Work

Ill-Posedness of the Boltzmann–BGK Model in the Exponential Class

Year: 2026

Type: article

Abstract: The BGK (Bhatnagar–Gross–Krook) model is a relaxation-type model of the Boltzmann equation, which is popularly used in place of the Boltzmann equation in physics and engineering. In this paper, we address the ill-posedness problem for the BGK model, in which the solution instantly escapes the initial solution space. For this, we propose two ill-posedness scenarios, namely, the homogeneous and the inhomogeneous ill-posedness mechanisms. In the former case, we find a class of spatially homogeneous solutions to the BGK model, where removing the small velocity part of the initial data triggers ill-posedness by increasing temperature. For the latter, we construct a spatially inhomogeneous solution to the BGK model such that the local temperature constructed from the solution has a polynomial growth in spatial variable. These ill-posedness properties for the BGK model pose a stark contrast with the Boltzmann equation for which the solution map is, at least for a finite time, stable in the corresponding solution spaces.

Source: Archive for Rational Mechanics and Analysis

DOI: 10.1007/s00205-026-02176-6

Authors: Donghyun Lee, Sungbin Park, Seok-Bae Yun

Institutions: Pohang University of Science and Technology, Sungkyunkwan University

Language: English

Topic: Gas Dynamics and Kinetic Theory

Subfield: Applied Mathematics

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,090

Funders: National Research Foundation, National Research Foundation of Korea, Ministry of Science and ICT, South Korea