论文标题
关于3D Euler方程中旋转的稳定效果
On the stabilizing effect of rotation in the 3d Euler equations
论文作者
论文摘要
尽管众所周知，恒定旋转在各种流体模型中诱导线性分散效应，但我们在这里研究其对无粘性环境中长时间非线性动力学的影响。更确切地说，我们研究了3D旋转Euler方程的稳定性，$ \ Mathbb {r}^3 $，旋转速度固定。我们表明，对于任何$ m> 0 $，轴对称的初始数据足够小$ \ varepsilon $导致长期存在的解决方案至少$ \ varepsilon^{ - m} $和分散。这是旋转稳定效应的表现，无论其速度如何。 为了实现这一目标，我们开发了一种自然建立在可用对称性的各向异性框架。这允许对非线性相互作用的几何形状进行精确的量化和控制，同时提供了足够的信息以通过适应的线性分散估计获得分散衰减。
While it is well known that constant rotation induces linear dispersive effects in various fluid models, we study here its effect on long time nonlinear dynamics in the inviscid setting. More precisely, we investigate stability in the 3d rotating Euler equations in $\mathbb{R}^3$ with a fixed speed of rotation. We show that for any $M>0$, axisymmetric initial data of sufficiently small size $\varepsilon$ lead to solutions that exist for a long time at least $\varepsilon^{-M}$ and disperse. This is a manifestation of the stabilizing effect of rotation, regardless of its speed. To achieve this we develop an anisotropic framework that naturally builds on the available symmetries. This allows for a precise quantification and control of the geometry of nonlinear interactions, while at the same time giving enough information to obtain dispersive decay via adapted linear dispersive estimates.
