Geodesic
On viscous limit solutions of the Riemann problem for the equations of isentropic gas dynamics in Eulerian coordinates
Sbornik. Mathematics, Tome 194 (2003) no. 6, pp. 793-811 
B. P. Andreianov. On viscous limit solutions of the Riemann problem for the equations of isentropic gas dynamics in Eulerian coordinates. Sbornik. Mathematics, Tome 194 (2003) no. 6, pp. 793-811. http://geodesic.mathdoc.fr/item/SM_2003_194_6_a0/
@article{SM_2003_194_6_a0,
     author = {B. P. Andreianov},
     title = {On viscous limit solutions of the {Riemann} problem for the equations of isentropic gas dynamics in {Eulerian} coordinates},
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     year = {2003},
     volume = {194},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2003_194_6_a0/}
}
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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.

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