论文标题
Delta Shock作为无压力的Euler流中的免费活塞
Delta Shock as Free Piston in Pressureless Euler Flows
论文作者
论文摘要
对于一维无压压缩欧拉方程，我们建立了自由活塞和三角冲击的等效性。奇异的黎曼问题中出现的三角洲冲击正好是可以在直线管中自由移动或向后移动的活塞，这是由无压力的欧拉在其两侧在管中的两个侧面驱动的。这个结果不仅有助于了解某种神秘的三角洲冲击的物理学，而且还提供了一种减少流体固定相互作用问题的方法，该问题包括几个初始有限的价值问题，再加上移动边界，以及更简单的库奇问题。我们从三种不同的角度表明了等效性。第一个是来自粘性颗粒，并通过直接应用动量保护定律来得出活塞轨迹的普通微分方程（ODE），这在物理上简单明了。第二个是研究无压力欧拉方程的耦合初始价值问题，而活塞是牛顿第二定律后的移动边界。这取决于可压缩欧拉方程的初始有限价值问题的ra量概念，该方程使我们能够计算流量给出的活塞上的力。最后一个是直接解决奇异的黎曼问题，并通过广义的兰金·霍尼奥特（Rankine-Hugoniot）条件获得Delta休克的颂歌。所有三种方法都导致相同的颂歌。
We establish the equivalence of free piston and delta shock, for the one-space-dimensional pressureless compressible Euler equations. The delta shock appearing in the singular Riemann problem is exactly the piston that may move freely forward or backward in a straight tube, driven by the pressureless Euler flows on two sides of it in the tube. This result not only helps to understand the physics of the somewhat mysterious delta shocks, but also provides a way to reduce the fluid-solid interaction problem, which consists of several initial-boundary value problems coupled with moving boundaries, to a simpler Cauchy problem. We show the equivalence from three different perspectives. The first one is from the sticky particles, and derives the ordinary differential equation (ODE) of the trajectory of the piston by a straightforward application of conservation law of momentum, which is physically simple and clear. The second one is to study a coupled initial-boundary value problem of pressureless Euler equations, with the piston as a moving boundary following the Newton's second law. It depends on a concept of Radon measure solutions of initial-boundary value problems of the compressible Euler equations which enables us to calculate the force on the piston given by the flow. The last one is to solve directly the singular Riemann problem and obtain the ODE of delta shock by the generalized Rankine-Hugoniot conditions. All the three methods lead to the same ODE.
