April 2017, 37(4): 1923-1939. doi: 10.3934/dcds.2017081

Regularity of 3D axisymmetric Navier-Stokes equations

School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, China

 

Received  April 2016 Revised  November 2016 Published  December 2016

In this paper, we study the three-dimensional axisymmetric Navier-Stokes system with nonzero swirl. By establishing a new key inequality for the pair $(\frac{ω^{r}}{r},\frac{ω^{θ}}{r})$, we get several Prodi-Serrin type regularity criteria based on the angular velocity, $u^θ$. Moreover, we obtain the global well-posedness result if the initial angular velocity $u_{0}^{θ}$ is appropriate small in the critical space $L^{3}(\mathbb{R}^{3})$. Furthermore, we also get several Prodi-Serrin type regularity criteria based on one component of the solutions, say $ω^3$ or $u^3$.

Citation: Hui Chen, Daoyuan Fang, Ting Zhang. Regularity of 3D axisymmetric Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 1923-1939. doi: 10.3934/dcds.2017081
References:
[1]

H. Abidi and P. Zhang, Global smooth axisymmetric solutions of 3-D inhomogeneous incompressible Navier-Stokes system, Calc. Var. Partial Differential Equations, 54 (2015), 3251-3276. doi: 10.1007/s00526-015-0902-6.

[2]

M. Badiale and G. Tarantello, A Sobolev-Hardy inequality with applications to a nonlinear elliptic equation arising in astrophysics, Arch. Ration. Mech. Anal., 163 (2002), 259-293. doi: 10.1007/s002050200201.

[3]

H. Beirão da Veiga, On the smoothness of a class of weak solutions to the Navier-Stokes equations, J. Math. Fluid Mech., 2 (2000), 315-323. doi: 10.1007/PL00000955.

[4]

L. CafferalliR. Kohn and L. Nirenberg, Partial regularity of suitable weak solutions of the Navier-Stokes equations, Comm. Pure Appl. Math., 35 (1982), 771-831. doi: 10.1002/cpa.3160350604.

[5]

C. Cao and E. S. Titi, Regularity criteria for the three-dimensional Navier-Stokes equations, Indiana Univ. Math. J., 57 (2008), 2643-2661. doi: 10.1512/iumj.2008.57.3719.

[6]

D. Chae and J. Lee, On the regularity of the axisymmetric solutions of the Navier-Stokes equations, Math. Z., 239 (2002), 645-671. doi: 10.1007/s002090100317.

[7]

C. Chen, R. M. Strain, H. Yau and T. Tsai, Lower bound on the blow-up rate of the axisymmetric Navier-Stokes equations Int. Math. Res. Not. IMRN 2008 (2008), 31 pp. doi: 10.1093/imrn/rnn016.

[8]

C. ChenR. M. StrainH. Yau and T. Tsai, Lower bounds on the blow-up rate of the axisymmetric Navier-Stokes equations Ⅱ, Comm. Partial Differential Equations, 34 (2009), 203-232. doi: 10.1080/03605300902793956.

[9]

Q. Chen and Z. Zhang, Regularity criterion of axisymmetric weak solutions to the 3D Navier-Stokes equations, J. Math. Anal. Appl., 331 (2007), 1384-1395. doi: 10.1016/j.jmaa.2006.09.069.

[10]

P. Constantin and C. Foias, Navier-Stokes Equation Chicago Lectures in Mathematics, University of Chicago Press, Chicago, 1988.

[11]

L. EscauriazaG. Seregin and V. Šverák, Backward uniqueness for parabolic equations, Arch. Ration. Mech. Anal., 169 (2003), 147-157. doi: 10.1007/s00205-003-0263-8.

[12]

E. B. FabesB. F. Jones and N. M. Rivieére, The initial value problem for the Navier-Stokes equations with data in Lp, Arch. Rational Mech. Anal., 45 (1972), 222-240. doi: 10.1007/BF00281533.

[13]

D. Fang and C. Qian, The regularity criterion for 3D Navier-Stokes equations involving one velocity gradient component, Nonlinear Anal., 78 (2013), 86-103. doi: 10.1016/j.na.2012.09.019.

[14]

D. Fang and C. Qian, Some new regularity criteria for the 3D Navier-Stokes equations, preprint, arXiv: 1212.2335.

[15]

D. Fang and C. Qian, Several almost critical regularity conditions based on one component of the solutions for 3D NS Equations, preprint, arXiv: 1312.7378.

[16]

J. P. García Azorero and I. Peral Alonso, Hardy inequalities and some critical elliptic and parabolic problems, J. Differential Equations, 144 (1998), 441-476. doi: 10.1006/jdeq.1997.3375.

[17]

Y. Giga, Solutions for semilinear parabolic equations in Lp and regularity of weak solutions of the Navier-Stokes system, J. Differential Equations, 62 (1986), 186-212. doi: 10.1016/0022-0396(86)90096-3.

[18]

G. H. Hardy, J. E. Littlewood and G. Pólya, Inequalities 2nd edition, Cambridge, at the University Press, 1952.

[19]

E. Hopf, Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen, Math. Nachr., 4 (1951), 213-231.

[20]

T. Hou and C. Li, Dynamic stability of the three-dimensional axisymmetric Navier-Stokes equations with swirl, Comm. Pure Appl. Math., 61 (2008), 661-697. doi: 10.1002/cpa.20212.

[21]

N. Kim, Remarks for the axisymmetric Navier-Stokes equations, J. Differential Equations, 187 (2003), 226-239. doi: 10.1016/S0022-0396(02)00077-3.

[22]

G. KochN. NadirashviliG. A. Seregin and V. Šverák, Liouville theorems for the Navier-Stokes equations and applications, Acta Math., 203 (2009), 83-105. doi: 10.1007/s11511-009-0039-6.

[23]

O. Kreml and M. Pokorný, A regularity criterion for the angular velocity component in axisymmetric Navier-Stokes equations, Electron. J. Differential Equations 2007 (2007), 10 pp.

[24]

A. KubicaM. Pokorný and W. Zajaczkowski, Remarks on regularity criteria for axially symmetric weak solutions to the Navier-Stokes equations, Math. Methods Appl. Sci., 35 (2012), 360-371. doi: 10.1002/mma.1586.

[25]

O. A. Ladyženskaja, Unique global solvability of the three-dimensional Cauchy problem for the Navier-Stokes equations in the presence of axial symmetry, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov., 7 (1968), 155-177.

[26]

Z. Lei and Qi S. Zhang, A Liouville theorem for the axially-symmetric Navier-Stokes equations, J. Funct. Anal., 261 (2011), 2323-2345. doi: 10.1016/j.jfa.2011.06.016.

[27]

S. LeonardiJ. MálekJ. Nečas and M. Pokorný, On axially symmetric flows in R3, Z. Anal. Anwendungen, 18 (1999), 639-649. doi: 10.4171/ZAA/903.

[28]

J. Leray, Étude de diverses équations intégrales non linéaires et de quelques problémes que pose l'Hydrodynamique, Journal de Mathématiques Pures et Appliquées, 12 (1933), 1-82.

[29]

F. Lin, A new proof of the Caffarelli-Kohn-Nirenberg theorem, Comm. Pure Appl. Math., 51 (1998), 241-257. doi: 10.1002/(SICI)1097-0312(199803)51:3<241::AID-CPA2>3.0.CO;2-A.

[30]

J. Liu and W. Wang, Energy and helicity preserving schemes for hydro-and magnetohydro-dynamics flows with symmetry, J. Comput. Phys., 200 (2004), 8-33. doi: 10.1016/j.jcp.2004.03.005.

[31]

J. Liu and W. Wang, Convergence analysis of the energy and helicity preserving scheme for axisymmetric flows, SIAM J. Numer. Anal., 44 (2006), 2456-2480. doi: 10.1137/050639314.

[32]

J. Liu and W. Wang, Characterization and regularity for axisymmetric solenoidal vector fields with application to Navier-Stokes equation, SIAM J. Math. Anal., 41 (2009), 1825-1850. doi: 10.1137/080739744.

[33]

C. Miao and X. Zheng, On the global well-posedness for the Boussinesq system with horizontal dissipation, Comm. Math. Phys., 321 (2013), 33-67. doi: 10.1007/s00220-013-1721-2.

[34]

J. Neustupa and M. Pokorný, Axisymmetrc flow of Navier-Stokes fluid in the whole space with non-zero angular velocity component, Math. Bohem., 126 (2001), 469-481.

[35]

P. Penel and M. Pokorný, Some new regularity criteria for the Navier-Stokes equations containing gradient of the velocity, Appl. Math., 49 (2004), 483-493. doi: 10.1023/B:APOM.0000048124.64244.7e.

[36]

G. Prodi, Un teorema di unicitá per le equazioni di Navier-Stokes, Ann. Mat. Pura Appl., 48 (1959), 173-182. doi: 10.1007/BF02410664.

[37]

J. Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations, Arch. Rational Mech. Anal., 9 (1962), 187-195.

[38]

M. Struwe, On partial regularity results for the Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 437-458. doi: 10.1002/cpa.3160410404.

[39]

S. Takahashi, On interior regularity criteria for weak solutions of the Navier-Stokes equations, Manuscripta Math., 69 (1990), 237-254. doi: 10.1007/BF02567922.

[40]

M. R. Ukhovskii and V. I. Iudovich, Axially symmetric flows of ideal and viscous fluids filling the whole space, J. Appl. Math. Mech., 32 (1968), 52-61. doi: 10.1016/0021-8928(68)90147-0.

[41]

P. Zhang and T. Zhang, Global axisymmetric solutions to three-dimensional Navier-Stokes system, Int. Math. Res. Not., 2014 (2014), 610-642.

[42]

Y. Zhou and M. Pokorný, On the regularity of the solutions of the Navier-Stokes equations via one velocity component, Nonlinearity, 23 (2010), 1097-1107. doi: 10.1088/0951-7715/23/5/004.

show all references

References:
[1]

H. Abidi and P. Zhang, Global smooth axisymmetric solutions of 3-D inhomogeneous incompressible Navier-Stokes system, Calc. Var. Partial Differential Equations, 54 (2015), 3251-3276. doi: 10.1007/s00526-015-0902-6.

[2]

M. Badiale and G. Tarantello, A Sobolev-Hardy inequality with applications to a nonlinear elliptic equation arising in astrophysics, Arch. Ration. Mech. Anal., 163 (2002), 259-293. doi: 10.1007/s002050200201.

[3]

H. Beirão da Veiga, On the smoothness of a class of weak solutions to the Navier-Stokes equations, J. Math. Fluid Mech., 2 (2000), 315-323. doi: 10.1007/PL00000955.

[4]

L. CafferalliR. Kohn and L. Nirenberg, Partial regularity of suitable weak solutions of the Navier-Stokes equations, Comm. Pure Appl. Math., 35 (1982), 771-831. doi: 10.1002/cpa.3160350604.

[5]

C. Cao and E. S. Titi, Regularity criteria for the three-dimensional Navier-Stokes equations, Indiana Univ. Math. J., 57 (2008), 2643-2661. doi: 10.1512/iumj.2008.57.3719.

[6]

D. Chae and J. Lee, On the regularity of the axisymmetric solutions of the Navier-Stokes equations, Math. Z., 239 (2002), 645-671. doi: 10.1007/s002090100317.

[7]

C. Chen, R. M. Strain, H. Yau and T. Tsai, Lower bound on the blow-up rate of the axisymmetric Navier-Stokes equations Int. Math. Res. Not. IMRN 2008 (2008), 31 pp. doi: 10.1093/imrn/rnn016.

[8]

C. ChenR. M. StrainH. Yau and T. Tsai, Lower bounds on the blow-up rate of the axisymmetric Navier-Stokes equations Ⅱ, Comm. Partial Differential Equations, 34 (2009), 203-232. doi: 10.1080/03605300902793956.

[9]

Q. Chen and Z. Zhang, Regularity criterion of axisymmetric weak solutions to the 3D Navier-Stokes equations, J. Math. Anal. Appl., 331 (2007), 1384-1395. doi: 10.1016/j.jmaa.2006.09.069.

[10]

P. Constantin and C. Foias, Navier-Stokes Equation Chicago Lectures in Mathematics, University of Chicago Press, Chicago, 1988.

[11]

L. EscauriazaG. Seregin and V. Šverák, Backward uniqueness for parabolic equations, Arch. Ration. Mech. Anal., 169 (2003), 147-157. doi: 10.1007/s00205-003-0263-8.

[12]

E. B. FabesB. F. Jones and N. M. Rivieére, The initial value problem for the Navier-Stokes equations with data in Lp, Arch. Rational Mech. Anal., 45 (1972), 222-240. doi: 10.1007/BF00281533.

[13]

D. Fang and C. Qian, The regularity criterion for 3D Navier-Stokes equations involving one velocity gradient component, Nonlinear Anal., 78 (2013), 86-103. doi: 10.1016/j.na.2012.09.019.

[14]

D. Fang and C. Qian, Some new regularity criteria for the 3D Navier-Stokes equations, preprint, arXiv: 1212.2335.

[15]

D. Fang and C. Qian, Several almost critical regularity conditions based on one component of the solutions for 3D NS Equations, preprint, arXiv: 1312.7378.

[16]

J. P. García Azorero and I. Peral Alonso, Hardy inequalities and some critical elliptic and parabolic problems, J. Differential Equations, 144 (1998), 441-476. doi: 10.1006/jdeq.1997.3375.

[17]

Y. Giga, Solutions for semilinear parabolic equations in Lp and regularity of weak solutions of the Navier-Stokes system, J. Differential Equations, 62 (1986), 186-212. doi: 10.1016/0022-0396(86)90096-3.

[18]

G. H. Hardy, J. E. Littlewood and G. Pólya, Inequalities 2nd edition, Cambridge, at the University Press, 1952.

[19]

E. Hopf, Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen, Math. Nachr., 4 (1951), 213-231.

[20]

T. Hou and C. Li, Dynamic stability of the three-dimensional axisymmetric Navier-Stokes equations with swirl, Comm. Pure Appl. Math., 61 (2008), 661-697. doi: 10.1002/cpa.20212.

[21]

N. Kim, Remarks for the axisymmetric Navier-Stokes equations, J. Differential Equations, 187 (2003), 226-239. doi: 10.1016/S0022-0396(02)00077-3.

[22]

G. KochN. NadirashviliG. A. Seregin and V. Šverák, Liouville theorems for the Navier-Stokes equations and applications, Acta Math., 203 (2009), 83-105. doi: 10.1007/s11511-009-0039-6.

[23]

O. Kreml and M. Pokorný, A regularity criterion for the angular velocity component in axisymmetric Navier-Stokes equations, Electron. J. Differential Equations 2007 (2007), 10 pp.

[24]

A. KubicaM. Pokorný and W. Zajaczkowski, Remarks on regularity criteria for axially symmetric weak solutions to the Navier-Stokes equations, Math. Methods Appl. Sci., 35 (2012), 360-371. doi: 10.1002/mma.1586.

[25]

O. A. Ladyženskaja, Unique global solvability of the three-dimensional Cauchy problem for the Navier-Stokes equations in the presence of axial symmetry, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov., 7 (1968), 155-177.

[26]

Z. Lei and Qi S. Zhang, A Liouville theorem for the axially-symmetric Navier-Stokes equations, J. Funct. Anal., 261 (2011), 2323-2345. doi: 10.1016/j.jfa.2011.06.016.

[27]

S. LeonardiJ. MálekJ. Nečas and M. Pokorný, On axially symmetric flows in R3, Z. Anal. Anwendungen, 18 (1999), 639-649. doi: 10.4171/ZAA/903.

[28]

J. Leray, Étude de diverses équations intégrales non linéaires et de quelques problémes que pose l'Hydrodynamique, Journal de Mathématiques Pures et Appliquées, 12 (1933), 1-82.

[29]

F. Lin, A new proof of the Caffarelli-Kohn-Nirenberg theorem, Comm. Pure Appl. Math., 51 (1998), 241-257. doi: 10.1002/(SICI)1097-0312(199803)51:3<241::AID-CPA2>3.0.CO;2-A.

[30]

J. Liu and W. Wang, Energy and helicity preserving schemes for hydro-and magnetohydro-dynamics flows with symmetry, J. Comput. Phys., 200 (2004), 8-33. doi: 10.1016/j.jcp.2004.03.005.

[31]

J. Liu and W. Wang, Convergence analysis of the energy and helicity preserving scheme for axisymmetric flows, SIAM J. Numer. Anal., 44 (2006), 2456-2480. doi: 10.1137/050639314.

[32]

J. Liu and W. Wang, Characterization and regularity for axisymmetric solenoidal vector fields with application to Navier-Stokes equation, SIAM J. Math. Anal., 41 (2009), 1825-1850. doi: 10.1137/080739744.

[33]

C. Miao and X. Zheng, On the global well-posedness for the Boussinesq system with horizontal dissipation, Comm. Math. Phys., 321 (2013), 33-67. doi: 10.1007/s00220-013-1721-2.

[34]

J. Neustupa and M. Pokorný, Axisymmetrc flow of Navier-Stokes fluid in the whole space with non-zero angular velocity component, Math. Bohem., 126 (2001), 469-481.

[35]

P. Penel and M. Pokorný, Some new regularity criteria for the Navier-Stokes equations containing gradient of the velocity, Appl. Math., 49 (2004), 483-493. doi: 10.1023/B:APOM.0000048124.64244.7e.

[36]

G. Prodi, Un teorema di unicitá per le equazioni di Navier-Stokes, Ann. Mat. Pura Appl., 48 (1959), 173-182. doi: 10.1007/BF02410664.

[37]

J. Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations, Arch. Rational Mech. Anal., 9 (1962), 187-195.

[38]

M. Struwe, On partial regularity results for the Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 437-458. doi: 10.1002/cpa.3160410404.

[39]

S. Takahashi, On interior regularity criteria for weak solutions of the Navier-Stokes equations, Manuscripta Math., 69 (1990), 237-254. doi: 10.1007/BF02567922.

[40]

M. R. Ukhovskii and V. I. Iudovich, Axially symmetric flows of ideal and viscous fluids filling the whole space, J. Appl. Math. Mech., 32 (1968), 52-61. doi: 10.1016/0021-8928(68)90147-0.

[41]

P. Zhang and T. Zhang, Global axisymmetric solutions to three-dimensional Navier-Stokes system, Int. Math. Res. Not., 2014 (2014), 610-642.

[42]

Y. Zhou and M. Pokorný, On the regularity of the solutions of the Navier-Stokes equations via one velocity component, Nonlinearity, 23 (2010), 1097-1107. doi: 10.1088/0951-7715/23/5/004.

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