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.

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关于等熵欧拉系统测度值解的选择

流体方程的测量值解自然会出现，例如消失的粘度极限，但在很大程度上表现出非唯一性。在本文中，我们展示了二维等熵可压缩欧拉方程的一些测量值解，虽然它们是能量允许的，但可以作为非物理性的，因为它们不会作为消失的粘度极限而出现。事实上，与不可压缩的情况相反，这些测度值解也不是由欧拉方程的一系列弱解产生的。Chiodaroli、Feireisl、Kreml 和 Wiedemann 已经使用 $\mathcal{A}$-free 刚性论证观察到了这种现象，但仅限于非确定性初始数据。我们将它们的刚性结果发展到非常态的情况，并将其与由 Chiodaroli、De Lellis 和 Kreml 构建的压缩波解演化为无限多个弱解相结合。因此，我们证明存在无限多个具有二次压力定律的二维等熵欧拉系统的广义测度值解，这些解在一定时间内具有确定性的行为，并且不能由具有有界能量的弱解或粘度为零产生序列。