Global regularity for a family of 3D models of the axisymmetric Navier-Stokes equations
Thomas Y Hou Pengfei Liu Fei Wang

t-Structures on stable derivators and Grothendieck hearts
    Manuel Saorín[The first named author was supported by the research projects from
the Ministerio de Economía y Competitividad of Spain (MTM2016-77445-P) and the Fundación `Séneca' of
Murcia (19880/GERM/15), both with a part of FEDER funds.] Jan Šťovíček[The second named author was supported by Neuron Fund for Support of Science.] Simone Virili[The third named author was supported by the Ministerio de Economía y Competitividad of
Spain via a grant `Juan de la Cierva-formación'. He was also supported by the
Fundación `Séneca' of Murcia (19880/GERM/15) with a part of FEDER funds.]
    
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	We consider a family of 3D models for the axi-symmetric incompressible Navier-Stokes equations. The models are derived by changing the strength of the convection terms in the axisymmetric Navier-Stokes equations written using a set of transformed variables. We prove the global regularity of the family of models in the case that the strength of convection is slightly stronger than that of the original Navier-Stokes equations, which demonstrates the potential stabilizing effect of convection. 



§ INTRODUCTION AND MAIN RESULT

The three-dimensional (3D) Euler and Navier-Stokes equations govern the motion of ideal incompressible fluid in the absence of external forcing:

    𝐮_t+𝐮·∇𝐮=-∇ p+νΔ𝐮,  ∇·𝐮=0.

Here 𝐮(x,t): 𝐑^3× [0, T)→𝐑^3 is the 3D velocity vector of the fluid, and p(x,t): 𝐑^3× [0, T)→𝐑 describes the scalar pressure.  The viscous term νΔ𝐮 models the viscous forcing in the fluid. In the case that ν=0, equations (<ref>) are referred to as the Euler equations, and in the case that ν>0, equations (<ref>) are referred to as the Navier-Stokes equations. The divergence-free condition ∇·𝐮=0 guarantees the incompressibility of the fluid. The Euler and Navier-Stokes equations are among the most fundamental nonlinear partial differential equations (PDEs) in nature yet far from being fully understood. The fundamental question regarding the global regularity of the Euler and Navier-Stokes equations with smooth initial data in the 3D setting remains open, and it is generally viewed as one of the most important open questions in mathematical fluid mechanics; see the surveys <cit.>.  

The Euler equations have the following scaling-invariance 

    𝐮(x, t)→λ/τ𝐮(x/λ, t/τ),     p(x,t)→λ^2/τ^2p(x/λ, t/τ),

and for the Navier-Stokes equations, due to the viscous term, the two-parameter symmetry group in (<ref>) is restricted to the following one-parameter group

    𝐮(x, t)→1/τ^1/2𝐮(x/τ^1/2, t/τ),     p(x,t)→1/τp(x/τ^1/2, t/τ).


Smooth solutions to the Euler/Navier-Stokes equations (<ref>) enjoy the following energy identity,

	

    1/2∫ |𝐮(x,t)|^2dx+ν∫_0^t∫_𝐑^3|∇𝐮(x,s)|^2dxds=1/2∫ |𝐮(x,0)|^2dx,

which implies the following a priori estimates for Navier-Stokes:

    1/2∫_𝐑^3|𝐮(x,t)|^2dx,  ∫_0^t∫_𝐑^3|∇𝐮(x,s)|^2dxds≤ C.

The above estimates seem to be the only known coercive a priori estimates for smooth solutions to the Navier-Stokes (<ref>). The main difficulty for the global regularity problem of the 3D Navier-Stokes equations lies in the fact that these known a priori estimates (<ref>) are supercritical with respect to the invariant scaling of the equations (<ref>); see <cit.> for more discussion about this supercritical barrier. For the 3D Euler equations, due to the lack of regularization mechanism (there is no viscosity), to prove the global regularity of the solutions becomes even more challenging.







In this work, we consider a family of 3D models for the Navier-Stokes equations with axial symmetry, which is proposed in <cit.>,



    u_1,t+u^ru_1,r+u^zu_1,z   =ν(∂_r^2+3/r∂_r+∂_z^2)u_1+2u_1ϕ_1,z,
    ω_1,t+u^rω_1,r+u^zω_1,z   =ν(∂_r^2+3/r∂_r+∂_z^2)ω_1+(u_1^2)_z,
    
    	-(∂_r^2+3/r∂_r+∂_z^2)ϕ_1   =ω_1,

with the Biot-Savart law given by

    u^r=-ϵ rϕ_1,z,    u^z=2ϵϕ_1+ϵ rϕ_1,r.


We give the derivation of this model in Section <ref> for the sake of completeness.

In (<ref>), the parameter ϵ characterizes the strength of the convection. The case that ϵ=1 corresponds to the original axi-symmetric Euler/Navier-Stokes equations, and the case ϵ=0 corresponds to the 3D model investigated in <cit.>. This family of models was proposed in <cit.> to study the effect of convection on the depletion of nonlinearity or formation of finite-time singularities. This family of models share several regularity results with the original Euler and Navier-Stokes equations, including an energy identity and two well-known non-blowup criteria. The numerical results in <cit.> suggest that the inviscid models with weak convection can develop self-similar singularity and such singularity scenario does not seem to persist as the strength of the convection terms increases, specifically for the original axisymmetric Euler. 

For the family of viscous models with ϵ∈ [1,2), we can obtain a maximum principle for a modified circulation quantity 
Γ^ϵ=u_1r^2/ϵ, i.e.,

    Γ^ϵ_L^∞≤Γ^ϵ_0_L^∞,

which is subcritical with respect to the invariant scaling for ϵ>1.

The models (<ref>) can also be written in a velocity-pressure form as 

	
 
    𝐯_t+ϵ𝐯·∇𝐯=-∇ p+νΔ𝐯+(2ϵ-2)v^θ v^r 𝐞_θ/r,

where the velocity 𝐯(x,t) is a rescaling of the velocity in model (<ref>):

    𝐯=u^r/ϵ𝐞_𝐫+u^z/ϵ𝐞_𝐳+u^θ/ϵ^3/2𝐞_θ,   u^θ=ru_1.




The viscous models (<ref>) enjoy the same scaling-invariance as the Navier-Stokes equations, and smooth solutions enjoy a similar energy identity as (<ref>) for ϵ∈ [0,2). To be specific, the  a priori energy estimate is supercritical. 

We also have the following maximum principle for a modified total circulation, Γ^ϵ, in the case that ϵ∈ [1, 2):

    Γ^ϵ=u_1r^2/ϵ,   Γ^ϵ_L^∞≤Γ^ϵ_0_L^∞.

The a priori estimate for Γ^ϵ_L^∞ is subcritical with respect to the invariant scaling of the models, which is the key in our proof of the global regularity. 


Next we state our main result.

	
	Consider the viscous models (<ref>) with ϵ∈ (20/19,2) and data 
	
    𝐯(·, 0)∈ H^4( 𝐑^3).

	Then the solution 𝐯(x,t) is globally regular in time.


This result further demonstrates the potential stabilizing effect of the convection terms, which has been demonstrated in the numerical results in <cit.> that the self-similar singularity of the inviscid models with weak convection does not persist as the strength of the convection terms increases.

To prove the main result Theorem <ref>, we can use L^p estimate for ω_1 and L^q estimate for u_1. To control the nonlinear vortex stretching term in the equation of ω_1 using the viscous term, we only need to use the subcritical a priori estimate (<ref>) and the Hardy inequality, under the condition that q=2p-p/2ϵ' for some ϵ'<ϵ. However, for the nonlinear term in the equation of u_1, the subcritical a priori estimate (<ref>) seems insufficient, because it can only control the angular component of the velocity. We use a combination of the supercritical energy estimate (<ref>) and the subcritical estimate of Γ^ϵ (<ref>) in the nonlinear term in the equation of u_1. To bound the nonlinear term using the viscous term, we need the condition ϵ > 20/19 in (<ref>). 

In our proof of the main result in section <ref>, we only conduct L^2 estimate for ω_1, and using any L^p estimate for ω_1 with p∈ (1, +∞) will lead to the same result under the condition ϵ > 20/19. 

The rest of this paper is organized as follows. In section <ref>, we derive the family of the models that we investigate in this work and list some regularity results for these models. We also give a brief review of recent regularity results for the Navier-Stokes equations with axial symmetry. In section <ref>, we prove our main result Theorem <ref>.



§ DERIVATION OF THE MODELS AND REVIEW OF THE LITERATURE


Recently the Euler and Navier-Stokes equations with axial symmetry have attracted a lot of interests. The global regularity problem in this setting remains open although a lot of progress has been made. Let 𝐞_𝐫, 𝐞_θ and 𝐞_𝐳 be the standard orthonormal vectors defining the cylindrical coordinates,

    𝐞_𝐫=(x_1/r, x_2/r, 0)^T,  𝐞_θ=(x_2/r, -x_1/r, 0)^T,  𝐞_𝐳=(0,0,1)^T,

where r=√(x_1^2+x_2^2) and z=x_3. Then the 3D velocity field 𝐮(x,t) is called axi-symmetric if it can be written as 

    𝐮(x,t)=u^r(r, z, t)𝐞_𝐫+u^θ(r, z, t)𝐞_θ+u^z(r, z, t)𝐞_𝐳,

where u^r, u^θ and u^z do not depend on the θ coordinate.

We denote the axi-symmetric vorticity field ω as,

    ω(x,t)=∇×𝐮(x,t)=ω^r(r, z, t)𝐞_𝐫+ω^θ(r, z, t)𝐞_θ+ω^z(r, z, t)𝐞_𝐳,

and then the Euler and Navier-Stokes equations with axial symmetry can be written using the cylindrical coordinates as 

	
	
    u^θ_t+u^ru^θ_r+u^zu_z^θ   =ν(Δ-1/r^2)u^θ-u^ru^θ/r,
    ω^θ_t+u^rω_r^θ+u^zω_z^θ   =ν(Δ-1/r^2)ω^θ+2/ru^θ u^θ_z+u^rω^θ/r,
    
    		-[Δ-1/r^2]ϕ^θ   =ω^θ,

where the radial and angular velocity fields u^r(r,z,t) and u^z(r,z,t) are recovered from the stream function ϕ^θ based on the Biot-Savart law

    u^r=-∂_zϕ^θ,   u^z=r^-1∂_r(rϕ^θ).



Note that the equations for angular velocity (<ref>), axial vorticity (<ref>), and the Biot-Savart law (<ref>)-(<ref>) form a closed system. Equations (<ref>) have a formal singularity on the axis r=0 due to the 1/r terms. Using the fact that the angular component u^θ(r,z), ω^θ(r,z) and ϕ^θ(r,z) can all be viewed as odd functions of r <cit.>, Hou and Li introduced the following transformed variables in <cit.>,

    u_1=u^θ/r,  ω_1=ω^θ/r,  ϕ_1=ϕ^θ/r,

to remove the formal singularity in (<ref>). This leads to the following reformulated axi-symmetric Navier-Stokes equations:



    u_1,t+u^ru_1,r+u^zu_1,z   =ν(∂_r^2+3/r∂_r+∂_z^2)u_1+2u_1ϕ_1,z,
    ω_1,t+u^rω_1,r+u^zω_1,z   =ν(∂_r^2+3/r∂_r+∂_z^2)ω_1+(u_1^2)_z,
    
    	-[∂_r^2+3/r∂_r+∂_z^2]ϕ_1   =ω_1,

with the Biot-Savart law given by

    u^r=-rϕ_1,z,    u^z=2ϕ_1+rϕ_1,r.



In <cit.>, a family of 3D models for axi-symmetric Euler and Navier-Stokes equations was proposed by changing the Biot-Savart law (<ref>):

    u^r=-ϵ rϕ_1,z,   u^z=2ϵϕ_1+ϵ rϕ_1,r,

to study the potential stabilizing effect of the convection terms.

The viscous model (<ref>) enjoys the following scaling-invariance:

    u_1(r, z,t)→1/τu_1(r/τ^1/2, z/τ^1/2, t/τ),   ω_1(r, z,t)→1/τ^3/2ω_1(r/τ^1/2,z/τ^1/2, t/τ).


The modified velocity field (<ref>) is still divergence-free

    ∇·𝐯=1/ϵ((u^rr)_r+(u^zr)_z)=0.


It was proved in <cit.> that the models (<ref>) with ϵ∈ [0, 2) share several regularity results with the original Euler and Navier-Stokes equations, including an energy identity, the conservation of a modified circulation quantity, the BKM non-blowup criterion, and the Prodi-Serrin non-blowup criterion.

	Smooth solutions to the models (<ref>) with ϵ∈[0,2) enjoy the following energy identity with u^θ=ru^1:

	

    1/2d/dt∫ (u^r)^2+(u^z)^2+1/2-ϵ(u^θ)^2 rdrdz
       =-ν∫ |∇ u^r|^2+|∇ u^z|^2+(u^r)^2/r^2+1/2-ϵ(u^θ)^2/r^2 rdrdz.


Note that the modified energy functional in (<ref>), 

    E_ϵ=d/dt∫ (u^r)^2+(u^z)^2+1/2-ϵ(u^θ)^2 rdrdz

is equivalent to that of the original Euler and Navier-Stokes equations, E_1, 

    min(1, 1/2-ϵ) E_1≤ E_ϵ≤max(1, 1/2-ϵ) E_1.


Based on (<ref>), we have the  following a priori estimates of the solutions 

    u^θ(r, z, t)_L^2,  u^r(r, z, t)_L^2,
       ∫_0^t ϕ_1,z(s)_L^2^2ds=∫_0^t 1/ϵ^2u^r(s)/r_L^2^2ds≤ C.



We define the modified total circulation Γ^ϵ as 

	

    Γ^ϵ=u_1r^2/ϵ,

and then Γ^ϵ satisfies the following equation 

    Γ^ϵ_t+u^rΓ^ϵ_r+u^zΓ^ϵ_z=ν(Δ -2/r(2/ϵ-1)∂_r+1/r^22/ϵ(2/ϵ-2))Γ^ϵ.

Then for the inviscid model with ν=0, or the viscous model with ν>0, ϵ≥ 1, we have the following maximum principle

    Γ^ϵ(r, z, t)_L^∞≤Γ^ϵ(r, z, 0)_L^∞=Γ^ϵ_0_L^∞.



For the viscous models with ϵ>1, the quantity Γ^ϵ is indeed subcritical with respect to the invariant scaling of the equations in (<ref>), which is the key in our proof of the global regularity result for the models in this paper.

Both the inviscid models and the viscous models enjoy the following BKM type criterion for smooth initial data with decay at infinity. If

	

    ∫_0^T∇×𝐯(x, t)_𝐁𝐌𝐎dt<+∞,

then

    𝐯(x,t)∈ L^∞( H^4(𝐑^3), [0, T]).



The viscous models also enjoy the Prodi-Serrin type of regularity criterion for smooth initial data with decay at infinity. If 

	

    𝐯(x,t)∈ L^q(L^p(𝐑^3), (0, T)), 3/p+2/q=1,  p∈ (3, +∞], q∈ [2, +∞),

then

    𝐯(x,t)∈ L^∞( H^4(𝐑^3), [0, T]).



In <cit.>, convincing numerical evidence is presented to show that  the inviscid models with weak convection could develop stable self-similar singularity on the symmetric axis. The singularity scenario in <cit.> is different from that at the boundary described in <cit.> in the sense that the center of the singularity region is not stationary but traveling along the symmetric axis. As the strength of the convection terms increases, the self-similar singularity scenario becomes less stable. Such finite-time singularity scenario does not seem to persist for the models with strong convection (ϵ≥ϵ_0 for some ϵ_0 >0), specifically the original axi-symmetric Euler equations. These results demonstrate the potential stabilizing effect of the convection terms. In this work, we prove the global regularity of the viscous models when the strength of convection slightly stronger than the original Navier-Stokes, i.e. ϵ∈ (20/19,2). The result proved in this work further demonstrates the potential stabilizing effect of convection in axi-symmetric Navier-Stokes equations.

The modified total circulation Γ^ϵ (<ref>) is subcritical with respect to the scaling (<ref>) for all ϵ>1. However, the estimate (<ref>) can only control the angular component of the velocity, and using the technique presented in this work we can only prove the regularity of the models for ϵ∈ (20/19,2), not (1,2).

Some important progress has been made regarding the regularity of the axi-symmetric Navier-Stokes equations recently; see <cit.>, and we mention a few related works below. In <cit.>, Hou and Li proposed a 1D model by restricting the equations (<ref>) to the symmetric axis. Using a cancellation property in the equation for u_1,z, they proved the global regularity of the 1D model with or without viscosity. In <cit.>, the cancellation property used in <cit.> was further exploited, and several critical regularity criteria concerning only the angular velocity are proved. In particular, the authors of <cit.> showed that if 
r^du^θ∈ L^q(L^p(𝐑^3), (0, T)) with 

    d∈ [0, 1),   (p, q)∈{(3/1-d,∞]×[2/1-d,∞]},  3/p+2/q≤ 1-d,
 then the solutions can be smoothly extended beyond T.

In <cit.>, the global regularity was obtained if |Γ|≤ C|ln r|^-2, and this result was later improved to |Γ|≤  C|ln r|^-3/2 in <cit.>. The cancellation property in the equation of u_1,z is crucial for the results in <cit.>. However, for the family of models (<ref>) that we study in this paper, this cancellation is destroyed due to the change of strength in the convection terms in (<ref>).



§ PROOF OF THE MAIN RESULT



In this section we prove the main result Theorem <ref>. We need the following Hardy inequality in 1D, see <cit.>.

If λ>1, σ≠ 1, f(r) is a nonnegative measurable function, and 

    F(r)=∫_0^r f(t)dt,  for  σ>1,   F(r)=-∫_r^∞ f(t)dt,  for  σ<1,

then

    ∫_0^∞ r^-σF^λ dr≤ (λ/|σ-1|)^λ∫_0^∞ r^-σ(rf)^λ dr.
 


We also need the following elliptic estimates <cit.>

	For axi-symmetric smooth functions ϕ_1(r,z) and ω_1(r,z) in 𝐑^3, which satisfy the elliptic equation
	
    -Δϕ_1-2/r∂_rϕ_1=ω_1,
 
	we have the following estimates
	
    ∇^2 ϕ_1_L^2≤ Cω_1_L_2,  ∇^2 ϕ_1,z_L^2≤ C∇ω_1_L^2.




	For smooth solution of the model (<ref>), u_1 and ϵ'∈(1, ϵ), 
	
    ∫ |u_1(r, z)|^ϵ' f(r)^2rdrdz≤ C_1(r)∫ |∂_rf|^2rdrdz+Cr_1^-2ϵ'/ϵ∫_r≥ r_1f^2rdrdz,

	with
	
    C_1(r_1)=CΓ_0^ϵ^ϵ'_L^∞r_1^2-2ϵ'/ϵ(ϵ/ϵ-ϵ')^2,  lim_r_1→ 0^+C_1(r_1)=0 .



 Let ψ(r) be a radial cutoff function such that
	
    ψ(r)∈ C^∞ (R), ψ(r)=
    			1,   r≤ 1
    
    			0,   r≥ 2
    		, 0≤ψ(r)≤ 1, |ψ_r(r)|≤ 2.

		Denote ψ_r_1(r) as ψ(r/r_1), then we have 
		
    ∫ |u_1| ^ϵ'f(r)^2rdrdz=∫ |u_1|^ϵ'(f(r)ψ_r_1(r)+f(r)(1-ψ_r_1(r)))^2 rdrdz
       ≤ 2∫_r≤ 2r_1 |u_1|^ϵ'f(r)^2|ψ_r_1(r)|^2rdrdz+2∫_r≥ r_1|u_1|^ϵ'f(r)^2(1-ψ_r_1(r))^2rdrdz.


Using the maximum principle (<ref>), we have 
	
    |u_1(r,z,t)|≤Γ^ϵ_L^∞r^-2/ϵ≤Γ_0^ϵ_L^∞r^-2/ϵ.


	Putting (<ref>) in the first term on the RHS of (<ref>), and using the Hardy inequality (<ref>), we get
	
    ∫_r≤ 2r_1 |u_1|^ϵ'f(r)^2|ψ_r_1(r)|^2rdrdz 
       ≤∫_r≤ 2r_1Γ_0^ϵ_L^∞^ϵ'r^-2ϵ'/ϵ|f(r)ψ_r_1(r)|^2rdrdz
       ≤ CΓ_0^ϵ^ϵ'_L^∞(ϵ/ϵ-ϵ')^2∫_r≤ 2r_1 r^2-2ϵ'/ϵ|∂_r(f(r)ψ_r_1(r))|^2rdrdz
       ≤ CΓ_0^ϵ^ϵ'_L^∞(ϵ/ϵ-ϵ')^2[∫_r≤ 2r_1r^2-2ϵ'/ϵ|∂_r f|^2rdrdz+∫_r≥ r_1r^2-2ϵ'/ϵ|f|^2|∂_rψ_r_1|^2rdrdz]
       ≤ CΓ_0^ϵ^ϵ'_L^∞(ϵ/ϵ-ϵ')^2r_1^2-2ϵ'/ϵ∂_rf^2_L^2+CΓ_0^ϵ^ϵ'_L^∞(ϵ/ϵ-ϵ')^2r_1^-2ϵ'/ϵ∫_r≥ r_1f^2rdrdz.


	For the second term in (<ref>), using the estimate (<ref>), we have 
	
    ∫_r≥ r_1|u_1|^ϵ'f^2(r)(1-ψ_r_1(r))^2rdrdz≤ CΓ_0^ϵ^ϵ'_L^∞r_1^-2ϵ'/ϵ∫_r≥ r_1f^2rdrdz.


	Adding up estimates (<ref>) and (<ref>), we prove (<ref>).  
	

Next we give the proof for the Theorem <ref>. Without loss of generality, we assume that ν=1 in our proof.

	We denote ϵ'=20/19<ϵ and consider the following two quantities: 
	
    ∫ |ω_1|^2rdrdz,  ∫ |u_1|^qrdrdz,   q=4-ϵ'.


	Multiplying the equation of ω_1 (<ref>) by ω_1, we get
	
    d/dt1/2∫ω_1^2rdrdz+1/2∫ u^r (ω_1^2)_r+u^z(ω_1^2)_zrdrdz
       =∫ 2u_1u_1,zω_1rdrdz+∫ (Δω_1+2/rω_1,r)ω_1rdrdz.


	Using integration by part, we can show that the convection terms vanish due to the incompressibility condition (u^rr)_r+(u^zr)_z=0:
	
    ∫ u^r (ω_1^2)_r+u^z(ω_1^2)_zrdrdz=-∫ω_1^2((u^rr)_r+(u^zr)_z)drdz=0.


	For the viscous term on the RHS of (<ref>), we have
	
    ∫Δω_1 ω_1rdrdz=-∫|∇ω_1|^2rdrdz.


	Next we treat the first order derivative term on the RHS of (<ref>) as
	
    ∫2/rω_1,rω_1rdrdz=∫ (ω_1^2)_rdrdz=-∫ω_1(0, z, t)^2dz≤ 0.


Using integration by part and Young's inequality leads to

    |∫ 2u_1u_1,zω_1rdrdz|=|∫ u_1^2ω_1,zrdrdz|
       ≤1/2∫ u_1^4 rdrdz+1/2∫ |∇ω_1|^2rdrdz.


For the first term on RHS of (<ref>), using Lemma <ref> and q=4-ϵ',  we get

    1/2∫ u_1^4rdrdz=1/2∫ |u_1|^ϵ'|u_1|^qrdrdz
       ≤ C(r_1)∫ |∇(|u_1|^q/2)|^2rdrdz+C∫ |u_1|^qrdrdz.

	
Adding up estimates (<ref>) and (<ref>) in (<ref>), we have 

    d/dt∫ω_1^2rdrdz+∫ |∇ω_1|^2rdrdz
       ≤ C(r_1) ∫ |∇(|u_1|^q/2)|^2rdrdz+C∫ |u_1|^q rdrdz.


Next we consider the equation of u_1, (<ref>) and multiply both sides by |u_1|^q-2u_1 to obtain

    d/dt1/q∫ |u_1|^qrdrdz+1/q∫ u^r|u_1|^q_r+u^z|u_1|^q_zrdrdz
    
    	=   ∫ 2|u_1|^qϕ_1,zrdrdz+∫ (Δ u_1+2/ru_1,r)|u_1|^q-2u_1rdrdz.


	Again the convection terms vanish due to incompressibility. For the diffusion term, using estimates similar to those for the ω_1 equation, we arrive at
	
    ∫ (Δ u_1+2/ru_1,r)|u_1|^q-2u_1rdrdz
       =-4(q-1)/q^2∫ |∇ (|u_1|^q/2)|^2rdrdz-q/2∫ |u_1(0, z)|^qdz
       ≤ -4(q-1)/q^2∫ |∇ (|u_1|^q/2)|^2rdrdz.


	Next we decompose the nonlinear term on the RHS of (<ref>) into two parts 
	
    ∫ 2|u_1|^qϕ_1,zrdrdz=2∫ |u_1|^qϕ_1,zψ(r)+|u_1|^qϕ_1,z(1-ψ(r))rdrdz,

	where ψ(r) is the cut-off function defined in (<ref>) that satisfies ψ(r)=1 for r≤ 1.

	For the second term on the RHS of (<ref>),  Young's inequality implies,
	
    ∫_r≥ 1 |u_1|^qϕ_1,z(1-ψ(r))rdrdz
       ≤∫_r≥ 1 |u_1|^2q+|ϕ_1,z|^2rdrdz≤∫_r≥ 1 |ru_1|^2 r^-2|u_1|^2q-2+|ϕ_1,z|^2rdrdz
       ≤Γ_0^ϵ^2q-2_L^∞∫_r≥ 1 |ru_1|^2rdrdz+1/ϵ^2∫_r≥ 1 |u^r/r|^2rdrdz≤ C,

	where we have used the a priori estimates (<ref>) in the last step. 

	As for the first term on the RHS of (<ref>), we denote 

    g(r,z)=ϕ_1,z(r,z)ψ(r)

and have 
	
    |∫ 2|u_1|^qg(r,z) rdrdz|	≤ 2∫ |u_1|^α |u_1|^β |g(r,z)|rdrdz,

	where the exponents α, β are
	
    α=16-8ϵ'/4-ϵ',  β=(ϵ')^2/4-ϵ',  α+β=q=4-ϵ'.

	
	Then applying Young's inequality with 
	
    L_1=4-ϵ'/4-2ϵ',   L_2=4-ϵ'/ϵ',  1/L_1+1/L_2=1,

	we obtain 		

    ∫ 2|u_1|^q|g|rdrdz   ≤1/2∫ |u_1|^L_1α rdrdz+C∫ |u_1|^L_2β |g|^L_2rdrdz
       =1/2∫ |u_1|^4rdrdz+C∫ |u_1|^ϵ' |g(r,z)|^4-ϵ'/ϵ'rdrdz.

	
	The first term in (<ref>) is treated as in the estimate (<ref>). For the second term on the RHS of (<ref>), using Lemma <ref> with r_1=2 and the fact that g(r,z)=0 for r≥ 2, we obtain   
	
    ∫ |u_1|^ϵ'|g(r,z)|^4-ϵ'/ϵ'rdrdz≤ C∫ |∂_r(|g(r,z)|^4-ϵ'/2ϵ')|^2rdrdz
        =C ∫ |g_r(r,z)|^2|g(r,z)|^4-3ϵ'/ϵ'rdrdz ≤ Cg(r,z)_L^∞^4-3ϵ'/ϵ'∇ g(r,z)_L^2^2 .



Using the following interpolation inequality,

    ∇ g(r,z)_L^2   ≤ Cg(r,z)_L^2^1/2∇^2g(r,z)_L^2^1/2,
    g(r,z)_L^∞   ≤ Cg(r,z)_L^2^1/4∇^2g(r,z)_L^2^3/4,

we have 






    ∇ g(r,z)_L^2^11ϵ'-4/6ϵ' g(r,z)_L^∞^4-3ϵ'/ϵ'≤ Cg(r,z)_L^2^ϵ'+4/6ϵ'∇^2 g(r,z)_L^2^8-4ϵ'/3ϵ'.


Using (<ref>) in (<ref>), we have 

    ∫ |u_1|^ϵ'|g(r,z)|^4-ϵ'/ϵ'rdrdz≤  Cg(r,z)_L^∞^4-3ϵ'/ϵ'∇ g(r,z)_L^2^2 
       = C[g(r,z)_L^∞^4-3ϵ'/ϵ'∇ g(r,z)_L^2^11ϵ'-4/6ϵ'] ∇ g(r,z)_L^2^4+ϵ'/6ϵ'
       ≤ Cg(r,z)_L^2^4+ϵ'/6ϵ'∇ g(r,z)_L^2^4+ϵ'/6ϵ'∇^2 g(r,z)_L^2^8-4ϵ'/3ϵ'.


In deriving the above estimate, we have used the interpolation inequality (<ref>) such that the exponents for g(r,z)_L^2 and ∇ g(r,z)_L^2 are the same in the RHS of (<ref>).

Since ϵ'=20/19, using Young's inequality with 

    L_1=12ϵ'/4+ϵ',    L_2=6ϵ'/8-4ϵ',  1/L_1+1/L_2=1,

	in (<ref>), we have 

    ∫ |u_1|^ϵ'|g(r,z)|^4-ϵ'/ϵ'rdrdz
       ≤ C(δ) (g(r,z)_L^2^4+ϵ'/6ϵ'∇ g(r,z)_L^2^4+ϵ'/6ϵ')^L_1+δ( ∇^2 g(r,z)_L^2^8-4ϵ'/3ϵ')^L_2
       =C(δ)g(r,z)_L^2^2∇ g(r,z)_L^2^2+δ∇^2 g(r,z)_L^2^2.


	Since g(r,z)=ϕ_1,zψ(r) and ψ(r) is constant for r≤ 1, we have 
	
		
		
    g(r,z)_L^2^2≤ Cϕ_1,z^2_L^2,

	
    ∇ g(r,z)_L^2^2≤ C∇ϕ_1,z_L^2^2+C1_r≥ 1ϕ_1,z_L^2^2,


    ∇^2 g(r,z)_L^2^2≤ C∇^2ϕ_1,z^2+C1_r≥ 1∇ϕ_1,z_L^2^2+C1_r≥ 1ϕ_1,z_L^2^2.


	
By the a priori estimate (<ref>), we have

    1_r≥ 1ϕ_1,z_L^2^2=1_r≥ 1u^r/r_L^2^2≤u^r_L^2^2≤ C.

In view of (<ref>), we get from (<ref>) and (<ref>)

    ∫ |u_1|^ϵ'|g(r,z)|^4-ϵ'/ϵ'rdrdz ≤ Cδ∇^2ϕ_1,z_L^2^2+C∇ϕ_1,z_L^2^2ϕ_1,z_L^2^2
       +C∇ϕ_1,z_L^2^2+Cϕ_1,z_L^2+C.

Employing the elliptic estimate (<ref>) in (<ref>), we deduce 

    ∫ |u_1|^ϵ'|g(r,z)|^4-ϵ'/ϵ'rdrdz ≤ Cδω_1,z_L^2^2+Cω_1_L^2^2ϕ_1,z_L^2^2
       +Cω_1_L^2^2+Cϕ_1,z_L^2+C.

Putting the estimates (<ref>), (<ref>), (<ref>) in (<ref>), we get 

    d/dt∫ |u_1|^qrdrdz+ C(1-C_1(r_1))∇(|u_1|^q/2)^2_L^2-Cδ∇ω_1_L^2^2
       ≤ 
    	Cϕ_1,z_L^2^2ω_1_L^2^2+Cϕ_1,z_L^2^2+Cω_1_L^2^2+C.


At last, choosing r_1, δ small enough, and adding up (<ref>) with (<ref>) give

    d/dt∫ |ω_1|^2+|u_1|^qrdrdz+C∫ |∇ω_1|^2+|∇(|u_1|^q/2)|^2rdrdz
       ≤ C(ϕ_1,z_L^2^2+1) (∫ |ω_1|^2+|u_1|^q rdrdz)+Cϕ_1,z^2+C,

which together with the a priori estimate (<ref>) implies 

    ∫ω_1^2(T)+|u_1(T)|^qrdrdz+∫_0^T∫ |∇ω_1|^2+|∇(|u_1|^q/2)|^2rdrdzds≤ C

where the constant C may depend on the initial data and T.

To prove the global regularity of the solutions, we consider 

    𝐯(x)_L^4≤ Cu^θ_L^4 +Cu^r_L^4+Cu^z_L^4 .

For the u^θ_L^4 term in (<ref>), using estimates (<ref>), (<ref>), and (<ref>), we obtain

    ∫ u_1^4r^4rdrdz   =∫_r≤ 1 |u_1|^q |u_1|^ϵ'r^4 rdrdz+∫_r≥ 1 (u^θ)^2 |Γ^ϵ|^2/r^4/ϵ-2rdrdz
       ≤Γ_0^ϵ_L^∞^ϵ'∫ |u_1|^qrdrdz+Γ^ϵ_0_L^∞^2∫ (u^θ)^2rdrdz≤ C.


Then we consider the equation for ω^θ=rω_1, which is 

    ω^θ_t+u^rω^θ_r+u^zω^θ_z=u^r/rω^θ+(u^θ)^2_z/r+(Δ-1/r^2)ω^θ.

Multiplying both sides of (<ref>) by ω^θ and integrating, we get 

    1/2d/dt∫ (ω^θ)^2rdrdz+1/2∫ u^r(ω^θ)^2_r+u_z(ω^θ)^2_zrdrdz
        =∫u^r/r(ω^θ)^2rdrdz+∫(u^θ)^2_z ω^θ/rrdrdz-∫ |∇ω^θ|^2+(ω^θ)^2/r^2rdrdz.


The convection terms vanish due to the incompressibility condition, and for the first nonlinear term in (<ref>), we have 

    ∫u^r/r(ω^θ)^2rdrdz≤u^r/r_L^∞∫(ω^θ)^2rdrdz=ϵϕ_1,z_L^∞ω^θ_L^2^2
       ≤ C(ϕ_1,z_L^2+∇^2ϕ_1,z_L^2)ω^θ_L^2^2≤ C(ϕ_1,z_L^2+∇ω_1_L^2)ω^θ_L^2^2,

where we have used the Biot-Savart law (<ref>) ϵϕ_1,z(r,z)=u^r/r, the Sobolev embedding, and the elliptic estimate (<ref>) in the last step.

For the second nonlinear term in (<ref>), we have 

    |∫(u^θ)^2_zω^θ/rrdrdz|=|∫(u^θ)^2/rω^θ_zrdrdz|
       ≤1/2∫(u^θ)^4/r^2rdrdz+1/2∇ω^θ^2.

The first integral term in (<ref>) is estimated as 

    ∫(u^θ)^4/r^2rdrdz   =∫_r≥ 1(u^θ)^4/r^2rdrdz+∫_r≤ 1(u^θ)^4/r^2rdrdz
       ≤∫ (u^θ)^4rdrdz+∫_r≤ 1 u_1^4r^2rdrdz
       ≤∫ (u^θ)^4rdrdz+∫_r≤ 1 |u_1|^4-ϵ'|u_1|^ϵ'r^2rdrdz
       ≤ C+CΓ^ϵ_0_L^∞^ϵ'≤ C.

		
Adding up the estimates (<ref>), (<ref>) and (<ref>) in (<ref>), and using the Gronwell's inequality, we get that 
		
    ω^θ(t)_L^2≤ C.

		Then since 
		
    u^r𝐞_𝐫+u^z𝐞_𝐳=ϵ∇× (-Δ)^-1(ω^θ𝐞_θ),

		using Sobolev embedding, we have 
		
    u^r𝐞_𝐫+u^z𝐞_𝐳_L^6≤∇ (u^r𝐞_𝐫+u^z𝐞_𝐳)_L^2≤ Cω^θ_L^2≤ C.

		Then based on the a priori estimate (<ref>), we have 
		
    u^r𝐞_𝐫+u^z𝐞_𝐳_L^4≤u^r𝐞_𝐫+u^z𝐞_𝐳^3/4_L^6u^r𝐞_𝐫+u^z𝐞_𝐳^1/4_L^2≤ C.

		
		This together with the estimate (<ref>) and the Prodi-Serrin criterion (<ref>) implies the global regularity of the solutions. 



	We proved our main result Theorem <ref> using the L^2 estimate for ω_1 and the L^4-ϵ' estimate for u_1. And we can also use the L^p estimate for ω_1 and L^q estimate for u_1 with q=2p-pϵ'/2 for p>1 to get the same result.


Acknowledgments. The research was in part supported by the NSF Grants No. DMS-1613861 and DMS-1318377.
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