Existence results for viscous polytropic fluids with degenerate viscosities and far field vacuum

In this paper, we considered the isentropic Navier-Stokes equations for compressible fluids with density-dependent viscosities in $\mathbb{R}^3$. These systems come from the Boltzmann equations through the Chapman-Enskog expansion to the second order, cf.\cite{tlt}, and are degenerate when vacuum appears. We firstly establish the existence of the unique local regular solution (see Definition \ref{d1} or \cite{sz3}) when the initial data are arbitrarily large with vacuum at least appearing in the far field. Moreover it is interesting to show that we could't obtain any global regular solution that the $L^\infty$ norm of $u$ decays to zero as time $t$ goes to infinity.

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