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September  2013, 6(3): 545-556. doi: 10.3934/krm.2013.6.545

Logarithmically improved regularity criteria for the generalized Navier-Stokes and related equations

Jishan Fan 1, , Yasuhide Fukumoto 2, and  Yong Zhou 3,

1. 

Department of Applied Mathematics, Nanjing Forestry University, Nanjing, 210037
2. 

Faculty of Mathematics and and Mathematical Research Center, for Industrial Technology, Kyushu University, 744 Motooka, Nishi-ku, Fukuoka 819-0395, Japan
3. 

Department of Mathematics, Zhejiang Normal University, Jinhua 321004, Zhejiang

Received  November 2012 Revised  February 2013 Published  May 2013

In this paper, logarithmically improved regularity criteria for the generalized Navier-Stokes equations are established in terms of the velocity, vorticity and pressure, respectively. Here $BMO$, the Triebel-Lizorkin and Besov spaces are used, which extend usual Sobolev spaces much. Similar results for the quasi-geostrophic flows and the generalized MHD equations are also listed.
Keywords: related equations., Generalized Navier-Stokes equations, Besov spaces, regularity criterion.
Mathematics Subject Classification: Primary: 35Q35, 35B65; Secondary: 76D0.
Citation: Jishan Fan, Yasuhide Fukumoto, Yong Zhou. Logarithmically improved regularity criteria for the generalized Navier-Stokes and related equations. Kinetic & Related Models, 2013, 6 (3) : 545-556. doi: 10.3934/krm.2013.6.545
References:
[1]

M. Cannone and G. Karch, Incompressible Navier-Stokes equations in abstract Banach spaces, in "Tosio Kato's Method and Principle for Evolution Equations in Mathematical Physics" (Sapporo, 2001), Sūrikaisekikenkyūsho Kōkyūroku, No. 1234, (2001), 27-41.  Google Scholar

[2]

D. Chae and J. Lee, On the global well-posedness and stability of the Navier-Stokes and the related equations, in "Contributions to Current Challenges in Mathematical Fluid Mechanics," Adv. Math. Fluid Mech., Birkhäuser, Basel, (2004), 31-51.  Google Scholar

[3]

A. Córdoba and D. Córdoba, A maximum principle applied to quasi-geostrophic equations, Comm. Math. Phys., 249 (2004), 511-528. doi: 10.1007/s00220-004-1055-1.  Google Scholar

[4]

J. Fan, S. Jiang, G. Nakamura and Y. Zhou, Logarithmically improved regularity criteria for the Navier-Stokes and MHD equations, J. Math. Fluid Mech., 13 (2011), 557-571. doi: 10.1007/s00021-010-0039-5.  Google Scholar

[5]

J. Fan and T. Ozawa, Regularity criteria for the generalized Navier-Stokes and related equations, Differential Integral Equations, 21 (2008), 681-691.  Google Scholar

[6]

J. Fan and T. Ozawa, On the regularity criteria for the generalized Navier-Stokes equations and Lagrangian averaged Euler equations, Differential Integral Equations, 21 (2008), 443-457.  Google Scholar

[7]

J. Jiménez, Hyperviscous vortices, J. Fluid Mech., 279 (1994), 169-176. Google Scholar

[8]

T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 891-907. doi: 10.1002/cpa.3160410704.  Google Scholar

[9]

D. L. Koch and J. F. Brady, Anomalous diffusion in heterogeneous porous media, Phys. Fluids, 31 (1988), 965-973. doi: 10.1063/1.866716.  Google Scholar

[10]

H. Kozono and Y. Shimada, Bilinear estimates in homogeneous Triebel-Lizorkin spaces and the Navier-Stokes equations, Math. Nachr., 276 (2004), 63-74. doi: 10.1002/mana.200310213.  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), 251-278. doi: 10.1007/s002090100332.  Google Scholar

[12]

H. Kozono and Y. Taniuchi, Bilinear estimates in BMO and the Navier-Stokes equations, Math. Z., 235 (2000), 173-194. doi: 10.1007/s002090000130.  Google Scholar

[13]

J.-L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites Nonlinéaires," Dunod, Paris, 1969. Google Scholar

[14]

S. Tourville, Existence and uniqueness of solutions for the Navier-Stokes equations with hyperdissipation, J. Math. Anal. Appl., 281 (2003), 62-75. doi: 10.1016/S0022-247X(02)00453-5.  Google Scholar

[15]

H. Triebel, "Theory of Function Spaces," Monographs in Mathematics, 78, Birkhäuser Verlag, Basel, 1983. doi: 10.1007/978-3-0346-0416-1.  Google Scholar

[16]

J. Wu, The generalized incompressible Navier-Stokes equations in Besov spaces, Dyn. Partial Differ. Equ., 1 (2004), 381-400.  Google Scholar

[17]

J. Wu, Regularity criteria for the generalized MHD equations, Comm. Partial Differential Equations, 33 (2008), 285-306. doi: 10.1080/03605300701382530.  Google Scholar

[18]

Y. Zhang, D. A. Benson and D. M. Reeves, Time and space nonlocalities underlying fractional-derivative models: Distinction and literature review of field applications, Adv. Water Resour., 32 (2009), 561-581. doi: 10.1016/j.advwatres.2009.01.008.  Google Scholar

[19]

Y. Zhou, Regularity criteria for the generalized viscous MHD equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 491-505. doi: 10.1016/j.anihpc.2006.03.014.  Google Scholar

[20]

Y. Zhou, Decay rate of higher order derivatives for solutions to the 2-D dissipative quasi-geostrophic flows, Discrete Contin. Dyn. Syst., 14 (2006), 525-532. doi: 10.3934/dcds.2006.14.525.  Google Scholar

[21]

Y. Zhou, Asymptotic behaviour of the solutions to the 2D dissipative quasi-geostrophic flows, Nonlinearity, 21 (2008), 2061-2071. doi: 10.1088/0951-7715/21/9/008.  Google Scholar

[22]

Y. Zhou and J. Fan, Logarithmically improved regularity criteria for the 3D viscous MHD equations, Forum Math., 24 (2012), 691-708. doi: 10.1515/form.2011.079.  Google Scholar

[23]

Y. Zhou and S. Gala, A new regularity criterion for weak solutions to the viscous MHD equations in terms of the vorticity field, Nonlinear Anal., 72 (2010), 3643-3648. doi: 10.1016/j.na.2009.12.045.  Google Scholar

[24]

Y. Zhou and S. Gala, Regularity criteria for the solutions to the 3D MHD equations in the multiplier space, Z. Angew. Math. Phys., 61 (2010), 193-199. doi: 10.1007/s00033-009-0023-1.  Google Scholar

show all references

References:
[1]

M. Cannone and G. Karch, Incompressible Navier-Stokes equations in abstract Banach spaces, in "Tosio Kato's Method and Principle for Evolution Equations in Mathematical Physics" (Sapporo, 2001), Sūrikaisekikenkyūsho Kōkyūroku, No. 1234, (2001), 27-41.  Google Scholar

[2]

D. Chae and J. Lee, On the global well-posedness and stability of the Navier-Stokes and the related equations, in "Contributions to Current Challenges in Mathematical Fluid Mechanics," Adv. Math. Fluid Mech., Birkhäuser, Basel, (2004), 31-51.  Google Scholar

[3]

A. Córdoba and D. Córdoba, A maximum principle applied to quasi-geostrophic equations, Comm. Math. Phys., 249 (2004), 511-528. doi: 10.1007/s00220-004-1055-1.  Google Scholar

[4]

J. Fan, S. Jiang, G. Nakamura and Y. Zhou, Logarithmically improved regularity criteria for the Navier-Stokes and MHD equations, J. Math. Fluid Mech., 13 (2011), 557-571. doi: 10.1007/s00021-010-0039-5.  Google Scholar

[5]

J. Fan and T. Ozawa, Regularity criteria for the generalized Navier-Stokes and related equations, Differential Integral Equations, 21 (2008), 681-691.  Google Scholar

[6]

J. Fan and T. Ozawa, On the regularity criteria for the generalized Navier-Stokes equations and Lagrangian averaged Euler equations, Differential Integral Equations, 21 (2008), 443-457.  Google Scholar

[7]

J. Jiménez, Hyperviscous vortices, J. Fluid Mech., 279 (1994), 169-176. Google Scholar

[8]

T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 891-907. doi: 10.1002/cpa.3160410704.  Google Scholar

[9]

D. L. Koch and J. F. Brady, Anomalous diffusion in heterogeneous porous media, Phys. Fluids, 31 (1988), 965-973. doi: 10.1063/1.866716.  Google Scholar

[10]

H. Kozono and Y. Shimada, Bilinear estimates in homogeneous Triebel-Lizorkin spaces and the Navier-Stokes equations, Math. Nachr., 276 (2004), 63-74. doi: 10.1002/mana.200310213.  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), 251-278. doi: 10.1007/s002090100332.  Google Scholar

[12]

H. Kozono and Y. Taniuchi, Bilinear estimates in BMO and the Navier-Stokes equations, Math. Z., 235 (2000), 173-194. doi: 10.1007/s002090000130.  Google Scholar

[13]

J.-L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites Nonlinéaires," Dunod, Paris, 1969. Google Scholar

[14]

S. Tourville, Existence and uniqueness of solutions for the Navier-Stokes equations with hyperdissipation, J. Math. Anal. Appl., 281 (2003), 62-75. doi: 10.1016/S0022-247X(02)00453-5.  Google Scholar

[15]

H. Triebel, "Theory of Function Spaces," Monographs in Mathematics, 78, Birkhäuser Verlag, Basel, 1983. doi: 10.1007/978-3-0346-0416-1.  Google Scholar

[16]

J. Wu, The generalized incompressible Navier-Stokes equations in Besov spaces, Dyn. Partial Differ. Equ., 1 (2004), 381-400.  Google Scholar

[17]

J. Wu, Regularity criteria for the generalized MHD equations, Comm. Partial Differential Equations, 33 (2008), 285-306. doi: 10.1080/03605300701382530.  Google Scholar

[18]

Y. Zhang, D. A. Benson and D. M. Reeves, Time and space nonlocalities underlying fractional-derivative models: Distinction and literature review of field applications, Adv. Water Resour., 32 (2009), 561-581. doi: 10.1016/j.advwatres.2009.01.008.  Google Scholar

[19]

Y. Zhou, Regularity criteria for the generalized viscous MHD equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 491-505. doi: 10.1016/j.anihpc.2006.03.014.  Google Scholar

[20]

Y. Zhou, Decay rate of higher order derivatives for solutions to the 2-D dissipative quasi-geostrophic flows, Discrete Contin. Dyn. Syst., 14 (2006), 525-532. doi: 10.3934/dcds.2006.14.525.  Google Scholar

[21]

Y. Zhou, Asymptotic behaviour of the solutions to the 2D dissipative quasi-geostrophic flows, Nonlinearity, 21 (2008), 2061-2071. doi: 10.1088/0951-7715/21/9/008.  Google Scholar

[22]

Y. Zhou and J. Fan, Logarithmically improved regularity criteria for the 3D viscous MHD equations, Forum Math., 24 (2012), 691-708. doi: 10.1515/form.2011.079.  Google Scholar

[23]

Y. Zhou and S. Gala, A new regularity criterion for weak solutions to the viscous MHD equations in terms of the vorticity field, Nonlinear Anal., 72 (2010), 3643-3648. doi: 10.1016/j.na.2009.12.045.  Google Scholar

[24]

Y. Zhou and S. Gala, Regularity criteria for the solutions to the 3D MHD equations in the multiplier space, Z. Angew. Math. Phys., 61 (2010), 193-199. doi: 10.1007/s00033-009-0023-1.  Google Scholar

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