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

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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

Abstract
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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 and 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.
[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.
[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.
[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.
[5]	J. Fan and T. Ozawa, Regularity criteria for the generalized Navier-Stokes and related equations, Differential Integral Equations, 21 (2008), 681-691.
[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.
[7]	J. Jiménez, Hyperviscous vortices, J. Fluid Mech., 279 (1994), 169-176.
[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.
[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.
[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.
[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 Y. Taniuchi, Bilinear estimates in BMO and the Navier-Stokes equations, Math. Z., 235 (2000), 173-194. doi: 10.1007/s002090000130.
[13]	J.-L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites Nonlinéaires," Dunod, Paris, 1969.
[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.
[15]	H. Triebel, "Theory of Function Spaces," Monographs in Mathematics, 78, Birkhäuser Verlag, Basel, 1983. doi: 10.1007/978-3-0346-0416-1.
[16]	J. Wu, The generalized incompressible Navier-Stokes equations in Besov spaces, Dyn. Partial Differ. Equ., 1 (2004), 381-400.
[17]	J. Wu, Regularity criteria for the generalized MHD equations, Comm. Partial Differential Equations, 33 (2008), 285-306. doi: 10.1080/03605300701382530.
[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.
[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.
[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.
[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.
[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.
[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.
[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.

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.
[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.
[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.
[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.
[5]	J. Fan and T. Ozawa, Regularity criteria for the generalized Navier-Stokes and related equations, Differential Integral Equations, 21 (2008), 681-691.
[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.
[7]	J. Jiménez, Hyperviscous vortices, J. Fluid Mech., 279 (1994), 169-176.
[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.
[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.
[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.
[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 Y. Taniuchi, Bilinear estimates in BMO and the Navier-Stokes equations, Math. Z., 235 (2000), 173-194. doi: 10.1007/s002090000130.
[13]	J.-L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites Nonlinéaires," Dunod, Paris, 1969.
[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.
[15]	H. Triebel, "Theory of Function Spaces," Monographs in Mathematics, 78, Birkhäuser Verlag, Basel, 1983. doi: 10.1007/978-3-0346-0416-1.
[16]	J. Wu, The generalized incompressible Navier-Stokes equations in Besov spaces, Dyn. Partial Differ. Equ., 1 (2004), 381-400.
[17]	J. Wu, Regularity criteria for the generalized MHD equations, Comm. Partial Differential Equations, 33 (2008), 285-306. doi: 10.1080/03605300701382530.
[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.
[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.
[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.
[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.
[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.
[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.
[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.

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