High regularity of solutions of compressible Navier-Stokes equations

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We study the barotropic compressible Navier-Stokes equations in a bounded or an unbounded domain $\Omega $ of $ \mathbf{R}^3$. The initial density may vanish in an open subset of $\Omega$ or be positive but vanish at space infinity. We first prove the local existence of solutions $(\rho^{(j)}, u^{(j)})$ in $C([0,T_* ]; H^{2(k-j)+3} \times D_0^1 \cap D^{2(k-j)+3} (\Omega ) )$, $0 \le j \le k, k \ge 1$ 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 $|\rho|_{L^\infty}$.