The specular reflective boundary problem for the Boltzmann equation with soft potentials

In this paper, we consider the specular reflective boundary problem for the one-dimensional Boltzmann equation with soft potentials. It is shown that the solution converges to a global Maxwellian M+M+ under certain initial conditions. In order to prove this, we construct a local Maxwellian M̄ which is very close to M+M+ as an ansatz of MM and obtain the stability of M̄. Our proof is based on elementary energy estimates and detailed properties of Burnett functions.