论文标题
部分可观测时空混沌系统的无模型预测
Illposedness of $C^{2}$ vortex patches
论文作者
论文摘要
众所周知，如果$ 0 <α<1 $ $ 0 <α<1 $，则涡旋补丁在$ c^{1，α} $中得到充分量。在本文中，我们证明了$ c^{2} $ Vortex补丁的不良性。该设置是考虑Sobolev Spaces中的Vortex补丁$ W^{2，p} $，其中边界的曲率为$ l^p $。在这种情况下，我们显示$ w^{2，p} $的持久性当$ 1 <p <\ infty $并构造$ c^{2} $初始补丁数据，贴片边界的曲率以$ t> 0 $而无限。关键成分是曲率的进化方程，其中主要项是线性和分散的。
It is well known that vortex patches are wellposed in $C^{1,α}$ if $0<α<1$. In this paper, we prove the illposedness of $C^{2}$ vortex patches. The setup is to consider the vortex patches in Sobolev spaces $W^{2,p}$ where the curvature of the boundary is $L^p$ integrable. In this setting, we show the persistence of $W^{2,p}$ regularity when $1<p <\infty$ and construct $C^{2}$ initial patch data for which the curvature of the patch boundary becomes unbounded immediately for $t>0$. The key ingredient is the evolution equation for the curvature, the dominant term in which turns out to be linear and dispersive.
