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

Bin  Han; Changhua  Wei

doi:10.3934/dcds.2016101

Article Contents

2016, Volume 36, Issue 12: 6921-6941. Doi: 10.3934/dcds.2016101

This issue Previous Article Global existence and general decay estimates for the viscoelastic equation with acoustic boundary conditions Next Article Random data Cauchy problem for the nonlinear Schrödinger equation with derivative nonlinearity

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
2.
Department of Mathematics, Zhejiang Sci-Tech University, Hangzhou 310018

Received: September 30, 2015

Revised: July 31, 2016

Published: May 2017

Abstract Related Papers Cited by

Abstract

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:

Mathematics Subject Classification: 76D03, 76D05.

Citation:

Related Papers

Cited by

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.