Une approche algébrique du problème de Riemann. (An algebraic approach for the Riemann problem). (French) Zbl 0691.35060

Homothetic transformations of the (x,t) plane belong to the invariance group of quasilinear first order systems of conservation laws \(u_ t+f(u)_ x=0\). Finding self-similar solutions with a prescribed initial data \((u_{\ell},u_ r)\) is the well known Riemann problem. Solving it, is essentially to invert a complicated function from \({\mathbb{R}}^ n\) to the phase space, knowing the precise shape of the wave curves, including rarefaction, shock and contact discontinuities.
The article presents a new approach. It proves the equivalence of the Riemann problem with a family of algebraic inequalities.
Consequences are given concerning the piecewise smoothness, which shows that the construction of the solution necessarily goes back to Lax’s method.
The algebra works for multi-d systems, too. It does not need any information like strict hyperbolicity or genuine non-linearity. It only requires a non-degenerate convex entropy, which occurs in physical examples. One hope is to use it for numerical computations in 2-d Riemann problems.
Reviewer: A.Heibig


35L65 Hyperbolic conservation laws
35Q15 Riemann-Hilbert problems in context of PDEs