De Giorgi Argument for non-cutoff Boltzmann equation with soft potentials

In this paper, we consider the global well-posedness to the non-cutoff Boltzmann equation with soft potential in the $L^\infty$ setting. We show that when the initial data is close to equilibrium and the perturbation is small in $L^2 \cap L^\infty$ polynomial weighted space, the Boltzmann equation has a global solution in the weighted $L^2 \cap L^\infty$ space. The ingredients of the proof lie in strong averaging lemma, new polynomial weighted estimate for the non-cutoff Boltzmann equation and the $L^2$ level set Di Giorgi iteration method developed in \cite{AMSY2}. The convergence to the equilibrium state in both $L^2$ and $L^\infty$ spaces is also proved.