On the singularities of the global small solutions of the full Boltzmann equation
The authors consider global solutions of initial value problem for Boltzmann equation describing the expansion of a rare gas cloud in vacuum (i.e., the mean free path has to be sufficiently large, or the initial value has to be sufficiently small), solutions which are known to exist since the mid 1980s. Here these solutions are shown to have a representation \[ f(t,x,v)=f_0(x-tv,v) \Gamma_1(t,x,v) + \Gamma_2(t,x,v) \] with \(\Gamma_1,\Gamma_2 \in H^{\alpha}_{\text{loc}}( \Re_+\times \Re^3_x \times \Re_v^3)\), \(\alpha \in (0,1/25)\). This says in essence that initial singularities propagate with the free flow and decay (because \(\Gamma_1\) decays exponentially in time). The proof of this result is based on Duhamel principle and on regularity estimates for various components of collision operator. A result on the propagation of regularity is also given. The original reference for global existence of solutions of Boltzmann equation for a rare gas cloud is misquoted as [\textit{S. Kaniel} and \textit{M. Shinbrot}, Commun. Math. Phys. 58, 65-84 (1978; Zbl 0371.76061)] (a paper in which local existence and uniqueness were proved). The correct reference is [\textit{R. Illner} and \textit{M. Shinbrot}, Commun. Math. Phys. 95, 217-226 (1984; Zbl 0599.76088)].
scientific article; zbMATH DE number 1563301