On viscous limit solutions of the Riemann problem for the equations of isentropic gas dynamics in Eulerian coordinates

For the problem $\rho_t+(\rho u)_x=0$, $(\rho u)_t+(\rho u^2+p(\rho))_x=0$, $(\rho,u)\big|_{t=0,\,x<0}=(\rho_-,u_-)$, $(\rho,u)\big|_{t=0,\,x>0}=(\rho_+,u_+)$ one shows the existence and uniqueness of a solution obtainable as a limit as $\varepsilon$ tends to zero of the bounded self-similar solutions of the regularized problem with additional viscosity term $\varepsilon tu_{xx}$, $\varepsilon>0$, in the second equation. The structure of the solutions is described in detail, in particular, when they contain vacuum states.