  Following pioneering work by Fan and Slemrod who studied the effect of
artificial viscosity terms, we consider the system of conservation laws arising
in liquid-vapor phase dynamics with {\sl physical} viscosity and capillarity
effects taken into account. Following Dafermos we consider self-similar
solutions to the Riemann problem and establish uniform total variation bounds,
allowing us to deduce new existence results. Our analysis cover both the
hyperbolic and the hyperbolic-elliptic regimes and apply to arbitrarily large
Riemann data.
  The proofs rely on a new technique of reduction to two coupled scalar
equations associated with the two wave fans of the system. Strong $L^1$
convergence to a weak solution of bounded variation is established in the
hyperbolic regime, while in the hyperbolic-elliptic regime a stationary
singularity near the axis separating the two wave fans, or more generally an
almost-stationary oscillating wave pattern (of thickness depending upon the
capillarity-viscosity ratio) are observed which prevent the solution to have
globally bounded variation.

Analysis of PDEs;Numerical Analysis