We show uniqueness and stability in $L^2$ and for all time for piecewise-smooth solutions to hyperbolic balance laws. We have in mind applications to gas dynamics, the isentropic Euler system and the full Euler system for a polytropic gas in particular. We assume the discontinuity in the piecewise smooth solution is an extremal shock. We use only mild hypotheses on the system. Our techniques and result hold without smallness assumptions on the solutions. We can handle shocks of any size. We work in the class of bounded, measurable solutions satisfying a single entropy condition. We also assume a strong trace condition on the solutions, but this is weaker than $BV_{\text{loc}}$. We use the theory of a-contraction (see Kang and Vasseur [Arch. Ration. Mech. Anal., 222(1):343--391, 2016]) developed for the stability of pure shocks in the case without source.

中文翻译：

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双曲平衡定律的分段平滑解的 $a$-收缩和稳定性的标准

我们展示了 $L^2$ 的唯一性和稳定性，并且一直是双曲平衡定律的分段平滑解。我们考虑在气体动力学、等熵欧拉系统和多方气体的全欧拉系统中的应用。我们假设分段平滑解的不连续性是极值冲击。我们只对系统使用温和的假设。我们的技术和结果在解决方案上没有小假设的情况下成立。我们可以处理任何大小的冲击。我们在满足单个熵条件的有界、可测量的解决方案类中工作。我们还假设解具有强跟踪条件，但这比 $BV_{\text{loc}}$ 弱。我们使用收缩理论（参见 Kang 和 Vasseur [Arch. Ration. Mech. Anal., 222(1):343--391,