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Mean oscillation and boundedness of Toeplitz Type operators associated to pseudo-differential operators
Logarithmic and improved regularity criteria for the 3D nematic liquid crystals models, Boussinesq system, and MHD equations in a bounded domain
1. | Dipartimento di Matematica, Via F. Buonarroti 2, 56126 Pisa, Italy |
2. | Department of Applied Mathematics, Nanjing Forestry University, Nanjing, 210037 |
References:
[1] |
R. A. Adams and J. F. Fournier, Sobolev Spaces, 2nd ed., Elsevier/Academic Press, Amsterdam, 2003. |
[2] |
H. Beir ao da Veiga, Existence and asymptotic behavior for strong solutions of the Navier-Stokes equations in the whole space, Indiana Univ. Math. J., 36 (1987), 149-166.
doi: 10.1512/iumj.1987.36.36008. |
[3] |
H. Beir ao da Veiga and F. Crispo, Sharp inviscid limit results under Navier type boundary condition. An $L^p$ theory, J. Math. Fluid Mech., 12 (2010), 397-411.
doi: 10.1007/s00021-009-0295-4. |
[4] |
L. C. Berselli, On a regularity criterion for the solutions to the 3D Navier-Stokes equations, Differential Integral Equations, 15 (2002), 1129-1137. |
[5] |
L. C. Berselli, Some criteria concerning the vorticity and the problem of global regularity for the 3D Navier-Stokes equations, Ann. Univ. Ferrara Sez. VII Sci. Mat., 55 (2009), 209-224.
doi: 10.1007/s11565-009-0076-2. |
[6] |
L. C. Berselli and R. Manfrin, On a theorem by Sohr for the Navier-Stokes equations, J. Evol. Equ., 4 (2004), 193-211.
doi: 10.1007/s00028-003-1135-2. |
[7] |
L. C. Berselli and S. Spirito, On the Boussinesq system: Regularity criteria and singular limits, Methods Appl. Anal., 18 (2011), 391-416.
doi: 10.4310/MAA.2011.v18.n4.a3. |
[8] |
S. Chandrasekhar, Liquid Crystals, Cambridge University Press, 2nd edition, 1992. |
[9] |
R. Danchin, Density-dependent incompressible fluids in bounded domains, J. Math. Fluid Mech., 8 (2006), 333-381.
doi: 10.1007/s00021-004-0147-1. |
[10] |
J. L. Ericksen, Hydrostatic theory of liquid crystals, Arch. Ration. Mech. Anal., 9 (1962), 371-378. |
[11] |
J. Fan, Y. Fukumoto and Y. Zhou, Logarithmically improved regularity criteria for the generalized Navier-Stokes and related equations, Kinet. Relat. Models, 6 (2013), 545-556.
doi: 10.3934/krm.2013.6.545. |
[12] |
J. Fan, H. Gao and B. Guo, Regularity criteria for the Navier-Stokes-Landau-Lifshitz system, J. Math. Anal. Appl., 363 (2010), 29-37.
doi: 10.1016/j.jmaa.2009.07.047. |
[13] |
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. |
[14] |
J. Fan and T. Ozawa, Regularity criteria for the 3D density-dependent Boussinesq equations, Nonlinearity, 22 (2009), 553-568.
doi: 10.1088/0951-7715/22/3/003. |
[15] |
F. Guillén-González, M. A. Rojas-Medar and M. A. Rodríguez-Bellido, Sufficient conditions for regularity and uniqueness of a 3D nematic liquid crystal model, Math. Nachr., 282 (2009), 846-867.
doi: 10.1002/mana.200610776. |
[16] |
H. Kim, A blow-up criterion for the nonhomogeneous incompressible Navier-Stokes equations, SIAM J. Math. Anal., 37 (2006), 1417-1434.
doi: 10.1137/S0036141004442197. |
[17] |
F. M. Leslie, Some constitutive equations for liquid crystals, Arch. Ration. Mech. Anal., 28 (1968), 265-283.
doi: 10.1007/BF00251810. |
[18] |
X. Li and D. Wang, Global strong solution to the density-dependent incompressible flow of liquid crystals,, \emph{Trans Amer. Math. Soc.}, ().
|
[19] |
F.-H. Lin and C. Liu, Nonparabolic dissipative systems modeling the flow of liquid crystals, Comm. Pure Appl. Math, 48 (1995), 501-537.
doi: 10.1002/cpa.3160480503. |
[20] |
A. Lunardi, Interpolation Theory, Lecture Notes. Scuola Normale Superiore di Pisa (New Series), 2nd ed., Edizioni della Normale, Pisa, 2009. |
[21] |
T. Ogawa, Sharp Sobolev inequality of logarithmic type and the limiting regularity condition to the harmonic heat flow, SIAM J. Math. Anal., 34 (2003), 1318-1330.
doi: 10.1137/S0036141001395868. |
[22] |
T. Ogawa and Y. Taniuchi, A note on blow-up criterion to the 3D Euler equations in a bounded domain, J. Diff. Equations, 190 (2003), 39-63.
doi: 10.1016/S0022-0396(03)00013-5. |
[23] |
H. Sohr, A regularity class for the Navier-Stokes equations in Lorentz spaces, J. Evol. Equ., 1 (2001), 441-467.
doi: 10.1007/PL00001382. |
[24] |
H. Triebel, Theory of Function Spaces, Birkhäuser, Basel, 1983.
doi: 10.1007/978-3-0346-0416-1. |
[25] |
Y. Zhou and S. Gala, Logarithmically improved regularity criteria for the Navier-Stokes equations in multiplier spaces, J. Math. Anal. Appl., 356 (2009), 498-501.
doi: 10.1016/j.jmaa.2009.03.038. |
[26] |
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. |
show all references
References:
[1] |
R. A. Adams and J. F. Fournier, Sobolev Spaces, 2nd ed., Elsevier/Academic Press, Amsterdam, 2003. |
[2] |
H. Beir ao da Veiga, Existence and asymptotic behavior for strong solutions of the Navier-Stokes equations in the whole space, Indiana Univ. Math. J., 36 (1987), 149-166.
doi: 10.1512/iumj.1987.36.36008. |
[3] |
H. Beir ao da Veiga and F. Crispo, Sharp inviscid limit results under Navier type boundary condition. An $L^p$ theory, J. Math. Fluid Mech., 12 (2010), 397-411.
doi: 10.1007/s00021-009-0295-4. |
[4] |
L. C. Berselli, On a regularity criterion for the solutions to the 3D Navier-Stokes equations, Differential Integral Equations, 15 (2002), 1129-1137. |
[5] |
L. C. Berselli, Some criteria concerning the vorticity and the problem of global regularity for the 3D Navier-Stokes equations, Ann. Univ. Ferrara Sez. VII Sci. Mat., 55 (2009), 209-224.
doi: 10.1007/s11565-009-0076-2. |
[6] |
L. C. Berselli and R. Manfrin, On a theorem by Sohr for the Navier-Stokes equations, J. Evol. Equ., 4 (2004), 193-211.
doi: 10.1007/s00028-003-1135-2. |
[7] |
L. C. Berselli and S. Spirito, On the Boussinesq system: Regularity criteria and singular limits, Methods Appl. Anal., 18 (2011), 391-416.
doi: 10.4310/MAA.2011.v18.n4.a3. |
[8] |
S. Chandrasekhar, Liquid Crystals, Cambridge University Press, 2nd edition, 1992. |
[9] |
R. Danchin, Density-dependent incompressible fluids in bounded domains, J. Math. Fluid Mech., 8 (2006), 333-381.
doi: 10.1007/s00021-004-0147-1. |
[10] |
J. L. Ericksen, Hydrostatic theory of liquid crystals, Arch. Ration. Mech. Anal., 9 (1962), 371-378. |
[11] |
J. Fan, Y. Fukumoto and Y. Zhou, Logarithmically improved regularity criteria for the generalized Navier-Stokes and related equations, Kinet. Relat. Models, 6 (2013), 545-556.
doi: 10.3934/krm.2013.6.545. |
[12] |
J. Fan, H. Gao and B. Guo, Regularity criteria for the Navier-Stokes-Landau-Lifshitz system, J. Math. Anal. Appl., 363 (2010), 29-37.
doi: 10.1016/j.jmaa.2009.07.047. |
[13] |
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. |
[14] |
J. Fan and T. Ozawa, Regularity criteria for the 3D density-dependent Boussinesq equations, Nonlinearity, 22 (2009), 553-568.
doi: 10.1088/0951-7715/22/3/003. |
[15] |
F. Guillén-González, M. A. Rojas-Medar and M. A. Rodríguez-Bellido, Sufficient conditions for regularity and uniqueness of a 3D nematic liquid crystal model, Math. Nachr., 282 (2009), 846-867.
doi: 10.1002/mana.200610776. |
[16] |
H. Kim, A blow-up criterion for the nonhomogeneous incompressible Navier-Stokes equations, SIAM J. Math. Anal., 37 (2006), 1417-1434.
doi: 10.1137/S0036141004442197. |
[17] |
F. M. Leslie, Some constitutive equations for liquid crystals, Arch. Ration. Mech. Anal., 28 (1968), 265-283.
doi: 10.1007/BF00251810. |
[18] |
X. Li and D. Wang, Global strong solution to the density-dependent incompressible flow of liquid crystals,, \emph{Trans Amer. Math. Soc.}, ().
|
[19] |
F.-H. Lin and C. Liu, Nonparabolic dissipative systems modeling the flow of liquid crystals, Comm. Pure Appl. Math, 48 (1995), 501-537.
doi: 10.1002/cpa.3160480503. |
[20] |
A. Lunardi, Interpolation Theory, Lecture Notes. Scuola Normale Superiore di Pisa (New Series), 2nd ed., Edizioni della Normale, Pisa, 2009. |
[21] |
T. Ogawa, Sharp Sobolev inequality of logarithmic type and the limiting regularity condition to the harmonic heat flow, SIAM J. Math. Anal., 34 (2003), 1318-1330.
doi: 10.1137/S0036141001395868. |
[22] |
T. Ogawa and Y. Taniuchi, A note on blow-up criterion to the 3D Euler equations in a bounded domain, J. Diff. Equations, 190 (2003), 39-63.
doi: 10.1016/S0022-0396(03)00013-5. |
[23] |
H. Sohr, A regularity class for the Navier-Stokes equations in Lorentz spaces, J. Evol. Equ., 1 (2001), 441-467.
doi: 10.1007/PL00001382. |
[24] |
H. Triebel, Theory of Function Spaces, Birkhäuser, Basel, 1983.
doi: 10.1007/978-3-0346-0416-1. |
[25] |
Y. Zhou and S. Gala, Logarithmically improved regularity criteria for the Navier-Stokes equations in multiplier spaces, J. Math. Anal. Appl., 356 (2009), 498-501.
doi: 10.1016/j.jmaa.2009.03.038. |
[26] |
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. |
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