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.