The vanishing viscosity method is a way to construct and select physically relevant solutions for hyperbolic conservation laws. In this talk, we will consider a class of hyperbolic systems in one spatial dimension with a non-constant viscosity matrix. First, we discuss the global-in-time existence of smooth solutions to the hyperbolic-parabolic equation for initial data with a small total variation. We show that the solution to the parabolic equation converges to a semigroup solution of the hyperbolic system as viscosity goes to zero. This talk is based on joint work with Boris Haspot.
