Ill-posedness of the Boltzmann-BGK model in the exponential class

TL;DR

This paper demonstrates that the BGK model, a simplified version of the Boltzmann equation, exhibits ill-posedness where solutions can instantly leave the initial function space, contrasting with the stability observed in the Boltzmann equation.

Contribution

It introduces two ill-posedness scenarios for the BGK model, revealing fundamental stability issues not present in the Boltzmann equation.

Findings

Identifies a class of homogeneous solutions where temperature increases cause ill-posedness.

Constructs inhomogeneous solutions with polynomial growth in temperature.

Shows stark contrast between BGK model and Boltzmann equation stability.

Abstract

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.