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A pointwise regularity criterion for axisymmetric Navier–Stokes system
Accepted Manuscript A pointwise regularity criterion for axisymmetric Navier-Stokes system
Zujin Zhang
PII: DOI: Reference:
S0022-247X(18)30004-0 https://doi.org/10.1016/j.jmaa.2017.12.069 YJMAA 21930
To appear in:
Journal of Mathematical Analysis and Applications
Received date:
9 May 2017
Please cite this article in press as: Z. Zhang, A pointwise regularity criterion for axisymmetric Navier-Stokes system, J. Math. Anal. Appl. (2018), https://doi.org/10.1016/j.jmaa.2017.12.069
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
A pointwise regularity criterion for axisymmetric Navier-Stokes system Zujin Zhanga,1,˚ a
School of Mathematics and Computer Science, Gannan Normal University Ganzhou 341000, Jiangxi, P.R. China
Abstract In this paper, we consider the axisymmetric Navier-Stokes equations with non-zero swirl component. It is proved that if rur ě M for some M ą ´2, then the solution actually is smooth. This extends the result in [X.H. Pan, A regularity condition of 3d axisymmetric Navier-Stokes equations, Acta Appl. Math., (2017), 150 (2017), 103–109]. Keywords: Axisymmetric Navier-Stokes equations, regularity criterion 2010 MSC: 35B65, 35Q35, 76D03 1. Introduction The three-dimensional Navier-Stokes equations read as $ ’ ’ B u ` pu ¨ ∇qu ´ Δu ` ∇π “ 0, ’ & t ∇ ¨ u “ 0, ’ ’ ’ % u| “ u , t“0 0
(1)
where u denotes the fluid velocity field, π is a scalar pressure, and u0 is the initial data satisfying ∇ ¨ u0 “ 0 in the sense of distributions. It is well-known that (1) possesses a global weak solution for initial data of finite energy, see [6, 11]. However, the issue of its regularity/uniqueness is an challenging ˚ 1
Corresponding author, tel: (86) 07978393663 [email protected]
Preprint submitted to Mathematical Journal for Possible Publication
January 3, 2018
open problem. Leaded by Serrin [17] and Prodi [16], there are many sufficient conditions to ensure the smoothness of the solution. In particular, we have the following classical regularity criterion (see [4, 16, 17] for example) u P Lα p0, T ; Lβ pR3 qq,
2 3 ` “ 1, α β
3 ď β ď 8.
(2)
In this paper, we shall concern axisymmetric solutions to (1). If the initial data u0 P H 2 pR3 q is axially symmetric, that is, u0 “ ur0 pr, zqer ` uθ0 pr, zqeθ ` uz0 pr, zqez with r, θ, z being the cylindrical coordinates: x “ px1 , x2 , x3 q “ pr cos θ, r sin θ, zq, and
x2 ¯ , 0 “ pcos θ, sin θ, 0q , r r ´ x x ¯ 2 1 eθ “ ´ , , 0 “ p´ sin θ, cos θ, 0q , r r er “
´x
1
,
ez “ p0, 0, 1q being the corresponding basis vectors, then as in [12, Section 3], we see the NavierStokes system (1) possess a unique local axisymmetric strong solution u “ ur pt, r, zqer ` uθ pt, r, zqeθ ` uz pt, r, zqez , where ur , uθ and uz are called the radial, swirl (or azimuthal) and axial components of the velocity field respectively. Direct calculations show that ur , uθ and uz satisfy the following equations: $ ` ˘ θ 2 ˜ r D ’ ’ u ´ Br2 ` Bz2 ` 1r Br ´ r12 ur ´ pur q ` Br π “ 0, ’ Dt ’ ’ ’ ’ D˜ uθ ´ `B 2 ` B 2 ` 1 B ´ 1 ˘ uθ ` ur uθ “ 0, ’ r z ’ r r r2 r & Dt ` ˘ ˜ D z 1 2 2 z u ´ Br ` Bz ` r Br u ` Bz π “ 0, ’ Dt ’ ’ ’ ’ Br prur q ` Bz pruz q “ 0, ’ ’ ’ ’ % pur , uθ , uz q| “ pur , uθ , uz q, t“0
0
0
2
0
(3)
where
˜ D “ Bt ` ur Br ` uz Bz Dt
(4)
denotes the convection derivative (or material derivative). When the initial swirl component uθ0 vanishes, the following system $ ` ˘ ˜ r D ’ ’ u ´ Br2 ` Bz2 ` 1r Br ´ r12 ur ` Br π “ 0, ’ Dt ’ ’ ’ & D˜ uz ´ `B 2 ` B 2 ` 1 B ˘ uz ` B π “ 0, r
Dt
z
r r
z
’ ’ Br prur q ` Bz pruz q “ 0, ’ ’ ’ ’ % pur , uz q| “ pur , uz q t“0 0 0
(5)
has two unknowns ur , uz (with the pressure still playing the role of Lagrange multiplier associated with the isochoricity constraint ∇ ¨ u “ 0) in two spatial domains. Argued as in [9, 12, 18], we can get its global regularity. Once the global unique strong solution pur , uz q of (5) is obtained, we may then apply the uniqueness argument to the Navier-Stokes equations to see that pur , 0, uz q is a global solution to (3). Notice that in (3)2 , each term has uθ , so that uθ “ 0 satisfies (3)2 in a trivial way. The case of non-zero swirl component is much more difficult, vortex stretching still appears (in the vorticity equation), and it is open for the global well-posedness of (3). Tremendous efforts and interesting progresses have been made on the regularity problem, see [1, 2, 3, 5, 7, 8, 10, 13, 15, 19, 20, 21, 22, 23] and references therein. Let us recall here some recent advances. Chen-Fang-Zhang [2] established the following two weighted regularity criteria (see [2, Theorem 1.1, Remark 1.3]) rd uθ P Lα p0, T ; Lβ pR3 qq,
2 3 3 ` “ 1 ´ d, ă β ď 8, ´1 ď d ă 1 α β 1´d
(6)
and (see [2, Theorem 1.4, Corollary 1.5]) rd uz P Lα p0, T ; Lβ pR3 qq,
3 3 2 ` “ 1 ´ d, ă β ď 8, 0 ď d ă 1. α β 1´d
(7)
Zhang [21, 22] (see some partial results in [1, 2, 8]) showed the following three weighted regularity criteria rd ur P Lα p0, T ; Lβ pR3 qq,
3 2 ` “ 1 ´ d, α β 3
3 ă β ď 8, 1´d
´1 ď d ă 1,
(8)
rd ω z P Lα p0, T ; Lβ pR3 qq,
3 2 ` “ 2 ´ d, α β
3 ă β ă 8, 2´d
´2 ď d ă 2,
(9)
3 2 ` “ 2 ´ d, α β
3 ă β ă 8, 2´d
0 ď d ă 2.
(10)
as well as rd ω θ P Lα p0, T ; Lβ pR3 qq,
Very recently, Pan [14] applied the maximum principle to the uθ equation, and derived the following pointwise regularity criterion rur ě ´1.
(11)
An interesting open problem “whether or not ´1 in (11) can be lowered down” was stated in [14, Remark 1.2]. The purpose of the present paper is to give an affirmative answer to this question. We will show that as long as rur ě M for some M ą ´2, the solution actually is smooth. Now, the precise result reads Theorem 1. Let ´2 ă M ă ´1 be an arbitrary real number. Assume that u0 P H 2 pR3 q be axially symmetric and be divergence-free, ruθ0 P L8 pR3 q, and u P Cpr0, T q; H 2 pR3 qq X L2 p0, T ; H 3 pR3 qq be the unique axisymmetric classical solution of (3). If rur ě M
(12)
then the solution can be extended smoothly beyond T . 2. Proof of Theorem 1 In this section, we shall prove Theorem 1. First, let us show an unexpected weighted regularity criterion based on uθ , which could have its own interest. Proposition 2. Assume as in Theorem 1. If rd uθ P Lα p0, T ; Lβ pR3 qq,
3 2 ` “ 1 ´ d, α β
3 ă β ď 8, 1´d
then the solution can be extended smoothly beyond T . 4
d ă 1,
(13)
Proof. The case 0 ď d ă 1 was already shown in [2, Theorem 1.1]. We now treat the case d ă 0. By the H¨older inequality and the well-known a priori bound (see [1, Proposition 1]) › θ› ›ru ›
L8
we deduce
› › ď ›ruθ0 ›L8 ď C,
› › › θ ›p1´dqα ´d ›p1´dqα 1 θ 1´d ›u › p1´dqβ “ ››prd uθ q 1´d ¨ pru q › p1´dqβ L L › › › ıp1´dqα ”› ´d › 1 › › › ď ›prd uθ q 1´d › p1´dqβ ›pruθ q 1´d › L L8 j „ p1´dqα ´d › 1 › θ › 1´d › ›ru › 8 ď ›rd uθ › 1´d β L
L
›α › ›´dα › “ ›rd uθ ›Lβ ›ruθ ›L8 ›α › ď C ›rd uθ › β . L
It then follows from (13) that uθ P Lp1´dqα p0, T ; Lp1´dqβ pR3 qq,
3 2 ` “ 1, p1 ´ dqα p1 ´ dqβ
3 ă p1 ´ dqβ ď 8.
By invoking regularity criterion (6), we complete the proof of Proposition 2. Remark 3. From Proposition 2, we obtain the following two regularity criteria: (1) the first one is assuming H¨ older continuity of r|uθ |: r|uθ | ď Crα ,
α ą 0.
(14)
Notice that in [2, Remark 1.2], the α should be 0 ă α ď 2. Also, (14) supplements the following regularity conditions in [14]: r|ur | ď Crα , α P p0, 1s;
r|uz | ď Crβ , β P r0, 1s.
(2) The second one is to add conditions on ω z “ Br uθ `
(15)
uθ , the axial component of r
the vorticity: rd ω z P Lα p0, T ; Lβ pR3 qq,
3 2 ` “ 2 ´ d, α β 5
3 ă β ď 8, 2´d
d ă 2.
(16)
The proof follows similarly as the proof of [21, Theorem 1.1 (1.17)]. In particular, we obtained in [21, Page 193] › d{2 θ › ›r u ›
L2α p0,T ;L2β pR3 qq
›1{2 › ď C ›rd ω z ›Lα p0,T ;Lβ pR3 qq .
We now show Theorem 1. From (3)2 , we can easily deduce the governing equations of Γ ” rd uθ for any d P R as ˆ ˙ ˜ D 1 rur ` 1 ` d 2d 2 2 Γ “ 0. Γ ´ Br ` Bz ` Br Γ ` Br Γ ` p1 ´ dq Dt r r r2
(17)
In fact, direct computations show dur r pu Br ` u Bz qu “ pu Br ` u Bz qΓ ´ Γ, ˆ ˙ ˙r ˆ 1 1 2d Γ ´rd Br2 ` Bz2 ` Br uθ “ ´ Br2 ` Bz2 ` Br Γ ´ d2 2 ` Br Γ, r r r r ˙ ˆ r θ r 1 θ uu u Γ rd u ` “ 2 ` Γ. 2 r r r r d
r
z
θ
r
z
Now, taking d “ ´M ´ 1, we have 0 ă d ă 1. By the maximum principle as [14], we obtain
› › › d θ› ›r u › 8 ď ›rd uθ0 › 8 L L › θ ›1´d › θ ›d ď ›u0 ›L8 ›ru0 ›L8 pby interpolation inequalityq › θ ›d ›ru0 › 8 pby Sobolev inequalityq ď C }u0 }1´d 2 H
L
ă 8. This completes the proof of Theorem 1 by invoking Proposition 2. Acknowledgements We thank the anonymous referees’ remarks on the relaxation of assumptions in Theorem 1 of the first draft. This work is partially supported by the National Natural Science Foundation of China (grant nos. 11761009,11501125) and the Natural Science Foundation of Jiangxi (grant no. 20171BAB201004).
6
References [1] D. Chae, J. Lee, On the regularity of the axisymmetric solutions of the NavierStokes equations, Math. Z., 239 (2002), 645–671. [2] H. Chen, D.Y. Fang, T. Zhang, Regularity of 3D axisymmetric Navier-Stokes equations, Discrete Contin. Dyn. Syst., 37 (2017), 1923–1939. [3] Q.L. Chen, Z.F. Zhang, Regularity criterion of axisymmetric weak solutions to the 3D Navier-Stokes equations, J. Math. Anal. Appl., 331 (2007), 1384–1395. ˇ ak, L3,8 -solutions of Navier-Stokes equations [4] L. Eskauriaza, G.A. Ser¨egin, V. Sver´ and backward uniqueness, Russ. Math. Surv., 58 (2003), 211–250. [5] S. Gala, On the regularity criterion of axisymmetric weak solutions to the 3D Navier-Stokes equations, Nonlinear Anal., 74 (2011), 775–782. ¨ [6] E. Hopf, Uber die Anfangswertaufgabe f¨ ur die hydrodynamischen Grundgleichungen, Math. Nachr., 4 (1951). 213–231. [7] O. Kreml, M. Pokorn´ y, A regularity criterion for the angular velocity component in axisymmetric Navier-Stokes equations, Electron J. Differential Equations, 08 (2007), 1–10. [8] A. Kubica, M. Pokorn´ y, W. Zajaczkowski, Remarks on regularity criteria for axially symmetric weak solutions to the Navier-Stokes equations, Math. Meth. Appl. Sci., 35 (2012), 360–371. [9] O.A. Ladyˇzhenskaya, On unique solvability “in the large” of three-dimensional Cauchy problem for Navier-Stokes equations with axial symmetry, Zap. Nauchn. Sem. LOMI, 7, (1968), 155–177. [10] L. Zhen, Q.S. Zhang, A Liouville theorem for the axially-symmetric NavierStokes equations, J. Funct. Anal., 261 (2011), 2323–2345. 7
[11] J. Leray, Sur le mouvement d’un liquide visqueux emplissant l’espace, Acta Math., 63 (1934), 193–248. [12] S. Leonardi, J.M´alek, J.Neˇcas, M. Pokorn´ y, On axially symmetric flows in R3 , Z. Anal. Anwendungen, 18 (1999), 639–649. [13] J. Neustupa, M. Pokorn´ y, Axisymmetric flow of Navier-Stokes fluid in the whole space with non-zero angular velocity component, Math. Bohem, 126, (2001): 469–481. [14] X.H. Pan, A regularity condition of 3d axisymmetric Navier-Stokes equations, Acta Appl. Math., 150 (2017), 103–109. [15] M. Pokorn´ y, A regularity criterion for the angular velocity component in the case of axisymmetric Navier-Stokes equations, Proceedings of the 4th European Congress on Elliptic and Parabolic Problems, Rolduc and Gaeta 2001, World Scientific (2002), 233–242. [16] G. Prodi, Un teorema di unicit´a per le equazioni di Navier-Stokes, Ann. Mat. Pura Appl., 48 (1959), 173–182. [17] J. Serrin, The initial value problem for the Navier-Stokes equations, Nonlinear Problems, Proc. Symposium, Madison, Wisconsin, University of Wisconsin Press, Madison, Wisconsin, 1963, pp. 69–98. [18] M.R. Ukhovskii, V.I. Yudovich, Axially symmetric flows of ideal and viscous fluids filling the whole space, J. Appl. Math. Mech., 32 (1968), 52–62. [19] D.Y. Wei, Regularity criterion to the axially symmetric Navier-Stokes equations, J. Math. Anal. Appl., 435 (2016), 402–413. [20] P. Zhang, T. Zhang, Global axisymmetric solutions to three-dimensional NavierStokes system, Int. Math. Res. Not., 3 (2014), 610–642.
8
[21] Z.J. Zhang, Remarks on regularity criteria for the Navier-Stokes equations with axisymmetric data, Ann. Polon. Math., 117 (2016), 181–196. [22] Z.J. Zhang, On weighted regularity criteria for the axisymmetric Navier-Stokes equations, Appl. Math. Comput., 296 (2017), 18–22. [23] Z.J. Zhang, X.Q. Ouyang, X. Yang, Refined a priori estimates for the axisymmetric Navier-Stokes equations, J. Appl. Anal. Comput., 7 (2017), 554–558.
9
Zujin Zhang
PII: DOI: Reference:
S0022-247X(18)30004-0 https://doi.org/10.1016/j.jmaa.2017.12.069 YJMAA 21930
To appear in:
Journal of Mathematical Analysis and Applications
Received date:
9 May 2017
Please cite this article in press as: Z. Zhang, A pointwise regularity criterion for axisymmetric Navier-Stokes system, J. Math. Anal. Appl. (2018), https://doi.org/10.1016/j.jmaa.2017.12.069
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
A pointwise regularity criterion for axisymmetric Navier-Stokes system Zujin Zhanga,1,˚ a
School of Mathematics and Computer Science, Gannan Normal University Ganzhou 341000, Jiangxi, P.R. China
Abstract In this paper, we consider the axisymmetric Navier-Stokes equations with non-zero swirl component. It is proved that if rur ě M for some M ą ´2, then the solution actually is smooth. This extends the result in [X.H. Pan, A regularity condition of 3d axisymmetric Navier-Stokes equations, Acta Appl. Math., (2017), 150 (2017), 103–109]. Keywords: Axisymmetric Navier-Stokes equations, regularity criterion 2010 MSC: 35B65, 35Q35, 76D03 1. Introduction The three-dimensional Navier-Stokes equations read as $ ’ ’ B u ` pu ¨ ∇qu ´ Δu ` ∇π “ 0, ’ & t ∇ ¨ u “ 0, ’ ’ ’ % u| “ u , t“0 0
(1)
where u denotes the fluid velocity field, π is a scalar pressure, and u0 is the initial data satisfying ∇ ¨ u0 “ 0 in the sense of distributions. It is well-known that (1) possesses a global weak solution for initial data of finite energy, see [6, 11]. However, the issue of its regularity/uniqueness is an challenging ˚ 1
Corresponding author, tel: (86) 07978393663 [email protected]
Preprint submitted to Mathematical Journal for Possible Publication
January 3, 2018
open problem. Leaded by Serrin [17] and Prodi [16], there are many sufficient conditions to ensure the smoothness of the solution. In particular, we have the following classical regularity criterion (see [4, 16, 17] for example) u P Lα p0, T ; Lβ pR3 qq,
2 3 ` “ 1, α β
3 ď β ď 8.
(2)
In this paper, we shall concern axisymmetric solutions to (1). If the initial data u0 P H 2 pR3 q is axially symmetric, that is, u0 “ ur0 pr, zqer ` uθ0 pr, zqeθ ` uz0 pr, zqez with r, θ, z being the cylindrical coordinates: x “ px1 , x2 , x3 q “ pr cos θ, r sin θ, zq, and
x2 ¯ , 0 “ pcos θ, sin θ, 0q , r r ´ x x ¯ 2 1 eθ “ ´ , , 0 “ p´ sin θ, cos θ, 0q , r r er “
´x
1
,
ez “ p0, 0, 1q being the corresponding basis vectors, then as in [12, Section 3], we see the NavierStokes system (1) possess a unique local axisymmetric strong solution u “ ur pt, r, zqer ` uθ pt, r, zqeθ ` uz pt, r, zqez , where ur , uθ and uz are called the radial, swirl (or azimuthal) and axial components of the velocity field respectively. Direct calculations show that ur , uθ and uz satisfy the following equations: $ ` ˘ θ 2 ˜ r D ’ ’ u ´ Br2 ` Bz2 ` 1r Br ´ r12 ur ´ pur q ` Br π “ 0, ’ Dt ’ ’ ’ ’ D˜ uθ ´ `B 2 ` B 2 ` 1 B ´ 1 ˘ uθ ` ur uθ “ 0, ’ r z ’ r r r2 r & Dt ` ˘ ˜ D z 1 2 2 z u ´ Br ` Bz ` r Br u ` Bz π “ 0, ’ Dt ’ ’ ’ ’ Br prur q ` Bz pruz q “ 0, ’ ’ ’ ’ % pur , uθ , uz q| “ pur , uθ , uz q, t“0
0
0
2
0
(3)
where
˜ D “ Bt ` ur Br ` uz Bz Dt
(4)
denotes the convection derivative (or material derivative). When the initial swirl component uθ0 vanishes, the following system $ ` ˘ ˜ r D ’ ’ u ´ Br2 ` Bz2 ` 1r Br ´ r12 ur ` Br π “ 0, ’ Dt ’ ’ ’ & D˜ uz ´ `B 2 ` B 2 ` 1 B ˘ uz ` B π “ 0, r
Dt
z
r r
z
’ ’ Br prur q ` Bz pruz q “ 0, ’ ’ ’ ’ % pur , uz q| “ pur , uz q t“0 0 0
(5)
has two unknowns ur , uz (with the pressure still playing the role of Lagrange multiplier associated with the isochoricity constraint ∇ ¨ u “ 0) in two spatial domains. Argued as in [9, 12, 18], we can get its global regularity. Once the global unique strong solution pur , uz q of (5) is obtained, we may then apply the uniqueness argument to the Navier-Stokes equations to see that pur , 0, uz q is a global solution to (3). Notice that in (3)2 , each term has uθ , so that uθ “ 0 satisfies (3)2 in a trivial way. The case of non-zero swirl component is much more difficult, vortex stretching still appears (in the vorticity equation), and it is open for the global well-posedness of (3). Tremendous efforts and interesting progresses have been made on the regularity problem, see [1, 2, 3, 5, 7, 8, 10, 13, 15, 19, 20, 21, 22, 23] and references therein. Let us recall here some recent advances. Chen-Fang-Zhang [2] established the following two weighted regularity criteria (see [2, Theorem 1.1, Remark 1.3]) rd uθ P Lα p0, T ; Lβ pR3 qq,
2 3 3 ` “ 1 ´ d, ă β ď 8, ´1 ď d ă 1 α β 1´d
(6)
and (see [2, Theorem 1.4, Corollary 1.5]) rd uz P Lα p0, T ; Lβ pR3 qq,
3 3 2 ` “ 1 ´ d, ă β ď 8, 0 ď d ă 1. α β 1´d
(7)
Zhang [21, 22] (see some partial results in [1, 2, 8]) showed the following three weighted regularity criteria rd ur P Lα p0, T ; Lβ pR3 qq,
3 2 ` “ 1 ´ d, α β 3
3 ă β ď 8, 1´d
´1 ď d ă 1,
(8)
rd ω z P Lα p0, T ; Lβ pR3 qq,
3 2 ` “ 2 ´ d, α β
3 ă β ă 8, 2´d
´2 ď d ă 2,
(9)
3 2 ` “ 2 ´ d, α β
3 ă β ă 8, 2´d
0 ď d ă 2.
(10)
as well as rd ω θ P Lα p0, T ; Lβ pR3 qq,
Very recently, Pan [14] applied the maximum principle to the uθ equation, and derived the following pointwise regularity criterion rur ě ´1.
(11)
An interesting open problem “whether or not ´1 in (11) can be lowered down” was stated in [14, Remark 1.2]. The purpose of the present paper is to give an affirmative answer to this question. We will show that as long as rur ě M for some M ą ´2, the solution actually is smooth. Now, the precise result reads Theorem 1. Let ´2 ă M ă ´1 be an arbitrary real number. Assume that u0 P H 2 pR3 q be axially symmetric and be divergence-free, ruθ0 P L8 pR3 q, and u P Cpr0, T q; H 2 pR3 qq X L2 p0, T ; H 3 pR3 qq be the unique axisymmetric classical solution of (3). If rur ě M
(12)
then the solution can be extended smoothly beyond T . 2. Proof of Theorem 1 In this section, we shall prove Theorem 1. First, let us show an unexpected weighted regularity criterion based on uθ , which could have its own interest. Proposition 2. Assume as in Theorem 1. If rd uθ P Lα p0, T ; Lβ pR3 qq,
3 2 ` “ 1 ´ d, α β
3 ă β ď 8, 1´d
then the solution can be extended smoothly beyond T . 4
d ă 1,
(13)
Proof. The case 0 ď d ă 1 was already shown in [2, Theorem 1.1]. We now treat the case d ă 0. By the H¨older inequality and the well-known a priori bound (see [1, Proposition 1]) › θ› ›ru ›
L8
we deduce
› › ď ›ruθ0 ›L8 ď C,
› › › θ ›p1´dqα ´d ›p1´dqα 1 θ 1´d ›u › p1´dqβ “ ››prd uθ q 1´d ¨ pru q › p1´dqβ L L › › › ıp1´dqα ”› ´d › 1 › › › ď ›prd uθ q 1´d › p1´dqβ ›pruθ q 1´d › L L8 j „ p1´dqα ´d › 1 › θ › 1´d › ›ru › 8 ď ›rd uθ › 1´d β L
L
›α › ›´dα › “ ›rd uθ ›Lβ ›ruθ ›L8 ›α › ď C ›rd uθ › β . L
It then follows from (13) that uθ P Lp1´dqα p0, T ; Lp1´dqβ pR3 qq,
3 2 ` “ 1, p1 ´ dqα p1 ´ dqβ
3 ă p1 ´ dqβ ď 8.
By invoking regularity criterion (6), we complete the proof of Proposition 2. Remark 3. From Proposition 2, we obtain the following two regularity criteria: (1) the first one is assuming H¨ older continuity of r|uθ |: r|uθ | ď Crα ,
α ą 0.
(14)
Notice that in [2, Remark 1.2], the α should be 0 ă α ď 2. Also, (14) supplements the following regularity conditions in [14]: r|ur | ď Crα , α P p0, 1s;
r|uz | ď Crβ , β P r0, 1s.
(2) The second one is to add conditions on ω z “ Br uθ `
(15)
uθ , the axial component of r
the vorticity: rd ω z P Lα p0, T ; Lβ pR3 qq,
3 2 ` “ 2 ´ d, α β 5
3 ă β ď 8, 2´d
d ă 2.
(16)
The proof follows similarly as the proof of [21, Theorem 1.1 (1.17)]. In particular, we obtained in [21, Page 193] › d{2 θ › ›r u ›
L2α p0,T ;L2β pR3 qq
›1{2 › ď C ›rd ω z ›Lα p0,T ;Lβ pR3 qq .
We now show Theorem 1. From (3)2 , we can easily deduce the governing equations of Γ ” rd uθ for any d P R as ˆ ˙ ˜ D 1 rur ` 1 ` d 2d 2 2 Γ “ 0. Γ ´ Br ` Bz ` Br Γ ` Br Γ ` p1 ´ dq Dt r r r2
(17)
In fact, direct computations show dur r pu Br ` u Bz qu “ pu Br ` u Bz qΓ ´ Γ, ˆ ˙ ˙r ˆ 1 1 2d Γ ´rd Br2 ` Bz2 ` Br uθ “ ´ Br2 ` Bz2 ` Br Γ ´ d2 2 ` Br Γ, r r r r ˙ ˆ r θ r 1 θ uu u Γ rd u ` “ 2 ` Γ. 2 r r r r d
r
z
θ
r
z
Now, taking d “ ´M ´ 1, we have 0 ă d ă 1. By the maximum principle as [14], we obtain
› › › d θ› ›r u › 8 ď ›rd uθ0 › 8 L L › θ ›1´d › θ ›d ď ›u0 ›L8 ›ru0 ›L8 pby interpolation inequalityq › θ ›d ›ru0 › 8 pby Sobolev inequalityq ď C }u0 }1´d 2 H
L
ă 8. This completes the proof of Theorem 1 by invoking Proposition 2. Acknowledgements We thank the anonymous referees’ remarks on the relaxation of assumptions in Theorem 1 of the first draft. This work is partially supported by the National Natural Science Foundation of China (grant nos. 11761009,11501125) and the Natural Science Foundation of Jiangxi (grant no. 20171BAB201004).
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