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