Elementary Solutions for a model Boltzmann Equation in one-dimension and the connection to Grossly Determined Solutions

摘要

The Fourier-transformed version of the BGK model in one-dimension is solvedin order to determine the general solution's asymptotics. The ultimate goal ofthis paper is to demonstrate that the solution to the model Boltzmann possessesa special property that was conjectured by Truesdell and Muncaster: thatsolutions decay to a subclass of the solution set uniquely determined by theinitial first moment of the gas. First we determine the spectrum andeigendistributions of the associated homogeneous equation. Then, using Case'smethod of elementary solutions, we find analytic time-dependent solutions tothe original problem. In doing so, we show that the spectrum separates thesolutions into two distinct parts; one that behaves as a set of transientsolutions and the other limiting to a stable subclass of solutions. Thisdemonstrates that in time all gas flows for the one-dimensional BGK modelBoltzmann act as grossly determined solutions.