Hopf-type global weak solution , given by (12), to the initial-boundary value problem (2) is globally smooth in for any and any smooth domain .

Remark 3. The assumptions and are key in the proofs of Theorems 1 and 2. The key point of the proof of Theorem 1 is to establish the a priori estimate on the quality and then to use the special geometry structure (12) of the solutions , which guarantees that there exist some kinds of cancelation regimes so that we can deal with the vortex stretching term in the vorticity equation for . The present method used in this paper cannot be extended to the case of or . The global well-posedness problem on the 3D incompressible Boussinesq system with partial viscosity or diffusivity and without swirl in spherical coordinates is complex because each component of the velocity field in spherical coordinates in the Boussinesq system given by the classical Biot-Savart law is very complex, which will be discussed in the future. The classical Biot-Savart law expresses the velocity field that transports the vorticity in terms of the vorticity itself; see [38] and the references therein. The assumption in Theorem 2 on the domain with the special geometry structure given by (9) is key for one to prove our global regularity for the strong solution and global interior regularity for the smooth solution in time for 3D Boussinesq equations with large smooth initial data, which yields to one inequality of Ladyzhenskaya’s type (see [3] and Lemma 6 for details), close to a two-dimensional case, for the function having the special geometry structure (12) for . Also, if we replace the domain in Theorem 2 by one smooth domain satisfying that there exists one positive constant , and such that with then the global strong solution obtained in Theorem 2 is also smooth in .

Remark 4. The axisymmetric flow makes the 3D flow close to the 2D flow; that is, all velocity components (radial, angular (or swirl) and component) as well as the pressure are independent of the angular variable in the cylindrical coordinates. As a kind of fluid with special geometry structure, we know that the 1D parabolic Hausdorff measure of the set of possible singular points to the suitable weak solutions of the incompressible Navier-Stokes or Boussinesq system is zero; see [7, 8, 39] for details. This implies that the incompressible axisymmetric Navier-Stokes or Boussinesq equations cannot develop finite time singularities away from the symmetry axis. Based on this fact, it is not clear whether the potential finite-time-blow-up set for 3D incompressible Boussinesq equations in spherical coordinates is only one point set, where the flow is a special variant of axisymmetric, i.e., spherically symmetric, in . This is the main motivation of the current paper.

The rest of this paper is organized as follows. In Section 2, we introduce some technical lemmas used for the proof of the main theorems. In Section 3, we prove Theorems 1 and 2.

2. Preliminaries

In this section, we provide some lemmas used for the proof of the main theorems.

Lemma 5. (see [40]). Let be a velocity field with its divergence free and vorticity ; then, the inequality holds for any , where the constant depends only on .

Lemma 6 (see [3]). Let ; then, there exists a constant such that, for any

Lemma 7 (see [41]). Suppose that the initial data with in (1); then, any Leray-Hopf weak solution of 3D incompressible Boussinesq equation (1) is also a smooth solution in if there holds that in which and satisfy the conditions

Lemma 8 (see [42]). Suppose that is smooth and the initial data in (2) satisfies with and ; then, any Leray-Hopf weak solution of 3D incompressible Boussinesq equation (2) is also a smooth solution in if there holds that in which and satisfy the conditions

3. Proof of Main Results

In this section, we give the proofs of Theorems 1 and 2.

Proof of Theorem 1. From (1), for any we have the energy inequality By the existence and uniqueness of the local smooth solution to the Cauchy problem (1) for the 3D Boussinesq equations, it is easy to get that for the case of no swirl initial data . In this kind of case of no swirl, the velocity and vorticity satisfy the following special form: and hence, equation (23) for is simplified as Multiplying (34) by and then letting , , or and by the existence and uniqueness of the local smooth solution to the Cauchy problem (1) or (15)–(17) for the 3D Boussinesq equations, it is easy to see that Similarly, we have Taking , i.e., satisfying , then we have Now putting into (34) and using (37)–(38), we obtain the following equation for : To deal with the more singular second term in the right-hand side of (39), we decompose into ; then, satisfies and the following equation which implies that Multiplying equation (42) by and integrating the resulting equation on , we have where and are defined by and can be estimated as follows: Putting (44) and (45) into (43), we get which, together with (32), yields to the following estimate for : Thus, we have Next, we obtain the estimate for the vorticity , given by (33) in the case of no swirl for the 3D incompressible Boussinesq equation in the spherical coordinate system.
It is known that the vorticity equation for the vorticity for the 3D incompressible Boussinesq equation is the following: Multiplying equation (49) by and integrating the resulting equation on , we have, for any , , where and can be estimated as follows by using the special structure (33) of the velocity and the vorticity . Using (33), with the help of the Hölder inequality, Gagliardo-Nirenberg inequality, and Young inequality, we have Putting (51) and (52) into (50), we have which, by applying Gronwall’s inequality and by using (32), yields to, for any , Using Lemma 5, we get from (54) that, for any , and hence, by Sobolev’s imbedding theorem, we have, for any , Now, the desired regularity estimate for the 3D incompressible Boussinesq equation (1) is obtained; hence, by applying Lemma 7, we obtain the results stated in Theorem 1.
The proof of Theorem 1 is complete.

Proof of Theorem 2. We take in Theorem 2, where is given by (9) having one special geometry structure. Also, in Theorem 2 is given by (12), which satisfies that by using the orthogonality of three spherical coordinate unit vectors. Firstly, for the system (2), we have the following basic energy estimates, for , for some constant and any , which, together with Poincare’s inequality for and , yield the energy estimate, for , for some constants and .
Next, we give the estimates of .
Differentiating (2) with respect to , one gets where and satisfy, by using (2), that It is easy to get that In fact, multiplying (64) by and integrating the resulting equation on , applying the Hölder inequality, Gagliardo-Nirenberg inequality, and Young inequality, we have which implies (65). Similarly, we have (66).
Multiplying the first equation in (62) by and integrating the resulting equation on , with the help of the Hölder inequality, we get Multiplying the second equation in (62) by and integrating the resulting equation on , with the help of the Hölder inequality, we get In the following, we use the special geometry structure (9) of the domain and the special geometry structure (12) of the functions in spherical coordinates in to obtain the following inequality for defined in with