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Well-posedness results for the Navier-Stokes equations in the rotational framework
1. | Fachbereich Mathematik, Angewandte Analysis, Technische Universität Darmstadt, Schlossgartenstr. 7, 64289 Darmstadt |
2. | LATP UMR 6632, CMI, Technopôle de Château-Gombert, 39 rue Frédéric Joliot-Curie, 13453 Marseille Cedex 13, France |
References:
[1] |
A. Babin, A. Mahalov and B. Nicolaenko, Regularity and integrability for the 3D Euler and Navier-Stokes equations for uniformly rotating fluids, Asympt. Anal., 15 (1997), 103-150. |
[2] |
A. Babin, A. Mahalov and B. Nicolaenko, 3D Navier-Stokes and Euler equations with initial data characterized by uniformly large vorticity, Indiana Univ. Math. J., 50 (2001), 1-35.
doi: 10.1512/iumj.2001.50.2155. |
[3] |
I. Bejenaru and T. Tao, Sharp well-posedness and ill-posedness results for a quadratic non-linear Schrödinger equation, J. Funct. Anal., 223 (2006), 228-259.
doi: 10.1016/j.jfa.2005.08.004. |
[4] |
J. Bourgain and N. Pavlović, Ill-posedness of the Navier-Stokes equations in a critical space in 3D, J. Funct. Anal., 255 (2008), 2233-2247.
doi: 10.1016/j.jfa.2008.07.008. |
[5] |
M. Cannone, Harmonic analysis tools for solving the incompressible Navier-Stokes equations, Handbook of Mathematical Fluid Dynamics, (eds. S. Friedlander and D. Serre), Elsevier, 3 (2003). |
[6] |
C. Cao and E. Titi, Global wellposedness of the three dimensional viscous primitive equations of large scale ocean and atmosphere dynamics, Annals of Math., 166 (2007), 245-267.
doi: 10.4007/annals.2007.166.245. |
[7] |
J.-Y. Chemin, B. Desjardins, I. Gallagher and E. Grenier, "Mathematical Geophysics," Oxford University Press, 2006. |
[8] |
J. A. Goldstein, "Semigroups of Operators and Applications," Oxford University Press, 1985. |
[9] |
Y. Giga, K. Inui and S. Matsui, On the Cauchy problem for the Navier-Stokes equations with nondecaying initial data, Quaderni di Math., 4 (1999), 28-68. |
[10] |
Y. Giga, K. Inui, A. Mahalov and S. Matsui, Navier-Stokes equations in a rotating frame in $R^3$ with initial data nondecreasing at infinity, Hokkaido Math. J., 35 (2006), 321-364. |
[11] |
Y. Giga, K. Inui, A. Mahalov and J. Saal, Uniform global solvability of the Navier-Stokes equations for nondecaying data, Indiana Univ. Math. J., 57 (2008), 2775-2792. 321-364.
doi: 10.1512/iumj.2008.57.3795. |
[12] |
Y. Giga, K. Inui, A. Mahalov, S. Matsui and J. Saal, Rotating NS-equations in $\mathbbR^3_+$ with initial data nondecreasing at infinity: The Ekman boundary layer problem, Arch. Rat. Mech. Anal., (2007). |
[13] |
M. Hieber and S. Monniaux, Global solutions of the Navier-Stokes-Coriolis system in domains, preprint, 2012. |
[14] |
M. Hieber and O. Sawada, The Navier-Stokes equations in $\mathbb{R}^{N}$ with linearly growing initial data, Arch. Rational Mech. Anal., 175 (2005), 269-285.
doi: 10.1007/s00205-004-0347-0. |
[15] |
M. Hieber and Y. Shibata, The Fujita-Kato approach to the Navier-Stokes equations in the rotational framework, Math. Z., 265 (2010), 481-491.
doi: 10.1007/s00209-009-0525-8. |
[16] |
T. Iwabuchi and R. Takada, Global well-posedness and ill-posedness for the Navier-Stokes equations with the Coriolis force in function spaces of Besov type, preprint, 2011.
doi: 10.1016/j.jmaa.2011.02.010. |
[17] |
T. Iwabuchi and R. Takada, Dispersive effects of the Coriolis force and the local well-posedness for the Navier-Stokes equations in the rotational framework, preprint, 2011. |
[18] |
T. Kato, Strong $L^p$-solutions of Navier-Stokes equations in $\mathbb{R}^{N}$ with applications to weak solutions, Math. Z., 187 (1984), 471-480.
doi: 10.1007/BF01174182. |
[19] |
H. Koch and D. Tataru, Wellposedness for the Navier-Stokes equations, Advances Math., 157 (2001), 22-35.
doi: 10.1006/aima.2000.1937. |
[20] |
P. Konieczny and T. Yoneda, On the dispersive effect of the Coriolis force for stationary Navier-Stokes equations, J. Diff. Equ., 250 (2011), 3859-3873.
doi: 10.1016/j.jde.2011.01.003. |
[21] |
A. Majda, "Introduction to PDEs and Waves for the Atmosphere and Ocean," Courant Lecture Notes in Math., 2003. |
[22] |
S. Monniaux, Navier-Stokes equations in arbitrary domains: The Fujita-Kato scheme, Math. Res. Lett., 13 (2006), 455-461. |
[23] |
T. Yoneda, Long-time solvability of the Navier-Stokes equations in a rotating frame with spatially almost periodic large data, Arch. Rational Mech. Anal., 200 (2011), 225-237.
doi: 10.1007/s00205-010-0360-4. |
show all references
References:
[1] |
A. Babin, A. Mahalov and B. Nicolaenko, Regularity and integrability for the 3D Euler and Navier-Stokes equations for uniformly rotating fluids, Asympt. Anal., 15 (1997), 103-150. |
[2] |
A. Babin, A. Mahalov and B. Nicolaenko, 3D Navier-Stokes and Euler equations with initial data characterized by uniformly large vorticity, Indiana Univ. Math. J., 50 (2001), 1-35.
doi: 10.1512/iumj.2001.50.2155. |
[3] |
I. Bejenaru and T. Tao, Sharp well-posedness and ill-posedness results for a quadratic non-linear Schrödinger equation, J. Funct. Anal., 223 (2006), 228-259.
doi: 10.1016/j.jfa.2005.08.004. |
[4] |
J. Bourgain and N. Pavlović, Ill-posedness of the Navier-Stokes equations in a critical space in 3D, J. Funct. Anal., 255 (2008), 2233-2247.
doi: 10.1016/j.jfa.2008.07.008. |
[5] |
M. Cannone, Harmonic analysis tools for solving the incompressible Navier-Stokes equations, Handbook of Mathematical Fluid Dynamics, (eds. S. Friedlander and D. Serre), Elsevier, 3 (2003). |
[6] |
C. Cao and E. Titi, Global wellposedness of the three dimensional viscous primitive equations of large scale ocean and atmosphere dynamics, Annals of Math., 166 (2007), 245-267.
doi: 10.4007/annals.2007.166.245. |
[7] |
J.-Y. Chemin, B. Desjardins, I. Gallagher and E. Grenier, "Mathematical Geophysics," Oxford University Press, 2006. |
[8] |
J. A. Goldstein, "Semigroups of Operators and Applications," Oxford University Press, 1985. |
[9] |
Y. Giga, K. Inui and S. Matsui, On the Cauchy problem for the Navier-Stokes equations with nondecaying initial data, Quaderni di Math., 4 (1999), 28-68. |
[10] |
Y. Giga, K. Inui, A. Mahalov and S. Matsui, Navier-Stokes equations in a rotating frame in $R^3$ with initial data nondecreasing at infinity, Hokkaido Math. J., 35 (2006), 321-364. |
[11] |
Y. Giga, K. Inui, A. Mahalov and J. Saal, Uniform global solvability of the Navier-Stokes equations for nondecaying data, Indiana Univ. Math. J., 57 (2008), 2775-2792. 321-364.
doi: 10.1512/iumj.2008.57.3795. |
[12] |
Y. Giga, K. Inui, A. Mahalov, S. Matsui and J. Saal, Rotating NS-equations in $\mathbbR^3_+$ with initial data nondecreasing at infinity: The Ekman boundary layer problem, Arch. Rat. Mech. Anal., (2007). |
[13] |
M. Hieber and S. Monniaux, Global solutions of the Navier-Stokes-Coriolis system in domains, preprint, 2012. |
[14] |
M. Hieber and O. Sawada, The Navier-Stokes equations in $\mathbb{R}^{N}$ with linearly growing initial data, Arch. Rational Mech. Anal., 175 (2005), 269-285.
doi: 10.1007/s00205-004-0347-0. |
[15] |
M. Hieber and Y. Shibata, The Fujita-Kato approach to the Navier-Stokes equations in the rotational framework, Math. Z., 265 (2010), 481-491.
doi: 10.1007/s00209-009-0525-8. |
[16] |
T. Iwabuchi and R. Takada, Global well-posedness and ill-posedness for the Navier-Stokes equations with the Coriolis force in function spaces of Besov type, preprint, 2011.
doi: 10.1016/j.jmaa.2011.02.010. |
[17] |
T. Iwabuchi and R. Takada, Dispersive effects of the Coriolis force and the local well-posedness for the Navier-Stokes equations in the rotational framework, preprint, 2011. |
[18] |
T. Kato, Strong $L^p$-solutions of Navier-Stokes equations in $\mathbb{R}^{N}$ with applications to weak solutions, Math. Z., 187 (1984), 471-480.
doi: 10.1007/BF01174182. |
[19] |
H. Koch and D. Tataru, Wellposedness for the Navier-Stokes equations, Advances Math., 157 (2001), 22-35.
doi: 10.1006/aima.2000.1937. |
[20] |
P. Konieczny and T. Yoneda, On the dispersive effect of the Coriolis force for stationary Navier-Stokes equations, J. Diff. Equ., 250 (2011), 3859-3873.
doi: 10.1016/j.jde.2011.01.003. |
[21] |
A. Majda, "Introduction to PDEs and Waves for the Atmosphere and Ocean," Courant Lecture Notes in Math., 2003. |
[22] |
S. Monniaux, Navier-Stokes equations in arbitrary domains: The Fujita-Kato scheme, Math. Res. Lett., 13 (2006), 455-461. |
[23] |
T. Yoneda, Long-time solvability of the Navier-Stokes equations in a rotating frame with spatially almost periodic large data, Arch. Rational Mech. Anal., 200 (2011), 225-237.
doi: 10.1007/s00205-010-0360-4. |
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