# American Institute of Mathematical Sciences

February  2014, 34(2): 437-459. doi: 10.3934/dcds.2014.34.437

## 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

Received  February 2013 Revised  March 2013 Published  August 2013

In this paper, we study the Cauchy problem of the 3-dimensional (3D) generalized Navier-Stokes equations (gNS) in the Triebel-Lizorkin spaces $\dot{F}^{-\alpha,r}_{q_\alpha}$ with $(\alpha,r)\in(1,\frac{5}{4})\times[1,\infty]$ and $q_\alpha=\frac{3}{\alpha-1}$. Our work establishes a dichotomy of well-posedness and ill-posedness depending on $r$. Specifically, by combining the new endpoint bilinear estimates in $L^{\!q_\alpha}_x\!L^2_T$ and $L^\infty_T\dot{F}^{-\alpha,1}_{q_\alpha}$ and characterization of the Triebel-Lizorkin spaces via fractional semigroup, we prove well-posedness of the gNS in $\dot{F}^{-\alpha,r}_{q_\alpha}$ for $r\in[1,2]$. Meanwhile, for any $r\in(2,\infty]$, we show that the solution to the gNS can develop norm inflation in the sense that arbitrarily small initial data in $\dot{F}^{-\alpha,r}_{q_\alpha}$ can produce arbitrarily large solution after arbitrarily short time.
Citation: Chao Deng, Xiaohua Yao. Well-posedness and ill-posedness for the 3D generalized Navier-Stokes equations in $\dot{F}^{-\alpha,r}_{\frac{3}{\alpha-1}}$. Discrete and Continuous Dynamical Systems, 2014, 34 (2) : 437-459. doi: 10.3934/dcds.2014.34.437
##### 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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##### 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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