Generic Structural Stability for $2 \times 2$ Systems of Hyperbolic Conservation Laws

TL;DR

This work addresses the stability of Riemann solutions for generic $2\times 2$ hyperbolic conservation-law systems in one space dimension without diffusion. It introduces a geometric criterion—transverse intersections between Hugoniot loci and rarefaction curves—along with a regular-manifold assumption and a variant of Thom’s parametric transversality theorem to prove that double-wave entropy solutions are structurally stable under small perturbations to flux functions and the left/right states, for almost every data pair. The authors develop an analytic framework based on the implicit-function theorem on Banach spaces and a foliated parametric transversality theorem, and apply it to the $p$-system and to gravity-driven particle-laden thin-film models, including flux-function interpolation and polynomial approximations with numerical validation. The results guarantee robustness of solution structure under modeling and measurement perturbations, and offer practical guidance for flux-function approximation and numerical schemes in applications such as particle-laden flows. This has significant implications for reliability of Riemann-solution-based diagnostics and for designing numerics that preserve qualitative wave structures under realistic perturbations.

Abstract

This paper presents a proof of generic structural stability for Riemann solutions to $2 \times 2$ system of hyperbolic conservation laws in one spatial variable, without diffusive terms. This means that for almost every left and right state, shocks and rarefaction solutions of the same type are preserved via perturbations of the flux functions, the left state, and the right state. The main assumptions for this proof involve standard assumptions on strict hyperbolicity and genuine non-linearity, a technical assumption on directionality of rarefaction curves, and the regular manifold (submersion) assumption motivated by concepts in differential topology. We show that the structural stability of the Riemann solutions is related to the transversality of the Hugoniot loci and rarefaction curves in the state space. The regular manifold assumption is required to invoke a variant of a theorem from differential topology, Thom's parametric transversality theorem, to show the genericity of transversality of these curves. This in turn implies the genericity of structural stability. We then apply this theorem to two examples: the p-system and a $2 \times 2$ system governing the evolution of gravity-driven monodisperse particle-laden thin films. In particular, we illustrate how one can verify all the above assumptions for the former, and apply the theorem to different numerical and physical aspects of the system governing the latter.

Figures (6)

• Figure 1: A diagram illustrating the geometry of the wave curves and their constituent Hugoniot loci and rarefaction curves with a prescribed left and right state $(u_l,v_l)$ and $(u_r,v_r)$ respectively. We see that since $(u^*,v^*)$ lies along the intersection of the two Hugoniot loci, the solution to the Riemann shock is a double shock problem, with constant states going from $(u_l,v_l)$ to $(u^*,v^*)$ to $(u_r,v_r)$.
• Figure 1: A diagram illustrating the Implicit Function Theorem on Banach space on the product space $K \times C^2(K)^2 \times \mathbb{R}^4$. For each $(F,G,\tilde{u}_l,\tilde{v}_l,\tilde{u}_r,\tilde{v}_r) \in B_{\varepsilon_1^{(1)}}(F_0,G_0,u_l,v_l,u_r,v_r)$, the black curves represents the forward $1-$wave curve emanating from $(u_l,v_l)$, while the blue curves represents the backward $2-$wave curve originating from $(u_r,v_r)$. The red curve represents the $C^1$ map for the intermediate states $(u^*,v^*) = \mathbf{M}^{(1)}(F,G,\tilde{u}_l,\tilde{v}_l,\tilde{u}_r,\tilde{v}_r)$ obtained upon applying the Implicit Function Theorem on Banach space. The dashed curves for a perturbed pair of flux functions $(F,G)$ correspond to the wave curves when it was at $(F_0,G_0)$ with left and right states $(u_l,v_l)$ and $(u_r,v_r)$.
• Figure 1: Left: The submanifold $ULR$ of $\mathbb{R}^6$. Right: The punctured Hugoniot loci for each given states ${\color{blue}(u_l,v_l)}$ and ${\color{red}(u_r,v_r)}$ on the doubly-punctured open set $U_{({\color{blue}u_l},{\color{blue}v_l},{\color{red}u_r},{\color{red}v_r})}$ for the double-shock scenario (accounting for Lax entropy conditions). Intuitively, with the $C^1$ dependence of the loci on the given states, we can see that transversality is generic (recall that if the loci do not intersect, they are considered to be transverse too).
• Figure 1: Comparing the different types of polynomial approximation with the interpolation approximation for $f, f', g,$ and $g'$. The interpolated functions are labeled as 'true curve' for each of the plots above.
• Figure 2: A diagram illustrating a single shock solution. Since $(u_r,v_r)$ lies along the Hugoniot locus $\mathcal{H}^-_{1,(u_l,v_l)}$, then $(u^*,v^*) = (u_r,v_r)$. In this case, we will have a single shock solution, with constant states going from $(u_l,v_l)$ to $(u_r,v_r)$.

Theorems & Definitions (27)

• Definition 2.1
• Proposition 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Definition 3.1
• Definition 3.2
• Theorem 3.3
• Definition 3.4
• Definition 3.5