A new energy method for the Boltzmann equation

An energy method for the Boltzmann equation was proposed by Liu, Yang, and Yu [Physica D 188, 178-192 (2004)] based on the decomposition of the Boltzmann equation and its solution around the local Maxwellian. The main idea is to rewrite the Boltzmann equation as a fluid-type dynamics system with the nonfluid component appearing in the source terms, coupled with an equation for the time evolution of the nonfluid component. In this paper, we will elaborate this method and our main observation is that the microscopic projection of the local Maxwellian with respect to a given global Maxwellian is not linear but quadratic. Based on this and by analyzing the fluid-type system using the analytic techniques for the system of conservation laws, we can indeed control the conserved quantities ρ, ρu, and ρ(1/2 u²+E) of the Boltzmann equation by the microscopic projection of the solution of the Boltzmann equation with respect to the global Maxwellian, which is sufficient to deduce the energy estimates for the solution of the Boltzmann equation. The main purpose here is to show that there is no need to perform two sets of energy estimates with respect to the local and a global Maxwellian as in the previous works. In fact, one set of energy estimates with respect to the global Maxwellian is sufficient for closing the energy estimates. Therefore, it not only simplifies the analysis in the previous works, but also shed some light on the stability analysis in some complicated systems, such as the Vlasov-Poisson-Boltzmann and Vlasov-Maxwell-Boltzmann systems. © 2006 American Institute of Physics.