NONCUTOFF BOLTZMANN EQUATION WITH SOFT POTENTIALS IN THE WHOLE SPACE

We prove the existence, uniqueness and convergence of global solutions to the Boltzmann equation with noncutoff soft potentials in the whole space when the initial data is a small perturbation of a Maxwellian with polynomial decay in velocity. Our method is based in the decomposition of the desired solution into two parts: one with polynomial decay in velocity satisfying the Boltzmann equation with only a dissipative part of the linearized operator, the other with Gaussian decay in velocity verifying the Boltzmann equation with a coupling term.