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Positive solutions of nonhomogeneous fractional Laplacian problem with critical exponent
Regularity criterion for 3D Navier-Stokes equations in Besov spaces
1. | Department of Mathematics, Zhejiang University, Hangzhou 310027 |
2. | Department of Mathematics, Zhejiang University, Hangzhou, 310027, China |
References:
[1] |
H. Bahouri, R. Danchin and J. Y. Chemin, "Fourier Analysis and Nonlinear Partial Differential Equations, A Series of Comprehensive Studies in Mathematics,'' Springer Heidelberg Dordrecht London New York. Springer-Verlag Berlin Heidelberg, 2011.
doi: 10.1007/978-3-642-16830-7. |
[2] |
H. Beirão da Veiga, A new regularity class for the Navier-Stokes equations in $\mathbb{R}^{N}$, Chinese Ann. Math. Ser. B, 16 (1995), 407-412. |
[3] |
L. C. Berselli, On a regularity criterion for the solutions to the 3D Navier-Stokes equations, Dierential Integral Equations, 15 (2002), 1129-1137. |
[4] |
C. S. Cao and E. S. Titi, Global regularity criterion for the 3D Navier-Stokes equations involving one entry of the velocity gradient tensor, Arch. Rational Mech. Anal., 202 (2011), 919-932.
doi: 10.1007/s00205-011-0439-6. |
[5] |
Z. F. Zhang and Q. L. Chen, Regularity criterion via two components of vorticity on weak solutions to the Navier-Stokes equations in $\mathbb{R}^3$, J. Differential Equations, 216 (2005), 470-481.
doi: 10.1016/j.jde.2005.06.001. |
[6] |
A. Cheskidov and R. Shvydkoy, On the regularity of weak solutions of the 3D Navier-Stokes equations in $B^{-1}_{\infty,\infty}$, arXiv:0708.3067v2 [math.AP]. |
[7] |
G. P. Galdi, "An Introduction to the Mathematical Theory of the Navier-Stokes Equations,'' Vol. I, II. Springer, New York, 1994.
doi: 10.1007/978-0-387-09620-9. |
[8] |
S. Gala, A remark on the blow-up criterion of strong solutions to the Navier-Stokes equations, Applied Mathematics and Computation, 217 (2011), 9488-9491.
doi: 10.1016/j.amc.2011.03.156. |
[9] |
E. Hopf, Über die anfang swetaufgabe für die hydrodynamischer grundgleichungan, Math. Nach., 4 (1951), 213-231. |
[10] |
H. Kozono and Y. Taniuchi, Bilinear estimates in BMO and the Navier-Stokes equations, Math. Z., 235 (2000), 173-194.
doi: 10.1007/s002090000130. |
[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 N. Yatsu, Extension criterion via two-components of vorticity on strong solution to the 3D Navier-Stokes equations, Math. Z., 246 (2003), 55-68.
doi: 10.1007/s00209-003-0576-1. |
[13] |
I. Kukavica and M. Ziane, Navier-Stokes equations with regularity in one direction, J. Math Phys., 48 (2007), 065203.
doi: 10.1063/1.2395919. |
[14] |
I. Kukavica and M. Ziane, One component regularity for the Navier-Stokes equations, Nonlinearity, 19 (2006), 453-469.
doi: 10.1088/0951-7715/19/2/012. |
[15] |
O. A. Ladyzhenskaya, "The Boundary Value Problems of Mathematical Physics," Springer, Berlin, 1985. |
[16] |
J. Leray, Sur le mouvement d'um liquide visqieux emlissant l'space, Acta Math., 63 (1934), 193-248.
doi: 10.1007/BF02547354. |
[17] |
J. Neustupa and P. Penel, Anisotropic and geometric criteria for interior regularity of weak solutions to the 3D Navier-Stokes equations, In "11 Mathematical Fluid Mechanics: Recent Results and Open Problems"(J. Neustupa and P. Penel eds.), Advances in Mathematical Fluid Mechanics, pp. 239-267. Birkhäuser, Basel, 2001.
doi: 10.1007/978-3-0348-8243-9_10. |
[18] |
P. Penel and M. Pokorný, On anisotropic regularity criteria for the Solutions to 3D Navier-Stokes equations, J. Math. Fluid Mech., 13 (2011), 341-353.
doi: 10.1007/s00021-010-0038-6. |
[19] |
P. Penel and M. Pokorný, Some new regularity criteria for the Navier-Stokes equations containing gradient of the velocity, Appl. Math., 49 (2004), 483-493.
doi: 10.1023/B:APOM.0000048124.64244.7e. |
[20] |
M. Pokorný, On the result of He concerning the smoothness of solutions to the Navier-Stokes equations, Electron. J. Differ. Equ., 11 (2003), 1-8. |
[21] |
G. Prodi, Un teorema di unicità per el equazioni di Navier-Stokes, Ann. Mat. Pura Appl. IV, 48 (1959), 173-82. |
[22] |
J. Serrin, "The Initial Value Problems for the Navier-Stokes Equations, in Nonlinear Problems," edited by R. E. Langer, University of Wisconsin Press, Madison, WI, 1963. |
[23] |
H. Sohr, "The Navier-Stokes Equations, An Elementary Functional Analytic Approach," Birkhäuser Verlag, Basel, 2001.
doi: 10.1007/978-3-0348-0551-3. |
[24] |
B. Q. Yuan and B. Zhang, Blow-up criterion of strong solutions to the Navier-Stokes equations in Besov spaces with negative indices, J. Differential Equations, 242 (2007), 1-10.
doi: 0.1016/j.jde.2007.07.009. |
[25] |
Y. Zhou and M. Pokorný, On the regularity of the solutions of the Navier-Stokes equations via one velocity component, Nonlinearity, 23 (2010), 1097-1107.
doi: 10.1088/0951-7715/23/5/004. |
show all references
References:
[1] |
H. Bahouri, R. Danchin and J. Y. Chemin, "Fourier Analysis and Nonlinear Partial Differential Equations, A Series of Comprehensive Studies in Mathematics,'' Springer Heidelberg Dordrecht London New York. Springer-Verlag Berlin Heidelberg, 2011.
doi: 10.1007/978-3-642-16830-7. |
[2] |
H. Beirão da Veiga, A new regularity class for the Navier-Stokes equations in $\mathbb{R}^{N}$, Chinese Ann. Math. Ser. B, 16 (1995), 407-412. |
[3] |
L. C. Berselli, On a regularity criterion for the solutions to the 3D Navier-Stokes equations, Dierential Integral Equations, 15 (2002), 1129-1137. |
[4] |
C. S. Cao and E. S. Titi, Global regularity criterion for the 3D Navier-Stokes equations involving one entry of the velocity gradient tensor, Arch. Rational Mech. Anal., 202 (2011), 919-932.
doi: 10.1007/s00205-011-0439-6. |
[5] |
Z. F. Zhang and Q. L. Chen, Regularity criterion via two components of vorticity on weak solutions to the Navier-Stokes equations in $\mathbb{R}^3$, J. Differential Equations, 216 (2005), 470-481.
doi: 10.1016/j.jde.2005.06.001. |
[6] |
A. Cheskidov and R. Shvydkoy, On the regularity of weak solutions of the 3D Navier-Stokes equations in $B^{-1}_{\infty,\infty}$, arXiv:0708.3067v2 [math.AP]. |
[7] |
G. P. Galdi, "An Introduction to the Mathematical Theory of the Navier-Stokes Equations,'' Vol. I, II. Springer, New York, 1994.
doi: 10.1007/978-0-387-09620-9. |
[8] |
S. Gala, A remark on the blow-up criterion of strong solutions to the Navier-Stokes equations, Applied Mathematics and Computation, 217 (2011), 9488-9491.
doi: 10.1016/j.amc.2011.03.156. |
[9] |
E. Hopf, Über die anfang swetaufgabe für die hydrodynamischer grundgleichungan, Math. Nach., 4 (1951), 213-231. |
[10] |
H. Kozono and Y. Taniuchi, Bilinear estimates in BMO and the Navier-Stokes equations, Math. Z., 235 (2000), 173-194.
doi: 10.1007/s002090000130. |
[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 N. Yatsu, Extension criterion via two-components of vorticity on strong solution to the 3D Navier-Stokes equations, Math. Z., 246 (2003), 55-68.
doi: 10.1007/s00209-003-0576-1. |
[13] |
I. Kukavica and M. Ziane, Navier-Stokes equations with regularity in one direction, J. Math Phys., 48 (2007), 065203.
doi: 10.1063/1.2395919. |
[14] |
I. Kukavica and M. Ziane, One component regularity for the Navier-Stokes equations, Nonlinearity, 19 (2006), 453-469.
doi: 10.1088/0951-7715/19/2/012. |
[15] |
O. A. Ladyzhenskaya, "The Boundary Value Problems of Mathematical Physics," Springer, Berlin, 1985. |
[16] |
J. Leray, Sur le mouvement d'um liquide visqieux emlissant l'space, Acta Math., 63 (1934), 193-248.
doi: 10.1007/BF02547354. |
[17] |
J. Neustupa and P. Penel, Anisotropic and geometric criteria for interior regularity of weak solutions to the 3D Navier-Stokes equations, In "11 Mathematical Fluid Mechanics: Recent Results and Open Problems"(J. Neustupa and P. Penel eds.), Advances in Mathematical Fluid Mechanics, pp. 239-267. Birkhäuser, Basel, 2001.
doi: 10.1007/978-3-0348-8243-9_10. |
[18] |
P. Penel and M. Pokorný, On anisotropic regularity criteria for the Solutions to 3D Navier-Stokes equations, J. Math. Fluid Mech., 13 (2011), 341-353.
doi: 10.1007/s00021-010-0038-6. |
[19] |
P. Penel and M. Pokorný, Some new regularity criteria for the Navier-Stokes equations containing gradient of the velocity, Appl. Math., 49 (2004), 483-493.
doi: 10.1023/B:APOM.0000048124.64244.7e. |
[20] |
M. Pokorný, On the result of He concerning the smoothness of solutions to the Navier-Stokes equations, Electron. J. Differ. Equ., 11 (2003), 1-8. |
[21] |
G. Prodi, Un teorema di unicità per el equazioni di Navier-Stokes, Ann. Mat. Pura Appl. IV, 48 (1959), 173-82. |
[22] |
J. Serrin, "The Initial Value Problems for the Navier-Stokes Equations, in Nonlinear Problems," edited by R. E. Langer, University of Wisconsin Press, Madison, WI, 1963. |
[23] |
H. Sohr, "The Navier-Stokes Equations, An Elementary Functional Analytic Approach," Birkhäuser Verlag, Basel, 2001.
doi: 10.1007/978-3-0348-0551-3. |
[24] |
B. Q. Yuan and B. Zhang, Blow-up criterion of strong solutions to the Navier-Stokes equations in Besov spaces with negative indices, J. Differential Equations, 242 (2007), 1-10.
doi: 0.1016/j.jde.2007.07.009. |
[25] |
Y. Zhou and M. Pokorný, On the regularity of the solutions of the Navier-Stokes equations via one velocity component, Nonlinearity, 23 (2010), 1097-1107.
doi: 10.1088/0951-7715/23/5/004. |
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