High regularity of solutions of compressible Navier-Stokes equations. (English) Zbl 1146.35072

The author considers the barotropic compressible Navier-Stokes system in a domain (bounded or not) \(\Omega\subset {\mathbb R}^3\), where the density may be zero (vacuum) in an open subset of \(\Omega\) or, in the unbounded case, may be zero at infinity. He proves the local existence of solutions \((\rho_j,v_j)\in C([0,t_*]; H^{2(k-j)+3}\times D_0^1\cap D^{2(k-j)+3}(\Omega))\) for \(0\leq j\leq k\), \(k\geq 1\), provided that the data satisfy compatibility conditions and the initial density is small enough.


35Q30 Navier-Stokes equations
76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics