Power-like decreasing solutions of the linearized Boltzmann equation and conservation of mass or energy

We study the power-like solutions of the spatially homogeneous linearized Boltzmann equation for a class of binary cross-sections proportional to |g|1−4s, s=4 or s < - 1, g being the relative speed. We show that these solutions violate the physical requirement of conservation of energy. A similar study for the associate thermalization problem leads to a violation of the conservation law of mass. We study the asymptotic behaviours of the eigenfunctions associated to non-discrete eigenvalues and corresponding to the regular spectrum. The main point, which was already present in our previous study of the hard sphere case is the link between a critical power like decreasing behaviour and conservation of energy. We proved that there exists a solution (R≈v-(6−4s)) associated to this behaviour (as conjecture by Ernst, Hellesoe, Hauge) and it is the only one living outside the standard Hilbert space. A very interesting tool is provided by asymptotic kernels which carry the dominant part of the asymptotic behaviour of the solutions.