论文标题
在单调剪切流附近的非线性无粘性阻尼
Nonlinear inviscid damping near monotonic shear flows
论文作者
论文摘要
我们证明，在通道$ \ mathbb {t} \ times [0,1] $的2D Euler方程的解决方案中，一大型单调剪切流的非线性渐近稳定性。更确切地说，我们考虑剪切流$（b（y），0）$由函数$ b $给出的，这是gevrey光滑的，严格增加，并且在间隔$（0,1）$的紧凑子集之外（以避免与Indiscid Damping不符的边界贡献）。我们还假设相关的线性化操作员满足适当的光谱条件，这是证明线性触觉阻尼所需的。 在这些假设下，我们表明，如果$ u $是一种解决方案，它是这种剪切流量$（b（y），0）$的小而且平稳的扰动，那么时间$ t = 0 $，则速度field $ u $ u $ u $ u $ y $ $ $ a $ $ $ y $ a $ $ $ $ $ $ u $ a $ $ t = 0 $。这是在一般稳态溶液周围无法明确求解的一般稳定溶液周围的Euler方程的第一个非线性渐近稳定性。
We prove nonlinear asymptotic stability of a large class of monotonic shear flows among solutions of the 2D Euler equations in the channel $\mathbb{T}\times[0,1]$. More precisely, we consider shear flows $(b(y),0)$ given by a function $b$ which is Gevrey smooth, strictly increasing, and linear outside a compact subset of the interval $(0,1)$ (to avoid boundary contributions which are incompatible with inviscid damping). We also assume that the associated linearized operator satisfies a suitable spectral condition, which is needed to prove linear inviscid damping. Under these assumptions, we show that if $u$ is a solution which is a small and Gevrey smooth perturbation of such a shear flow $(b(y),0)$ at time $t=0$, then the velocity field $u$ converges strongly to a nearby shear flow as the time goes to infinity. This is the first nonlinear asymptotic stability result for Euler equations around general steady solutions for which the linearized flow cannot be explicitly solved.
