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

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.

💡 Research Summary

The paper investigates the well‑posedness of the BGK (Bhatnagar‑Gross‑Krook) model in function spaces endowed with exponential (Gaussian‑type) weights, and demonstrates that the model is fundamentally ill‑posed in these spaces. The authors begin by recalling the standard estimate used in BGK existence theory, namely the polynomial‑weighted bound
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