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

Authors: Yongxin Wang; Fan Geng; Shu Wang
Journal: Journal of Function Spaces
 (2022)

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## 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 spaceℝ3, 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.


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## Body

## 1. Introduction and Main Results

In this paper, we consider the Cauchy problem for the three-dimensional(3D) incompressible Boussinesq (ν,μ>0) equations
(1)ut+u·∇u+∇P=νΔu+ρe3,x∈ℝ3,t>0,ρt+u·∇ρ=μΔρ,x∈ℝ3,t>0,divu=0,x∈ℝ3,t>0,u0,x=u0x,ρ0,x=ρ0x,x∈ℝ3,and the initial boundary value problem for the 3D incompressible Boussinesq (ν,μ>0) equations in the bounded domain
(2)ut+u·∇u+∇P=νΔu+ρe3,x∈Ω,t>0,ρt+u·∇ρ=μΔρ,x∈Ω,t>0,divu=0,x∈Ω,t>0,u=0,ρ=0,x∈∂Ω,t>0,u0,x=u0x,ρ0,x=ρ0x,x∈Ω,respectively. Here, x=x1,x2,x3; the unknowns u=u1t,x,u2t,x,u3t,xT denote the fluid velocity vector field; P=Pt,x is the scalar pressure and ρ=ρt,x is the scalar density; ν,μ are viscosity and thermal diffusivity, respectively; n is the unit outer normal vector of bounded domain Ω; e3=0,0,1T is the unit vector in the vertical direction; and u0 and ρ0 are the given initial velocity and initial density, respectively, with divu0=0. It should be noted that, if ρ≡0, (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(3)A=a11a12a13a21a22a23a31a32a33,be a real orthogonal matrix, i.e., ATA=I, where I is an identity matrix and AT is a transpose of the matrix A. For the given
(4)α=α1α2α3∈ℝ3,and the constant a>0, introduce a class of variant spherical coordinatesr,θ,φ defined as
(5)x=x1x1x3=α1α2α3+aArsinθcosφrsinθsinφrcosθ,x∈ℝ3.Because the matrixA is an orthogonal one, we have
(6)r=1aξ2x+η2x+ζ2x=1ax1−α12+x2−α22+x3−α32≥0,0≤θ=arctanξ2x+η2xζx≤π,0≤φ=arctanηxξx<2π,where
(7)ξx=a11x1−α1+a21x2−α2+a31x3−α3,ηx=a12x1−α1+a22x2−α2+a32x3−α3,ζx=a13x1−α1+a23x2−α2+a33x3−α3.Note that, for variant spherical coordinatesr,θ,φ, the r coordinate is spherical symmetric in ℝ3, but the θ coordinate and φ coordinate are not axisymmetric with respect to the Cartesian coordinates x∈ℝ3 except that A=I. Denote
(8)er=Asinθcosφsinθsinφcosθ=1ξ2x+η2x+ζ2xx1−α1x2−α2x3−α3,eθ=Acosθcosφcosθsinφ−sinθ=a11ξ2x+a12ξxηx−a13ξ2x+η2xa21ξ2x+a22ξxηx−a23ξ2x+η2xa31ξ2x+a32ξxηx−a33ξ2x+η2xξ2x+η2x+ζ2xξ2x+η2x,eφ=A−sinφcosφ0=1ξ2x+η2x−a11ξx+a12ηx−a21ξx+a22ηx−a31ξx+a32ηx.Also, denote the special bounded domainΩ~ described by variant spherical coordinates by
(9)Ω~=x1,x2,x3=α1+aa11rsinθcosφ+aa12rsinθsinφ+aa13rcosθ,α2+aa21rsinθcosφ+aa22rsinθsinφ+aa23rcosθ,α3+aa31rsinθcosφ+aa32rsinθsinφ+aa33rcosθ∈ℝ3:0<r0≤r≤R0<∞,0<θ0≤θ≤θ1<π,0≤φ<2π,where r0,R0,θ0,θ1 are given fixed positive constants. Here, we give an explicit example for the domain
(10)Ω~=x1,x2,x3=2rsinθsinφ,−rsinθcosφ+rcosθ,rsinθcosφ+rcosθ∈ℝ3:1≤r≤10,π8≤θ≤3π4,0≤φ<2π,by taking
(11)a=2,α=0,A=010−1201212012,r0=1,R0=10,θ0=π8,θ1=3π4.Now, we consider the 3D incompressible Boussinesq equations (1) and (2) with the form
(12)ut,x=urt,r,θer+uθt,r,θeθ+uφt,r,θeφ,Pt,x=Pt,r,θ,ρt,x=ρt,r,θ,with
(13)u0x=u0rt,r,θer+u0θt,r,θeθ+u0φt,r,θeφ,ρ0x=ρ0r,θ.When the matrixA is an orthogonal matrix, the gradient operator ∇ and Laplacian Δ have the expression
(14)∇=er1a∂r+1areθ∂θ+1arsinθeφ∂φ,Δ=1a2∂r2+2r∂r+1r2∂θ2+cosθr2sinθ∂θ+1r2sin2θ∂φ2,respectively.Then, one can derive the evolution equations forur,uθ,uφρt,r,θ for 3D incompressible Boussinesq equations as follows:
(15)∂tur+u~·∇~ur+1a∂rP=νΔ~−2a2r2ur−2cosθa2r2sinθuθ−2a2r2∂θuθ+uθ2+uφ2ar+ρcosθ,∂tuθ+u~·∇~uθ+1ar∂θP=νΔ~−1a2r2sin2θuθ+2a2r2∂θur−uruθar+cosθarsinθuφ2−ρsinθ,∂tuφ+u~·∇~uφ=νΔ~−1a2r2sin2θuφ−uruφar−cosθarsinθuθuφ,∂tρ+u~·∇~ρ=νΔ~ρ,∂rur+2rur+1r∂θuθ+cosθrsinθuθ=0,where u~=urer+uθeθ, and
(16)∇~=er∂r+eθ1r∂θ,Δ~=∂r2+2r∂r+1r2∂θ2+cosθr2sinθ∂θ.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 Δ~ given by (16).We take the initial condition for system (15) as follows:
(17)ur,uθ,uφ,ρt=0,r,θ=u0r,u0θ,u0φr,θ.Moreover, the boundary conditionu∂Ω=0,t≥0 is equivalent to the following condition:
(18)ur,uθ,uφ,ρ∂Ω=0,t≥0.It is easy to know, by direct computation, that the vorticityω=∇×u can be expressed as
(19)ωt,x=ωrt,r,θer+ωθt,r,θeθ+ωφt,r,θeφ,with the initial vorticity
(20)ω0=ω0,x=ω0rr,θer+ω0θr,θeθ+ω0φr,θeφ,where
(21)ωr=1arsinθ∂θsinθuφ,ωθ=−1ar∂rruφ,ωφ=1a∂ruθ+uθr−∂θurr.It is clear that(22)divω=1a∂rωr+2rωr+1r∂θωθ+cosθrsinθωθ≡0.In addition, we can obtain the equation ofωφ from (15) as
(23)∂tωφ+u~·∇~ωφ=νΔ~−1a2r2sin2θωφ+urωφar+cosθarsinθuθωφ+cosθarsinθ∂r−1ar2∂θuφ2−sinθ∂rρ+cosθ1r∂θρ.We now state our main results as follows:Theorem 1 (the case of 3D incompressible Boussinesq equations inℝ3 without swirl in the sense of spherical coordinates).
Assume thatν>0 and μ>0. Let u0,ρ0t,x be given by (13) with u0φ=0. Let ω0φ=∂ru0θ+u0θ/r−∂θu0r/r. If u0,ρ0∈H2ℝ3 with divu0=0 and ω0φ/rsinθ∈L2ℝ3, then the Cauchy problems (15) and (17) have a unique global strong solution ur,uθ,uφ,P,ρt,r,θ,φ with uφ≡0 satisfying u∈L∞0,+∞;H1ℝ3, given by (12). Moreover, assume that u0x=u0rr,θer+u0θr,θeθ is smooth with u0r0,θ=u0θ0,θ=ρ00,θθ=0,π=0, and furthermore, with some compatibility conditions for the initial data with respect to θ=0,π and r=0, 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 ofℝ3 with swirl in the sense of spherical coordinates).
Assume thatν>0 and μ>0. Let Ω=Ω~⊂ℝ3 in (2), given by (9). Let u0,ρ0t,x be given by (13) with u0φ≢0. If u0,ρ0∈H2Ω with divu0=0 and u0,ρ0∂Ω=0, then the initial-boundary value problems (15), (17), and (18) to the incompressible Boussinesq equation (2) have a unique global strong solution ur,uθ,uφ,P,ρt,r,θ,φ satisfying ∂tiu,ρ∈L∞0,+∞;H1−iΩ, i=0,1, given by (12), and the exponential decay rate in time
(24)u,ρt,·H1Ω2+ut,ρtt,·L2Ω2≤Ce−αt,0≤t≤+∞,for some constants C=CΩ,ν,μ,u0,ρ0H2Ω>0 and α=αΩ,ν,μ>0, independent of t:0≤t≤∞. Moreover, any Leray-Hopf-type global weak solution u,ρ,P, given by (12), to the initial-boundary value problem (2) is globally smooth in 0,T×Ω1 for any 0≤T≤∞ and any smooth domain Ω1⊂⊂Ω⊂ℝ3.Remark 3.
The assumptionsν>0 and μ>0 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 ωφ/rsinθ−1/2νρ and then to use the special geometry structure (12) of the solutions u,ρt,x, which guarantees that there exist some kinds of cancelation regimes so that we can deal with the vortex stretching term ω·∇u in the vorticity equation for ω. The present method used in this paper cannot be extended to the case of ν=0 or μ=0. 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 Ω=Ω~ 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 u,ρt,x having the special geometry structure (12) for x∈Ω~⊂ℝ3. Also, if we replace the domain Ω~ in Theorem 2 by one smooth domain Ω2⊂ℝ3 satisfying that there exists one positive constant ϵ1,ϵ2,ϵ3, and ϵ4 such that Ω2⊂Ωϵ⊂ℝ3 with
(25)Ωϵ=x1,x2,x3=rsinθcosφ,rsinθsinφ,rcosθ∈ℝ3:0<ϵ1≤r≤ϵ2<∞,0<ϵ3≤θ≤ϵ4<π,0≤φ<2π,then the global strong solution obtained in Theorem 2 is also smooth in 0,∞×Ω2.Remark 4.
The axisymmetric flow makes the 3D flow close to the 2D flow; that is, all velocity components (radial, angular (or swirl) andx3 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 ℝ3. This is the main motivation of the current paper.The rest of this paper is organized as follows. In Section2, 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]).
Letu∈W1,pℝ3 be a velocity field with its divergence free and vorticity ω; then, the inequality
(26)∇uLp≤CpωLp,holds for any p∈1,∞, where the constant Cp depends only on p.Lemma 6 (see [3]).
LetD⊆ℝ2; then, there exists a constant CD such that, for any f∈H01D,(27)fL4D≤CDfL2D1/2∇fL2D1/2.Lemma 7 (see [41]).
Suppose that the initial datau0,ρ0∈H2ℝ3 with divu0=0 in (1); then, any Leray-Hopf weak solution u of 3D incompressible Boussinesq equation (1) is also a smooth solution in 0,T×ℝ3 if there holds that
(28)u∈Lp0,T;Lqℝ3,in which p and q satisfy the conditions
(29)2p+3q≤1with3<q<∞,2<p≤∞.Lemma 8 (see [42]).
Suppose thatΩ is smooth and the initial data u0,ρ0 in (2) satisfies u0,ρ0∈H2Ω with divu0=0 and u0,ρ0∂Ω=0; then, any Leray-Hopf weak solution u of 3D incompressible Boussinesq equation (2) is also a smooth solution in 0,T×Ω if there holds that
(30)u∈Lp0,T;LqΩ,in which p and q satisfy the conditions
(31)2p+3q≤1with3<q≤∞,2≤p≤∞.


## 3. Proof of Main Results

In this section, we give the proofs of Theorems1 and 2.Proof of Theorem 1.
From (1), for any T>0, we have the energy inequality
(32)sup0≤t≤TuL2ℝ32+∥ρ∥L2ℝ32+2ν∫0T∇uL2ℝ32dt+2μ∫0T∇ρL2ℝ32dt≤CT,u0L2ℝ3,ρ0L2ℝ3.
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 uφ≡0 for the case of no swirl initial data u0φ≡0. In this kind of case of no swirl, the velocity and vorticity satisfy the following special form:
(33)ut,x=urt,r,θer+uθt,r,θeθ,ωt,x=ωφt,r,θeφ,and hence, equation (23) for ωφ is simplified as
(34)∂tωφ+u~·∇~ωφ=νΔ~−1r2sin2θωφ+urωφr+cosθrsinθuθωφ−sinθ∂rρ+cosθ1r∂θρ.
Multiplying (34) by r2sin2θ and then letting r=0, θ=0, or θ=π 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
(35)ωφt,0,θ=ωφt,r,0=ωφt,r,π=0.
Similarly, we have(36)ωφt,r,θrsinθr=0=ωφt,r,θrsinθθ=0,π=0.
Takingωφt,r,θ=gt,r,θrsinθ, i.e., gt,r,θ=ωφt,r,θ/rsinθ, satisfying gt,0,θ=gt,r,0=gt,r,π=0, then we have
(37)u~·∇~grsinθ=ur∂r+uθr∂θgrsinθ=ursinθg+r∂rg+uθgcosθ+sinθ∂θg=rsinθu~·∇~g+urgsinθ+uθgcosθ,(38)Δ~−1r2sin2θgrsinθ=∂r2+2r∂r+1r2∂θ2+cosθr2sinθ∂θ−1r2sin2θgrsinθ=sinθ∂r2rg+2r∂rrg+1r∂θ2gsinθ+cosθrsinθ∂θgsinθ−grsinθ=sinθr∂r2g+4∂rg+2rg+1r3cosθ∂θg−gsinθ+sinθ∂θ2g+cos2θsinθg−grsinθ=rsinθ∂r2+4r∂rg+2sinθrg+rsinθ1r2∂θ2g+3cosθr2sinθ∂θg+cos2θrsinθ−sinθrg−grsinθ=rsinθ∂r2+4r∂r+1r2∂θ2+3cosθr2sinθ∂θg+2sinθr+1−2sin2θrsinθ−1rsinθg=rsinθΔ~+2r∂r+2cosθr2sinθ∂θg.
Now puttingωφt,r,θ=gt,r,θrsinθ into (34) and using (37)–(38), we obtain the following equation for gt,r,θ:
(39)∂tg+u~·∇~g−νΔ~g=2ν1r∂r+cosθr2sinθ∂θg−1rsinθsinθ∂rρ+cosθ1r∂θρ.
To deal with the more singular second term in the right-hand side of (39), we decompose g into g=G+1/2νρ; then, Gt,r,θ=gt,r,θ−1/2νρt,r,θ satisfies
(40)Gt,0,θ=Gt,r,0=Gt,r,π=0,and the following equation
(41)∂tG+12νρ+u~·∇~G+12νρ−νΔ~G+12νρ=2ν1r∂r+cosθr2sinθ∂θG+12νρ−1rsinθsinθ∂rρ+cos1r∂θρ,which implies that
(42)∂tG+u~·∇~G−νΔ~G=2ν1r∂r+cosθr2sinθ∂θG−μ−ν2νΔ~ρ.
Multiplying equation (42) by G and integrating the resulting equation on ℝ3, we have
(43)12ddtGL2ℝ32+ν∇~GL2ℝ32=ν∫ℝ31r∂rG2dx+∫ℝ3cosθr2sinθ∂θG2dx−μ−ν2ν∫ℝ3GΔ~ρdx=I1+I2,where I1 and I2 are defined by and can be estimated as follows:
(44)I1=ν∫ℝ31r∂rG2dx+∫ℝ3cosθr2sinθ∂θG2dx=ν∫02π∫0π∫0∞1r∂rG2r2sinθdrdθdφ+∫02π∫0π∫0∞cosθr2sinθ∂θG2r2sinθdrdθdφ=ν∫02π∫0π∫0∞rsinθ∂rG2drdθdφ+∫02π∫0π∫0∞cosθ∂θG2drdθdφ=−ν∫02π∫0π∫0∞G2sinθ∂rrdrdθdφ+∫02π∫0π∫0∞G2∂θcosθdrdθdφ≡0,(45)I2=−μ−ν2ν∫ℝ3GΔ~ρdx=μ−ν2ν∫ℝ3∇~G·∇~ρdx≤ν2∇~GL2ℝ32+C∇~ρL2ℝ32.
Putting (44) and (45) into (43), we get
(46)12ddtGL2ℝ32+ν∥∇~G∥L2ℝ32≤ν2∥∇~G∥L2ℝ32+C∥∇~ρ∥L2ℝ32,which, together with (32), yields to the following estimate for Gt,r,θ:
(47)Gt,r,θL2ℝ3≤CT,∥G0,r,θ∥L2ℝ3,u0L2ℝ3,ρ0L2ℝ3.
Thus, we have(48)gt,r,θL2ℝ3=G+12νρL2ℝ3≤GL2ℝ3+CρL2ℝ3≤CT,ω0φr,θrsinθL2ℝ3,u0L2ℝ3,ρ0L2ℝ3.
Next, we obtain the estimate for the vorticityω=ωφt,r,θeφ, 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 vorticityω=∇×u for the 3D incompressible Boussinesq equation is the following:
(49)∂tω+u·∇ω−νΔω=ω·∇u−sinθ∂rρ+cosθ1r∂θρeφ.
Multiplying equation (49) by ω and integrating the resulting equation on ℝ3, we have, for any T>0, 0≤t≤T,
(50)12ddtωL2ℝ32+ν∇ωL2ℝ32=∫ℝ3ω·∇u·ωdx−∫ℝ3ωφsinθ∂rρ+cosθ1r∂θρdx=J1+J2,where J1=∫ℝ3ω·∇u·ωdx and J2=−∫ℝ3ωφsinθ∂rρ+cosθ1/r∂θρdx can be estimated as follows by using the special structure (33) of the velocity u and the vorticity ω. Using (33), with the help of the Hölder inequality, Gagliardo-Nirenberg inequality, and Young inequality, we have
(51)J1=∫ℝ3ωφeφ·er∂r+eθ1r∂θ+eφ1rsinθ∂φurer+uθeθ·ωφeφdx=∫ℝ3ωφrsinθ∂φurer+uθeθ·ωφeφdx=∫ℝ3ωφrsinθursinθ+uθcosθeφ·ωφeφdx=∫ℝ31rurωφωφdx+∫ℝ3cosθrsinθuθωφωφdx=∫ℝ3urgωφsinθdx+∫ℝ3uθgωφcosθdx≤∫ℝ3urgωφdx+∫ℝ3uθgωφdx≤urL3ℝ3+uθL3ℝ3gL2ℝ3ωL6ℝ3≤CuL2ℝ31/2∇uL2ℝ31/2gL2ℝ3∇ωL2ℝ3≤CuH1ℝ32+ν2∇ωL2ℝ32,(52)J2≤∫ℝ3ωφ∂rρ+1r∂θρdx≤ωL2ℝ32+∇~ρL2ℝ32.
Putting (51) and (52) into (50), we have
(53)12ddtωL2ℝ32+ν2∇ωL2ℝ32≤CuH1ℝ32+ωL2ℝ32+∇~ρL2ℝ32,which, by applying Gronwall’s inequality and by using (32), yields to, for any T>0,
(54)ωt,·L2ℝ3≤CT,u0H1ℝ3,ω0φrsinθL2ℝ3,ρ0L2ℝ3,0≤t≤T,(55)∫0t∇ωs,·L2ℝ32ds≤CT,u0H1ℝ3,ω0φrsinθL2ℝ3,ρ0L2ℝ3,0≤t≤T.
Using Lemma5, we get from (54) that, for any 0≤T≤∞,
(56)∇u∈L∞0,T;L2ℝ3,and hence, by Sobolev’s imbedding theorem, we have, for any 0≤T≤∞,
(57)u∈L∞0,T;L6ℝ3.
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 Theorem1 is complete.Proof of Theorem 2.
We takeΩ=Ω~ in Theorem 2, where Ω~ is given by (9) having one special geometry structure. Also, u in Theorem 2 is given by (12), which satisfies that u2=u12+u22+u32=ur2+uθ2+uφ2 by using the orthogonality of three spherical coordinate unit vectors. Firstly, for the system (2), we have the following basic energy estimates, for 0≤t≤+∞,
(58)ddt∫Ωut,·2dx+2ν∫Ω∇ut,·2dx≤δ∫Ωut,·2dx+Cδ∫Ωρt,·2dx,(59)ddt∫Ωρt,·2dx+2μ∫Ω∇ρt,·2dx=0,for some constant Cδ>0 and any δ>0, which, together with Poincare’s inequality for u and ρ, yield the energy estimate, for 0≤t≤+∞,
(60)ut,·L2Ω2+ρt,·L2Ω2+2ν∫0t∇ut,·L2Ω2dt+2μ∫0t∇ρt,·L2Ω2dt≤CΩ,ν,μ,u0L2Ω,ρ0L2Ω,(61)u,ρt,·L2Ω2≤CΩ,ν,μ,u0L2Ω,ρ0L2Ωe−αt,for some constants C=CΩ,ν,μ,u0L2Ω,∥ρ0∥L2Ω>0 and α=αΩ,ν,μ>0.
Next, we give the estimates of∥∂tu,ρt,·∥L2Ω2.
Differentiating (2) with respect to t, one gets
(62)utt+ut·∇u+u·∇ut+∇Pt=νΔut+ρte3,ρtt+u·∇ρt+ut·∇ρ=μΔρt,divut=0,ut∂Ω=0,ρt∂Ω=0,ut0,x=v0x,ρt0,x=ρ0x,where v0 and ρ0 satisfy, by using (2), that
(63)v0+u0·∇u0+∇P0=νΔu0+ρ0e3,divv0=0,(64)ρ0+u0·∇ρ0=μΔρ0.
It is easy to get that(65)v0L2Ω≤Cu0H2Ω,ρ0H2Ω,(66)ρ0L2Ω≤Cu0H2Ω,ρ0H2Ω.
In fact, multiplying (64) by v0 and integrating the resulting equation on Ω, applying the Hölder inequality, Gagliardo-Nirenberg inequality, and Young inequality, we have
(67)v0L2Ω2=−∫Ωu0·∇u0·v0dx+ν∫ΩΔu0·v0dx+∫Ωρ0e3·v0dx≤Cu0L3Ω∇u0L6Ωv0L2Ω+CΔu0L2Ωv0L2Ω+ρ0L2Ωv0L2Ω≤Cu0H2Ω+∥ρ0∥L2Ωv0L2Ω,which implies (65). Similarly, we have (66).
Multiplying the first equation in (62) by ut and integrating the resulting equation on Ω, with the help of the Hölder inequality, we get
(68)12ddtutL2Ω2+ν∇utL2Ω2=∫Ωρte3·utdx−∫Ωut·∇u·utdx≤Cδ∥ρt∥L2Ω2+δ∥ut∥L2Ω2+CutL4Ω2∇uL2Ω.
Multiplying the second equation in (62) by ρt and integrating the resulting equation on Ω, with the help of the Hölder inequality, we get
(69)12ddtρtL2Ω2+μ∇ρtL2Ω2=−∫Ωut·∇ρ·ρtdx≤CutL4Ω∇ρL2Ω∥ρt∥L4Ω.
In the following, we use the special geometry structure (9) of the domain Ω=Ω~⊂ℝ3 and the special geometry structure (12) of the functions u,ρt,x in spherical coordinates in ℝ3 to obtain the following inequality for ut,ρtt,x defined in 0,t×Ω~⊂0,∞×⊂ℝ3 with u,ρ∂Ω=0: there exists a constant C=CΩ>0 such that
(70)utt,·L4Ω≤CΩutt,·L2Ω1/2∥∇utt,·∥L2Ω1/2,(71)ρtt,·L4Ω≤CΩρtt,·L2Ω1/2∥∇ρtt,·∥L2Ω1/2,where
(72)∇=er∂r+1reθ∂θ+1rsinθeφ∂φ.
We note that the inequalities (70) are the same as in the two-dimensional case, which are, in general, not true for the general functions ut,x or ρt,x when x∈Ω if Ω is the general bounded domain of ℝ3. However, the equalities (70) are true under the assumption of Theorem 2 because of the special geometry structures (9) and (12) for the domain Ω and the functions u,ρt,x, especially ρt,x=ρt,r,θ independent of φ and ut,x=ut,r,θ,φ depending upon φ only by the three orthogonal unit vectors in spherical coordinates (the combination of the functions cosφ and sinφ). In fact, the inequalities (70) and (71) has been proven by S. Wang and Y.X. Wang in [16] for the domain Ω∈ℝ3 having the special geometry structure (9) and for the function ut,x,x∈Ω⊂ℝ3, with ut,x∂Ω=0 having the special geometry structure (12).
For completeness, we give the proof of the inequalities (70) and (71). Taking ft,r,θ=ρtr1/2sin1/4θ in Lemma 6 with the domain D=r0,r1×θ0,θ1⊂ℝ2 and ρt,r,θ∂D=0, we get
(73)ρtL4Ω4=∫Ωρt4t,r,θdx=∫02π∫θ0θ1∫r0R0ρt4t,r,θr2sinθdrdθdφ=2π∫θ0θ1∫r0R0ρtr1/2sin1/4θ4drdθ≤C∫θ0θ1∫r0R0ρtr1/2sin1/4θ2drdθ∫θ0θ1∫r0R0∇r,θρtr1/2sin1/4θ2drdθ,where C=CΩ is a constant depending upon the domain Ω=Ω~, and ∇r,θ=∂r,∂θ. It is clear that
(74)∫θ0θ1∫r0R0ρtr1/2sin1/4θ2drdθ=12π∫02π∫θ0θ1∫r0R01rsinθρt2r2sinθdrdθdφ≤12π∫02π∫θ0θ1∫r0R01r0minθ∈θ0,θ1sinθρt2r2sinθdrdθdφ=12πr0minθ∈θ0,θ1sinθρtL2Ω2,(75)∫θ0θ1∫r0R0∇r,θρtr1/2sin1/4θ2drdθ=12π∫02π∫θ0θ1∫r0R0∂rρtr1/2sin1/4θ2+∂θρtr1/2sin1/4θ2drdθdφ≤1π∫02π∫θ0θ1∫r0R0rsin1/2θ∂rρt2+∂θρt2+14rsin1/2θ+cos2θ16rsin−3/2θρt2drdθdφ=K1+K2,where
(76)K1=1π∫02π∫θ0θ1∫r0R01rsinθ∂rρt2+rsinθ1r∂θρt2r2sinθdrdθdφ≤1π∫02π∫θ0θ1∫r0R01r0minθ∈θ0,θ1sinθ∂rρt2+R0minθ∈θ0,θ1sinθ1r∂θρt2r2sinθdrdθdφ≤1πminθ∈θ0,θ1sinθmax1r0,R0∫Ω∂rρt2+1r∂θρt2dx≤1πminθ∈θ0,θ1sinθmax1r0,R0∫Ω∂rρt2+1r∂θρt2+1rsinθ∂φρt2dx=1πminθ∈θ0,θ1sinθmax1r0,R0∇ρtL2Ω2,(77)K2=1π∫02π∫θ0θ1∫r0R014r3sinθ+116rsin2θsinθρt2r2sinθdrdθdφ≤1π∫02π∫θ0θ1∫r0R014r03minθ∈θ0,θ1sinθ+116r0minθ∈θ0,θ1sinθ2minθ∈θ0,θ1sinθρt2r2sinθdrdθdφ≤1π14r03minθ∈θ0,θ1sinθ+116r0minθ∈θ0,θ1sinθ2minθ∈θ0,θ1sinθ∫Ωρt2dx≤1π14r03minθ∈θ0,θ1sinθ+116r0minθ∈θ0,θ1sinθ2minθ∈θ0,θ1sinθρtL2Ω2≤CΩ14r03minθ∈θ0,θ1sinθ+116r0minθ∈θ0,θ1sinθ2minθ∈θ0,θ1sinθ∇ρtL2Ω2,with the help of Poincare’s inequality and the fact that ∇ρr,θ,φ,t2=∂rρ2+∂θρ/r2+∂φρ/rsinθ2.
Combining (75) together with (76)–(77), we have
(78)∫θ0θ1∫r0R0∇r,θρtr1/2sin1/4θ2drdθ≤CΩ∇ρtL2Ω2.
Thus, putting (74) and (78) into (73), we get (70) and (71).
Combining (68), (69), and (70), with the help of the Young inequality, we obtain
(79)12ddtutL2Ω2+M1ρtL2Ω2+ν∇utL2Ω2+M1μ∥∇ρt∥L2Ω2≤Cδ∥ρt∥L2Ω2+δ∥ut∥L2Ω2+CutL4Ω2∇uL2Ω+CutL4Ω∇ρL2Ω∥ρt∥L4Ω≤Cδ∥ρt∥L2Ω2+δ∥ut∥L2Ω2+CutL2Ω∇utL2Ω∇uL2Ω+CutL2Ω1/2∥∇ut∥L2Ω1/2∥∇ρ∥L2Ω1/2ρtL2Ω1/2∥∇ρt∥L2Ω1/2∥∇ρ∥L2Ω1/2≤Cδ∥ρt∥L2Ω2+δ∥ut∥L2Ω2+C∇uL2Ω2utL2Ω2+C∇ρL2Ω2utL2Ω2+C∇ρL2Ω2ρtL2Ω2+ν2∇utL2Ω2+μ2∇ρtL2Ω2for any δ>0, for suitably large constant M1>0 and for some constants Cδ>0 and C=CM1,Ω,ν,μ>0, which yields, by applying Gronwall’s inequality and Poincare’s inequality, using the estimate (60), to the decay exponentially in time
(80)∥utt,·∥L2Ω+∥ρtt,·∥L2Ω≤C∥u0∥H2Ω,∥ρ0∥H2Ωe−αt,0≤t≤+∞.
Finally, we obtain the estimateL∞0,T;∥∇u∥L2Ω2 for u and for any T>0.
Multiplying (2) by u and integrating the resulting equation on Ω, and by using the estimates (61) and (80), we have, for 0≤t≤+∞,
(81)∇ut,·L2Ω2=1ν∫Ωρe3·udx−1ν∫Ωu·utdx≤Cν∥ρ∥L2Ω2+∥u∥L2Ω2+utL2ΩuL2Ω≤Cu0H2Ω,∥ρ0∥L2Ωe−αt.
Combining (60) and (81) together, we have
(82)u∈L∞0,T;H01Ω,which gives, by Sobolev’s imbedding theorem, that
(83)u∈L∞0,T;L6Ω.
Thus, we can obtain the desired global regularity estimate for strong solutionu,ρ and the global smooth interior regularity for the solution u,ρ by choosing the suitable cutoff function with compact subset of the domain Ω, and we can conclude the regularity results on Theorem 2 by using Lemma 8.
Also, it is easy from (59) to get that
(84)∇ρt,·L2Ω2≤C∥ρt,·∥L2Ω∥ρtt,·∥L2Ω≤Ce−αt,0≤t≤+∞.
Thus, the decay rate (24) in Theorem 2 can be obtained from the estimates (61) and (80)–(84).
The proof of Theorem2 is complete.



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*Source: 9121813-2022-05-28.xml*