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May  2018, 38(5): 2609-2627. doi: 10.3934/dcds.2018110

## On the interior regularity criteria of the 3-D navier-stokes equations involving two velocity components

 1 School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China 2 School of Mathematical Sciences, UCAS and Institute of Mathematics, AMSS, Hua Loo-Keng Key Laboratory of Mathematics, Chinese Academy of Sciences, Beijing 100190, China 3 School of Mathematical Sciences, Peking University, Beijing 100871, China

* Corresponding author: Wendong Wang

Received  July 2017 Revised  November 2017 Published  March 2018

Fund Project: The first author is supported by NSF of China under Grant 11671067 and "the Fundamental Research Funds for the Central Universities". L. Zhang is partially supported by NSFC under grant 11471320 and 11631008. Z. Zhang is partially supported by NSF of China under Grant 11425103

We present some interior regularity criteria of the 3-D Navier-Stokes equations involving two components of the velocity. These results in particular imply that if the solution is singular at one point, then at least two components of the velocity have to blow up at the same point.

Citation: Wendong Wang, Liqun Zhang, Zhifei Zhang. On the interior regularity criteria of the 3-D navier-stokes equations involving two velocity components. Discrete & Continuous Dynamical Systems - A, 2018, 38 (5) : 2609-2627. doi: 10.3934/dcds.2018110
##### References:
 [1] L. Caffarelli, R. Kohn and L. Nirenberg, Partial regularity of suitable weak solutions of the Navier-Stokes equations, Comm. Pure Appl. Math., 35 (1982), 771-831. doi: 10.1002/cpa.3160350604. Google Scholar [2] C. Cao and E. S. Titi, Regularity criteria for the three dimensional Navier-Stokes equations, Indiana Univ. Math. J., 57 (2008), 2643-2661. doi: 10.1512/iumj.2008.57.3719. Google Scholar [3] C. Cao and E. S. Titi, Global regularity criterion of the 3D Navier-Stokes equations involving one entry of the velocity gradient tensor, Arch. Ration. Mech. Anal., 202 (2011), 919-932. doi: 10.1007/s00205-011-0439-6. Google Scholar [4] J. Y. Chemin and P. Zhang, On the critical one component regularity for 3-D Navier-Stokes system, Ann. Sci. Éc. Norm. Supér., 49 (2016), 131-167. doi: 10.24033/asens.2278. Google Scholar [5] L. Escauriaza, G. A. Seregin and V. Šverák, ${{L}^{3,\infty }}$ solutions to the Navier-Stokes equations and backward uniqueness, Russ. Math. Surveys, 58 (2003), 211-250. Google Scholar [6] Y. Giga, Solutions for semilinear parabolic equations in $L^p$ and regularity of weak solutions of the Navier-Stokes system, J. Differential Equations, 62 (1986), 186-212. doi: 10.1016/0022-0396(86)90096-3. Google Scholar [7] S. Gustafson, K. Kang and T.-P. Tsai, Interior regularity criteria for suitable weak solutions of the Navier-Stokes equations, Comm. Math. Phys., 273 (2007), 161-176. doi: 10.1007/s00220-007-0214-6. Google Scholar [8] I. Kukavica, On partial regularity for the Navier-Stokes equations, Discrete Contin. Dyn. Syst., 21 (2008), 717-728. doi: 10.3934/dcds.2008.21.717. Google Scholar [9] I. Kukavica and M. Ziane, One component regularity for the Navier-Stokes equation, Nonlinearity, 19 (2006), 453-469. doi: 10.1088/0951-7715/19/2/012. Google Scholar [10] O. A. Ladyzhenskaya and G. A. Seregin, On partial regularity of suitable weak solutions to the three-dimensional Navier-Stokes equations, J. Math. Fluid Mech., 1 (1999), 356-387. doi: 10.1007/s000210050015. Google Scholar [11] J. Leray, Sur le mouvement d'un liquids visqeux emplissant l'espace, Acta Math., 63 (1934), 193-248. doi: 10.1007/BF02547354. Google Scholar [12] F. H. Lin, A new proof of the Caffarelli-Kohn-Nirenberg theorem, Comm. Pure Appl. Math., 51 (1998), 241-257. doi: 10.1002/(SICI)1097-0312(199803)51:3<241::AID-CPA2>3.0.CO;2-A. Google Scholar [13] J. Nečas, M. Růžička and V. Šverák, On Leray's self-similar solutions of the Navier-Stokes equations, Acta Math., 176 (1996), 283-294. doi: 10.1007/BF02551584. Google Scholar [14] V. Scheffer, Partial regularity of solutions to the Navier-Stokes equations, Pacific J. Math., 66 (1976), 535-562. doi: 10.2140/pjm.1976.66.535. Google Scholar [15] V. Scheffer, Hausdorff measure and the Navier-Stokes equations, Commun. Math. Phy., 55 (1977), 97-112. doi: 10.1007/BF01626512. Google Scholar [16] V. Scheffer, The Navier-Stokes equations on a bounded domain, Commun. Math. Phy., 73 (1980), 1-42. doi: 10.1007/BF01942692. Google Scholar [17] G. A. Seregin, Estimate of suitable solutions to the Navier-Stokes equations in critical Morrey spaces, Journal of Mathematical Sciences, 143 (2007), 2961-2968. Google Scholar [18] J. Serrin, The initial value problem for the Navier-stokes equations, in Nonlinear problems(R. E. Langer Ed.), Univ. of Wisconsin Press, Madison, (1963), 69–98. Google Scholar [19] M. Struwe, On partial regularity results for the Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 437-458. doi: 10.1002/cpa.3160410404. Google Scholar [20] G. Tian and Z. Xin, Gradient estimation on Navier-Stokes equations, Comm. Anal. Geom., 7 (1999), 221-257. doi: 10.4310/CAG.1999.v7.n2.a1. Google Scholar [21] A. Vasseur, A new proof of partial regularity of solutions to Navier-Stokes equations, Nonlinear Differential Equations Appl., 14 (2007), 753-785. doi: 10.1007/s00030-007-6001-4. Google Scholar [22] W. Wang and Z. Zhang, On the interior regularity criterion and the number of singular points to the Navier-Stokes equations, J. Anal. Math., 123 (2014), 139-170. doi: 10.1007/s11854-014-0016-7. Google Scholar [23] Y. Zhou and M. Pokorný, On the regularity of the solutions of the Navier-Stokes equations via one velocity component, Nonlinearity, 23 (2010), 1097-1107. doi: 10.1088/0951-7715/23/5/004. Google Scholar

show all references

##### References:
 [1] L. Caffarelli, R. Kohn and L. Nirenberg, Partial regularity of suitable weak solutions of the Navier-Stokes equations, Comm. Pure Appl. Math., 35 (1982), 771-831. doi: 10.1002/cpa.3160350604. Google Scholar [2] C. Cao and E. S. Titi, Regularity criteria for the three dimensional Navier-Stokes equations, Indiana Univ. Math. J., 57 (2008), 2643-2661. doi: 10.1512/iumj.2008.57.3719. Google Scholar [3] C. Cao and E. S. Titi, Global regularity criterion of the 3D Navier-Stokes equations involving one entry of the velocity gradient tensor, Arch. Ration. Mech. Anal., 202 (2011), 919-932. doi: 10.1007/s00205-011-0439-6. Google Scholar [4] J. Y. Chemin and P. Zhang, On the critical one component regularity for 3-D Navier-Stokes system, Ann. Sci. Éc. Norm. Supér., 49 (2016), 131-167. doi: 10.24033/asens.2278. Google Scholar [5] L. Escauriaza, G. A. Seregin and V. Šverák, ${{L}^{3,\infty }}$ solutions to the Navier-Stokes equations and backward uniqueness, Russ. Math. Surveys, 58 (2003), 211-250. Google Scholar [6] Y. Giga, Solutions for semilinear parabolic equations in $L^p$ and regularity of weak solutions of the Navier-Stokes system, J. Differential Equations, 62 (1986), 186-212. doi: 10.1016/0022-0396(86)90096-3. Google Scholar [7] S. Gustafson, K. Kang and T.-P. Tsai, Interior regularity criteria for suitable weak solutions of the Navier-Stokes equations, Comm. Math. Phys., 273 (2007), 161-176. doi: 10.1007/s00220-007-0214-6. Google Scholar [8] I. Kukavica, On partial regularity for the Navier-Stokes equations, Discrete Contin. Dyn. Syst., 21 (2008), 717-728. doi: 10.3934/dcds.2008.21.717. Google Scholar [9] I. Kukavica and M. Ziane, One component regularity for the Navier-Stokes equation, Nonlinearity, 19 (2006), 453-469. doi: 10.1088/0951-7715/19/2/012. Google Scholar [10] O. A. Ladyzhenskaya and G. A. Seregin, On partial regularity of suitable weak solutions to the three-dimensional Navier-Stokes equations, J. Math. Fluid Mech., 1 (1999), 356-387. doi: 10.1007/s000210050015. Google Scholar [11] J. Leray, Sur le mouvement d'un liquids visqeux emplissant l'espace, Acta Math., 63 (1934), 193-248. doi: 10.1007/BF02547354. Google Scholar [12] F. H. Lin, A new proof of the Caffarelli-Kohn-Nirenberg theorem, Comm. Pure Appl. Math., 51 (1998), 241-257. doi: 10.1002/(SICI)1097-0312(199803)51:3<241::AID-CPA2>3.0.CO;2-A. Google Scholar [13] J. Nečas, M. Růžička and V. Šverák, On Leray's self-similar solutions of the Navier-Stokes equations, Acta Math., 176 (1996), 283-294. doi: 10.1007/BF02551584. Google Scholar [14] V. Scheffer, Partial regularity of solutions to the Navier-Stokes equations, Pacific J. Math., 66 (1976), 535-562. doi: 10.2140/pjm.1976.66.535. Google Scholar [15] V. Scheffer, Hausdorff measure and the Navier-Stokes equations, Commun. Math. Phy., 55 (1977), 97-112. doi: 10.1007/BF01626512. Google Scholar [16] V. Scheffer, The Navier-Stokes equations on a bounded domain, Commun. Math. Phy., 73 (1980), 1-42. doi: 10.1007/BF01942692. Google Scholar [17] G. A. Seregin, Estimate of suitable solutions to the Navier-Stokes equations in critical Morrey spaces, Journal of Mathematical Sciences, 143 (2007), 2961-2968. Google Scholar [18] J. Serrin, The initial value problem for the Navier-stokes equations, in Nonlinear problems(R. E. Langer Ed.), Univ. of Wisconsin Press, Madison, (1963), 69–98. Google Scholar [19] M. Struwe, On partial regularity results for the Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 437-458. doi: 10.1002/cpa.3160410404. Google Scholar [20] G. Tian and Z. Xin, Gradient estimation on Navier-Stokes equations, Comm. Anal. Geom., 7 (1999), 221-257. doi: 10.4310/CAG.1999.v7.n2.a1. Google Scholar [21] A. Vasseur, A new proof of partial regularity of solutions to Navier-Stokes equations, Nonlinear Differential Equations Appl., 14 (2007), 753-785. doi: 10.1007/s00030-007-6001-4. Google Scholar [22] W. Wang and Z. Zhang, On the interior regularity criterion and the number of singular points to the Navier-Stokes equations, J. Anal. Math., 123 (2014), 139-170. doi: 10.1007/s11854-014-0016-7. Google Scholar [23] Y. Zhou and M. Pokorný, On the regularity of the solutions of the Navier-Stokes equations via one velocity component, Nonlinearity, 23 (2010), 1097-1107. doi: 10.1088/0951-7715/23/5/004. Google Scholar
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