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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

 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.
Citation: Yi Zhou, Zhen Lei. Logarithmically improved criteria for Euler and Navier-Stokes equations. Communications on Pure & 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.   Google Scholar [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.   Google Scholar [3] C. H. Chan and A. Vasseur, Log improvement of the Prodi-Serrin criteria for Navier-Stokes equations,, available online at arXiv:0705.3659., ().   Google Scholar [4] J. Y. Chemin, "Perfect Incompressibe Fluids,", Oxford University Press, (1998).   Google Scholar [5] P. Constantin and C. Foias, "Navier-Stokes Equations,", Chicago University Press, (1988).   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.   Google Scholar [7] C. Fefferman, http://www.claymath.org/millennium/Navier-Stokes equations., preprint., ().   Google Scholar [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.   Google Scholar [9] H. Kozono and Y. Taniuchi, Bilinear estimates in BMO and the Navier-Stokes equations,, Math. Z., 235 (2000), 173.   Google Scholar [10] O. A. Ladyzhenskaya, "Mathematical Questions of the Dynamics of a Viscous Incompressible Fluid,", Nauka, (1970).   Google Scholar [11] A. J. Majda and A. L. Bertozzi, "Vorticity and Incompressible Flow,", Cambridge Texts in Applied Mathematics, (2002).   Google Scholar [12] G. Prodi, Un teorema di unicità per le equazioni di Navier-Stokes,, Ann. Mat. Pura Appl., 48 (1959), 173.   Google Scholar [13] G. Seregin and V. Sverak, Navier-Stokes equations with lower bounds on the pressure,, Arch. Ration. Mech. Anal., 163 (2002), 65.   Google Scholar [14] J. Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations,, Arch. Rational Mech. Anal., 9 (1962), 187.   Google Scholar [15] J. Serrin, The initial value problem for the Navier-Stokes equations,, Nonlinear Problems, (1963), 69.   Google Scholar [16] M. Struwe, On partial regularity results for the Navier-Stokes equations,, Comm. Pure Appl. Math., 41 (1988), 437.   Google Scholar [17] T. Tao, Nonlinear dispersive equations. Local and global analysis,, CBMS Regional Conference Series in Mathematics, (2006).   Google Scholar [18] R. Temam, "Navier-Stokes Equations,", Second Edition, (2001).   Google Scholar [19] H. Triebel, "Theory of Function Spaces,", Birkauser Verlag, (1983).   Google Scholar [20] Y. Zhou and S. Gala, Logarithmically improved Serrin's criterion to the Navier-Stokes equations in multiplier spaces,, preprint, (2008).   Google Scholar [21] Y. Zhou and Z. Lei, Logarithmically improved criteria for Euler and Navier-Stokes equations,, avaliable on http://arxiv.org/abs/0805.2784v1, ().   Google Scholar

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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.   Google Scholar [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.   Google Scholar [3] C. H. Chan and A. Vasseur, Log improvement of the Prodi-Serrin criteria for Navier-Stokes equations,, available online at arXiv:0705.3659., ().   Google Scholar [4] J. Y. Chemin, "Perfect Incompressibe Fluids,", Oxford University Press, (1998).   Google Scholar [5] P. Constantin and C. Foias, "Navier-Stokes Equations,", Chicago University Press, (1988).   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.   Google Scholar [7] C. Fefferman, http://www.claymath.org/millennium/Navier-Stokes equations., preprint., ().   Google Scholar [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.   Google Scholar [9] H. Kozono and Y. Taniuchi, Bilinear estimates in BMO and the Navier-Stokes equations,, Math. Z., 235 (2000), 173.   Google Scholar [10] O. A. Ladyzhenskaya, "Mathematical Questions of the Dynamics of a Viscous Incompressible Fluid,", Nauka, (1970).   Google Scholar [11] A. J. Majda and A. L. Bertozzi, "Vorticity and Incompressible Flow,", Cambridge Texts in Applied Mathematics, (2002).   Google Scholar [12] G. Prodi, Un teorema di unicità per le equazioni di Navier-Stokes,, Ann. Mat. Pura Appl., 48 (1959), 173.   Google Scholar [13] G. Seregin and V. Sverak, Navier-Stokes equations with lower bounds on the pressure,, Arch. Ration. Mech. Anal., 163 (2002), 65.   Google Scholar [14] J. Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations,, Arch. Rational Mech. Anal., 9 (1962), 187.   Google Scholar [15] J. Serrin, The initial value problem for the Navier-Stokes equations,, Nonlinear Problems, (1963), 69.   Google Scholar [16] M. Struwe, On partial regularity results for the Navier-Stokes equations,, Comm. Pure Appl. Math., 41 (1988), 437.   Google Scholar [17] T. Tao, Nonlinear dispersive equations. Local and global analysis,, CBMS Regional Conference Series in Mathematics, (2006).   Google Scholar [18] R. Temam, "Navier-Stokes Equations,", Second Edition, (2001).   Google Scholar [19] H. Triebel, "Theory of Function Spaces,", Birkauser Verlag, (1983).   Google Scholar [20] Y. Zhou and S. Gala, Logarithmically improved Serrin's criterion to the Navier-Stokes equations in multiplier spaces,, preprint, (2008).   Google Scholar [21] Y. Zhou and Z. Lei, Logarithmically improved criteria for Euler and Navier-Stokes equations,, avaliable on http://arxiv.org/abs/0805.2784v1, ().   Google Scholar
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