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

The Cauchy problem for the Boltzmann equation with soft potential, in the framework of small perturbation of an equilibrium state, has been studied in many spaces. The method of strongly continuous semigroup has been applied by Caflisch\cite{Caflisch1980a} and Ukai-Asano\cite{Ukai1982} for the case of soft potential, where they obtained the $L^\infty$ solution without requiring any velocity deviation. By generalizing the estimate on linearized collision operator $L$ to the case of very soft potential, we obtain a similar global existence result for $\gamma\in[0,d)$. For soft potential, the spectrum structure of the linearized Boltzmann operator couldn't give spectral gap, so we use the method of integration by parts and consider a weighted velocity space in order to obtain algebraic decay in time.