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

TL;DR

This paper proves the local existence of unique regular solutions for the isentropic Navier-Stokes equations with density-dependent viscosities and vacuum in the far field, highlighting the challenges in obtaining global solutions.

Contribution

It establishes the first local existence result for large initial data with vacuum in the far field for these degenerate viscous fluids.

Findings

Existence of unique local regular solutions for large initial data with vacuum.

Global regular solutions with decay of velocity are not obtainable.

The system models fluids derived from Boltzmann equations via Chapman-Enskog expansion.

Abstract

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.