Oscillatory solutions in conservation laws related to nonexistence of weak self-similar Riemann solutions
We study Riemann solutions of inviscid systems of conservation laws obtained as a viscous limit of an associated parabolic system. This limit depends on the positive definite viscosity matrix. Specifically, we consider Riemann problems with shock initial data, i.e., the initial data for which the right and left states correspond to a Lax admissible shock. We are particularly interested in what happens with a Riemann solution if this shock does not admit a viscous profile due to the presence of a Hopf bifurcation and limit cycles in the dynamical system associated to the viscous entropy criterion;We focus our study on two classes of models: the shallow water equations and a three-phase flow model arising in petroleum engineering. For these models with Riemann data in the strictly hyperbolic region, it is proved that there exists no weak self-similar Riemann solution. Instead, numerical simulations provide solutions exhibiting continuously generated oscillations. We prove that the limit of these oscillatory solutions, as the viscosity goes to zero, satisfies the system of conservation laws in a measure-valued sense. We conjecture that in the three-phase flow model this solution corresponds to interspersing of different phases.