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

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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

Abstract
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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 and 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), 407-412.
[3]	L. C. Berselli, On a regularity criterion for the solutions to the 3D Navier-Stokes equations, Dierential Integral Equations, 15 (2002), 1129-1137.
[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), 919-932. 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), 470-481. 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, II. Springer, New York, 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), 9488-9491. doi: 10.1016/j.amc.2011.03.156.
[9]	E. Hopf, Über die anfang swetaufgabe für die hydrodynamischer grundgleichungan, Math. Nach., 4 (1951), 213-231.
[10]	H. Kozono and Y. Taniuchi, Bilinear estimates in BMO and the Navier-Stokes equations, Math. Z., 235 (2000), 173-194. 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), 251-278. 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), 55-68. 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), 065203. doi: 10.1063/1.2395919.
[14]	I. Kukavica and M. Ziane, One component regularity for the Navier-Stokes equations, Nonlinearity, 19 (2006), 453-469. doi: 10.1088/0951-7715/19/2/012.
[15]	O. A. Ladyzhenskaya, "The Boundary Value Problems of Mathematical Physics," Springer, Berlin, 1985.
[16]	J. Leray, Sur le mouvement d'um liquide visqieux emlissant l'space, Acta Math., 63 (1934), 193-248. 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 "11 Mathematical Fluid Mechanics: Recent Results and Open Problems"(J. Neustupa and P. Penel eds.), Advances in Mathematical Fluid Mechanics, pp. 239-267. Birkhäuser, Basel, 2001. 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), 341-353. 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), 483-493. 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), 1-8.
[21]	G. Prodi, Un teorema di unicità per el equazioni di Navier-Stokes, Ann. Mat. Pura Appl. IV, 48 (1959), 173-82.
[22]	J. Serrin, "The Initial Value Problems for the Navier-Stokes Equations, in Nonlinear Problems," edited by R. E. Langer, University of Wisconsin Press, Madison, WI, 1963.
[23]	H. Sohr, "The Navier-Stokes Equations, An Elementary Functional Analytic Approach," Birkhäuser Verlag, Basel, 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), 1-10. 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), 1097-1107. 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), 407-412.
[3]	L. C. Berselli, On a regularity criterion for the solutions to the 3D Navier-Stokes equations, Dierential Integral Equations, 15 (2002), 1129-1137.
[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), 919-932. 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), 470-481. 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, II. Springer, New York, 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), 9488-9491. doi: 10.1016/j.amc.2011.03.156.
[9]	E. Hopf, Über die anfang swetaufgabe für die hydrodynamischer grundgleichungan, Math. Nach., 4 (1951), 213-231.
[10]	H. Kozono and Y. Taniuchi, Bilinear estimates in BMO and the Navier-Stokes equations, Math. Z., 235 (2000), 173-194. 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), 251-278. 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), 55-68. 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), 065203. doi: 10.1063/1.2395919.
[14]	I. Kukavica and M. Ziane, One component regularity for the Navier-Stokes equations, Nonlinearity, 19 (2006), 453-469. doi: 10.1088/0951-7715/19/2/012.
[15]	O. A. Ladyzhenskaya, "The Boundary Value Problems of Mathematical Physics," Springer, Berlin, 1985.
[16]	J. Leray, Sur le mouvement d'um liquide visqieux emlissant l'space, Acta Math., 63 (1934), 193-248. 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 "11 Mathematical Fluid Mechanics: Recent Results and Open Problems"(J. Neustupa and P. Penel eds.), Advances in Mathematical Fluid Mechanics, pp. 239-267. Birkhäuser, Basel, 2001. 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), 341-353. 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), 483-493. 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), 1-8.
[21]	G. Prodi, Un teorema di unicità per el equazioni di Navier-Stokes, Ann. Mat. Pura Appl. IV, 48 (1959), 173-82.
[22]	J. Serrin, "The Initial Value Problems for the Navier-Stokes Equations, in Nonlinear Problems," edited by R. E. Langer, University of Wisconsin Press, Madison, WI, 1963.
[23]	H. Sohr, "The Navier-Stokes Equations, An Elementary Functional Analytic Approach," Birkhäuser Verlag, Basel, 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), 1-10. 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), 1097-1107. doi: 10.1088/0951-7715/23/5/004.

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