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

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

Daoyuan Fang 1, and  Chenyin Qian 2,

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 $\partial_3u_3$.
Keywords: 3D Navier-Stokes equations, Leray-Hopf weak solution, Regularity criterion..
Mathematics Subject Classification: 35Q30, 76D0.
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.  Google Scholar

[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.   Google Scholar

[3]

L. C. Berselli, On a regularity criterion for the solutions to the 3D Navier-Stokes equations,, Dierential Integral Equations, 15 (2002), 1129.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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]}., ().   Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[9]

E. Hopf, Über die anfang swetaufgabe für die hydrodynamischer grundgleichungan,, Math. Nach., 4 (1951), 213.   Google Scholar

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H. Kozono and Y. Taniuchi, Bilinear estimates in BMO and the Navier-Stokes equations,, Math. Z., 235 (2000), 173.  doi: 10.1007/s002090000130.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

I. Kukavica and M. Ziane, Navier-Stokes equations with regularity in one direction,, J. Math Phys., 48 (2007).  doi: 10.1063/1.2395919.  Google Scholar

[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.  Google Scholar

[15]

O. A. Ladyzhenskaya, "The Boundary Value Problems of Mathematical Physics,", Springer, (1985).   Google Scholar

[16]

J. Leray, Sur le mouvement d'um liquide visqieux emlissant l'space,, Acta Math., 63 (1934), 193.  doi: 10.1007/BF02547354.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[21]

G. Prodi, Un teorema di unicità per el equazioni di Navier-Stokes,, Ann. Mat. Pura Appl. IV, 48 (1959), 173.   Google Scholar

[22]

J. Serrin, "The Initial Value Problems for the Navier-Stokes Equations, in Nonlinear Problems,", edited by R. E. Langer, (1963).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[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.   Google Scholar

[3]

L. C. Berselli, On a regularity criterion for the solutions to the 3D Navier-Stokes equations,, Dierential Integral Equations, 15 (2002), 1129.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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]}., ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

E. Hopf, Über die anfang swetaufgabe für die hydrodynamischer grundgleichungan,, Math. Nach., 4 (1951), 213.   Google Scholar

[10]

H. Kozono and Y. Taniuchi, Bilinear estimates in BMO and the Navier-Stokes equations,, Math. Z., 235 (2000), 173.  doi: 10.1007/s002090000130.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

I. Kukavica and M. Ziane, Navier-Stokes equations with regularity in one direction,, J. Math Phys., 48 (2007).  doi: 10.1063/1.2395919.  Google Scholar

[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.  Google Scholar

[15]

O. A. Ladyzhenskaya, "The Boundary Value Problems of Mathematical Physics,", Springer, (1985).   Google Scholar

[16]

J. Leray, Sur le mouvement d'um liquide visqieux emlissant l'space,, Acta Math., 63 (1934), 193.  doi: 10.1007/BF02547354.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[21]

G. Prodi, Un teorema di unicità per el equazioni di Navier-Stokes,, Ann. Mat. Pura Appl. IV, 48 (1959), 173.   Google Scholar

[22]

J. Serrin, "The Initial Value Problems for the Navier-Stokes Equations, in Nonlinear Problems,", edited by R. E. Langer, (1963).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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