Eternal solutions of the Boltzmann equation near travelling Maxwellians

It is shown in this paper that the Cauchy problem of the Boltzmann equation, with a cut-off soft potential and an initial datum close to a travelling Maxwellian, has a unique positive eternal solution. This eternal solution is exponentially decreasing at infinity for all t ∈ (−∞,∞), consequently the moments of any order are finite. This result gives a negative answer to the conjecture of Villani in the spatially inhomogeneous case.  2005 Elsevier Inc. All rights reserved.