Boltzmann Equation and Hard Sphere Gas with Singular Initial Data: Well-posedness, Kinetic Limit, and Weak Stability
Abstract
We develop a local well-posedness theory for the spatially inhomogeneous 3-d Boltzmann equation with hard sphere cross-section, accommodating a class of large initial data. This class is an interpolation between the space Lx∞Lv1 and Lv1Lx∞ (with weight in v). It is notable for including data that can have infinite Lx,v∞-norm, entropy density. The core idea of the proof is a homogenization method based on collision geometries. As a key application, we prove the kinetic limit of hard sphere gases with initial data in this space, having certain extra conditions. This is the first proof on the kinetic limit of hard sphere gases with singular initial data. We also derive weak stability estimates of Boltzmann equation and hard sphere gas within the same functional spaces.
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BibTeX
@misc{qi2026boltzmann,
title = {{Boltzmann Equation and Hard Sphere Gas with Singular Initial Data: Well-posedness, Kinetic Limit, and Weak Stability}},
author = {Chenjiayue Qi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.31570},
eprint = {2610.31570},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.31570},
}
RIS
TY - UNPB
TI - Boltzmann Equation and Hard Sphere Gas with Singular Initial Data: Well-posedness, Kinetic Limit, and Weak Stability
AU - Chenjiayue Qi
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.31570
UR - https://arxiv.org/abs/2610.31570
AB - We develop a local well-posedness theory for the spatially inhomogeneous 3-d Boltzmann equation with hard sphere cross-section, accommodating a class of large initial data. This class is an interpolation between the space L_x^∞L_v^1 and L_v^1L_x^∞ (with weight in v). It is notable for including data that can have infinite L_x,v^∞-norm, entropy density. The core idea of the proof is a homogenization method based on collision geometries. As a key application, we prove the kinetic limit of hard sphere gases with initial data in this space, having certain extra conditions. This is the first proof on the kinetic limit of hard sphere gases with singular initial data. We also derive weak stability estimates of Boltzmann equation and hard sphere gas within the same functional spaces.
ER -