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 sclf-similar Riemann solution does not exist. We show that this happens for the hyperbolic initi al data that is unstable in the sense studied by Majda and Pego. We prove t hat such initial data always exist for positive definite, symmetric, nondia gonal viscosity matrices. In the work that follows previous work by the aut hors (in press, Nonlinear Anal.) we show that in the situations presented i n this paper, measure-value solutions exhibiting continuously generated osc illations take place. The results of the present paper provide a new insigh t into the role of the viscous profile entropy criterion and the Majda-Pego instability in the existence of Riemann solutions for nonlinear conservati on laws. (C) 2001 Academic Press.