Logarithmically improved criteria for Euler and Navier-Stokes equations

Yi Zhou; Zhen Lei

doi:10.3934/cpaa.2013.12.2715

Article Contents

2013, Volume 12, Issue 6: 2715-2719. Doi: 10.3934/cpaa.2013.12.2715

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Logarithmically improved criteria for Euler and Navier-Stokes equations

Yi Zhou¹ and
Zhen Lei^2,

1.
Department and Institute of Mathematics, Fudan University, Shanghai 200433
2.
Shanghai Key Laboratory for Contemporary Applied Mathematics, School of Mathematical Sciences, Fudan University, Shanghai 200433

Received: August 31, 2012

Revised: January 31, 2013

Published: October 2013

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Abstract

In this paper we prove the logarithmically improved Serrin's criteria to the three-dimensional incompressible Navier-Stokes equations.

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Mathematics Subject Classification: Primary: 35Q35; Secondary: 76D03.

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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