Analysis of the Riemann Problem for a Shallow Water Model with Two Velocities

Some shallow water type models describing the vertical profile of the horizontal velocity with several degrees of freedom have been recently proposed. The question addressed in the current work is the hyperbolicity of a shallow water model with two velocities. The model is written in a nonconservative form and the analysis of its eigenstructure shows the possibility that two eigenvalues coincide. A definition of the nonconservative product is given which enables us to analyze the resonance and coalescence of waves. Eventually, we prove the well-posedness of the two-dimensional Riemann problem with initial condition constant by half-plane.