Nonuniqueness for the vanishing viscosity solution with fixed initial condition in a nonstrictly hyperbolic system of conservation laws

We consider the Riemann problem for a system of two decoupled, nonstrictly hyperbolic, Burgers-like conservation equations with added artificial viscosity. We analytically establish two different vanishing viscosity limits for the solution of this system, which correspond to the two cases where one of the viscosities vanishes much faster than the other. This is done without altering the initial condition as is necessary with travelling wave methods. Numerical evidence is then provided to show that when the two viscosities vanish at the same rate, the solution converges to a limit that lies strictly between the two previously established limits. Finally, we use control theory to explain the mechanism behind this nonuniqueness behavior, which indicates other systems of nonstrictly hyperbolic conservation laws where nonuniqueness will occur.