On the Cauchy problem of Boltzmann equation with a very soft potential

This work proves the global existence to Boltzmann equation in the whole space with very soft potential γ ∈ [0, d) and angular cutoff, in the framework of small perturbation of equilibrium state. In this article, we generalize the estimate on linearized collision operator L to the case of very soft potential and obtain the spectrum structure of the linearized Boltzmann operator correspondingly. The global classic solution can be derived by the method of strongly continuous semigroup. For soft potential, the linearized Boltzmann operator could not give spectral gap; hence, we have to consider a weighted velocity space in order to obtain algebraic decay in time.