双曲守恒律 $n \ imes n$ 方程组 Riemann 解的通用结构稳定性
Generic Structural Stability for Riemann Solutions to $n \times n$ Systems of Hyperbolic Conservation Laws
- UCLA（加州大学洛杉矶分校）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文证明了一维 $n \ imes n$ 双曲守恒律方程组 Riemann 解在几乎处处意义下具有结构稳定性，通过顺序横截性方法将稳定性条件转化为横截矩阵可逆性，并应用于 $p$-系统、薄膜及机器学习通量近似。
AI中文摘要：
本文证明了一维空间中 $n \ imes n$ 双曲守恒律方程组 Riemann 解的通用结构稳定性。在严格双曲性、真正非线性以及 Rankine-Hugoniot 映射的正则流形假设下，我们证明对于几乎所有的左、右状态对，由 Lax 可容许激波和稀疏波组成的任意 $n$-波 Riemann 解，在 $C^2$ 拓扑下对左状态、右状态以及通量函数的扰动是结构稳定的。核心新思想是顺序横截性，它通过中间状态将 $n$ 个波串联起来，并利用推前映射将其切向贡献传递到公共参考点，从而将结构稳定性条件归结为 $n \ imes n$ 横截矩阵的可逆性。我们将结果应用于 $p$-系统、多分散颗粒负载薄膜以及机器学习通量近似。
英文摘要：
This paper proves generic structural stability for Riemann solutions to $n \times n$ 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 $n \times n$ transversality matrix. We apply the results to the $p$-system, polydisperse particle-laden thin films, and machine-learned flux approximations.
