# American Institute of Mathematical Sciences

2014, 13(2): 585-603. doi: 10.3934/cpaa.2014.13.585

## Regularity criterion for 3D Navier-Stokes equations in Besov spaces

 1 Department of Mathematics, Zhejiang University, Hangzhou 310027 2 Department of Mathematics, Zhejiang University, Hangzhou, 310027, China

Received  October 2012 Revised  July 2013 Published  October 2013

Several regularity criterions of Leray-Hopf weak solutions $u$ to the 3D Navier-Stokes equations are obtained. The results show that a weak solution $u$ becomes regular if the gradient of velocity component $\nabla_{h}{u}$ (or $\nabla{u_3}$) satisfies the additional conditions in the class of $L^{q}(0,T; \dot{B}_{p,r}^{s}(\mathbb{R}^{3}))$, where $\nabla_{h}=(\partial_{x_{1}},\partial_{x_{2}})$ is the horizontal gradient operator. Besides, we also consider the anisotropic regularity criterion for the weak solution of Navier-Stokes equations in $\mathbb{R}^3$. Finally, we also get a further regularity criterion, when give the sufficient condition on $\partial_3u_3$.
Citation: Daoyuan Fang, Chenyin Qian. Regularity criterion for 3D Navier-Stokes equations in Besov spaces. Communications on Pure & Applied Analysis, 2014, 13 (2) : 585-603. doi: 10.3934/cpaa.2014.13.585
##### References:
 [1] H. Bahouri, R. Danchin and J. Y. Chemin, "Fourier Analysis and Nonlinear Partial Differential Equations, A Series of Comprehensive Studies in Mathematics,'', Springer Heidelberg Dordrecht London New York. Springer-Verlag Berlin Heidelberg, (2011). doi: 10.1007/978-3-642-16830-7. [2] H. Beirão da Veiga, A new regularity class for the Navier-Stokes equations in $\mathbbR^n$,, Chinese Ann. Math. Ser. B, 16 (1995), 407. [3] L. C. Berselli, On a regularity criterion for the solutions to the 3D Navier-Stokes equations,, Dierential Integral Equations, 15 (2002), 1129. [4] C. S. Cao and E. S. Titi, Global regularity criterion for the 3D Navier-Stokes equations involving one entry of the velocity gradient tensor,, Arch. Rational Mech. Anal., 202 (2011), 919. doi: 10.1007/s00205-011-0439-6. [5] Z. F. Zhang and Q. L. Chen, Regularity criterion via two components of vorticity on weak solutions to the Navier-Stokes equations in $\mathbbR^3$,, J. Differential Equations, 216 (2005), 470. doi: 10.1016/j.jde.2005.06.001. [6] A. Cheskidov and R. Shvydkoy, On the regularity of weak solutions of the 3D Navier-Stokes equations in $B^{-1}_{\infty,\infty}$,, \arXiv{0708.3067v2 [math.AP]}., (). [7] G. P. Galdi, "An Introduction to the Mathematical Theory of the Navier-Stokes Equations,'', Vol. I, (1994). doi: 10.1007/978-0-387-09620-9. [8] S. Gala, A remark on the blow-up criterion of strong solutions to the Navier-Stokes equations,, Applied Mathematics and Computation, 217 (2011), 9488. doi: 10.1016/j.amc.2011.03.156. [9] E. Hopf, Über die anfang swetaufgabe für die hydrodynamischer grundgleichungan,, Math. Nach., 4 (1951), 213. [10] H. Kozono and Y. Taniuchi, Bilinear estimates in BMO and the Navier-Stokes equations,, Math. Z., 235 (2000), 173. doi: 10.1007/s002090000130. [11] H. Kozono, T. Ogawa and Y. Taniuchi, The critical Sobolev inequalities in Besov spaces and regularity criterion to some semi-linear evolution equations,, Math. Z., 242 (2002), 251. doi: 10.1007/s002090100332. [12] H. Kozono and N. Yatsu, Extension criterion via two-components of vorticity on strong solution to the 3D Navier-Stokes equations,, Math. Z., 246 (2003), 55. doi: 10.1007/s00209-003-0576-1. [13] I. Kukavica and M. Ziane, Navier-Stokes equations with regularity in one direction,, J. Math Phys., 48 (2007). doi: 10.1063/1.2395919. [14] I. Kukavica and M. Ziane, One component regularity for the Navier-Stokes equations,, Nonlinearity, 19 (2006), 453. doi: 10.1088/0951-7715/19/2/012. [15] O. A. Ladyzhenskaya, "The Boundary Value Problems of Mathematical Physics,", Springer, (1985). [16] J. Leray, Sur le mouvement d'um liquide visqieux emlissant l'space,, Acta Math., 63 (1934), 193. doi: 10.1007/BF02547354. [17] J. Neustupa and P. Penel, Anisotropic and geometric criteria for interior regularity of weak solutions to the 3D Navier-Stokes equations,, In, (2001), 239. doi: 10.1007/978-3-0348-8243-9_10. [18] P. Penel and M. Pokorný, On anisotropic regularity criteria for the Solutions to 3D Navier-Stokes equations,, J. Math. Fluid Mech., 13 (2011), 341. doi: 10.1007/s00021-010-0038-6. [19] P. Penel and M. Pokorný, Some new regularity criteria for the Navier-Stokes equations containing gradient of the velocity,, Appl. Math., 49 (2004), 483. doi: 10.1023/B:APOM.0000048124.64244.7e. [20] M. Pokorný, On the result of He concerning the smoothness of solutions to the Navier-Stokes equations,, Electron. J. Differ. Equ., 11 (2003), 1. [21] G. Prodi, Un teorema di unicità per el equazioni di Navier-Stokes,, Ann. Mat. Pura Appl. IV, 48 (1959), 173. [22] J. Serrin, "The Initial Value Problems for the Navier-Stokes Equations, in Nonlinear Problems,", edited by R. E. Langer, (1963). [23] H. Sohr, "The Navier-Stokes Equations, An Elementary Functional Analytic Approach,", Birkh$\ddot{\mboxa}$user Verlag, (2001). doi: 10.1007/978-3-0348-0551-3. [24] B. Q. Yuan and B. Zhang, Blow-up criterion of strong solutions to the Navier-Stokes equations in Besov spaces with negative indices,, J. Differential Equations, 242 (2007), 1. doi: 0.1016/j.jde.2007.07.009. [25] Y. Zhou and M. Pokorný, On the regularity of the solutions of the Navier-Stokes equations via one velocity component,, Nonlinearity, 23 (2010), 1097. doi: 10.1088/0951-7715/23/5/004.

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##### References:
 [1] H. Bahouri, R. Danchin and J. Y. Chemin, "Fourier Analysis and Nonlinear Partial Differential Equations, A Series of Comprehensive Studies in Mathematics,'', Springer Heidelberg Dordrecht London New York. Springer-Verlag Berlin Heidelberg, (2011). doi: 10.1007/978-3-642-16830-7. [2] H. Beirão da Veiga, A new regularity class for the Navier-Stokes equations in $\mathbbR^n$,, Chinese Ann. Math. Ser. B, 16 (1995), 407. [3] L. C. Berselli, On a regularity criterion for the solutions to the 3D Navier-Stokes equations,, Dierential Integral Equations, 15 (2002), 1129. [4] C. S. Cao and E. S. Titi, Global regularity criterion for the 3D Navier-Stokes equations involving one entry of the velocity gradient tensor,, Arch. Rational Mech. Anal., 202 (2011), 919. doi: 10.1007/s00205-011-0439-6. [5] Z. F. Zhang and Q. L. Chen, Regularity criterion via two components of vorticity on weak solutions to the Navier-Stokes equations in $\mathbbR^3$,, J. Differential Equations, 216 (2005), 470. doi: 10.1016/j.jde.2005.06.001. [6] A. Cheskidov and R. Shvydkoy, On the regularity of weak solutions of the 3D Navier-Stokes equations in $B^{-1}_{\infty,\infty}$,, \arXiv{0708.3067v2 [math.AP]}., (). [7] G. P. Galdi, "An Introduction to the Mathematical Theory of the Navier-Stokes Equations,'', Vol. I, (1994). doi: 10.1007/978-0-387-09620-9. [8] S. Gala, A remark on the blow-up criterion of strong solutions to the Navier-Stokes equations,, Applied Mathematics and Computation, 217 (2011), 9488. doi: 10.1016/j.amc.2011.03.156. [9] E. Hopf, Über die anfang swetaufgabe für die hydrodynamischer grundgleichungan,, Math. Nach., 4 (1951), 213. [10] H. Kozono and Y. Taniuchi, Bilinear estimates in BMO and the Navier-Stokes equations,, Math. Z., 235 (2000), 173. doi: 10.1007/s002090000130. [11] H. Kozono, T. Ogawa and Y. Taniuchi, The critical Sobolev inequalities in Besov spaces and regularity criterion to some semi-linear evolution equations,, Math. Z., 242 (2002), 251. doi: 10.1007/s002090100332. [12] H. Kozono and N. Yatsu, Extension criterion via two-components of vorticity on strong solution to the 3D Navier-Stokes equations,, Math. Z., 246 (2003), 55. doi: 10.1007/s00209-003-0576-1. [13] I. Kukavica and M. Ziane, Navier-Stokes equations with regularity in one direction,, J. Math Phys., 48 (2007). doi: 10.1063/1.2395919. [14] I. Kukavica and M. Ziane, One component regularity for the Navier-Stokes equations,, Nonlinearity, 19 (2006), 453. doi: 10.1088/0951-7715/19/2/012. [15] O. A. Ladyzhenskaya, "The Boundary Value Problems of Mathematical Physics,", Springer, (1985). [16] J. Leray, Sur le mouvement d'um liquide visqieux emlissant l'space,, Acta Math., 63 (1934), 193. doi: 10.1007/BF02547354. [17] J. Neustupa and P. Penel, Anisotropic and geometric criteria for interior regularity of weak solutions to the 3D Navier-Stokes equations,, In, (2001), 239. doi: 10.1007/978-3-0348-8243-9_10. [18] P. Penel and M. Pokorný, On anisotropic regularity criteria for the Solutions to 3D Navier-Stokes equations,, J. Math. Fluid Mech., 13 (2011), 341. doi: 10.1007/s00021-010-0038-6. [19] P. Penel and M. Pokorný, Some new regularity criteria for the Navier-Stokes equations containing gradient of the velocity,, Appl. Math., 49 (2004), 483. doi: 10.1023/B:APOM.0000048124.64244.7e. [20] M. Pokorný, On the result of He concerning the smoothness of solutions to the Navier-Stokes equations,, Electron. J. Differ. Equ., 11 (2003), 1. [21] G. Prodi, Un teorema di unicità per el equazioni di Navier-Stokes,, Ann. Mat. Pura Appl. IV, 48 (1959), 173. [22] J. Serrin, "The Initial Value Problems for the Navier-Stokes Equations, in Nonlinear Problems,", edited by R. E. Langer, (1963). [23] H. Sohr, "The Navier-Stokes Equations, An Elementary Functional Analytic Approach,", Birkh$\ddot{\mboxa}$user Verlag, (2001). doi: 10.1007/978-3-0348-0551-3. [24] B. Q. Yuan and B. Zhang, Blow-up criterion of strong solutions to the Navier-Stokes equations in Besov spaces with negative indices,, J. Differential Equations, 242 (2007), 1. doi: 0.1016/j.jde.2007.07.009. [25] Y. Zhou and M. Pokorný, On the regularity of the solutions of the Navier-Stokes equations via one velocity component,, Nonlinearity, 23 (2010), 1097. doi: 10.1088/0951-7715/23/5/004.
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