Nonlinear inviscid damping near monotonic shear flows
Nonlinear inviscid damping near monotonic shear flows
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DOI：
10.4310/acta.2023.v230.n2.a2
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发表时间：
2020-01
期刊：
影响因子：
3.7
通讯作者：
A. Ionescu;H. Jia
中科院分区：
文献类型：
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作者：
A. Ionescu;H. Jia
我们证明了一个大类的单调剪切流的非线性渐近稳定性之间的二维欧拉方程的解决方案在通道$\mathbb{T}\times[0，1]$。更精确地说，我们考虑剪切流$（B（y），0）$，它由一个函数$B$给出，该函数是Gevrey光滑的，严格递增的，并且在区间$（0，1）$的一个紧致子集之外是线性的（以避免与无粘阻尼不相容的边界贡献）。我们还假设，相关的线性化运营商满足一个合适的频谱条件，这是需要证明线性无粘阻尼。在这些假设下，我们证明了如果u是这样一个剪切流（B（y），0）在t=0时的一个小的Gevrey光滑扰动的解，那么当时间趋于无穷大时，速度场u强烈地收敛到附近的剪切流。这是欧拉方程在一般定常解附近的第一个非线性渐近稳定性结果，其中线性化流不能显式求解。
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.