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| Consider the behavior of a rarefied gas whose initial state is such that the velocity distribution function (VDF) has a discontinuity. Though decaying, the discontinuity propagates in time for $t>0$. A similar propagation of the discontinuity on the boundary into a gas occurs in steady problems around convex bodies. These phenomena are well understood both theoretically and numerically.
On the other hand, according to recent mathematical studies, the infinite range intermolecular potential has a regularizing effect on the solution of the Boltzmann equation. Thus, starting from the initial data with a discontinuity, the velocity distribution function has no discontinuity immediately after the initial time, $t>0$.
The difference of the above two pictures is due to whether the collision frequency is finite or infinite. In the former case the gain and loss terms of the collision integral can be treated separately, while, in the latter case, they diverge individually and should be treated all together.
Hence, in practical applications, one often introduces the cutoff model for the latter case,
expecting little influence of the grazing collision effect on main physical picture.
In the present work, we would like to discuss the effect of grazing collision via comparisons between the cutoff and non-cutoff models. To this aim, we introduce a toy model, a two-dimensional Lorentz gas model, and study its spatially homogeneous initial-value problem with a discontinuous initial data. |
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