We consider the linearized Euler equations around a smooth, bilipschitz shear profile $U(y)$ on $\mathbb{T}_L \times \mathbb{R}$. We construct an explicit flow which exhibits linear inviscid damping for $L$ sufficiently small, but for which damping fails if $L$ is large. In particular, similar to the instability results for convex profiles for a shear flow being bilipschitz is not sufficient for linear inviscid damping to hold. Instead of an eigenvalue-based argument the underlying mechanism here is shown to be based on a new cascade of resonances moving to higher and higher frequencies in $y$, which is distinct from the echo chain mechanism in the nonlinear problem.

中文翻译：

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关于线性无粘阻尼的小条件：单调性和共振链

我们考虑在 $\mathbb{T}_L \times \mathbb{R}$ 上围绕平滑的 bilipschitz 剪切剖面 $U(y)$ 的线性化欧拉方程。我们构建了一个显式流，它在 $L$ 足够小时表现出线性无粘阻尼，但如果 $L$ 很大，则阻尼会失败。特别是，类似于剪切流的凸轮廓的不稳定性结果是 bilipschitz 不足以保持线性无粘阻尼。这里显示的潜在机制不是基于特征值的论证，而是基于在 y 中移动到越来越高频率的新共振级联，这与非线性问题中的回声链机制不同。