Effects of concavity on the motion of a body immersed in a Vlasov gas

We consider a body immersed in a perfect gas, moving under the action of a constant force $E$ along the $x$ axis . We assume the gas to be described by the mean-field approximation and interacting elastically with the body. Such a dynamic was studied in previous papers In these studies the asymptotic trend showed no sensitivity whatsoever to the shape of the object moving through the gas. In this work we investigate how a simple concavity in the shape of the body can affect its asymptotic behavior; we thus consider the case of hollow cylinder in three dimensions or a box-like body in two dimensions. We study the approach of the body velocity $V(t)$ to the limiting velocity $V_{\mathrm{\infty }}$ and prove that, under suitable smallness assumptions, the approach to equilibrium is $\mathrm{\mid }{V}_{\mathrm{\infty }}-V(t)\mathrm{\mid }\approx C{t}^{-3}$ both in two or three dimensions, being $C$ a positive constant. This approach is not exponential, as typical in friction problems, and even slower than for the simple disk and the convex body in ${\mathbb{R}}^2$ or ${\mathbb{R}}^3$.