论文标题
关于国际象棋台球流和庞加莱问题的平稳性和规律性
On the Smoothness and Regularity of the Chess Billiard Flow and the Poincaré Problem
论文作者
论文摘要
庞加莱问题是稳定分层流体中二维内波的模型。典型台球流的变体的国际棋子台球流动驱动了背后的形成并描述了这些内部波的演变，并且其轨迹可以表示为围绕给定域边界的旋转。我们发现，对于正方形中的足够不合理的旋转，或者当旋转数$ r（λ）$是二磷酸时，进化问题的解决方案$ u（t）$的规律性直接与强迫函数$ f（x）$的规律性直接相关。此外，我们表明当$ f $平滑时，$ u $也很光滑。这些结果扩展了研究象棋台球流的合理旋转的奇异点或缺乏规律性的研究。我们还介绍了各种几何形状的数值模拟，这些模拟分析了$ r（λ）$中的平稳形成和分形维度，并猜测我们的结果的扩展。我们的结果可以应用于建模二维海洋波，并且还将经典量子对应与流体研究联系起来。
The Poincaré problem is a model of two-dimensional internal waves in stable-stratified fluid. The chess billiard flow, a variation of a typical billiard flow, drives the formation behind and describes the evolution of these internal waves, and its trajectories can be represented as rotations around the boundary of a given domain. We find that for sufficiently irrational rotation in the square, or when the rotation number $r(λ)$ is Diophantine, the regularity of the solution $u(t)$ of the evolution problem correlates directly to the regularity of the forcing function $f(x)$. Additionally, we show that when $f$ is smooth, then $u$ is also smooth. These results extend studies that have examined singularity points, or the lack of regularity, in rational rotations of the chess billiard flow. We also present numerical simulations in various geometries that analyze plateau formation and fractal dimension in $r(λ)$ and conjecture an extension of our results. Our results can be applied in modeling two dimensional oceanic waves, and they also relate the classical quantum correspondence to fluid study.
