The Riemann Problem Near a Hyperbolic Singularity II

The Riemann Problem near a Hyperbolic Singularity : the Classification of Solutions of Quadratic Riemann Problems

The purpose of this paper is to classify the solutions of Riemann problems near a hyperbolic singularity in a nonlinear system of conservation laws. Hyperbolic singularities play the role in the theory of Riemann problems that rest points play in the theory of ordinary differential equations: Indeed, generically, only a finite number of structures can appear in a neighborhood of such a singular...

Hyperbolic balance laws: Riemann invariants and the generalized Riemann problem

Abstract. The Generalized Riemann Problem (GRP) for nonlinear hyperbolic system of m balance laws (or alternatively “quasi-conservative” laws) in one space dimension is formulated as follows: Given initial-data which are analytic on two sides of a discontinuity, determine the time evolution of the solution at the discontinuity. In particular, the GRP numerical scheme (second-order high resoluti...

The hyperbolic/elliptic transition in the multi-dimensional Riemann Problem

For a continuous self-similar solution to a system of conservation laws, genuine nonlinearity yields Lipschitz continuity at points where the type of the governing system changes. This is a well-known fact in one space dimension, where a constant state ū bifurcates towards a rarefaction wave at a point x/t that equals an eigenvalue λj(ū). We extend this observation to several space dimensions. ...

On the Riemann Problem for Non-conservative Hyperbolic Systems

A Riemann Singularity Theorem for Integral Curves

We prove results generalizing the classical Riemann Singularity Theorem to the case of integral, singular curves. The main result is a computation of the multiplicity of the theta divisor of an integral, nodal curve at an arbitrary point. We also suggest a general formula for the multiplicity of the theta divisor of a singular, integral curve at a point and present some evidence that this formu...