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On the existence and asymptotic stability of solutions for unsteady mixing-layer models
Well-posedness and ill-posedness for the 3D generalized Navier-Stokes equations in $\dot{F}^{-\alpha,r}_{\frac{3}{\alpha-1}}$
1. | School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, Jiangsu 221116, China |
2. | School of Mathematics and Statistics, Central China Normal University, Wuhan, Hubei 430079, China |
References:
[1] |
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. |
[2] |
C. Calderón, Existence of weak solutions for the Navier-Stokes equations with initial data in $L^p$, Trans. Amer. Math. Soc., 318 (1990), 179-200.
doi: 10.2307/2001234. |
[3] |
M. Cannone, "Ondelettes, Paraproduits et Navier-Stokes," Dierot Editeur, Paris, 1995. |
[4] |
M. Cannone, Harmonic analysis tools for solving the incompressible Navier-Stokes equations, in "Handbook of Mathematical Fluid Dynamics. Vol. III" (eds. S. Friedlander and D. Serre), North Holland, Amsterdam, (2004), 161-244. |
[5] |
A. Cheskidov and M. Dai, Norm inflation for generalized Navier-Stokes equations, arXiv:1212.3801v2. |
[6] |
A. Cheskidov and R. Shvydkoy, Ill-posedness for subcritical hyperdissipative Navier-Stokes equations in the largest critical spaces, J. Math. Phys., 53 (2012), 115620.
doi: 10.1063/1.4765332. |
[7] |
C. Deng and X. Yao, Ill-posedness of the incompressible Navier-Stokes equations in Triebel-Lizorkin spaces $\dotF^{-1,q>2}_{\infty}(\mathbbR^3)$, arXiv:1302.7084. |
[8] |
Q. Deng, Y. Ding and X. Yao, Characterizations of Hardy spaces associated to higher order elliptic operators, J. Funct. Anal., 263 (2012), 604-674.
doi: 10.1016/j.jfa.2012.05.001. |
[9] |
H. Fujita and T. Kato, On the Navier-Stokes initial value problem. I, Arch. Rat. Mech. Anal., 16 (1964), 269-315.
doi: 10.1007/BF00276188. |
[10] |
T. Kato, Strong $L^p$-solutions of the Navier-Stokes equation in $\mathbbR^m$, with applications to weak solutions, Math. Z., 187 (1984), 471-480.
doi: 10.1007/BF01174182. |
[11] |
H. Koch and D. Tataru, Well-posedness for the Navier-Stokes equations, Adv. Math., 157 (2001), 22-35.
doi: 10.1006/aima.2000.1937. |
[12] |
P. G. Lemarié-Rieusset, "Recent Developments in the Navier-Stokes Problem," Chapman & Hall/CRC Research Notes in Mathematics, 431, Chapman & Hall/CRC, Boca Raton, FL, 2002.
doi: 10.1201/9781420035674. |
[13] |
J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math., 63 (1934), 193-248.
doi: 10.1007/BF02547354. |
[14] |
P. Li and Z. C. Zhai, Well-posedness and regularity of generalized Navier-Stokes equations in some critical $Q$-spaces, J. Funct. Anal., 259 (2010), 2457-2519.
doi: 10.1016/j.jfa.2010.07.013. |
[15] |
J.-L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires," (French) Dunod; Gauthier-Villars, Paris, 1969. |
[16] |
R. May, Rôle de l'espace de Besov $B^{-1,\infty}_{\infty}$ dans le contrôle de l'explosion éventuelle en temps fini des solutions régulières équations de Navier-Stokes, C. R. Acad. Sci. Paris., 336 (2003), 731-734.
doi: 10.1016/S1631-073X(03)00155-9. |
[17] |
C. Miao, B. Yuan and B. Zhang, Well-posedness of the Cauchy problem for the fractional power dissipative equations, Nonlinear Anal., 68 (2008), 461-484.
doi: 10.1016/j.na.2006.11.011. |
[18] |
S. Montgomery-Smith, Finite time blow up for a Navier-Stokes like equation, Proc. Amer. Math. Soc., 129 (2001), 3025-3029.
doi: 10.1090/S0002-9939-01-06062-2. |
[19] |
F. Planchon, "Solutions globales et comportement asymptotique pour les équations de Navier-Stokes," Thèse, Ecole Polytechnique, 1996. |
[20] |
E. M. Stein, "Harmonic Analysis: real-Variable Methods, Orthogonality, and Oscillatory Integrals," Princeton Mathematical Series, 43, Monographs in Harmonic Analysis, III, Princeton University Press, Princeton, NJ, 1993. |
[21] |
H. Triebel, "Theory of Function Spaces. II," Monographs in Mathematics, 84, Birkhäuser Verlag, Basel, 1992.
doi: 10.1007/978-3-0346-0419-2. |
[22] |
J. H. Wu, Lower bounds for an integral involving fractional Laplacians and the generalized Navier-Stokes equations in Besov spaces, Comm. Math. Phys., 263 (2006), 803-831.
doi: 10.1007/s00220-005-1483-6. |
[23] |
J. Xiao, Homothetic variant of fractional Sobolev space with application to Navier-Stokes system, Dynamic of PDE, 4 (2007), 227-245. |
[24] |
X. Yu and Z. Zhai, Well-posedness for fractional Navier-Stokes equations in the largest critical spaces $\dotB^{1-2\beta}_{\infty,\infty}(\mathbbR^n),$ Math. Meth. Appl. Sci., 35 (2012), 676-683. |
[25] |
T. Yoneda, Ill-posedness of the 3D Navier-Stokes equations in a generalized Besov space near $BMO^{-1}$, J. Funct. Anal., 258 (2010), 3376-3387.
doi: 10.1016/j.jfa.2010.02.005. |
show all references
References:
[1] |
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. |
[2] |
C. Calderón, Existence of weak solutions for the Navier-Stokes equations with initial data in $L^p$, Trans. Amer. Math. Soc., 318 (1990), 179-200.
doi: 10.2307/2001234. |
[3] |
M. Cannone, "Ondelettes, Paraproduits et Navier-Stokes," Dierot Editeur, Paris, 1995. |
[4] |
M. Cannone, Harmonic analysis tools for solving the incompressible Navier-Stokes equations, in "Handbook of Mathematical Fluid Dynamics. Vol. III" (eds. S. Friedlander and D. Serre), North Holland, Amsterdam, (2004), 161-244. |
[5] |
A. Cheskidov and M. Dai, Norm inflation for generalized Navier-Stokes equations, arXiv:1212.3801v2. |
[6] |
A. Cheskidov and R. Shvydkoy, Ill-posedness for subcritical hyperdissipative Navier-Stokes equations in the largest critical spaces, J. Math. Phys., 53 (2012), 115620.
doi: 10.1063/1.4765332. |
[7] |
C. Deng and X. Yao, Ill-posedness of the incompressible Navier-Stokes equations in Triebel-Lizorkin spaces $\dotF^{-1,q>2}_{\infty}(\mathbbR^3)$, arXiv:1302.7084. |
[8] |
Q. Deng, Y. Ding and X. Yao, Characterizations of Hardy spaces associated to higher order elliptic operators, J. Funct. Anal., 263 (2012), 604-674.
doi: 10.1016/j.jfa.2012.05.001. |
[9] |
H. Fujita and T. Kato, On the Navier-Stokes initial value problem. I, Arch. Rat. Mech. Anal., 16 (1964), 269-315.
doi: 10.1007/BF00276188. |
[10] |
T. Kato, Strong $L^p$-solutions of the Navier-Stokes equation in $\mathbbR^m$, with applications to weak solutions, Math. Z., 187 (1984), 471-480.
doi: 10.1007/BF01174182. |
[11] |
H. Koch and D. Tataru, Well-posedness for the Navier-Stokes equations, Adv. Math., 157 (2001), 22-35.
doi: 10.1006/aima.2000.1937. |
[12] |
P. G. Lemarié-Rieusset, "Recent Developments in the Navier-Stokes Problem," Chapman & Hall/CRC Research Notes in Mathematics, 431, Chapman & Hall/CRC, Boca Raton, FL, 2002.
doi: 10.1201/9781420035674. |
[13] |
J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math., 63 (1934), 193-248.
doi: 10.1007/BF02547354. |
[14] |
P. Li and Z. C. Zhai, Well-posedness and regularity of generalized Navier-Stokes equations in some critical $Q$-spaces, J. Funct. Anal., 259 (2010), 2457-2519.
doi: 10.1016/j.jfa.2010.07.013. |
[15] |
J.-L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires," (French) Dunod; Gauthier-Villars, Paris, 1969. |
[16] |
R. May, Rôle de l'espace de Besov $B^{-1,\infty}_{\infty}$ dans le contrôle de l'explosion éventuelle en temps fini des solutions régulières équations de Navier-Stokes, C. R. Acad. Sci. Paris., 336 (2003), 731-734.
doi: 10.1016/S1631-073X(03)00155-9. |
[17] |
C. Miao, B. Yuan and B. Zhang, Well-posedness of the Cauchy problem for the fractional power dissipative equations, Nonlinear Anal., 68 (2008), 461-484.
doi: 10.1016/j.na.2006.11.011. |
[18] |
S. Montgomery-Smith, Finite time blow up for a Navier-Stokes like equation, Proc. Amer. Math. Soc., 129 (2001), 3025-3029.
doi: 10.1090/S0002-9939-01-06062-2. |
[19] |
F. Planchon, "Solutions globales et comportement asymptotique pour les équations de Navier-Stokes," Thèse, Ecole Polytechnique, 1996. |
[20] |
E. M. Stein, "Harmonic Analysis: real-Variable Methods, Orthogonality, and Oscillatory Integrals," Princeton Mathematical Series, 43, Monographs in Harmonic Analysis, III, Princeton University Press, Princeton, NJ, 1993. |
[21] |
H. Triebel, "Theory of Function Spaces. II," Monographs in Mathematics, 84, Birkhäuser Verlag, Basel, 1992.
doi: 10.1007/978-3-0346-0419-2. |
[22] |
J. H. Wu, Lower bounds for an integral involving fractional Laplacians and the generalized Navier-Stokes equations in Besov spaces, Comm. Math. Phys., 263 (2006), 803-831.
doi: 10.1007/s00220-005-1483-6. |
[23] |
J. Xiao, Homothetic variant of fractional Sobolev space with application to Navier-Stokes system, Dynamic of PDE, 4 (2007), 227-245. |
[24] |
X. Yu and Z. Zhai, Well-posedness for fractional Navier-Stokes equations in the largest critical spaces $\dotB^{1-2\beta}_{\infty,\infty}(\mathbbR^n),$ Math. Meth. Appl. Sci., 35 (2012), 676-683. |
[25] |
T. Yoneda, Ill-posedness of the 3D Navier-Stokes equations in a generalized Besov space near $BMO^{-1}$, J. Funct. Anal., 258 (2010), 3376-3387.
doi: 10.1016/j.jfa.2010.02.005. |
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