American Institute of Mathematical Sciences

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

 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.
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.
 [1] 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 [2] Igor Kukavica. On regularity for the Navier-Stokes equations in Morrey spaces. Discrete and Continuous Dynamical Systems, 2010, 26 (4) : 1319-1328. doi: 10.3934/dcds.2010.26.1319 [3] Xuanji Jia, Zaihong Jiang. An anisotropic regularity criterion for the 3D Navier-Stokes equations. Communications on Pure and Applied Analysis, 2013, 12 (3) : 1299-1306. doi: 10.3934/cpaa.2013.12.1299 [4] Hongjie Dong, Kunrui Wang. Interior and boundary regularity for the Navier-Stokes equations in the critical Lebesgue spaces. Discrete and Continuous Dynamical Systems, 2020, 40 (9) : 5289-5323. doi: 10.3934/dcds.2020228 [5] Hi Jun Choe, Bataa Lkhagvasuren, Minsuk Yang. Wellposedness of the Keller-Segel Navier-Stokes equations in the critical Besov spaces. Communications on Pure and Applied Analysis, 2015, 14 (6) : 2453-2464. doi: 10.3934/cpaa.2015.14.2453 [6] Zujin Zhang. A Serrin-type regularity criterion for the Navier-Stokes equations via one velocity component. Communications on Pure and Applied Analysis, 2013, 12 (1) : 117-124. doi: 10.3934/cpaa.2013.12.117 [7] Minghua Yang, Zunwei Fu, Jinyi Sun. Global solutions to Chemotaxis-Navier-Stokes equations in critical Besov spaces. Discrete and Continuous Dynamical Systems - B, 2018, 23 (8) : 3427-3460. doi: 10.3934/dcdsb.2018284 [8] Vittorino Pata. On the regularity of solutions to the Navier-Stokes equations. Communications on Pure and Applied Analysis, 2012, 11 (2) : 747-761. doi: 10.3934/cpaa.2012.11.747 [9] Igor Kukavica. On partial regularity for the Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 2008, 21 (3) : 717-728. doi: 10.3934/dcds.2008.21.717 [10] Hugo Beirão da Veiga. Navier-Stokes equations: Some questions related to the direction of the vorticity. Discrete and Continuous Dynamical Systems - S, 2019, 12 (2) : 203-213. doi: 10.3934/dcdss.2019014 [11] Chongsheng Cao. Sufficient conditions for the regularity to the 3D Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 2010, 26 (4) : 1141-1151. doi: 10.3934/dcds.2010.26.1141 [12] Zijin Li, Xinghong Pan. Some Remarks on regularity criteria of Axially symmetric Navier-Stokes equations. Communications on Pure and Applied Analysis, 2019, 18 (3) : 1333-1350. doi: 10.3934/cpaa.2019064 [13] Keyan Wang. On global regularity of incompressible Navier-Stokes equations in $\mathbf R^3$. Communications on Pure and Applied Analysis, 2009, 8 (3) : 1067-1072. doi: 10.3934/cpaa.2009.8.1067 [14] Hui Chen, Daoyuan Fang, Ting Zhang. Regularity of 3D axisymmetric Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 2017, 37 (4) : 1923-1939. doi: 10.3934/dcds.2017081 [15] Yukang Chen, Changhua Wei. Partial regularity of solutions to the fractional Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 2016, 36 (10) : 5309-5322. doi: 10.3934/dcds.2016033 [16] Houyu Jia, Xiaofeng Liu. Local existence and blowup criterion of the Lagrangian averaged Euler equations in Besov spaces. Communications on Pure and Applied Analysis, 2008, 7 (4) : 845-852. doi: 10.3934/cpaa.2008.7.845 [17] Pavel I. Plotnikov, Jan Sokolowski. Compressible Navier-Stokes equations. Conference Publications, 2009, 2009 (Special) : 602-611. doi: 10.3934/proc.2009.2009.602 [18] Jan W. Cholewa, Tomasz Dlotko. Fractional Navier-Stokes equations. Discrete and Continuous Dynamical Systems - B, 2018, 23 (8) : 2967-2988. doi: 10.3934/dcdsb.2017149 [19] Alessio Falocchi, Filippo Gazzola. Regularity for the 3D evolution Navier-Stokes equations under Navier boundary conditions in some Lipschitz domains. Discrete and Continuous Dynamical Systems, 2022, 42 (3) : 1185-1200. doi: 10.3934/dcds.2021151 [20] Minghua Yang, Jinyi Sun. Gevrey regularity and existence of Navier-Stokes-Nernst-Planck-Poisson system in critical Besov spaces. Communications on Pure and Applied Analysis, 2017, 16 (5) : 1617-1639. doi: 10.3934/cpaa.2017078

2021 Impact Factor: 1.398