论文标题
时空冻结：与PDE系统链接的功能方程
Freezing in Space-time: A functional equation linked with a PDE system
论文作者
论文摘要
我们分析了功能方程$$ f（x+f（x））= - f（x）$$，并揭示了其与偏微分方程系统的关系，该系统是在线上固定的台球球系统的流体动力学极限。球系统必须在某个时候冻结，即冻结时间后不会改变速度。末端速度和冻结时间曲线起着PDES边界条件的作用（尽管从这里追求的波动方程式是初始条件，但仍是最初的条件）。功能方程的解决方案一方面提供了冻结时间和终端速度曲线之间的链接，另一方面提供了对PDE的解决方案。
We analyze the functional equation $$F(x+F(x))=-F(x)$$ and reveal its relationship with a system of partial differential equations arising as the hydrodynamic limit of a system of pinned billiard balls on the line. The system of balls must freeze at some time, i.e., no velocity may change after the freezing time. The terminal velocity and the freezing time profiles play the role of boundary conditions for the PDEs (qua terminal conditions, despite being initial conditions from the wave equation perspective pursued here). Solutions to the functional equation provide the link between the freezing time and terminal velocity profiles on the one hand, and the solution to the PDE in the entirety of the space-time domain on the other.
