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November  2013, 12(6): 2715-2719. doi: 10.3934/cpaa.2013.12.2715

Logarithmically improved criteria for Euler and Navier-Stokes equations

Yi Zhou 1, and  Zhen Lei 2,

1. 

Department and Institute of Mathematics, Fudan University, Shanghai 200433, China
2. 

Shanghai Key Laboratory for Contemporary Applied Mathematics, School of Mathematical Sciences, Fudan University, Shanghai 200433

Received  September 2012 Revised  February 2013 Published  May 2013

In this paper we prove the logarithmically improved Serrin's criteria to the three-dimensional incompressible Navier-Stokes equations.
Keywords: global regularity., Serrin's Criterion, Navier-Stokes equations.
Mathematics Subject Classification: Primary: 35Q35; Secondary: 76D0.
Citation: Yi Zhou, Zhen Lei. Logarithmically improved criteria for Euler and Navier-Stokes equations. Communications on Pure and Applied Analysis, 2013, 12 (6) : 2715-2719. doi: 10.3934/cpaa.2013.12.2715
References:
[1]

J. T. Beale, T. Kato and A. Majda, Remarks on the breakdown of smooth solutions for the $3$-D Euler equations, Comm. Math. Phys., 94 (1984), 61-66.

[2]

J. M. Bony, Calcul symbolique et propagation des singularites pour les quations aux drivees partielles non lineaires, Ann. Sci. Ecole Norm. Sup., 14 (1981), 209-246.

[3]

C. H. Chan and A. Vasseur, Log improvement of the Prodi-Serrin criteria for Navier-Stokes equations, available online at arXiv:0705.3659.

[4]

J. Y. Chemin, "Perfect Incompressibe Fluids," Oxford University Press, New York, 1998.

[5]

P. Constantin and C. Foias, "Navier-Stokes Equations," Chicago University Press, Chicago, 1988.

[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]

C. Fefferman, http://www.claymath.org/millennium/Navier-Stokes equations. preprint.

[8]

L. Iskauriaza, G. A. Seregin and V. Shverak, $L_{3,\infty}$-solutions of Navier-Stokes equations and backward uniqueness, (Russian) Uspekhi Mat. Nauk, 58 (2003), 3-44; translation in Russian Math. Surveys, 58 (2003), 211-250

[9]

H. Kozono and Y. Taniuchi, Bilinear estimates in BMO and the Navier-Stokes equations, Math. Z., 235 (2000), 173-194.

[10]

O. A. Ladyzhenskaya, "Mathematical Questions of the Dynamics of a Viscous Incompressible Fluid," Nauka, Moscow, 1970.

[11]

A. J. Majda and A. L. Bertozzi, "Vorticity and Incompressible Flow," Cambridge Texts in Applied Mathematics, 27. Cambridge University Press, Cambridge, 2002.

[12]

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

[13]

G. Seregin and V. Sverak, Navier-Stokes equations with lower bounds on the pressure, Arch. Ration. Mech. Anal., 163 (2002), 65-86.

[14]

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

[15]

J. Serrin, The initial value problem for the Navier-Stokes equations, Nonlinear Problems, Univ. of Wisconsin Press, Madison, 1963, 69-98.

[16]

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

[17]

T. Tao, Nonlinear dispersive equations. Local and global analysis, CBMS Regional Conference Series in Mathematics, 106. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2006.

[18]

R. Temam, "Navier-Stokes Equations," Second Edition, AMS Chelsea Publishing, Providence, RI, 2001.

[19]

H. Triebel, "Theory of Function Spaces," Birkauser Verlag, Boston, 1983.

[20]

Y. Zhou and S. Gala, Logarithmically improved Serrin's criterion to the Navier-Stokes equations in multiplier spaces, preprint, 2008.

[21]

Y. Zhou and Z. Lei, Logarithmically improved criteria for Euler and Navier-Stokes equations, avaliable on http://arxiv.org/abs/0805.2784v1

show all references

References:
[1]

J. T. Beale, T. Kato and A. Majda, Remarks on the breakdown of smooth solutions for the $3$-D Euler equations, Comm. Math. Phys., 94 (1984), 61-66.

[2]

J. M. Bony, Calcul symbolique et propagation des singularites pour les quations aux drivees partielles non lineaires, Ann. Sci. Ecole Norm. Sup., 14 (1981), 209-246.

[3]

C. H. Chan and A. Vasseur, Log improvement of the Prodi-Serrin criteria for Navier-Stokes equations, available online at arXiv:0705.3659.

[4]

J. Y. Chemin, "Perfect Incompressibe Fluids," Oxford University Press, New York, 1998.

[5]

P. Constantin and C. Foias, "Navier-Stokes Equations," Chicago University Press, Chicago, 1988.

[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]

C. Fefferman, http://www.claymath.org/millennium/Navier-Stokes equations. preprint.

[8]

L. Iskauriaza, G. A. Seregin and V. Shverak, $L_{3,\infty}$-solutions of Navier-Stokes equations and backward uniqueness, (Russian) Uspekhi Mat. Nauk, 58 (2003), 3-44; translation in Russian Math. Surveys, 58 (2003), 211-250

[9]

H. Kozono and Y. Taniuchi, Bilinear estimates in BMO and the Navier-Stokes equations, Math. Z., 235 (2000), 173-194.

[10]

O. A. Ladyzhenskaya, "Mathematical Questions of the Dynamics of a Viscous Incompressible Fluid," Nauka, Moscow, 1970.

[11]

A. J. Majda and A. L. Bertozzi, "Vorticity and Incompressible Flow," Cambridge Texts in Applied Mathematics, 27. Cambridge University Press, Cambridge, 2002.

[12]

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

[13]

G. Seregin and V. Sverak, Navier-Stokes equations with lower bounds on the pressure, Arch. Ration. Mech. Anal., 163 (2002), 65-86.

[14]

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

[15]

J. Serrin, The initial value problem for the Navier-Stokes equations, Nonlinear Problems, Univ. of Wisconsin Press, Madison, 1963, 69-98.

[16]

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

[17]

T. Tao, Nonlinear dispersive equations. Local and global analysis, CBMS Regional Conference Series in Mathematics, 106. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2006.

[18]

R. Temam, "Navier-Stokes Equations," Second Edition, AMS Chelsea Publishing, Providence, RI, 2001.

[19]

H. Triebel, "Theory of Function Spaces," Birkauser Verlag, Boston, 1983.

[20]

Y. Zhou and S. Gala, Logarithmically improved Serrin's criterion to the Navier-Stokes equations in multiplier spaces, preprint, 2008.

[21]

Y. Zhou and Z. Lei, Logarithmically improved criteria for Euler and Navier-Stokes equations, avaliable on http://arxiv.org/abs/0805.2784v1

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