# American Institute of Mathematical Sciences

• Previous Article
Positive solutions of integral systems involving Bessel potentials
• CPAA Home
• This Issue
• Next Article
Elliptic equations involving linear and superlinear terms and critical Caffarelli-Kohn-Nirenberg exponent with sign-changing weight functions
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 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
 [1] 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 [2] Xuanji Jia, Zaihong Jiang. An anisotropic regularity criterion for the 3D Navier-Stokes equations. Communications on Pure and Applied Analysis, 2013, 12 (3) : 1299-1306. doi: 10.3934/cpaa.2013.12.1299 [3] Keyan Wang. On global regularity of incompressible Navier-Stokes equations in $\mathbf R^3$. Communications on Pure and Applied Analysis, 2009, 8 (3) : 1067-1072. doi: 10.3934/cpaa.2009.8.1067 [4] Daoyuan Fang, Chenyin Qian. Regularity criterion for 3D Navier-Stokes equations in Besov spaces. Communications on Pure and Applied Analysis, 2014, 13 (2) : 585-603. doi: 10.3934/cpaa.2014.13.585 [5] Vittorino Pata. On the regularity of solutions to the Navier-Stokes equations. Communications on Pure and Applied Analysis, 2012, 11 (2) : 747-761. doi: 10.3934/cpaa.2012.11.747 [6] Igor Kukavica. On regularity for the Navier-Stokes equations in Morrey spaces. Discrete and Continuous Dynamical Systems, 2010, 26 (4) : 1319-1328. doi: 10.3934/dcds.2010.26.1319 [7] Igor Kukavica. On partial regularity for the Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 2008, 21 (3) : 717-728. doi: 10.3934/dcds.2008.21.717 [8] Peter Constantin, Gregory Seregin. Global regularity of solutions of coupled Navier-Stokes equations and nonlinear Fokker Planck equations. Discrete and Continuous Dynamical Systems, 2010, 26 (4) : 1185-1196. doi: 10.3934/dcds.2010.26.1185 [9] Joel Avrin. Global existence and regularity for the Lagrangian averaged Navier-Stokes equations with initial data in $H^{1//2}$. Communications on Pure and Applied Analysis, 2004, 3 (3) : 353-366. doi: 10.3934/cpaa.2004.3.353 [10] Bo-Qing Dong, Juan Song. Global regularity and asymptotic behavior of modified Navier-Stokes equations with fractional dissipation. Discrete and Continuous Dynamical Systems, 2012, 32 (1) : 57-79. doi: 10.3934/dcds.2012.32.57 [11] Shuguang Shao, Shu Wang, Wen-Qing Xu. Global regularity for a model of Navier-Stokes equations with logarithmic sub-dissipation. Kinetic and Related Models, 2018, 11 (1) : 179-190. doi: 10.3934/krm.2018009 [12] Jishan Fan, Yasuhide Fukumoto, Yong Zhou. Logarithmically improved regularity criteria for the generalized Navier-Stokes and related equations. Kinetic and Related Models, 2013, 6 (3) : 545-556. doi: 10.3934/krm.2013.6.545 [13] Chongsheng Cao. Sufficient conditions for the regularity to the 3D Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 2010, 26 (4) : 1141-1151. doi: 10.3934/dcds.2010.26.1141 [14] Hongjie Dong, Kunrui Wang. Interior and boundary regularity for the Navier-Stokes equations in the critical Lebesgue spaces. Discrete and Continuous Dynamical Systems, 2020, 40 (9) : 5289-5323. doi: 10.3934/dcds.2020228 [15] Zijin Li, Xinghong Pan. Some Remarks on regularity criteria of Axially symmetric Navier-Stokes equations. Communications on Pure and Applied Analysis, 2019, 18 (3) : 1333-1350. doi: 10.3934/cpaa.2019064 [16] Hui Chen, Daoyuan Fang, Ting Zhang. Regularity of 3D axisymmetric Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 2017, 37 (4) : 1923-1939. doi: 10.3934/dcds.2017081 [17] Yukang Chen, Changhua Wei. Partial regularity of solutions to the fractional Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 2016, 36 (10) : 5309-5322. doi: 10.3934/dcds.2016033 [18] Xiaofeng Hou, Limei Zhu. Serrin-type blowup criterion for full compressible Navier-Stokes-Maxwell system with vacuum. Communications on Pure and Applied Analysis, 2016, 15 (1) : 161-183. doi: 10.3934/cpaa.2016.15.161 [19] Joanna Rencławowicz, Wojciech M. Zajączkowski. Global regular solutions to the Navier-Stokes equations with large flux. Conference Publications, 2011, 2011 (Special) : 1234-1243. doi: 10.3934/proc.2011.2011.1234 [20] Daoyuan Fang, Bin Han, Matthias Hieber. Local and global existence results for the Navier-Stokes equations in the rotational framework. Communications on Pure and Applied Analysis, 2015, 14 (2) : 609-622. doi: 10.3934/cpaa.2015.14.609

2021 Impact Factor: 1.273