Polynomial tail solutions of the non-cutoff Boltzmann equation near local Maxwellians

This paper aims to incorporate the Caflisch's decomposition into the macro-micro decomposition in Boltzmann theory for allowing the microscopic component to exhibit only the polynomial tail in large velocities. In particular, we treat the Cauchy problem on the non-cutoff Boltzmann equation under the compressible Euler scaling in case of three-dimensional whole space. Up to a finite time we construct the Boltzmann solution around a local Maxwellian corresponding to small-amplitude classical solutions of the full compressible Euler system around constant states. We design a new energy functional which can capture the convergence rate in the small Knudsen number and allow the microscopic part of solutions to decay polynomially in large velocities. Moreover, the energy norm of perturbations can be of the order ^1/2 which the usual method of Hilbert expansion fails to obtain. As a byproduct of the proof, our estimates immediately yield a global-in-time existence result when the Euler solutions are taken to be constant states.