On the Influence of Viscosity on Riemann Solutions

We show how the existence and uniqueness of Riemann solutions are affected by the precise form of viscosity which is used to select shock waves admitting a viscous profile. We study a complete list of codimension-1 bifurcations that depend on viscosity and distinguish between Lax shock waves with and without a profile. These bifurcations are the saddle–saddle heteroclinic bifurcation, the homoclinic bifurcation, and the nonhyperbolic periodic orbit bifurcation. We prove that these influence the existence and uniqueness of Riemann solutions and affect the number and type of waves comprising a Riemann solution. We present generic situations in which viscous Riemann solutions differ from Lax solutions.