Global Dynamic Patterns of Entropy Solutions for One-dimensional Pressureless Euler System
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 ρ0 is a locally finite Radon measure and the initial velocity u0∈ L∞ρ0. 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 ρ0 being locally finite Radon measures and u0∈ L∞ρ0.
Related
- Equivalence of entropy solutions and gradient flows for pressureless 1D Euler systems
- Unidirectional Entropic Solutions of the Pressureless Euler Alignment System
- On one-dimensional pressureless Euler--Poisson systems: existence for non-semiconvex interaction potentials
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BibTeX
@misc{cao2026global,
title = {{Global Dynamic Patterns of Entropy Solutions for One-dimensional Pressureless Euler System}},
author = {Gaowei Cao and Kai Hu and Feimin Huang and Guirong Tang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.05948},
eprint = {2609.05948},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.05948},
}
RIS
TY - UNPB
TI - Global Dynamic Patterns of Entropy Solutions for One-dimensional Pressureless Euler System
AU - Gaowei Cao
AU - Kai Hu
AU - Feimin Huang
AU - Guirong Tang
PY - 2026
DA - 2026/09/05
PB - arXiv
DO - 10.48550/arxiv.2609.05948
UR - https://arxiv.org/abs/2609.05948
AB - 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∈ L^∞_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∈ L^∞_ρ_0.
ER -