and BGK equations. In §\[sec:linBGK:1D\] we apply the second strategy to the linearized BGK model in 1D, proving exponential decay of the solution towards the spatially uniform Maxwellian, as stated in Theorem \[linBGK-decay\]. This is based on decomposing into spatial Fourier modes and introducing a Hermite function basis in velocity direction. In the Sections \[sec:linBGK:2D\] and \[sec:linBGK:3D\] we extend our result to 2D and 3D, respectively. But this is not straightforward, as it is already not obvious how to choose a convenient Hermite function basis in multi dimensions. Finally, in §\[sec:6\] we prove local exponential stability of the nonlinear BGK equation in 3D as stated in Theorem \[BGK-decay\](b).

Decay of hypocoercive ODEs {#sec:P-method}
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The local convergence result in Theorem \[BGK-decay\](b) is obtained from the global convergence result in Theorem \[linBGK-decay\] and a relatively straightforward control of the errors involved in linearization. Therefore, the essential content of the paper concerns the linearized BGK equations. To this end we shall rewrite them as ODEs – of infinite dimension – in fact. We therefore begin this section with a discussion of the hypocoercivity structure of ODEs and review (from [@AAC16]) a Lyapunov-type method that yields their sharp decay rate.

Lyapunov’s direct method
