We consider instability of the total variation in shock-only solutions to the one-dimensional isentropic Euler system. The main results concern the possibility of immediate blowup in variation of the density field $\rho $ when the data approaches vacuum. In the case of an initially bounded velocity field, it is verified that immediate variation blowup in $\rho $ can occur whenever the adiabatic exponent satisfies $\gamma>3$, whereas no such instability occurs for $1<\gamma \leq 3$. When the initial velocity field is unbounded, variation blowup in $\rho $ can occur for any $\gamma>1$.

中文翻译：

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一维等熵流中近真空的变化不稳定性

我们考虑一维等熵欧拉系统的仅冲击解的总变化的不稳定性。主要结果涉及当数据接近真空时密度场 $\rho $ 的变化立即爆炸的可能性。在初始有界速度场的情况下，验证了当绝热指数满足 $\gamma>3$ 时，$\rho $ 中的立即变化爆发可能发生，而对于 $1<\gamma \leq 3，则不会发生这种不稳定性美元。当初始速度场无界时，对于任何 $\gamma>1$，$\rho $ 中的变化爆炸都可能发生。