AI 中文总结
本文通过解公式分析一维无压力欧拉系统熵解的精细性质，建立了一般初始数据下熵解的全局动态模式，涵盖特征、局部结构、分界、不变量及渐近行为。
AI 中文摘要
本文研究一维无压力欧拉系统柯西问题熵解的精细性质，其中初始密度 $\rho_0$ 是局部有限的 Radon 测度，初始速度 $u_0\in L^\infty_{\rho_0}$。我们采用 [F.M. Huang 和 Z. Wang, Comm. Math. Phys. 222(1) (2001), 117--146.] 为该柯西问题引入的解公式来分析熵解，并获得了熵解的多种新的精细性质；这些性质可归纳为四个方面：（i）柯西问题的特征和初始波；（ii）熵解的精细局部结构；（iii）熵解的分界和全局结构；（iv）熵解的不变量和渐近行为，包括渐近轮廓和相应的衰减率。通过这些结果（i）-（iv），我们建立了初始数据 $\rho_0$ 为局部有限 Radon 测度且 $u_0\in L^\infty_{\rho_0}$ 的一般初始数据下，一维无压力欧拉系统柯西问题熵解的全局动态模式。
英文摘要
In this paper, we are concerned with the fine properties of entropy solutions of the Cauchy problem for the one-dimensional pressureless Euler system, wherein the initial density $ρ_0$ is a locally finite Radon measure and the initial velocity $u_0\in L^\infty_{ρ_0}$. We employ the solution formula introduced by [F.M. Huang and Z. Wang, Comm. Math. Phys. 222(1) (2001), 117--146.] for this Cauchy problem to analyze the entropy solutions and obtain various new fine properties of entropy solutions; these can be summarized in four aspects: (i) Characteristics and initial waves for the Cauchy problem; (ii) Fine local structures of entropy solutions; (iii) Divides and global structures of entropy solutions; (iv) Invariants and asymptotic behaviors of entropy solutions including the asymptotic profile and the corresponding decay rates. Through these results (i)-(iv), we establish the global dynamic patterns of entropy solutions of the Cauchy problem for the $1$-D pressureless Euler system with general initial data $ρ_0$ being locally finite Radon measures and $u_0\in L^\infty_{ρ_0}$.
