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arXiv · 2609.05948

Global Dynamic Patterns of Entropy Solutions for One-dimensional Pressureless Euler System

Gaowei Cao·Kai Hu·Feimin Huang·Guirong Tang
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Abstract

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 $\rho_0$ is a locally finite Radon measure and the initial velocity $u_0\in L^\infty_{\rho_0}$. 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 $\rho_0$ being locally finite Radon measures and $u_0\in L^\infty_{\rho_0}$.

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Gaowei Cao, Kai Hu, Feimin Huang, Guirong Tang. 2026-09-05. Global Dynamic Patterns of Entropy Solutions for One-dimensional Pressureless Euler System. https://arxiv.org/abs/2609.05948

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