## Abstract and Applied Analysis

### Global Well-Posedness and Long Time Decay of Fractional Navier-Stokes Equations in Fourier-Besov Spaces

#### Abstract

We study the Cauchy problem of the fractional Navier-Stokes equations in critical Fourier-Besov spaces $F{\stackrel{˙}{B}}_{p,q}^{1-2\beta +3/{p}^{\prime }}$. Some properties of Fourier-Besov spaces have been discussed, and we prove a general global well-posedness result which covers some recent works in classical Navier-Stokes equations. Particularly, our result is suitable for the critical case $\beta =1/2$. Moreover, we prove the long time decay of the global solutions in Fourier-Besov spaces.

#### Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2014), Article ID 463639, 11 pages.

Dates
First available in Project Euclid: 6 October 2014

https://projecteuclid.org/euclid.aaa/1412607115

Digital Object Identifier
doi:10.1155/2014/463639

Mathematical Reviews number (MathSciNet)
MR3246334

Zentralblatt MATH identifier
07022430

#### Citation

Xiao, Weiliang; Chen, Jiecheng; Fan, Dashan; Zhou, Xuhuan. Global Well-Posedness and Long Time Decay of Fractional Navier-Stokes Equations in Fourier-Besov Spaces. Abstr. Appl. Anal. 2014, Special Issue (2014), Article ID 463639, 11 pages. doi:10.1155/2014/463639. https://projecteuclid.org/euclid.aaa/1412607115

#### References

• J. Leray, “Sur le mouvement d'un liquide visqueux emplissant l'espace,” Acta Mathematica, vol. 63, no. 1, pp. 193–248, 1934 (French).
• H. Fujita and T. Kato, “On the Navier-Stokes initial value problem. I,” Archive for Rational Mechanics and Analysis, vol. 16, pp. 269–315, 1964.
• T. Kato, “Strong ${L}^{p}$-solutions of the Navier-Stokes equation in ${R}^{m}$, with applications to weak solutions,” Mathematische Zeitschrift, vol. 187, no. 4, pp. 471–480, 1984.
• M. Cannone, Wavelets, Paraproducts and Navier-Stokes: With a Preface by Yves Meyer, Diderot, Paris, France, 1995, (French).
• H. Koch and D. Tataru, “Well-posedness for the Navier-Stokes equations,” Advances in Mathematics, vol. 157, no. 1, pp. 22–35, 2001.
• J. Bourgain and N. Pavlović, “Ill-posedness of the Navier-Stokes equations in a critical space in 3D,” Journal of Functional Analysis, vol. 255, no. 9, pp. 2233–2247, 2008.
• A. Cheskidov and R. Shvydkoy, “Ill-posedness of the basic equations of fluid dynamics in Besov spaces,” Proceedings of the American Mathematical Society, vol. 138, no. 3, pp. 1059–1067, 2010.
• J.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod, Paris, France, 1969.
• J. Wu, “Generalized MHD equations,” Journal of Differential Equations, vol. 195, no. 2, pp. 284–312, 2003.
• J. Wu, “The generalized incompressible Navier-Stokes equations in Besov spaces,” Dynamics of Partial Differential Equations, vol. 1, no. 4, pp. 381–400, 2004.
• J. Wu, “Lower bounds for an integral involving fractional Laplacians and the generalized Navier-Stokes equations in Besov spaces,” Communications in Mathematical Physics, vol. 263, no. 3, pp. 803–831, 2006.
• J. Xiao, “Homothetic variant of fractional Sobolev space with application to Navier-Stokes system,” Dynamics of Partial Differential Equations, vol. 4, no. 3, pp. 227–245, 2007.
• P. Li and Z. Zhai, “Well-posedness and regularity of generalized Navier-Stokes equations in some critical $Q$-spaces,” Journal of Functional Analysis, vol. 259, no. 10, pp. 2457–2519, 2010.
• P. Li and Z. Zhai, “Generalized Navier-Stokes equations with initial data in local $Q$-type spaces,” Journal of Mathematical Analysis and Applications, vol. 369, no. 2, pp. 595–609, 2010.
• Z. Zhai, “Well-posedness for fractional Navier-Stokes equations in critical spaces close to ${\dot{B}}_{\infty ,\infty }^{-2\beta -1}({R}^{n})$,” Dynamics of Partial Differential Equations, vol. 7, pp. 25–44, 2010.
• X. Yu and Z. Zhai, “Well-posedness for fractional Navier-Stokes equations in the largest critical spaces ${\dot{B}}_{\infty ,\infty }^{-(\text{2}\beta -1)} ({R}^{n})$,” Mathe-matical Methods in the Applied Sciences, vol. 35, no. 6, pp. 676–683, 2012.
• A. Cheskidov and R. Shvydkoy, “Ill-posedness for subcritical hyperdissipative Navier-Stokes equations in the largest critical spaces,” Journal of Mathematical Physics, vol. 53, Article ID 115620, 2012.
• C. Deng and X. Yao, “Well-posedness and ill-posedness for the 3D generalized Navier-Stokes equations in ${\dot{F}}_{(3/\alpha -1)}^{-\alpha ,r}$,” Discrete and Continuous Dynamical Systems. Series A, vol. 34, no. 2, pp. 437–459, 2014.
• M. Cannone and G. Karch, “Smooth or singular solutions to the Navier-Stokes system?” Journal of Differential Equations, vol. 197, no. 2, pp. 247–274, 2004.
• A. Biswas and D. Swanson, “Gevrey regularity of solutions to the 3-D Navier-Stokes equations with weighted $l\setminus sbp$ initial data,” Indiana University Mathematics Journal, vol. 56, no. 3, pp. 1157–1188, 2007.
• P. Konieczny and T. Yoneda, “On dispersive effect of the Coriolis force for the stationary Navier-Stokes equations,” Journal of Differential Equations, vol. 250, no. 10, pp. 3859–3873, 2011.
• D. Fang, B. Han, and M. Hieber, “Global existence results for the Navier-Stokes equations in the rotational frameworkčommentComment on ref. [23?]: Please update the information of this reference, if possible.,” http://arxiv.org/abs/1205.1561.
• Z. Lei and F. Lin, “Global mild solutions of Navier-Stokes equations,” Communications on Pure and Applied Mathematics, vol. 64, no. 9, pp. 1297–1304, 2011.
• M. Cannone and G. Wu, “Global well-posedness for Navier-Stokes equations in critical Fourier-Herz spaces,” Nonlinear Analysis. Theory, Methods & Applications, vol. 75, no. 9, pp. 3754–3760, 2012.
• Z. Zhang and Z. Yin, “Global well-posedness for the generalized Navier-Stokes sysytemčommentComment on ref. [25?]: Please update the information of this reference, if possible.,” http://arxiv.org/abs/1306.3735.
• J. Benameur, “Long time decay to the Lei-Lin solution of 3D Navier-Stokes equationsčommentComment on ref. [26?]: Please update the information of this reference, if possible.,” http://arxiv.org/abs/1312.2136.
• I. Gallagher, D. Iftimie, and F. Planchon, “Non-explosion en temps grand et stabilité de solutions globales des équations de Navier–Stokes,” Comptes Rendus Mathematique, vol. 334, no. 4, pp. 289–292, 2002. \endinput