Structurally stable Riemann solutions
This paper is the first in a series that studies Riemann problem for systems of two hyperbolic conservation laws. The analysis is grounded in the viscous profile (traveling wave) criterion for shocks, leading to consideration of the theory of bifurcation of vector fields. In this paper, the authors identify Riemann problems whose solutions are structurally stable. Such solutions can include transitional waves but not overcompressive shocks. The paper includes a description of how a transitional wave that is sonic (i.e., travels at characteristic speed) on both sides can arise in a structurally stable Riemann problem solution. As part of the systematic analysis the authors introduce a new index, the Riemann number, defined on individual waves. A Riemann problem solution is then shown to be structurally stable if the Riemann numbers across waves in the solution sum to two, and additional nondegeneracy conditions hold.
scientific article; zbMATH DE number 880559