# High regularity of solutions of compressible Navier-Stokes equations

Preprint Series # 776
Cho, Yonggeun High regularity of solutions of compressible Navier-Stokes equations. (2006);

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## Abstract

We study the Navier-Stokes equations for compressible {\it barotropic} fluids in a bounded or unbounded domain $\Omega$ of $\mathbf{R}^3$. The initial density may vanish in an open subset of $\Omega$ or to 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 that the initial density is sufficiently small. To control the nonnegativity or decay at infinity of density, we need to establish a boundary value problem of $(k+1)$-coupled elliptic system which may not be in general solvable. The smallness condition of initial density is necessary for the solvability, which is not necessary in case that the initial density has positive lower bound. Secondly, we prove the global existence of smooth radial solutions of {\it isentropic} compressible Navier-Stokes equations on a bounded annulus or a domain which is the exterior of a ball under a smallness condition of initial density.

Item Type: Preprint 5 viscous compressible fluids, compressible Navier-Stokes equations, vacuum 35-xx PARTIAL DIFFERENTIAL EQUATIONS 1518