We aim to incorporate Caflisch’s decomposition into the macro-micro decomposition
in Boltzmann theory to allow 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 the 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
,
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.
