论文标题
单调，非阴性和紧凑型涡度的稳定性，斑块的无限周长生长
Stability of monotone, non-negative, and compactly supported vorticities in the half cylinder and infinite perimeter growth for patches
论文作者
论文摘要
我们考虑半缸$ \ Mathbb {r} _ {> 0} \ times \ Mathbb {t} $中的不可压缩的Euler方程。在此域中，任何独立于$ x_2 $的涡度定义了固定解决方案。我们证明，如果涡度是非负的，并且在$ x_1 $中，则这种固定解决方案在加权$ l^{1} $ norm中是非线性稳定的，涉及水平冲动。这包括圆柱贴剂的稳定性$ \ {x_ {1} <α\}，\; α> 0 $。稳定性结果是基于以下事实：这种轮廓是所有具有相同分布函数的函数之间水平冲动的独特最小化器。基于稳定性，我们证明在半圆柱体中存在涡流斑块，在无限时间内表现出无限周围的生长。
We consider the incompressible Euler equations in the half cylinder $ \mathbb{R}_{>0}\times\mathbb{T}$. In this domain, any vorticity which is independent of $x_2$ defines a stationary solution. We prove that such a stationary solution is nonlinearly stable in a weighted $L^{1}$ norm involving the horizontal impulse, if the vorticity is non-negative and non-increasing in $x_1$. This includes stability of cylindrical patches $\{x_{1}<α\},\; α>0$. The stability result is based on the fact that such a profile is the unique minimizer of the horizontal impulse among all functions with the same distribution function. Based on stability, we prove existence of vortex patches in the half cylinder that exhibit infinite perimeter growth in infinite time.
