Elementary solutions for a model Boltzmann equation in one dimension and the connection to grossly determined solutions
 Citation data:

Physica D: Nonlinear Phenomena, ISSN: 01672789, Vol: 347, Page: 111
 Publication Year:
 2017

 EBSCO 4
 Repository URL:
 http://arxiv.org/abs/1608.03510
 DOI:
 10.1016/j.physd.2017.02.008
 Author(s):
 Publisher(s):
 Tags:
 Physics and Astronomy; Mathematics  Analysis of PDEs; Mathematical Physics; 35Q35, 76P99, 46F12
article description
The Fouriertransformed version of the time dependent slipflow model Boltzmann equation associated with the linearized BGK model is solved in order to determine the solution’s asymptotics. The ultimate goal of this paper is to demonstrate that there exists a robust set of solutions to this model Boltzmann equation that possess a special property that was conjectured by Truesdell and Muncaster: that solutions decay to a subclass of the solution set uniquely determined by the initial mass density of the gas called the grossly determined solutions. First we determine the spectrum and eigendistributions of the associated homogeneous equation. Then, using Case’s method of elementary solutions, we find analytic timedependent solutions to the model Boltzmann equation for initial data with a specialized compact support condition under the Fourier transform. In doing so, we show that the spectrum separates the solutions into two distinct parts: one that behaves as a set of transient solutions and the other limiting to a stable subclass of solutions. Thus, we demonstrate that for gas flows with this specialized initial density condition, in time all gas flows for the one dimensional model Boltzmann equation act as grossly determined solutions.