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Elliptic equations involving linear and superlinear terms and critical Caffarelli-Kohn-Nirenberg exponent with sign-changing weight functions
Logarithmically improved criteria for Euler and Navier-Stokes equations
1. | Department and Institute of Mathematics, Fudan University, Shanghai 200433, China |
2. | Shanghai Key Laboratory for Contemporary Applied Mathematics, School of Mathematical Sciences, Fudan University, Shanghai 200433 |
References:
[1] |
J. T. Beale, T. Kato and A. Majda, Remarks on the breakdown of smooth solutions for the $3$-D Euler equations,, Comm. Math. Phys., 94 (1984), 61.
|
[2] |
J. M. Bony, Calcul symbolique et propagation des singularites pour les quations aux drivees partielles non lineaires,, Ann. Sci. Ecole Norm. Sup., 14 (1981), 209.
|
[3] |
C. H. Chan and A. Vasseur, Log improvement of the Prodi-Serrin criteria for Navier-Stokes equations,, available online at arXiv:0705.3659., ().
|
[4] |
J. Y. Chemin, "Perfect Incompressibe Fluids,", Oxford University Press, (1998).
|
[5] |
P. Constantin and C. Foias, "Navier-Stokes Equations,", Chicago University Press, (1988).
|
[6] |
P. Constantin and C. Fefferman, Direction of vorticity and the problem of global regularity for the Navier-Stokes equations,, Indiana Univ. Math. J., 42 (1993), 775.
|
[7] |
C. Fefferman, http://www.claymath.org/millennium/Navier-Stokes equations., preprint., (). Google Scholar |
[8] |
L. Iskauriaza, G. A. Seregin and V. Shverak, $L_{3,\infty}$-solutions of Navier-Stokes equations and backward uniqueness,, (Russian) Uspekhi Mat. Nauk, 58 (2003), 3.
|
[9] |
H. Kozono and Y. Taniuchi, Bilinear estimates in BMO and the Navier-Stokes equations,, Math. Z., 235 (2000), 173.
|
[10] |
O. A. Ladyzhenskaya, "Mathematical Questions of the Dynamics of a Viscous Incompressible Fluid,", Nauka, (1970).
|
[11] |
A. J. Majda and A. L. Bertozzi, "Vorticity and Incompressible Flow,", Cambridge Texts in Applied Mathematics, (2002).
|
[12] |
G. Prodi, Un teorema di unicità per le equazioni di Navier-Stokes,, Ann. Mat. Pura Appl., 48 (1959), 173.
|
[13] |
G. Seregin and V. Sverak, Navier-Stokes equations with lower bounds on the pressure,, Arch. Ration. Mech. Anal., 163 (2002), 65.
|
[14] |
J. Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations,, Arch. Rational Mech. Anal., 9 (1962), 187.
|
[15] |
J. Serrin, The initial value problem for the Navier-Stokes equations,, Nonlinear Problems, (1963), 69.
|
[16] |
M. Struwe, On partial regularity results for the Navier-Stokes equations,, Comm. Pure Appl. Math., 41 (1988), 437.
|
[17] |
T. Tao, Nonlinear dispersive equations. Local and global analysis,, CBMS Regional Conference Series in Mathematics, (2006).
|
[18] |
R. Temam, "Navier-Stokes Equations,", Second Edition, (2001).
|
[19] |
H. Triebel, "Theory of Function Spaces,", Birkauser Verlag, (1983).
|
[20] |
Y. Zhou and S. Gala, Logarithmically improved Serrin's criterion to the Navier-Stokes equations in multiplier spaces,, preprint, (2008). Google Scholar |
[21] |
Y. Zhou and Z. Lei, Logarithmically improved criteria for Euler and Navier-Stokes equations,, avaliable on http://arxiv.org/abs/0805.2784v1, (). Google Scholar |
show all references
References:
[1] |
J. T. Beale, T. Kato and A. Majda, Remarks on the breakdown of smooth solutions for the $3$-D Euler equations,, Comm. Math. Phys., 94 (1984), 61.
|
[2] |
J. M. Bony, Calcul symbolique et propagation des singularites pour les quations aux drivees partielles non lineaires,, Ann. Sci. Ecole Norm. Sup., 14 (1981), 209.
|
[3] |
C. H. Chan and A. Vasseur, Log improvement of the Prodi-Serrin criteria for Navier-Stokes equations,, available online at arXiv:0705.3659., ().
|
[4] |
J. Y. Chemin, "Perfect Incompressibe Fluids,", Oxford University Press, (1998).
|
[5] |
P. Constantin and C. Foias, "Navier-Stokes Equations,", Chicago University Press, (1988).
|
[6] |
P. Constantin and C. Fefferman, Direction of vorticity and the problem of global regularity for the Navier-Stokes equations,, Indiana Univ. Math. J., 42 (1993), 775.
|
[7] |
C. Fefferman, http://www.claymath.org/millennium/Navier-Stokes equations., preprint., (). Google Scholar |
[8] |
L. Iskauriaza, G. A. Seregin and V. Shverak, $L_{3,\infty}$-solutions of Navier-Stokes equations and backward uniqueness,, (Russian) Uspekhi Mat. Nauk, 58 (2003), 3.
|
[9] |
H. Kozono and Y. Taniuchi, Bilinear estimates in BMO and the Navier-Stokes equations,, Math. Z., 235 (2000), 173.
|
[10] |
O. A. Ladyzhenskaya, "Mathematical Questions of the Dynamics of a Viscous Incompressible Fluid,", Nauka, (1970).
|
[11] |
A. J. Majda and A. L. Bertozzi, "Vorticity and Incompressible Flow,", Cambridge Texts in Applied Mathematics, (2002).
|
[12] |
G. Prodi, Un teorema di unicità per le equazioni di Navier-Stokes,, Ann. Mat. Pura Appl., 48 (1959), 173.
|
[13] |
G. Seregin and V. Sverak, Navier-Stokes equations with lower bounds on the pressure,, Arch. Ration. Mech. Anal., 163 (2002), 65.
|
[14] |
J. Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations,, Arch. Rational Mech. Anal., 9 (1962), 187.
|
[15] |
J. Serrin, The initial value problem for the Navier-Stokes equations,, Nonlinear Problems, (1963), 69.
|
[16] |
M. Struwe, On partial regularity results for the Navier-Stokes equations,, Comm. Pure Appl. Math., 41 (1988), 437.
|
[17] |
T. Tao, Nonlinear dispersive equations. Local and global analysis,, CBMS Regional Conference Series in Mathematics, (2006).
|
[18] |
R. Temam, "Navier-Stokes Equations,", Second Edition, (2001).
|
[19] |
H. Triebel, "Theory of Function Spaces,", Birkauser Verlag, (1983).
|
[20] |
Y. Zhou and S. Gala, Logarithmically improved Serrin's criterion to the Navier-Stokes equations in multiplier spaces,, preprint, (2008). Google Scholar |
[21] |
Y. Zhou and Z. Lei, Logarithmically improved criteria for Euler and Navier-Stokes equations,, avaliable on http://arxiv.org/abs/0805.2784v1, (). Google Scholar |
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