论文标题
在粘性放松下，冲击悬挂的行进波的光谱稳定性
Spectral stability of shock-fronted travelling waves under viscous relaxation
论文作者
论文摘要
反应非线性扩散部分微分方程可以表现出冲击的行进波解决方案。 Yi等人的先前工作。 al。 （2021）证明了两类正规化（包括粘性松弛）的存在。他们的分析使用几何奇异扰动理论：对于参数$ \ varepsilon的足够小值> 0 $表征正则化的“强度”，将波浪构造为奇异杂色轨道的扰动。在这里，我们严格地表明，对于粘性放松的情况，这些波是频谱稳定的。我们的方法是表明，对于足够小的$ \ varepsilon> 0 $，正规化系统的“完整”特征值问题由$ \ varepsilon = 0 $定义的（减少）缓慢的特征值问题控制。在证明过程中，我们研究了这种几何结构补充和与其他减少的特征值问题的构建方式不同的方式，这些问题在波浪稳定性文献中已知。
Reaction-nonlinear diffusion partial differential equations can exhibit shock-fronted travelling wave solutions. Prior work by Yi et. al. (2021) has demonstrated the existence of such waves for two classes of regularisations, including viscous relaxation. Their analysis uses geometric singular perturbation theory: for sufficiently small values of a parameter $\varepsilon > 0$ characterising the `strength' of the regularisation, the waves are constructed as perturbations of a singular heteroclinic orbit. Here we show rigorously that these waves are spectrally stable for the case of viscous relaxation. Our approach is to show that for sufficiently small $\varepsilon>0$, the `full' eigenvalue problem of the regularised system is controlled by a (reduced) slow eigenvalue problem defined for $\varepsilon = 0$. In the course of our proof, we examine the ways in which this geometric construction complements and differs from constructions of other reduced eigenvalue problems that are known in the wave stability literature.
