Generic Structural Stability for Riemann Solutions to n × n Systems of Hyperbolic Conservation Laws
Abstract
This paper proves generic structural stability for Riemann solutions to n × n systems of hyperbolic conservation laws in one spatial dimension. Under assumptions of strict hyperbolicity, genuine nonlinearity, and a regular manifold hypothesis on the Rankine-Hugoniot map, we show that for almost every pair of left and right states, any n-wave Riemann solution consisting of Lax-admissible shocks and rarefactions is structurally stable under perturbations of the left state, the right state, and the flux function in the C2 topology. The central new idea is sequential transversality, which chains the n waves through intermediate states and transports their tangent contributions to a common reference point via pushforward maps, reducing the structural stability condition to the invertibility of an n × n transversality matrix. We apply the results to the p-system, polydisperse particle-laden thin films, and machine-learned flux approximations.
Related
- Time-asymptotic stability of generic Riemann solutions for viscous-dispersive heat-conducting compressible fluids
- Time-asymptotic stability of generic Riemann solutions for the system of heat-conductive ideal gas without viscosity
- Two global stability theorems for small-data 3D compressible Euler solutions: Unique maximal globally hyperbolic developments with gradient-blowup & smooth global existence in the entire spacetime
- Overcompressive shocks in a 2×2 hyperbolic system of conservation laws
- Entropy bounded compact stencil Runge-Kutta discontinuous Galerkin method for hyperbolic conservation laws
BibTeX
@misc{tan2026generic,
title = {{Generic Structural Stability for Riemann Solutions to $n \times n$ Systems of Hyperbolic Conservation Laws}},
author = {Hong Kiat Tan and Andrea L. Bertozzi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.28714},
eprint = {2609.28714},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.28714},
}
RIS
TY - UNPB
TI - Generic Structural Stability for Riemann Solutions to n × n Systems of Hyperbolic Conservation Laws
AU - Hong Kiat Tan
AU - Andrea L. Bertozzi
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.28714
UR - https://arxiv.org/abs/2609.28714
AB - This paper proves generic structural stability for Riemann solutions to n × n systems of hyperbolic conservation laws in one spatial dimension. Under assumptions of strict hyperbolicity, genuine nonlinearity, and a regular manifold hypothesis on the Rankine-Hugoniot map, we show that for almost every pair of left and right states, any n-wave Riemann solution consisting of Lax-admissible shocks and rarefactions is structurally stable under perturbations of the left state, the right state, and the flux function in the C^2 topology. The central new idea is sequential transversality, which chains the n waves through intermediate states and transports their tangent contributions to a common reference point via pushforward maps, reducing the structural stability condition to the invertibility of an n × n transversality matrix. We apply the results to the p-system, polydisperse particle-laden thin films, and machine-learned flux approximations.
ER -