Global existence of solutions for compressible Navier-Stokes equations with vacuum (Q2467762)

A free boundary value problem for one-dimensional Navier-Stokes gas with density-dependent viscosity is considered. This dependence is of the form of a power function of density. A new global existence theorem is obtained for the density exponent between 0 and 1, when the initial density is degenerate in the interior of gas. Lagrangian coordinates are introduced, involving a density's integral. The new obtained system was studed before only for a strictly positive density. An approximative Navier-Stokes equation is considered, by using the Friedrich's mollifiers. The approximate system is studied and some new a priori estimates are given. The main tools are some embedding theorems, Cauchy-Schwartz, Young and Gronwall inequalities. The limit process is studied in the last part, and the main global existence result is obtained.