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 talk, 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. This is a joint work with Donghyun

Lee (POSTECH) and Sungbin Park (POSTECH).