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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

Zujin Zhang 1,

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.
Keywords: regularity., Navier-Stokes equations.
Mathematics Subject Classification: Primary: 35Q30, 35B65; Secondary: 76D0.
Citation: Zujin Zhang. A Serrin-type regularity criterion for the Navier-Stokes equations via one velocity component. Communications on Pure & 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.  Google Scholar

[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.  Google Scholar

[3]

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

[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, ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

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

[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.  Google Scholar

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[18]

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

[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).  Google Scholar

[20]

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

[21]

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

[22]

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

[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.  Google Scholar

[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.  Google Scholar

[25]

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

[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. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

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

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.  Google Scholar

[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.  Google Scholar

[3]

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

[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, ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

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

[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.  Google Scholar

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[18]

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

[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).  Google Scholar

[20]

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

[21]

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

[22]

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

[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.  Google Scholar

[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.  Google Scholar

[25]

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

[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. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

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

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