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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

 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 $h(\xi)=|\xi|^{\frac{5}{4}}$.
Citation: Bin Han, Changhua Wei. Global well-posedness for inhomogeneous Navier-Stokes equations with logarithmical hyper-dissipation. Discrete and 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. [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. [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. [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. [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. [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. [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. [8] R. Danchin, Local and global well-posedness results for flows of inhomogeneous viscous fluids, Advances in Differential Equations, 9 (2004), 353-386. [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. [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. [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. [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. [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. [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. [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. [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. [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.

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