一类$n\times n$双曲系统的适定性与粘性近似的激波稳定性
Well-posedness for a class of $n\times n$ hyperbolic systems and shock stability for the viscous approximation
- University of Stuttgart（斯图加特大学）
- Indian Institute of Technology Kharagpur（印度理工学院卡拉格普尔分校）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文针对齐次通量的多维Keyfitz-Kranzer型系统，通过新颖粘性近似获得紧致性估计，证明弱解全局适定性，并利用相对熵方法分析粘性激波大扰动的$L^2$时间衰减。
AI中文摘要：
我们建立了一类具有齐次通量的多维$n\times n$ Keyfitz-Kranzer型系统柯西问题弱解的全局适定性。我们的证明依赖于一种新颖的粘性近似，该近似提供了必要的紧致性估计。通过利用所提出近似的特定结构，我们推导出先验一致$L^{\infty}$界，并证明了近似解序列的$L^1_{loc}$预紧性。这一框架使我们能够严格证明消失粘性极限。此外，在初始数据的适当假设下，我们采用相对熵方法分析粘性激波大扰动的$L^2$-时间衰减，直至一个动态位移。
英文摘要：
We establish the global well-posedness of weak solutions to the Cauchy problem for a broad class of multi-dimensional $n\times n$ Keyfitz-Kranzer type systems with homogeneous flux. Our proof relies on a novel viscous approximation that yields the necessary compactness estimates. By exploiting the specific structure of this proposed approximation, we derive a priori uniform $L^{\infty}$ bounds and prove the $L^1_{loc}$ precompactness of the sequence of approximate solutions. This framework allows us to rigorously justify the vanishing viscosity limit. Furthermore, under suitable assumptions on the initial data, we employ the relative entropy method to analyze the $L^2$-time decay of large perturbations of the viscous shock, up to a dynamical shift.Evan
