论文标题
在选择等级欧拉系统的测量值解决方案上
On the Selection of Measure-Valued Solutions for the Isentropic Euler System
论文作者
论文摘要
流体方程的测量值解决方案自然出现，例如消失的粘度限制，但在很大程度上表现出了非唯一性。在本文中，我们表明，二维等凝压压缩欧拉方程的某些测量解决方案虽然是能允许的，但可以被丢弃为无理的，因为它们并不是由于消失的粘度限制而产生的。实际上，与不可压缩的情况相比，这些度量值值的解决方案也不是由Euler方程的一系列弱解。 Chiodaroli，Feireisl，Kreml和Wiedemann已经使用了$ \ Mathcal {a} $ - 免费的刚性参数，但仅针对非确定性初始基准，已经观察到了这种现象。我们将它们的刚性结果发展为非恒定状态的情况，并将其与由Chiodaroli，de Lellis和Kreml构建的无限弱解决方案相结合。在此，我们表明，存在具有二维等压力定律的二维等质欧拉系统的无限广泛性测量解决方案，这些解决方案的表现是确定性的，直到一定时间，并且无法由有界能量的弱解决方案或通过有界能量或消失的粘度序列产生。
Measure-valued solutions to fluid equations arise naturally, for instance as vanishing viscosity limits, yet exhibit non-uniqueness to a vast extent. In this paper, we show that some measurevalued solutions to the two-dimensional isentropic compressible Euler equations, although they are energy admissible, can be discarded as unphysical, as they do not arise as vanishing viscosity limits. In fact, these measure-valued solutions also do not arise from a sequence of weak solutions of the Euler equations, in contrast to the incompressible case. Such a phenomenon has already been observed by Chiodaroli, Feireisl, Kreml, and Wiedemann using an $\mathcal{A}$-free rigidity argument, but only for non-deterministic initial datum. We develop their rigidity result to the case of nonconstant states and combine this with a compression wave solution evolving into infinitely many weak solutions, constructed by Chiodaroli, De Lellis, and Kreml. Hereby, we show that there exist infinitely many generalized measure-valued solutions to the two-dimensional isentropic Euler system with quadratic pressure law, which behave deterministically up to a certain time and which cannot be generated by weak solutions with bounded energy or by vanishing viscosity sequences.
