On the Cauchy problem for the Boltzmann equation in the whole space: Global existence and uniform stability in L_ξ² (H_x^N)

Based on a refined energy method, in this paper we prove the global existence and uniform-in-time stability of solutions in the space L_ξ² (H_x^N) to the Cauchy problem for the Boltzmann equation around a global Maxwellian in the whole space R³. Compared with the solution space used by the spectral analysis and the classical energy method, the velocity weight functions or time derivatives need not be included in the norms of L_ξ² (H_x^N), which is realized by introducing some temporal interactive energy functionals to estimate the macroscopic dissipation rate. The key proof is carried out in terms of the macroscopic equations together with the local conservation laws. It is also found that the perturbed macroscopic variables actually satisfy the linearized compressible Navier-Stokes equations with remaining terms only related to the microscopic part. © 2007 Elsevier Inc. All rights reserved.