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

TL;DR

This paper develops a new approach for the non-cutoff Boltzmann equation that allows the microscopic component to decay polynomially at large velocities, providing improved energy estimates and global existence results.

Contribution

It introduces a novel energy functional incorporating Caflisch's decomposition, enabling polynomial tail decay and better convergence rates in the Boltzmann equation analysis.

Findings

Constructed solutions around local Maxwellians for small-amplitude Euler solutions.

Achieved polynomial decay of microscopic components in large velocities.

Obtained global-in-time existence for constant state Euler solutions.

Abstract

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 $\varepsilon$ and allow the microscopic part of solutions to decay polynomially in large velocities. Moreover, the energy norm of perturbations can be of the order $\varepsilon^{1/2}$ 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.