最小作用量可容许性原理
The Least Action Admissibility Principle
AI总结：
本文提出最小作用量可容许性原理，用于在初值问题解不唯一时选择物理相关弱解，并证明该原理在正压欧拉方程中更偏好经典双激波解而非凸积分解。
摘要：
本文为拉格朗日力学和连续介质力学方程在初值问题解不唯一时，提供了一种新的可容许性准则，用于选择物理相关的弱解。该准则受到经典最小作用量原理的启发，但现在被应用于表现出非唯一解的初值问题。文中给出了拉格朗日力学和正压流体力学欧拉方程的例子。特别地，我们证明了最小作用量可容许性原理更倾向于黎曼初值问题的经典双激波解，而不是由凸积分生成的某些解。另一方面，Dafermos 的熵准则更倾向于凸积分解，而不是双激波解。此外，当压强由 $p(ρ)=ρ^2$ 给出时，我们证明了只要凸积分解对相同的初始数据有定义，双激波解就总是更受偏好。
英文摘要：
This paper provides a new admissibility criterion for choosing physically relevant weak solutions of the equations of Lagrangian and continuum mechanics when non-uniqueness of solutions to the initial value problem occurs. The criterion is motivated by the classical least action principle but is now applied to initial value problems which exhibit non-unique solutions. Examples are provided to Lagrangian mechanics and the Euler equations of barotropic fluid mechanics. In particular, we show the least action admissibility principle prefers the classical two shock solution to the Riemann initial value problem to certain solutions generated by convex integration. On the other hand, Dafermos's entropy criterion prefers convex integration solutions to the two shock solutions. Furthermore, when the pressure is given by $p(ρ)=ρ^2$, we show that the two shock solution is always preferred whenever the convex integration solutions are defined for the same initial data.
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