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Extending work of Carty, we show that $H^1$ solutions of a simplified 1D BGKmodel decay exponentially in $L^2$ to a subclass of the class of grosslydetermined solutions as defined by Truesdell and Muncaster. In the process, wedetermine the spectrum and generalized eigenfunctions of the associatednon-selfadjoint linearized operator and derive the associated generalizedFourier transform and Parseval's identity. Notably, our analysis makes use ofrigged space techniques originating from quantum mechanics, as adapted byLjance and others to the nonselfadjoint case.

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