On local strong solutions to the Cauchy problem of 3D isentropic compressible Navier-Stokes equations with degenerate viscosities and far field vacuum

Abstract

This paper concerns the Cauchy problem of the isentropic compressible Navier-Stokes equations with degenerate viscosities and far-field vacuum. When the shear and the bulk viscosity are power functions of the density (${\rho }^{\delta }$ with $0<\delta <1$), it is proved that the three-dimensional Cauchy problem of the compressible Navier-Stokes equations admits a unique strong solution provided the initial density decays as $\left|x{|}^{-\alpha }\right({(2\gamma -1-\delta )}^{-1}<\alpha <2(3-2\epsilon )/(1-\delta )(3-\epsilon ))$ at infinity with $\gamma >1$ and $\epsilon \in (0,1]$. Notably, the choice of δ can be made independently of γ, broadening the range of permissible δ values compared to previous studies. Consequently, these findings have broader applicability to a diverse array of physical models, including those involving Maxwellian molecules.