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December  2016, 36(12): 6921-6941. doi: 10.3934/dcds.2016101

Global well-posedness for inhomogeneous Navier-Stokes equations with logarithmical hyper-dissipation

Bin Han 1, and  Changhua Wei 2,

1. 

Department of Mathematics, Hangzhou Dianzi University, Hangzhou 310018, China
2. 

Department of Mathematics, Zhejiang Sci-Tech University, Hangzhou 310018, China

Received  October 2015 Revised  August 2016 Published  October 2016

This paper is devoted to studying the global well-posedness for 3D inhomogeneous logarithmical hyper-dissipative Navier-Stokes equations with dissipative terms $D^2u$. Here we consider the supercritical case, namely, the symbol of the Fourier multiplier $D$ takes the form $h(\xi)=|\xi|^{\frac{5}{4}}/g(\xi)$, where $g(\xi)=\log^{\frac{1}{4}}(2+|\xi|^2)$. This generalizes the work of Tao [17] to the inhomogeneous system, and can also be viewed as a generalization of Fang and Zi [12], in which they considered the critical case $h(\xi)=|\xi|^{\frac{5}{4}}$.
Keywords: Critical spaces, energy estimates., global solution.
Mathematics Subject Classification: 76D03, 76D0.
Citation: Bin Han, Changhua Wei. Global well-posedness for inhomogeneous Navier-Stokes equations with logarithmical hyper-dissipation. Discrete & Continuous Dynamical Systems, 2016, 36 (12) : 6921-6941. doi: 10.3934/dcds.2016101
References:
[1]

H. Bahouri, J. Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der mathematischen Wissenschaften, 343, 2011. doi: 10.1007/978-3-642-16830-7.  Google Scholar

[2]

D. Barbato, F. Morandin and M. Romito, Global regularity for a slightly supercritical hyperdissipative Navier-Stokes system, Analysis and PDE, 7 (2014), 2009-2027. doi: 10.2140/apde.2014.7.2009.  Google Scholar

[3]

J. M. Bony, Calcul symbolique et propagation des singularités pour équations aux dérivées partielles nonlinéaires, Annales Scinentifiques de l'école Normale Supérieure, 14 (1981), 209-246.  Google Scholar

[4]

J. Y. Chemin, Localization in Fourier space and Navier-Stokes system, Phase Space Analysis of partial Differential Equations, CRM series, Pisa, 1 (2004), 53-136.  Google Scholar

[5]

J. Y. Chemin and N. Lerner, Flot de champs de vecteurs non Lipschitziens et équations de Navier-Stokes, J. Differential Equations, 121 (1995), 314-328. doi: 10.1006/jdeq.1995.1131.  Google Scholar

[6]

R. Danchin, Density-dependent incompressible Viscous fluids in critical spaces, Proc. Roy. Soc. Edinburgh Sect.A, 133 (2003), 1311-1334. doi: 10.1017/S030821050000295X.  Google Scholar

[7]

R. Danchin, The inviscid limit for density-dependent incompressible fluids, Ann. Fac. Sci. Toulouse Math. Ser., 15 (2006), 637-688. doi: 10.5802/afst.1133.  Google Scholar

[8]

R. Danchin, Local and global well-posedness results for flows of inhomogeneous viscous fluids, Advances in Differential Equations, 9 (2004), 353-386.  Google Scholar

[9]

R. Danchin, Well-posedness in critical spaces for barotropic viscous fluids with truly not constant density, Communications in Partial Differential Equations, 32 (2007), 1373-1397. doi: 10.1080/03605300600910399.  Google Scholar

[10]

R. J. DiPerna and P. L. Lions, Ordinary differential equations transport theory and Sobolev spaces, Invent. Math., 98 (1989), 511-547. doi: 10.1007/BF01393835.  Google Scholar

[11]

H. Fujita and T. Kato, On the Navier-Stokes initial value problem I, Arch. Roational Mech. Anal. 16 (1964), 269-315. doi: 10.1007/BF00276188.  Google Scholar

[12]

D. Fang and Rui. Z. Zi, On the well-posedness of inhomogeneous hyperdissipative Navier-Stokes equations, Discrete and continuous Dynamical systems, 33 (2013), 3517-3541. doi: 10.3934/dcds.2013.33.3517.  Google Scholar

[13]

S. Itoh and A. Tani, Solvability of nonstationary problems for nonhomogeneous incompressible uids and the convergence with vanishing viscosity, Tokyo Joural of Mathematics, 22 (1999), 17-42. doi: 10.3836/tjm/1270041610.  Google Scholar

[14]

N. Katz and N. Pavlović, A cheap Caffarelli-Kohn-Nirenberg inequality for the Navier-Stokes equaiton with hyper-dissipation, Geom. Funct. Anal., 12 (2002), 355-379. doi: 10.1007/s00039-002-8250-z.  Google Scholar

[15]

P. L. Lions, Mathematical Topics in Fluid Mechanics. Vol. 1. Incompressible Models, Oxford Lecture Series in Mathematics and its Applications, 3. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1996.  Google Scholar

[16]

O. Ladyzhenskaja and V. Solonnikov, The unique solvability of an initial-boundary value problem for viscous incompressible inhomogeneous fluids, Journal of Soviet Mathematics, 9 (1978), 697-749. Google Scholar

[17]

T. Tao, Global regularity for a logarithmically supercritical hyperdissipative Navier-Stokes equation, Anal. PDE, 2 (2009), 361-366. doi: 10.2140/apde.2009.2.361.  Google Scholar

show all references

References:
[1]

H. Bahouri, J. Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der mathematischen Wissenschaften, 343, 2011. doi: 10.1007/978-3-642-16830-7.  Google Scholar

[2]

D. Barbato, F. Morandin and M. Romito, Global regularity for a slightly supercritical hyperdissipative Navier-Stokes system, Analysis and PDE, 7 (2014), 2009-2027. doi: 10.2140/apde.2014.7.2009.  Google Scholar

[3]

J. M. Bony, Calcul symbolique et propagation des singularités pour équations aux dérivées partielles nonlinéaires, Annales Scinentifiques de l'école Normale Supérieure, 14 (1981), 209-246.  Google Scholar

[4]

J. Y. Chemin, Localization in Fourier space and Navier-Stokes system, Phase Space Analysis of partial Differential Equations, CRM series, Pisa, 1 (2004), 53-136.  Google Scholar

[5]

J. Y. Chemin and N. Lerner, Flot de champs de vecteurs non Lipschitziens et équations de Navier-Stokes, J. Differential Equations, 121 (1995), 314-328. doi: 10.1006/jdeq.1995.1131.  Google Scholar

[6]

R. Danchin, Density-dependent incompressible Viscous fluids in critical spaces, Proc. Roy. Soc. Edinburgh Sect.A, 133 (2003), 1311-1334. doi: 10.1017/S030821050000295X.  Google Scholar

[7]

R. Danchin, The inviscid limit for density-dependent incompressible fluids, Ann. Fac. Sci. Toulouse Math. Ser., 15 (2006), 637-688. doi: 10.5802/afst.1133.  Google Scholar

[8]

R. Danchin, Local and global well-posedness results for flows of inhomogeneous viscous fluids, Advances in Differential Equations, 9 (2004), 353-386.  Google Scholar

[9]

R. Danchin, Well-posedness in critical spaces for barotropic viscous fluids with truly not constant density, Communications in Partial Differential Equations, 32 (2007), 1373-1397. doi: 10.1080/03605300600910399.  Google Scholar

[10]

R. J. DiPerna and P. L. Lions, Ordinary differential equations transport theory and Sobolev spaces, Invent. Math., 98 (1989), 511-547. doi: 10.1007/BF01393835.  Google Scholar

[11]

H. Fujita and T. Kato, On the Navier-Stokes initial value problem I, Arch. Roational Mech. Anal. 16 (1964), 269-315. doi: 10.1007/BF00276188.  Google Scholar

[12]

D. Fang and Rui. Z. Zi, On the well-posedness of inhomogeneous hyperdissipative Navier-Stokes equations, Discrete and continuous Dynamical systems, 33 (2013), 3517-3541. doi: 10.3934/dcds.2013.33.3517.  Google Scholar

[13]

S. Itoh and A. Tani, Solvability of nonstationary problems for nonhomogeneous incompressible uids and the convergence with vanishing viscosity, Tokyo Joural of Mathematics, 22 (1999), 17-42. doi: 10.3836/tjm/1270041610.  Google Scholar

[14]

N. Katz and N. Pavlović, A cheap Caffarelli-Kohn-Nirenberg inequality for the Navier-Stokes equaiton with hyper-dissipation, Geom. Funct. Anal., 12 (2002), 355-379. doi: 10.1007/s00039-002-8250-z.  Google Scholar

[15]

P. L. Lions, Mathematical Topics in Fluid Mechanics. Vol. 1. Incompressible Models, Oxford Lecture Series in Mathematics and its Applications, 3. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1996.  Google Scholar

[16]

O. Ladyzhenskaja and V. Solonnikov, The unique solvability of an initial-boundary value problem for viscous incompressible inhomogeneous fluids, Journal of Soviet Mathematics, 9 (1978), 697-749. Google Scholar

[17]

T. Tao, Global regularity for a logarithmically supercritical hyperdissipative Navier-Stokes equation, Anal. PDE, 2 (2009), 361-366. doi: 10.2140/apde.2009.2.361.  Google Scholar

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