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

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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

Abstract
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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 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, ().
[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 $\R^3$, 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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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, ().
[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 $\R^3$, 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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