AI 中文总结
研究与快速常微分方程耦合的粘性一维守恒律，利用平均策略和谱理论方法，证明对称初始条件下小正粘性会选择唯一线性稳定的周期解，数值模拟验证结果。
AI 中文摘要
我们研究了一个与快速常微分方程耦合的粘性一维守恒律。对于零粘性，该系统有一族无穷维的时间周期解，对应于松弛振荡。我们表明，对于对称初始条件，一个小的正粘性会选择一个唯一的周期解，且证明其线性稳定。证明通过平均策略和谱理论方法利用了快慢结构。结果由数值模拟说明。
英文摘要
We study a viscous one-dimensional conservation law, coupled to a fast ordinary differential equation. For a vanishing viscosity, the system has an infinite-dimensional family of time-periodic solutions, corresponding to relaxation oscillations. We show that for symmetric initial conditions, a small positive viscosity selects a unique periodic solution, which we prove to be linearly stable. The proof exploits the slow-fast structure through an averaging strategy, as well as spectral-theoretic methods. The results are illustrated by numerical simulations.
Comments53 pages, 13 figures. Some minor edits
