论文标题
自由边界单数传输方程作为离散动力系统的正式限制
A free boundary singular transport equation as a formal limit of a discrete dynamical system
论文作者
论文摘要
我们研究了一个名为Open Mancala的离散游戏的双曲线恢复的连续版本。结果PDE原来是一个单数传输方程，强迫术语以$ \ {0,1 \} $为单位，而在解决方案本身中不连续。我们证明了问题的某种表述的存在和唯一性，基于由自由边界所满足的非本地方程，这些方程将强迫为一个（活动区域）的区域和没有强迫（尾部区域）的区域。与奇点形成有关的几个例子，最著名的是Riemann问题。有趣的是，可以通过多功能的合适垂直重排获得解决方案。此外，PDE承认了Lyapunov的功能。
We study the continuous version of a hyperbolic rescaling of a discrete game, called open mancala. The resulting PDE turns out to be a singular transport equation, with a forcing term taking values in $\{0,1\}$, and discontinuous in the solution itself. We prove existence and uniqueness of a certain formulation of the problem, based on a nonlocal equation satisfied by the free boundary dividing the region where the forcing is one (active region) and the region where there is no forcing (tail region). Several examples, most notably the Riemann problem, are provided, related to singularity formation. Interestingly, the solution can be obtained by a suitable vertical rearrangement of a multi-function. Furthermore, the PDE admits a Lyapunov functional.
