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August  2013, 33(8): 3517-3541. doi: 10.3934/dcds.2013.33.3517

On the well-posedness of inhomogeneous hyperdissipative Navier-Stokes equations

Daoyuan Fang 1, and  Ruizhao Zi 2,

1. 

Department of Mathematics, Zhejiang University, Hangzhou 310027
2. 

Department of Mathematics, Zhejiang University, Hangzhou, 310027, China

Received  September 2012 Revised  November 2012 Published  January 2013

This paper is devoted to the study of the inhomogeneous hyperdissipative Navier-Stokes equations on the whole space $\mathbb{R}^N,N\geq3$. Compared with the classical inhomogeneous Navier-Stokes, the dissipative term $- Δ u$ here is replaced by $D^2u$, where $D$ is a Fourier multiplier whose symbol is $m(\xi)=|\xi|^{\frac{N+2}{4}}$. For arbitrary small positive constants $ε$ and $δ$, global well-posedness is showed for the data $(\rho_0, u_0)$ such that $(ρ_{0} - 1, u_0)∈ H^{\frac{N}{2}+ε} × H^{δ}$ with $\inf_{x\in \mathbb{R}^N}\rho_0>0$. To our best knowledge, this is the first result on the inhomogeneous hyperdissipative Navier-Stokes equations, and it can also be viewed as the high-dimensional generalization of the 2D result for classical inhomogeneous Navier-Stokes equations given by Danchin [Local and global well-posedness results for flows of inhomogeneous viscous fluids. Adv. Differential Equations 9 (2004), 353--386.]
Keywords: inhomogeneous, existence, uniqueness., Navier-Stokes equations, hyperdissipative.
Mathematics Subject Classification: Primary: 76D03, 76D0.
Citation: Daoyuan Fang, Ruizhao Zi. On the well-posedness of inhomogeneous hyperdissipative Navier-Stokes equations. Discrete & Continuous Dynamical Systems, 2013, 33 (8) : 3517-3541. doi: 10.3934/dcds.2013.33.3517
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show all references

References:
[1]

Comm. Pure Appl. Math., 64 (2011), 832-881. doi: 10.1002/cpa.20351.  Google Scholar

[2]

Arch. Rational Mech. Anal., 204 (2012), 189-230. doi: 10.1007/s00205-011-0473-4.  Google Scholar

[3]

Studies in Mathematics and its Applications, 22, North-Holland Publishing Co., Amsterdam, 1990.  Google Scholar

[4]

Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 343, Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-16830-7.  Google Scholar

[5]

Annales Scinentifiques de l'École Normale Supérieure, 14 (1981), 209-246.  Google Scholar

[6]

Vol. I, Pubbl. Cent. Ric. Mat. Ennio Giorgi, Scuola Norm. Sup., Pisa, (2004), 53-135.  Google Scholar

[7]

Journal of Differential Equations, 121 (1995), 314-228. doi: 10.1006/jdeq.1995.1131.  Google Scholar

[8]

Proc. Roy. Soc. Edinburgh Sect. A, 133 (2003), 1311-1334. doi: 10.1017/S030821050000295X.  Google Scholar

[9]

Adv. Differential Equations, 9 (2004), 353-386.  Google Scholar

[10]

Ann. Fac. Sci. Toulouse Math., 15 (2006), 637-688.  Google Scholar

[11]

J. Hyperbolic Differ. Equ., 4 (2007), 1-17. doi: 10.1142/S021989160700101X.  Google Scholar

[12]

Differential Integral Equations, 10 (1997), 587-598.  Google Scholar

[13]

Differential Integral Equations, 10 (1997), 577-586.  Google Scholar

[14]

Arch. Rational Mech. Anal., 137 (1997), 135-158. Google Scholar

[15]

Invent. Math., 98 (1989), 511-547. doi: 10.1007/BF01393835.  Google Scholar

[16]

Arch. Rational Mech. Anal., 16 (1964), 269-315.  Google Scholar

[17]

J. Funct. Anal., 261 (2011), 3181-3210. doi: 10.1016/j.jfa.2011.07.026.  Google Scholar

[18]

Chin. Ann. Math. Ser. B, 30 (2009), 607-630. doi: 10.1007/s11401-009-0027-3.  Google Scholar

[19]

Tokyo Joural of Mathematics, 22 (1999), 17-42. doi: 10.3836/tjm/1270041610.  Google Scholar

[20]

Geom. Funct. Anal., 12 (2002), 355-379. doi: 10.1007/s00039-002-8250-z.  Google Scholar

[21]

Journal of Soviet Mathematics, 9 (1978), 697-749.  Google Scholar

[22]

Oxford Lecture Series in Mathematics and its Applications, 3, Oxford University Press, The Clarendon Press, Oxford University Press, New York, 1996.  Google Scholar

[23]

Anal. PDE, 2 (2009), 361-366. doi: 10.2140/apde.2009.2.361.  Google Scholar

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