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

Heibig, Arnaud; Serre, Denis

C. R. Acad. Sci., Paris, Sér. I 309, No. 3, 157-162 (1989).

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

MSC:

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

Keywords:

self-similar; Riemann problem; algebraic inequalities; convex entropy