论文标题
高频弱稳定的准线性边界价值问题的横向不稳定
Transverse instability of high frequency weakly stable quasilinear boundary value problems
论文作者
论文摘要
这项工作旨在证明，当强迫边界项受到横向频率的小振幅振荡函数的扰动时，可能会出现强大的不稳定性。由于边界频率位于所谓的lopatinskii决定因素为零的位点，因此边界上的扩增产生了配置文件的高度耦合方程系统。使用Cauchy-Kovalevskaya定理在分析框架中解决了该系统的简化模型，以及它的版本，以确保解决方案的时空分析性。然后证明，通过共鸣和放大，该阶段的特定配置可能会产生不稳定性，从某种意义上说，在解决方案的渐近扩展中，强迫项的小扰动会干扰领先顺序。最后，我们研究了在第三个空间维度中使用等肌欧拉方程发生这种频率配置的可能性。
This work intends to prove that strong instabilities may appear for high order geometric optics expansions of weakly stable quasilinear hyperbolic boundary value problems, when the forcing boundary term is perturbed by a small amplitude oscillating function, with a transverse frequency. Since the boundary frequencies lie in the locus where the so-called Lopatinskii determinant is zero, the amplifications on the boundary give rise to a highly coupled system of equations for the profiles. A simplified model for this system is solved in an analytical framework using the Cauchy-Kovalevskaya theorem as well as a version of it ensuring analyticity in space and time for the solution. Then it is proven that, through resonances and amplification, a particular configuration for the phases may create an instability, in the sense that the small perturbation of the forcing term on the boundary interferes at the leading order in the asymptotic expansion of the solution. Finally we study the possibility for such a configuration of frequencies to happen for the isentropic Euler equations in space dimension three.
