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

Daoyuan Fang; Ruizhao Zi

doi:10.3934/dcds.2013.33.3517

Article Contents

2013, Volume 33, Issue 8: 3517-3541. Doi: 10.3934/dcds.2013.33.3517

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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

Received: August 31, 2012

Revised: October 31, 2012

Published: July 2013

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Abstract

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.]

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Mathematics Subject Classification: Primary: 76D03, 76D05.

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