AI 中文总结
本文研究可压缩Navier-Stokes方程中同族两道相互作用激波的消失黏性极限，通过反导数、近似碰撞时间、加权相对熵等方法，证明了小强度小黏性下解的收敛性及显式收敛速率。
AI 中文摘要
我们研究一维可压缩Navier-Stokes方程在来自基础Euler方程同一特征族的两道相互作用激波区域中的消失黏性极限。与来自不同族的两道激波相互作用会产生各自原族的两道出射激波不同，两道同族激波的碰撞会产生同一族的一道出射激波以及另一族的一道稀疏波。该构型更为奇异，因为碰撞发生在与波强度成反比的更大时间尺度上，这使得不仅在激波碰撞前后的过程中，更关键的是在碰撞点处，都难以证明消失黏性极限的合理性。此外，碰撞后激波与稀疏波的同时出现引入了额外的困难。为克服这些障碍，我们在碰撞前首先采用反导数方法，该方法允许我们精确固定两道黏性激波的位置直至碰撞点。然而，关于黏性的一致估计无法在碰撞时间前闭合；因此我们引入一个精心构造的近似碰撞时间以获得所需的高阶能量界。碰撞后，我们应用加权相对熵方法并结合时变偏移来处理由激波与稀疏波共存产生的复合波结构。我们的主要结果表明，对于适当小的波强度和黏性系数，存在一族Navier-Stokes方程的全局光滑解，它们以显式收敛速率收敛到Euler方程的熵解。这些技术有望应用于消失黏性理论中的其他相关问题。
英文摘要
We investigate the vanishing viscosity limit for the one-dimensional compressible Navier-Stokes equations in the regime of two interacting shock waves from the same characteristic family of the underlying Euler equations. Unlike the interaction of two shocks from distinct families, which produces two outgoing shocks in their respective original families, the collision of two same-family shocks generates an outgoing shock in the same family together with a rarefaction wave in the other family. This configuration is more singular because the collision occurs on a larger time scale that depends inversely on the wave strength, making it challenging to justify the vanishing viscosity limit not only in the processes before and after the shock collision but, more crucially, at the collision point. Furthermore, the emergence of both shock and rarefaction waves after the collision introduces an additional difficulty. To overcome these obstacles, we first employ the anti-derivative method before the collision, which allows us to fix the locations of both viscous shocks precisely up to the collision point. However, uniform estimates with respect to the viscosity cannot be closed up to the collision time; we therefore introduce a carefully constructed approximate collision time to obtain the required higher-order energy bounds. After the collision, we apply a weighted relative entropy method, combined with time-dependent shifts, to handle the composite wave structure arising from the coexistence of shock and rarefaction waves. Our main result establishes that, for suitably small wave strengths and viscosity coefficients, there exists a family of global smooth solutions to the Navier-Stokes equations that converge to the entropy solution of the Euler equations with an explicit convergence rate. The techniques are expected to be applicable to other related problems in vanishing viscosity theory.
