Global mild solution with polynomial tail for the Boltzmann equation in the whole space

Abstract

We are concerned with the Cauchy problem on the Boltzmann equation in the whole space. The goal is to construct global-in-time bounded mild solutions near Maxwellians with the perturbation admitting a polynomial tail in large velocities. The proof is based on the Caflisch's decomposition together with the $L^2-L^{\infty }$ interplay technique developed by Guo. The full range of both hard and soft potentials under the Grad's cutoff assumption can be covered. The main difficulty to be overcome in case of the whole space is the polynomial time decay of solutions which is much slower than the exponential rate in contrast with the torus case.

35 pages. An extra note added, proofs of Theorem 3.1 for an induction argument in hard potential case slightly modified according to helpful comments by an anonymous referee, and references updated and added. All comments are welcome