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

TL;DR

This paper analytically solves the Fourier-transformed 1D BGK model to show that solutions decay to a subclass determined by initial conditions, confirming a conjecture about grossly determined solutions in kinetic theory.

Contribution

It provides an explicit spectral analysis and solution construction for the 1D BGK model, demonstrating the decay to grossly determined solutions based on initial moments.

Findings

Solutions split into transient and stable subclasses.

Solutions decay to a subclass determined by initial first moment.

Confirmed the conjecture that solutions are grossly determined.

Abstract

The Fourier-transformed version of the BGK model in one-dimension is solved in order to determine the general solution's asymptotics. The ultimate goal of this paper is to demonstrate that the solution to the model Boltzmann possesses 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 first moment of the gas. First we determine the spectrum and eigendistributions of the associated homogeneous equation. Then, using Case's method of elementary solutions, we find analytic time-dependent solutions to the original problem. 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. This demonstrates that in time all gas flows for the one-dimensional BGK model Boltzmann act as grossly determined solutions.