We prove the uniqueness and stability of entropy shocks to the isentropic Euler systems among all vanishing viscosity limits of solutions to associated Navier-Stokes systems. To take into account the vanishing viscosity limit, we show a contraction property for any large perturbations of viscous shocks to the Navier-Stokes system. The contraction estimate does not depend on the strength of the viscosity. This provides a good control on the inviscid limit process. We prove that, for any initial value, there exists a vanishing viscosity limit to solutions of the Navier-Stokes system. The convergence holds in a weak topology. However, this limit satisfies some stability estimates measured by the relative entropy with respect to an entropy shock. In particular, our result provides the uniqueness of entropy shocks to the shallow water equation in a class of inviscid limits of solutions to the viscous shallow water equations.

中文翻译：

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等熵欧拉系统熵冲击的唯一性和稳定性在来自一大类 Navier-Stokes 系统的一类无粘性极限中

我们证明了熵冲击对等熵欧拉系统的唯一性和稳定性，在所有相关 Navier-Stokes 系统解决方案的消失粘度极限中。为了考虑消失的粘度极限，我们展示了对 Navier-Stokes 系统的任何大的粘性冲击扰动的收缩特性。收缩估计不取决于粘度的强度。这提供了对无粘性极限过程的良好控制。我们证明，对于任何初始值，Navier-Stokes 系统的解都存在一个消失的粘度极限。收敛在弱拓扑中成立。然而，这个限制满足一些稳定性估计，由相对于熵冲击的相对熵来衡量。特别是，