January  2013, 12(1): 117-124. doi: 10.3934/cpaa.2013.12.117

A Serrin-type regularity criterion for the Navier-Stokes equations via one velocity component

1. 

School of Mathematics and Computer Science, Gannan Normal University, Ganzhou, 341000 Jiangxi, China

Received  September 2010 Revised  March 2011 Published  September 2012

We study the Cauchy problem for the 3D Navier-Stokes equations, and prove some scalaring-invariant regularity criteria involving only one velocity component.
Citation: Zujin Zhang. A Serrin-type regularity criterion for the Navier-Stokes equations via one velocity component. Communications on Pure and Applied Analysis, 2013, 12 (1) : 117-124. doi: 10.3934/cpaa.2013.12.117
References:
[1]

H. Beirão da Veiga, A new regularity class for the Navier-Stokes equations in $R^n$, Chinese Ann. Math. Ser. B, 16 (1995), 407-412.

[2]

H. Beirão da Veiga and L. C. Berselli, On the regularizing effect of the vorticity direction in incompressible viscous flows, Differential Integral Equations, 15 (2002), 345-356.

[3]

C. S. Cao, Sufficient conditions for the regularity to the $3$D Navier-Stokes equations, Discrete Contin. Dyn. Syst., 26 (2010), 1141-1151.

[4]

C. S. Cao and E. S. Titi, Global regularity criterion for the $3$D Navier-Stokes equations involving one entry of the velocity gradient tensor, preprint, arXiv:math/1005.4463.

[5]

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

[6]

P. Constantin and C. Fefferman, Direction of vorticity and the problem of global regularity for the Navier-Stokes equations, Indiana Univ. Math. J., 42 (1993), 775-789.

[7]

L. Escauriaza, G. Seregin and V. Sverák, Backward uniqueness for parabolic equations, Arch. Ration. Mech. Anal., 169 (2003), 147-157.

[8]

J. S. Fan, S. Jiang and G. X. Ni, On regularity criteria for the $n$-dimensional Navier-Stokes equations in terms of the pressure, J. Differential Equations, 244 (2008), 2963-2979.

[9]

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

[10]

J. M. Kim, On regularity criteria of the Navier-Stokes equations in bounded domains, J. Math. Phys., 51 (2010), 053102.

[11]

I. Kukavica and M. Ziane, Navier-Stokes equations with regularity in one direction, J. Math. Phys., 48 (2007), 065203.

[12]

I. Kukavica and M. Ziane, One component regularity for the Navier-Stokes equations, Nonlinearity, 19 (2006), 453-469.

[13]

J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math., 63 (1934), 193-248.

[14]

J. Neustupa, A. Novotný and P. Penel, An interior regularity of a weak solution to the Navier-Stokes equations in dependence on one component of velocity, Topics in Mathematical Fluid Mechanics, Quaderni di Matematica, Dept. Math., Seconda University, Napoli, Caserta, Vol. 10, (2002) 163-183.

[15]

J. Neustupa and P. Penel, Anisotropic and geometric criteria for interior regularity of weak solutions to the $3$D Navier–Stokes equations, in Mathematical Fluid Mechanics (Recent Results and Open Problems), Advances in Mathematical Fluid Mechanics, edited by J. Neustupa, and P. Penel (Birkhäuser, Basel-Boston-Berlin, (2001), 239-267.

[16]

P. Penel and M. Pokorný, Some new regularity criteria for the Navier-Stokes equations containing the gradient of velocity, Appl. Math., 49 (2004), 483-493.

[17]

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

[18]

J. Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations, Arch. Rational Mech. Anal., 9 (1962), 187-191.

[19]

J. Serrin, The initial value problems for the Navier-Stokes equations, in "Nonlinear Problems" (ed. R. E. Langer), University of Wisconsin Press, Madison, WI, (1963).

[20]

R. Temam, "Navier-Stokes Equations, Theory and Numerical Analysis," North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977.

[21]

X. C. Zhang, A regularity criterion for the solutions of $3$D Navier-Stokes equations, J. Math. Anal. Appl., 346 (2008), 336-339.

[22]

Z. F. Zhang and Q. L. Chen, Regularity criterion via two components of vorticity on weak solutions to the Navier-Stokes equations in $mathbb{R}^{3}$, J. Differential Equations, 216 (2005), 470-481.

[23]

Z. J. Zhang, Z. A. Yao, P. Li, C. C. Guo and M. Lu, Two new regularity criteria for the 3D Navier-Stokes equations via two entries of the velocity gradient tensor, Acta Appl. Math.. doi: doi: 10.1007/s10440-012-9712-4.

[24]

Y, Zhou, A new regularity criterion for the Navier-Stokes equations in terms of the direction of vorticity, Monatsh. Math., 144 (2005), 251-257.

[25]

Y. Zhou, A new regularity criterion for weak solutions to the Navier-Stokes equations, J. Math. Pures Appl., 84 (2005), 1496-1514.

[26]

Y. Zhou, On a regularity criterion in terms of the gradient of pressure for the Navier-Stokes equations in $R^n$, Z. Angew. Math. Phys., 57 (2006), 384-392.

[27]

Y. Zhou and M. Pokorný, On a regularity criterion for the Navier-Stokes equations involving gradient of one velocity component, J. Math. Phys., 50 (2009), 123514.

[28]

Y. Zhou and M. Pokorný, On the regularity to the solutions of the Navier-Stokes equations via one velocity component, Nonlinearity, 23 (2010), 1097-1107.

[29]

Y. Zhou, Regularity criteria in terms of pressure for the $3$D Navier-Stokes equations in a generic domain, Math. Ann., 328 (2004), 173-192.

[30]

Y. Zhou, Weighted regularity criteria for the three-dimensional Navier-Stokes equations, Proc. Roy. Soc. Edinburgh, Sect. A: Math., 139 (2009), 661-671.

show all references

References:
[1]

H. Beirão da Veiga, A new regularity class for the Navier-Stokes equations in $R^n$, Chinese Ann. Math. Ser. B, 16 (1995), 407-412.

[2]

H. Beirão da Veiga and L. C. Berselli, On the regularizing effect of the vorticity direction in incompressible viscous flows, Differential Integral Equations, 15 (2002), 345-356.

[3]

C. S. Cao, Sufficient conditions for the regularity to the $3$D Navier-Stokes equations, Discrete Contin. Dyn. Syst., 26 (2010), 1141-1151.

[4]

C. S. Cao and E. S. Titi, Global regularity criterion for the $3$D Navier-Stokes equations involving one entry of the velocity gradient tensor, preprint, arXiv:math/1005.4463.

[5]

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

[6]

P. Constantin and C. Fefferman, Direction of vorticity and the problem of global regularity for the Navier-Stokes equations, Indiana Univ. Math. J., 42 (1993), 775-789.

[7]

L. Escauriaza, G. Seregin and V. Sverák, Backward uniqueness for parabolic equations, Arch. Ration. Mech. Anal., 169 (2003), 147-157.

[8]

J. S. Fan, S. Jiang and G. X. Ni, On regularity criteria for the $n$-dimensional Navier-Stokes equations in terms of the pressure, J. Differential Equations, 244 (2008), 2963-2979.

[9]

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

[10]

J. M. Kim, On regularity criteria of the Navier-Stokes equations in bounded domains, J. Math. Phys., 51 (2010), 053102.

[11]

I. Kukavica and M. Ziane, Navier-Stokes equations with regularity in one direction, J. Math. Phys., 48 (2007), 065203.

[12]

I. Kukavica and M. Ziane, One component regularity for the Navier-Stokes equations, Nonlinearity, 19 (2006), 453-469.

[13]

J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math., 63 (1934), 193-248.

[14]

J. Neustupa, A. Novotný and P. Penel, An interior regularity of a weak solution to the Navier-Stokes equations in dependence on one component of velocity, Topics in Mathematical Fluid Mechanics, Quaderni di Matematica, Dept. Math., Seconda University, Napoli, Caserta, Vol. 10, (2002) 163-183.

[15]

J. Neustupa and P. Penel, Anisotropic and geometric criteria for interior regularity of weak solutions to the $3$D Navier–Stokes equations, in Mathematical Fluid Mechanics (Recent Results and Open Problems), Advances in Mathematical Fluid Mechanics, edited by J. Neustupa, and P. Penel (Birkhäuser, Basel-Boston-Berlin, (2001), 239-267.

[16]

P. Penel and M. Pokorný, Some new regularity criteria for the Navier-Stokes equations containing the gradient of velocity, Appl. Math., 49 (2004), 483-493.

[17]

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

[18]

J. Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations, Arch. Rational Mech. Anal., 9 (1962), 187-191.

[19]

J. Serrin, The initial value problems for the Navier-Stokes equations, in "Nonlinear Problems" (ed. R. E. Langer), University of Wisconsin Press, Madison, WI, (1963).

[20]

R. Temam, "Navier-Stokes Equations, Theory and Numerical Analysis," North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977.

[21]

X. C. Zhang, A regularity criterion for the solutions of $3$D Navier-Stokes equations, J. Math. Anal. Appl., 346 (2008), 336-339.

[22]

Z. F. Zhang and Q. L. Chen, Regularity criterion via two components of vorticity on weak solutions to the Navier-Stokes equations in $mathbb{R}^{3}$, J. Differential Equations, 216 (2005), 470-481.

[23]

Z. J. Zhang, Z. A. Yao, P. Li, C. C. Guo and M. Lu, Two new regularity criteria for the 3D Navier-Stokes equations via two entries of the velocity gradient tensor, Acta Appl. Math.. doi: doi: 10.1007/s10440-012-9712-4.

[24]

Y, Zhou, A new regularity criterion for the Navier-Stokes equations in terms of the direction of vorticity, Monatsh. Math., 144 (2005), 251-257.

[25]

Y. Zhou, A new regularity criterion for weak solutions to the Navier-Stokes equations, J. Math. Pures Appl., 84 (2005), 1496-1514.

[26]

Y. Zhou, On a regularity criterion in terms of the gradient of pressure for the Navier-Stokes equations in $R^n$, Z. Angew. Math. Phys., 57 (2006), 384-392.

[27]

Y. Zhou and M. Pokorný, On a regularity criterion for the Navier-Stokes equations involving gradient of one velocity component, J. Math. Phys., 50 (2009), 123514.

[28]

Y. Zhou and M. Pokorný, On the regularity to the solutions of the Navier-Stokes equations via one velocity component, Nonlinearity, 23 (2010), 1097-1107.

[29]

Y. Zhou, Regularity criteria in terms of pressure for the $3$D Navier-Stokes equations in a generic domain, Math. Ann., 328 (2004), 173-192.

[30]

Y. Zhou, Weighted regularity criteria for the three-dimensional Navier-Stokes equations, Proc. Roy. Soc. Edinburgh, Sect. A: Math., 139 (2009), 661-671.

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