Relativistic collision geometry and global near-vacuum solutions at critical transverse regularity
Abstract
We establish global well-posedness near vacuum for the relativistic Boltzmann equation with soft and hard potentials in the range γ∈(−2,1]. The initial data have critical Besov regularity of order 2/r in the transverse spatial variables for an admissible range of r depending on γ, together with weighted integrability in the longitudinal and momentum variables. The proof combines a fixed-direction div--curl lemma with bilinear collision estimates. In particular, to control the gain term, we combine localization to spherical caps with energy-weighted angular estimates. The key relativistic estimate controls the singularity associated with the first component of the post-collisional relative velocity. These bounds yield the required time decay.
Related
- Regularity of the Time-to-Collision Map and Convergence of Learned Hamilton-Jacobi Solutions
- Multiple non-collision periodic solutions for second order singular differential systems with parameters
- Self-similar Imploding Solutions of the Relativistic Euler Equations
- Regularity for convex viscosity solutions of σ2 Equation
- Global large-data solutions for the BGK equation with physical collision frequency in a slab
BibTeX
@misc{hu2026relativistic,
title = {{Relativistic collision geometry and global near-vacuum solutions at critical transverse regularity}},
author = {Xinfeng Hu and Shuangqian Liu and Haoran Peng and Yi Zhou},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.40322},
eprint = {2610.40322},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.40322},
}
RIS
TY - UNPB
TI - Relativistic collision geometry and global near-vacuum solutions at critical transverse regularity
AU - Xinfeng Hu
AU - Shuangqian Liu
AU - Haoran Peng
AU - Yi Zhou
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.40322
UR - https://arxiv.org/abs/2610.40322
AB - We establish global well-posedness near vacuum for the relativistic Boltzmann equation with soft and hard potentials in the range γ∈(−2,1]. The initial data have critical Besov regularity of order 2/r in the transverse spatial variables for an admissible range of r depending on γ, together with weighted integrability in the longitudinal and momentum variables. The proof combines a fixed-direction div--curl lemma with bilinear collision estimates. In particular, to control the gain term, we combine localization to spherical caps with energy-weighted angular estimates. The key relativistic estimate controls the singularity associated with the first component of the post-collisional relative velocity. These bounds yield the required time decay.
ER -