High regularity of solutions of compressible Navier-Stokes equations

We study the barotropic compressible Navier-Stokes equations in a bounded or an unbounded domain ω of R³. The initial density may vanish in an open subset of ω or be positive but vanish at space infinity. We first prove the local existence of solutions (ρ^(j), u^(j)) in C([0, T_*];H^2(k-j)+3 × D¹ ₀ ∩ D^2(k-j)+3(ω)), 0 ≤ j ≤ k, k ≥ 1 under the assumptions that the data satisfy compatibility conditions and the initial density is sufficiently small. To control the non-negativity or decay at infinity of density, we need to establish a boundary-value problem of a (k + 1)-coupled elliptic system which may not be, in general, solvable. The smallness condition of the initial density is necessary for the solvability of the elliptic system; this is not necessary when the initial density has positive lower bound. Secondly, we prove the global existence of smooth radially symmetric solutions of isentropic compressible Navier-Stokes equations by controlling every regularity with |ρ|L_∞.