Global large-data solutions for the BGK equation with physical collision frequency in a slab
Abstract
We establish a global large-data existence theory for the Bhatnagar--Gross--Krook equation in a slab with the physical collision frequency given by the local density. We consider both inflow and Maxwell boundary conditions and assume only finite mass, kinetic energy, and entropy of the initial data, together with the corresponding inflow flux bounds. The data need not be small or pointwise bounded, and we impose neither positive lower bounds on the density or temperature nor prescribed velocity moments above order two. To our knowledge, this is the first nonperturbative global existence result for this initial and boundary value problem under these assumptions. The central obstacle is the simultaneous loss of energy compactness and integrability of the unrenormalized collision operator. Finite energy alone does not give uniform integrability of the second velocity moment, while the density dependent collision frequency prevents separate local L1 control of the gain and loss terms. We overcome these difficulties through a compensated estimate uniform in the truncation of the collision frequency, a de la Vallée Poussin weight determined by the data, and two successive renormalizations using entropy production. This yields the compactness of the macroscopic energy, local Maxwellians, and nonlinear collision term, and hence global renormalized solutions.
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BibTeX
@misc{choi2026global,
title = {{Global large-data solutions for the BGK equation with physical collision frequency in a slab}},
author = {Young-Pil Choi and Sihyun Song},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.08658},
eprint = {2610.08658},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.08658},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Global large-data solutions for the BGK equation with physical collision frequency in a slab
AU - Young-Pil Choi
AU - Sihyun Song
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.08658
UR - https://arxiv.org/abs/2610.08658
AB - We establish a global large-data existence theory for the Bhatnagar--Gross--Krook equation in a slab with the physical collision frequency given by the local density. We consider both inflow and Maxwell boundary conditions and assume only finite mass, kinetic energy, and entropy of the initial data, together with the corresponding inflow flux bounds. The data need not be small or pointwise bounded, and we impose neither positive lower bounds on the density or temperature nor prescribed velocity moments above order two. To our knowledge, this is the first nonperturbative global existence result for this initial and boundary value problem under these assumptions. The central obstacle is the simultaneous loss of energy compactness and integrability of the unrenormalized collision operator. Finite energy alone does not give uniform integrability of the second velocity moment, while the density dependent collision frequency prevents separate local L^1 control of the gain and loss terms. We overcome these difficulties through a compensated estimate uniform in the truncation of the collision frequency, a de la Vallée Poussin weight determined by the data, and two successive renormalizations using entropy production. This yields the compactness of the macroscopic energy, local Maxwellians, and nonlinear collision term, and hence global renormalized solutions.
ER -