Nonexistence of Riemann Solutions and Majda–Pego Instability

We study Riemann problems for the shallow water equations. We consider weak self-similar Riemann solutions consisting of constant states, rarefaction waves, and/or jump discontinuities that satisfy the viscous profile entropy criterion, with a positive definite, symmetric viscosity matrix. We prove that for a “generic” symmetric, positive definite viscosity matrix there is an open set of Riemann initial data for which a weak self-similar Riemann solution does not exist. We show that this happens for the hyperbolic initial data that is unstable in the sense studied by Majda and Pego. We prove that such initial data always exist for positive definite, symmetric, nondiagonal viscosity matrices. In the work that follows previous work by the authors (in press, Nonlinear Anal.) we show that in the situations presented in this paper, measure-value solutions exhibiting continuously generated oscillations take place. The results of the present paper provide a new insight into the role of the viscous profile entropy criterion and the Majda–Pego instability in the existence of Riemann solutions for nonlinear conservation laws.