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

Authors: Fan Geng; Shu Wang; Yongxin Wang
Journal: Journal of Function Spaces
 (2020)

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## 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 componentω3 and one a priori estimate on the solution that H3u3L∞0,T;Lpℝ3 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.


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## Body

## 1. Introduction

In this paper, we investigate the regularity problem on the following three-dimensional (3D) incompressible Navier-Stokes equations(1)∂tu+u⋅∇u+∇P=Δu,x∈ℝ3,t>0,divu=0,x∈ℝ3,t>0,u0,x=u0x,x∈ℝ3in general orthogonal curvilinear coordinate systems. Here, x=x1,x2,x3∈ℝ3, u=u1,u2,u3t,x denotes the velocity fields, P=Pt,x is the scalar pressure, and u0 is a given initial velocity with divu0=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 u0∈L2ℝ3 with divu0=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 [1–6]. 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 u will be smooth as long as
(2)u∈Lp0,T;LqR3,2p+3q≤1,2≤p≤∞,3≤q≤∞.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) andx3-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 ℝ3 is denoted by x=ξ1,ξ2,ξ3, where ξ1,ξ2,ξ3 is general orthogonal curvilinear coordinates with a line element ds given by
(3)ds=H1dξ1eξ1+H2dξ2eξ2+H3dξ3eξ3.Here,eξi=eξiξi=1,2,3 are orthogonal, of unit length and parallel to the coordinate lines with ξi increasing; the nonnegative functions Hii=1,2,3 are the G.Lamé coefficients corresponding to the vectors eξ1,eξ2,eξ3, respectively. Throughout this paper, we assume Hii=1,2,3 are independent of ξ3, i.e., Hi=Hiξ1,ξ2i=1,2,3, and the measure of the set ξ1,ξ2∈D:H1H2H3ξ1,ξ2=0 is zero in the sense of Lebesgue measure in ℝ2.In this paper, we consider the solutionu,P to the 3D incompressible Navier-Stokes (1) of having the form
(4)ut,x=u1t,ξ1,ξ2eξ1+u2t,ξ1,ξ2eξ2+u3t,ξ1,ξ2eξ3,Pt,x=Pt,ξ1,ξ2,with the initial data
(5)u0x=u01ξ1,ξ2eξ1+u02ξ1,ξ2eξ2+u03ξ1,ξ2eξ3.Our main results are the a priori estimates and the regularity criteria involving only in a vorticity componentω3 on 3D incompressible Navier-Stokes equations in general orthogonal curvilinear coordinate systems.Theorem 1 (the a priori estimate ofH3u3L∞0,T;Lpℝ3).
Suppose thatu be a smooth solution of system (1) with the form (4) and the initial data (5) satisfying divu0=0 and H3u03∈Lpℝ3 for p∈1,∞. Then, we have for any T>0,
(6)H3u3∈L∞0,T;L2ℝ3andH3u3∈L20,T;H1ℝ3,and, moreover, if assume that the G.Lamé coefficient H3ξ1,ξ2 satisfies
(7)ΔlnH3≤C<∞,it holds, for any T>0 and any p∈1,2∪2,∞, that
(8)H3u3∈L∞0,T;Lpℝ3andH3u3p/2∈L20,T;H1ℝ3.Theorem 2 (the regularity criteria involvingω3).
Letu be a weak solution of system (1) with the form (4) and the initial data (5) satisfying u0∈H2ℝ3 and divu0=0. Then, the solution u is smooth in 0,T×ℝ3, if
(9)ω3∈Lp0,T;Lqℝ3,where 2/p+3/q≤2,1≤p≤∞,3/2≤q≤∞.Remark 3.
The assumption (7) comes from the geometry on the harmonic mapping in some sense. It is easy to see that
(10)ΔlnH3=Δ~lnH3in (7), based on the notation Δ~ introduced in Section 2, because H3 is independent of ξ3. Thus, if H3 is a radial function in ℝ2, i.e.,
(11)H3=H3rwith
(12)r=x12+x22orr=x22+x32orr=x12+x32,then
(13)ΔlnH3≡0.
In this case, the assumption (7) is naturally true because the function lnH3 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 ℝ3 are be covered in Theorem 1 and Theorem 2, i.e., let
(14)ξ1=r=x12+x32,ξ2=x2,ξ3=θ=arctanx3x1;u1t,ξ1,ξ2=urt,r,x2,u2t,ξ1,ξ2=u3t,r,x2,u3t,ξ1,ξ2=uθt,r,x2;eξ1=er=x1r,x2r,0,eξ2=e3=0,0,1,eξ3=eθ=−x2r,x1r,0;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=xξ1,ξ2,ξ3=ξ1cosξ3,ξ2,ξ1sinξ3, and G.Lamé coefficients are H1=H2=1,H3=ξ1, 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 Section2, 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.


## 2. Navier-Stokes Equations in Orthogonal Curvilinear Coordinate Systems

In this section, we will first derive the incompressible Navier-Stokes equations in orthogonal curvilinear coordinate systemsξ1,ξ2,ξ3, given by Section 1.We assume thatx=xξ, being one-to-one and onto mapping, transforms ξ∈Ω~⊂ℝ3 into x∈Ω⊂ℝ3 with Ω~=D×α,β⊂ℝ3 and
(15)Ω=x1,x2,x3=x1,x2,x3ξ∈ℝ3:ξ1,ξ2∈D⊂ℝ2,−∞<α≤ξ3≤β<∞,where the domain D is the bounded or unbounded domain of ℝ2 with the smooth boundary ∂D if D is bounded, and the constants α and β satisfy −∞<α≤β<∞.SinceHi=Hiξ1,ξ2i=1,2,3, by the derivatives of the unit vectors eξ1,eξ2 and eξ3, we have
(16)∂eξ1∂ξ1=−eξ2H2∂H1∂ξ2,∂eξ1∂ξ2=eξ2H1∂H2∂ξ1,∂eξ1∂ξ3=eξ3H1∂H3∂ξ1,∂eξ2∂ξ1=eξ1H2∂H1∂ξ2,∂eξ2∂ξ2=−eξ1H1∂H2∂ξ1,∂eξ2∂ξ3=eξ3H2∂H3∂ξ2,∂eξ3∂ξ1=∂eξ3∂ξ2=0,∂eξ3∂ξ3=−eξ1H1∂H3∂ξ1−eξ2H2∂H3∂ξ2.Using the definition of gradient, we get the expression of the gradient operator∇ in orthogonal curvilinear coordinate systems:
(17)∇=eξ11H1∂∂ξ1+eξ21H2∂∂ξ2+eξ31H3∂∂ξ3,we also obtain the expression of Laplacian operator Δ in orthogonal curvilinear coordinate systems:
(18)Δ=1H1H2H3∂∂ξ1H2H3H1∂∂ξ1+∂∂ξ2H1H3H2∂∂ξ2+∂∂ξ3H1H2H3∂∂ξ3.Furthermore, for a vector fieldV=Vξ1,ξ2,ξ3=V1eξ1+V2eξ2+V3eξ3, we get the expressions of divV and rotV in orthogonal curvilinear coordinate systems:
(19)divV=1H1H2H3∂∂ξ1H2H3V1+∂∂ξ2H1H3V2+∂∂ξ3H1H2V3,(20)rotV=1H1H2H3H1eξ1H2eξ2H3eξ3∂∂ξ1∂∂ξ2∂∂ξ3H1V1H2V2H3V3.By the above expressions (17)–(19), then taking the inner product of equation (1)1with eξ1,eξ2, eξ3, respectively, we can derive the Navier-Stokes equations in orthogonal curvilinear coordinate systems as follows:
(21)∂u1∂t+u1H1∂u1∂ξ1+u2H2∂u1∂ξ2+1H1∂P∂ξ1=L1u1−1H1H32∂H3∂ξ12u1+L2u2−1H1H2H32∂H3∂ξ1∂H3∂ξ2u2−u1u2H1H2∂H1∂ξ2+u22H1H2∂H2∂ξ1+u32H1H3∂H3∂ξ2,∂u2∂t+u1H1∂u2∂ξ1+u2H2∂u2∂ξ2+1H2∂P∂ξ2=L1u2−1H2H32∂H3∂ξ22u2−L2u1−1H1H2H32∂H3∂ξ1∂H3∂ξ2u1−u1u2H1H2∂H2∂ξ1+u12H1H2∂H1∂ξ2+u32H1H3∂H3∂ξ1,∂u3∂t+u1H1∂u3∂ξ1+u2H2∂u3∂ξ2=L3u3−u3H3u1H1∂H3∂ξ1+u2H2∂H3∂ξ2,where ξ1,ξ2∈D⊂ℝ2,t>0 and
(22)∇ξ1,ξ2=eξ11H1∂∂ξ1+eξ21H2∂∂ξ2,Δ~=Δξ1,ξ2=1H1H2H3∂∂ξ1H2H3H1∂∂ξ1+∂∂ξ2H1H3H2∂∂ξ2,L1=Δξ1,ξ2−1H1H22∂H1∂ξ22+∂H2∂ξ12,L2=1H1H2H32H3H1∂H1∂ξ2∂∂ξ1+∂∂ξ1H3H1∂H1∂ξ2−2H3H2∂H2∂ξ1∂∂ξ2−∂∂ξ2H3H2∂H2∂ξ1,L3=Δξ1,ξ2−1H321H1∂H3∂ξ12+1H2∂H3∂ξ22.The incompressible constraint is(23)∂∂ξ1H2H3u1+∂∂ξ2H1H3u2=0.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:
(24)u1,u2,u3t=0,ξ1,ξ2=u01,u02,u03ξ1,ξ2.Moreover, the boundary conditionu→0 as ∣x∣→∞,t≥0 is equivalent to the following condition
(25)u1,u2,u3∂D=0,t≥0,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
(26)ωt,x=ω1t,ξ1,ξ2eξ1+ω2t,ξ1,ξ2eξ2+ω3t,ξ1,ξ2eξ3,with the initial vorticity
(27)ω0=ωt=0,x=ω01ξ1,ξ2eξ1+ω02ξ1,ξ2eξ2+ω03ξ1,ξ2eξ3,Where(28)ω1=1H2H3∂∂ξ2H3u3,ω2=−1H1H3∂∂ξ1H3u3,ω3=1H1H2∂∂ξ1H2u2−∂∂ξ2H1u1.Moreover, with the help of (28), we can get the equation of ω3 from (21) as follows:
(29)∂ω3∂t+u1H1∂ω3∂ξ1+u2H2∂ω3∂ξ2=L3ω3+ω3H3u1H1∂H3∂ξ1+u2H2∂H3∂ξ2+1H1H2H32∂H3∂ξ2∂H3u32∂ξ1−∂H3∂ξ1∂H3u32∂ξ2.


## 3. Some Useful Estimates

To study the main estimates of Theorem1 and Theorem 2, we need to introduce two basic lemmas and one estimate relates to ω3 in orthogonal curvilinear coordinate systems.Lemma 6 (see [6–8]).
Suppose that the initial datau0∈H2ℝ3 in (1), then any Leray-Hopf weak solution u of 3D incompressible Navier-Stokes equations (1) is also a smooth solution in 0,T×ℝ3, if there holds that
(30)u∈Lp0,T;Lqℝ3,in which p and q satisfy the conditions
(31)2p+3q≤1with3<q≤∞,2≤p<∞or3≤q<∞,p=∞.And we would like to recall the well-known relation between the velocity and vorticity of 3D flow.Lemma 7 (see [33]).
Letu∈W1,pℝ3 be a velocity field with its vorticity ω=∇×u, then the inequality
(32)∇uLpℝ3≤CpωLpℝ3+divuLpℝ3holds for any p∈1,∞, where the constant Cp 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 thatu∈W1,pℝ3 with the form (4) be a field with zero divergence, then the estimate
(33)∇u~Lpℝ3≤C~pω3Lpℝ3holds for any p∈1,∞, where
(34)u~=u1t,ξ1,ξ2eξ1+u2t,ξ1,ξ2eξ2and the constant C~p depends only on p.Proof.
Sincedivu=0 and we have
(35)0=∇⋅u=1H1H2H3∂∂ξ1H2H3u1+∂∂ξ2H1H3u2+∂∂ξ3H1H2u3=1H1H2H3∂∂ξ1H2H3u1+∂∂ξ2H1H3u2.
Thus, we obtain(36)divu~=1H1H2H3∂∂ξ1H2H3u1+∂∂ξ2H1H3u2=0.
On the other hand, by the expressions (20), (34), and (28), we get
(37)∇×u~=1H2H3−∂∂ξ3H2u2eξ1+1H1H3∂∂ξ3H1u1eξ2+1H1H2∂∂ξ1H2u2−∂∂ξ2H1u1eξ3=1H1H2∂∂ξ1H2u2−∂∂ξ2H1u1eξ3=ω3eξ3.
Consequently, by Lemma7 and using (36) and (37), one has
(38)∇u~Lpℝ3≤C~pω3eξ3Lpℝ3=C~pω3Lpℝ3.
This finishes the proof of the proposition.


## 4. Proof of Theorem 1

In this section, we prove Theorem1.Proof of Theorem 1.
LetFt,ξ1,ξ2=H3ξ1,ξ2u3t,ξ1,ξ2, then putting u3t,ξ1,ξ2=Ft,ξ1,ξ2/H3ξ1,ξ2 into (21)3, we can obtain the equation for F(39)∂tF+u1H1∂F∂ξ1+u2H2∂F∂ξ2=Δξ1,ξ2F−1H1H2∂∂ξ1H2H1H3∂H3∂ξ1F+2H2H1H3∂F∂ξ1∂H3∂ξ1+∂∂ξ2H1H2H3∂H3∂ξ2F+2H1H2H3∂F∂ξ2∂H3∂ξ2−FH321H1∂H3∂ξ12+1H2∂H3∂ξ22,with the help of the following calculations
(40)∂tu3=1H3∂tF,u1H1∂u3∂ξ1+u2H2∂u3∂ξ2=u1H11H3∂F∂ξ1−FH32∂H3∂ξ1+u2H21H3∂F∂ξ2−FH32∂H3∂ξ2=1H3u1H1∂F∂ξ1+u2H21H3∂F∂ξ2−FH32u1H1∂H3∂ξ1+u2H2∂H3∂ξ2,L3u3=1H31H1H2H3∂∂ξ1H2H3H1∂F∂ξ1+∂∂ξ2H1H3H2∂F∂ξ2+H2H3H1∂2F∂ξ12+H1H3H2∂2F∂ξ22−FH321H1∂H3∂ξ12+1H2∂H3∂ξ22+1H1H2H3F∂∂ξ1H2H3H1∂∂ξ1+2H2H3H1∂F∂ξ1∂∂ξ1+F∂∂ξ2H1H3H2∂∂ξ2+2H1H3H2∂F∂ξ2∂∂ξ21H3=1H3Δξ1,ξ2F−1H31H1H2H3∂∂ξ1H2H1H3∂H3∂ξ1F+2H2H1H3∂F∂ξ1∂H3∂ξ1+∂∂ξ2H1H2H3∂H3∂ξ2F+2H1H2H3∂F∂ξ2∂H3∂ξ2−FH321H1∂H3∂ξ12+1H2∂H3∂ξ22.
Multiplying the both sides of equation (39) by Fp−2F and integrating over ℝ3, we have
(41)1pddtFLpℝ3p+4p−1p2∇~Fp2L2ℝ32+∫ℝ3Fp−1u1H1∂F∂ξ1+u2H2∂F∂ξ2dx=−∫ℝ31H1H2∂∂ξ1H2H1H3∂H3∂ξ1Fp+2pH2H1H3∂H3∂ξ1∂Fp∂ξ1+∂∂ξ2H1H2H3∂H3∂ξ2Fp+2pH1H2H3∂H3∂ξ2∂Fp∂ξ2dx−∫ℝ3FpH321H1∂H3∂ξ12+1H2∂H3∂ξ22dx=−∫ℝ31H1H2∂∂ξ1H2H1H3∂H3∂ξ1Fp+H2H1H3∂Fp∂ξ1∂H3∂ξ1+∂∂ξ2H1H2H3∂H3∂ξ2Fp+H1H2H3∂H3∂ξ2∂Fp∂ξ2+FpH321H1∂H3∂ξ12+1H2∂H3∂ξ22dx−∫ℝ32p−11H1H2H2H1H3∂H3∂ξ1∂Fp∂ξ1+H1H2H3∂H3∂ξ2∂Fp∂ξ2dx=I1+I2.
Here, thanks to (15) and (25) and the incompressibility condition (23), by the fact of dx=H1H2H3dξ1ξ2ξ3, we have
(42)∫ℝ3Fp−1u1H1∂F∂ξ1+u2H2∂F∂ξ2dx=1pβ−α∫Du1H1∂∂ξ1+u2H2∂∂ξ2Fpt,ξ1,ξ2H1H2H3dξ1dξ2=1pβ−α∫DH2H3u1∂∂ξ1+H1H3u2∂∂ξ2Fpt,ξ1,ξ2dξ1dξ2=−1pβ−α∫D∂∂ξ1H2H3u1+∂∂ξ2H1H3u2Fpt,ξ1,ξ2dξ1dξ2≡0
For theI1 and I2 on the right of (41), by simple integration, one has
(43)I1=−∫ℝ31H1H2Fp∂∂ξ1H2H1H3∂H3∂ξ1+H2H1H3∂Fp∂ξ1∂H3∂ξ1+Fp∂∂ξ2H1H2H3∂H3∂ξ2+H1H2H3∂H3∂ξ2∂Fp∂ξ2+FpH321H1∂H3∂ξ12+1H2∂H3∂ξ22dx=−β−α∫DH3∂∂ξ1H2H1H3∂H3∂ξ1Fp+H2H1H3∂Fp∂ξ1∂H3∂ξ1+∂∂ξ2H1H2H3∂H3∂ξ2Fp+H1H2H3∂H3∂ξ2∂Fp∂ξ2dξ1dξ2−β−α∫DH1H2H31H1∂H3∂ξ12+1H2∂H3∂ξ22Fpdξ1dξ2=−β−α∫DH3∂∂ξ1H2H1H3∂H3∂ξ1Fp+∂∂ξ2H1H2H3∂H3∂ξ2Fpdξ1dξ2−β−α∫DH1H2H31H1∂H3∂ξ12+1H2∂H3∂ξ22Fpdξ1dξ2=β−α∫DH2H1H3∂H3∂ξ12+H1H2H3∂H3∂ξ22Fpdξ1dξ2−β−α∫DH2H1H3∂H3∂ξ12+H1H2H3∂H3∂ξ22Fpdξ1dξ2≡0(44)I2=−2p−1∫ℝ31H1H2H2H1H3∂H3∂ξ1∂Fp∂ξ1+H1H2H3∂H3∂ξ2∂Fp∂ξ2dx=−2p−1β−α∫DH2H1∂H3∂ξ1∂Fp∂ξ1+H1H2∂H3∂ξ2∂Fp∂ξ2dξ1dξ2=2p−1β−α∫DFp∂∂ξ1H2H1∂H3∂ξ1+∂∂ξ2H1H2∂H3∂ξ2dξ1dξ2=2p−1∫ℝ3FpH1H2H3∂∂ξ1H2H1∂H3∂ξ1+∂∂ξ2H1H2∂H3∂ξ2dx=2p−1∫ℝ3FpH1H2H3∂∂ξ1H2H3H11H3∂H3∂ξ1+∂∂ξ2H1H3H21H3∂H3∂ξ2dx=2p−1∫ℝ3FpH1H2H3∂∂ξ1H2H3H1∂lnH3∂ξ1+∂∂ξ2H1H3H2∂lnH3∂ξ2dx=2p−1∫ℝ3FpΔlnH3dx.
Combining (42)–(44) with (41), we have
(45)1pddtFLpℝ3p+4p−1p2∇ξ1,ξ2Fp2L2ℝ32=2p−1∫ℝ3FpΔlnH3dx.
Note that∇ξ1,ξ2Fp/2L2ℝ3=∇Fp/2L2ℝ3.
Ifp=2, it has, immediately, by Gronwall’s inequality,
(46)sup0≤t≤T∫ℝ3H3u3pdx+C∫0T∫ℝ3∇H3u3p22dxdt≤H3u03Lpℝ3p.
Ifp∈1,2∪2,∞, from (45) and (7), the right hand of (45) can be estimated by
(47)2p−1∫ℝ3FpΔlnH3dx≤C∫ℝ3FpΔlnH3dx≤CFLpℝ3.and, then, by Gronwall’s inequality again, one also has
(48)sup0≤t≤T∫ℝ3H3u3pdx+C∫0T∫ℝ3∇H3u3p22dxdt≤CT,H3u03Lpℝ3p.
The casep=∞ is immediate if we let p→∞ in (48). Thus, we finish the proof of Theorem 1.


## 5. Proof of Theorem 2

In this section, we prove Theorem2.Proof of Theorem 2.
From (1:1)1, we have the basic energy inequality, for any T>0,
(49)sup0≤t≤TuL2ℝ32+2∫0T∇uL2ℝ32≤u0L2ℝ32.
It is known that the vorticity equation for the vorticityω=∇×u for 3D incompressible Navier-Stokes equation is the following:
(50)∂tω+u⋅∇ω−Δω=ω⋅∇u.
Using the expressions (4), (17), and (26), we have
(51)ω⋅∇u=ω1eξ1+ω2eξ2+ω3eξ3⋅eξ11H1∂∂ξ1+eξ21H2∂∂ξ2+eξ31H3∂∂ξ3u1eξ1+u2eξ2+u3eξ3=ω11H1∂∂ξ1+ω21H2∂∂ξ2+ω31H3∂∂ξ3u1eξ1+u2eξ2+u3eξ3=ω11H1u1∂eξ1∂ξ1+eξ1∂u1∂ξ1+u2∂eξ2∂ξ1+eξ2∂u2∂ξ1+u3∂eξ3∂ξ1+eξ3∂u3∂ξ1+ω21H2u1∂eξ1∂ξ2+eξ1∂u1∂ξ2+u2∂eξ2∂ξ2+eξ2∂u2∂ξ2+u3∂eξ3∂ξ2+eξ3∂u3∂ξ2+ω31H3u1∂eξ1∂ξ3+u2∂eξ2∂ξ3+u3∂eξ3∂ξ3=ω1H1∂u1∂ξ1+ω2H2∂u1∂ξ2+ω1u2H1H2∂H1∂ξ2−ω2u2H1H2∂H2∂ξ1−ω3u3H1H3∂H3∂ξ1eξ1+ω1H1∂u2∂ξ1+ω2H2∂u2∂ξ2−ω1u1H1H2∂H1∂ξ2−ω2u1H1H2∂H2∂ξ1−ω3u3H2H3∂H3∂ξ2eξ2+ω1H1∂u3∂ξ1+ω2H2∂u3∂ξ2+ω3u1H1H3∂H3∂ξ1+ω3u2H2H3∂H3∂ξ2eξ3.
Then, by multiplyingω on the both sides of equation (50) and integrating over ℝ3, we get
(52)12ddtωL2ℝ32+∇ωL2ℝ32=∫ℝ3ω⋅∇u⋅ωdx=∫ℝ3ω1H1∂u1∂ξ1+ω2H2∂u1∂ξ2+ω1u2H1H2∂H1∂ξ2−ω2u2H1H2∂H2∂ξ1−ω3u3H1H3∂H3∂ξ1ω1+ω1H1∂u2∂ξ1+ω2H2∂u2∂ξ2−ω1u1H1H2∂H1∂ξ2−ω2u1H1H2∂H2∂ξ1−ω3u3H2H3∂H3∂ξ2ω2+ω1H1∂u3∂ξ1+ω2H2∂u3∂ξ2+ω3u1H1H3∂H3∂ξ1+ω3u2H2H3∂H3∂ξ2ω3dx=∫ℝ3ω1ω1H1∂u1∂ξ1+u2H2∂H1∂ξ2dx+∫ℝ3ω1ω2H1∂u2∂ξ1−u1H2∂H1∂ξ2dx+∫ℝ3ω1ω3H1∂u3∂ξ1dx+∫ℝ3ω1ω2H2∂u1∂ξ2−u2H1∂H2∂ξ1dx+∫ℝ3ω2ω2H2∂u2∂ξ2+u1H1∂H2∂ξ1dx+∫ℝ3ω2ω3H2∂u3∂ξ2dx−∫ℝ3ω2ω3u3H2H3∂H3∂ξ2dx−∫ℝ3ω1ω3u3H1H3∂H3∂ξ1dx+∫ℝ3ω3ω3H3u1H1∂H3∂ξ1+u2H2∂H3∂ξ2dx=∑9i=1Ji.
Just for the sake of clarity, we give the expressions as follows:(53)∇u=eξ11H1∂∂ξ1+eξ21H2∂∂ξ2+eξ31H3∂∂ξ3u1eξ1+u2eξ2+u3eξ3=eξ1⊗eξ1H1∂u1∂ξ1+u2H2∂H1∂ξ2+eξ1⊗eξ2H1∂u2∂ξ1−u1H2∂H1∂ξ2+eξ1⊗eξ3H1∂u3∂ξ1+eξ2⊗eξ1H2∂u1∂ξ2−u2H1∂H2∂ξ1+eξ2⊗eξ2H2∂u2∂ξ2+u1H1∂H2∂ξ1+eξ2⊗eξ3H2∂u3∂ξ2+eξ3⊗eξ1H3−u3H1∂H3∂ξ1+eξ3H3−u3H2∂H3∂ξ2+eξ3⊗eξ3H3u1H1∂H3∂ξ1+u2H2∂H3∂ξ2(54)∇u~=eξ11H1∂∂ξ1+eξ21H2∂∂ξ2+eξ31H3∂∂ξ3u1eξ1+u2eξ2=eξ1⊗eξ1H1∂u1∂ξ1+u2H2∂H1∂ξ2+eξ1⊗eξ2H1∂u2∂ξ1−u1H2∂H1∂ξ2+eξ2⊗eξ1H2∂u1∂ξ2−u2H1∂H2∂ξ1+eξ2⊗eξ2H2∂u2∂ξ2+u1H1∂H2∂ξ1.
Then, we estimate each integralJii=1,⋯,9 in the right-hand side of (52) by using the relation (53) and the relation (54), respectively.
On the one hand, by applying Proposition8, Hölder inequality, Sobolev’s imbedding inequality, and Young inequality, it follows, with the help of the relation (54), that
(55)J1=∫ℝ3ω1ω1H1∂u1∂ξ1+u2H2∂H1∂ξ2dx≤∫ℝ3ωω∇u~dx≤ωL2qq−1ℝ32∇u~Lqℝ3≤CωL2qq−1ℝ32ω3Lqℝ3≤CωL2ℝ32q−3q∇ωL2ℝ33qω3Lqℝ3≤Cω3Lqℝ32q2q−3ωL2ℝ32+118∇ωL2ℝ32.
The termJ2 can be estimated similarly to J1 as
(56)J2=∫ℝ3ω1ω2H1∂u2∂ξ1−u1H2∂H1∂ξ2dx≤∫ℝ3ωω∇u~dx≤Cω3Lqℝ32q2q−3ωL2ℝ32+118∇ωL2ℝ32.
Similarly,(57)J4=∫ℝ3ω1ω2H2∂u1∂ξ2−u2H1∂H2∂ξ1dx≤∫ℝ3ωω∇u~dx≤Cω3Lqℝ32q2q−3ωL2ℝ32+118∇ωL2ℝ32J5=∫ℝ3ω2ω2H2∂u2∂ξ2+u1H1∂H2∂ξ1dx≤C∫ℝ3ωω∇u~dx≤Cω3Lqℝ32q2q−3ωL2ℝ32+118∇ωL2ℝ32.
On the other hand, by using Lemma7, Hölder inequality, Sobolev imbedding inequality, and Young inequality, it yields, with the help of the relation (53), that
(58)J3=∫ℝ3ω1ω3H1∂u3∂ξ1dx≤∫ℝ3ω∇uω3dx≤ωL2qq−1ℝ3∇uL2qq−1ℝ3ω3Lqℝ3≤CωL2qq−1ℝ32ω3Lqℝ3≤CωL2ℝ32q−3q∇ωL2ℝ33qω3Lqℝ3≤Cω3Lqℝ32q2q−3ωL2ℝ32+118∇ωL2ℝ32.
The termsJ6,J7,J8 and J9 can be estimated similarly to J3as
(59)J6+J7+J8+J9≤C∫ℝ3ω∇uω3dx≤Cω3Lqℝ32q2q−3ωL2ℝ32+29∇ωL2ℝ32.
Combining all above estimates aboutJii=1,⋯,9, from (52), we obtain
(60)ddtωL2ℝ32+∇ωL2ℝ32≤Cω3Lqℝ32q2q−3ωL2ℝ32.
Applying Gronwall’s inequality, it yields, for any0≤t≤T(61)ω∈L∞0,T;L2ℝ3.
Together with Lemma7 and (49), it implies that, for any 0≤t≤T(62)u∈L∞0,T;H1ℝ3.
By Sobolev’s imbedding Theorem, we obtain that, for any0≤t≤T(63)u∈L∞0,T;L6ℝ3.
Hence, with the help of Lemma6, we finish the proof of Theorem 2.



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*Source: 2816183-2020-10-13.xml*