The Green's function and large‐time behavior of solutions for the one‐dimensional Boltzmann equation

Abstract

Abstract We study the pointwise behavior of the Green function of the Boltzmann equation. Our results reveal the particle and fluid aspects of the equation. The particle aspect is represented by singular waves. These waves are carried by transport equations and dominate the short‐time behavior of the solution. We devise a Picard‐type iteration for the construction of the increasingly regular particle‐like waves. The fluid‐like waves reveal the dissipative behavior of the type of Navier‐Stokes equations as seen usually by the Chapman‐Enskog expansion. These waves are constructed as part of the long waves expansion in the spectrum of the Fourier mode for the space variable. The fluid‐like waves represent the long‐time behavior of the solution. As an application, we obtain the pointwise description of the large‐time behavior of the convergence to the global Maxwellian when the initial perturbation is necessarily smooth. In our analysis of the exchanges of the microscopic velocity decay and space decay, we make essential uses of the hard sphere models. © 2004 Wiley Periodicals, Inc.