The two-dimensional Broadwell model of discrete kinetic theory is studied in order to clarify the physical relevance of its solutions in comparison to the solutions of the continuous Boltzmann equation. This is achieved by determining completely, in closed form, all non-stationary potential flows with steady limiting conditions and isotropic pressure tensor at infinity. Several classes of exact solutions are also constructed when some of the above hypotheses are dropped. Most results are made possible by suitable transformations, which reduce essentially a complicated overdetermined system of partial differential equations to solving explicitly a Liouville equation. The structure of the obtained solutions, and especially the unphysical features that they exhibit, are finally commented on. It is remarkable that, for the problem considered here, there is no solution showing the typical qualitative features which characterize the continuous Boltzmann equation.
Bobylev, A. V., Caraffini, G. L., & Spiga, G. (2000). Non-stationary two-dimensional potential flows by the Broadwell model equations. European Journal of Mechanics, B/Fluids, 19(2), 303–315. https://doi.org/10.1016/S0997-7546(90)00105-H