Spectral decomposition and decay to grossly determined solutions for a simplified BGK model

TL;DR

This paper demonstrates exponential decay of solutions in a simplified 1D BGK model to a specific class of solutions, using spectral analysis and advanced mathematical techniques from quantum mechanics.

Contribution

It extends previous work by explicitly determining the spectrum and eigenfunctions of the linearized operator, applying rigged space methods to analyze decay.

Findings

Solutions decay exponentially in $L^2$ norm

Spectrum and eigenfunctions of the linearized operator are explicitly characterized

Generalized Fourier transform and Parseval's identity are derived

Abstract

Extending work of Carty, we show that $H^1$ solutions of a simplified 1D BGK model decay exponentially in $L^2$ to a subclass of the class of grossly determined solutions as defined by Truesdell and Muncaster. In the process, we determine the spectrum and generalized eigenfunctions of the associated non-selfadjoint linearized operator and derive the associated generalized Fourier transform and Parseval's identity. Notably, our analysis makes use of rigged space techniques originating from quantum mechanics, as adapted by Ljance and others to the nonselfadjoint case.