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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 $\dot{F}^{-\alpha,r}_{\frac{3}{\alpha-1}}$

Chao Deng 1, and  Xiaohua Yao 2,

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.
Keywords: wellposedness, ill-posedness., Triebel-Lizorkin space, Generalized Navier-Stokes equations.
Mathematics Subject Classification: 76D03, 35Q3.
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 & 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.  Google Scholar

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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.  Google Scholar

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A. Cheskidov and M. Dai, Norm inflation for generalized Navier-Stokes equations,, , ().   Google Scholar

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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.  Google Scholar

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C. Deng and X. Yao, Ill-posedness of the incompressible Navier-Stokes equations in Triebel-Lizorkin spaces $\dotF^{-1,q>2}_{\infty}(\mathbbR^3)$,, , ().   Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math., 63 (1934), 193-248. doi: 10.1007/BF02547354.  Google Scholar

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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.  Google Scholar

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[19]

F. Planchon, "Solutions globales et comportement asymptotique pour les équations de Navier-Stokes," Thèse, Ecole Polytechnique, 1996.  Google Scholar

[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.  Google Scholar

[21]

H. Triebel, "Theory of Function Spaces. II," Monographs in Mathematics, 84, Birkhäuser Verlag, Basel, 1992. doi: 10.1007/978-3-0346-0419-2.  Google Scholar

[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.  Google Scholar

[23]

J. Xiao, Homothetic variant of fractional Sobolev space with application to Navier-Stokes system, Dynamic of PDE, 4 (2007), 227-245.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

M. Cannone, "Ondelettes, Paraproduits et Navier-Stokes," Dierot Editeur, Paris, 1995.  Google Scholar

[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.  Google Scholar

[5]

A. Cheskidov and M. Dai, Norm inflation for generalized Navier-Stokes equations,, , ().   Google Scholar

[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.  Google Scholar

[7]

C. Deng and X. Yao, Ill-posedness of the incompressible Navier-Stokes equations in Triebel-Lizorkin spaces $\dotF^{-1,q>2}_{\infty}(\mathbbR^3)$,, , ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math., 63 (1934), 193-248. doi: 10.1007/BF02547354.  Google Scholar

[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.  Google Scholar

[15]

J.-L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires," (French) Dunod; Gauthier-Villars, Paris, 1969.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

F. Planchon, "Solutions globales et comportement asymptotique pour les équations de Navier-Stokes," Thèse, Ecole Polytechnique, 1996.  Google Scholar

[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.  Google Scholar

[21]

H. Triebel, "Theory of Function Spaces. II," Monographs in Mathematics, 84, Birkhäuser Verlag, Basel, 1992. doi: 10.1007/978-3-0346-0419-2.  Google Scholar

[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.  Google Scholar

[23]

J. Xiao, Homothetic variant of fractional Sobolev space with application to Navier-Stokes system, Dynamic of PDE, 4 (2007), 227-245.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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