AI 中文总结
该研究针对无界几何中二维欧拉方程的Couette流，在Yudovich正则性下识别出非线性无粘阻尼机制，对不同几何的扰动动力学进行分类，通过Lyapunov泛函与哈密顿守恒的相互作用完成分类。
AI 中文摘要
我们研究无界几何中二维欧拉方程在Couette流附近的长时间动力学。在Yudovich正则性下，我们识别出一种非线性无粘阻尼机制，其与经典相位混合不同：由涡量的空间排空驱动的速度衰减。该机制对动力学进行了几何符号分类：在无限通道中，小的非负有界涡量扰动无需导数假设即可经历全局阻尼；在整个平面中，小的非负扰动在密度为1的一组时间上表现出增强的色散和阻尼，而非正扰动则被限制且不衰减，即使在Gevrey类中任意小也如此。此外，阻尼可与涡量导数的无限时间增长共存，即使在任意剪切调制下也是如此。该分类通过新的Lyapunov泛函与哈密顿守恒之间的相互作用得到。
英文摘要
We study the long-time dynamics near Couette flow for the 2D Euler equations in unbounded geometries. We identify a mechanism of nonlinear inviscid damping at Yudovich regularity, distinct from classical phase mixing: velocity decay driven by spatial evacuation of vorticity. The mechanism leads to a geometry-sign classification of the dynamics. In the infinite channel, small nonnegative bounded-vorticity perturbations undergo global damping without derivative assumptions. In the whole plane, small nonnegative perturbations exhibit enhanced dispersion and damping along a set of times of density one, while non-positive perturbations remain confined and do not damp, even when arbitrarily small in Gevrey classes. Moreover, damping can coexist with infinite-time growth of vorticity derivatives, even under arbitrary shear modulation. The classification is obtained through an interplay between new Lyapunov functionals and Hamiltonian conservation.
Comments53 pages
