Let $L=2\pi$ and let the initial data $f^I$ satisfy the normalization .

1.  \[BGK-decay:a\] For all $\gamma \ge 0$ there is an entropy functional ${\mathcal{E}}_\gamma(f)$ satisfying $$  \frac34 {\mathcal{E}}_\gamma(f)  \leq \|f-M_1\|_{\mathcal{H}_\gamma}^2 \leq \frac32 {\mathcal{E}}_\gamma(f)$$ such that, if $h$ is a solution of the linearized BGK equation in 3D with initial data $h^I$ and ${{\mathcal{E}}_\gamma(h^I+M_1)< \infty}$, then $$  {\mathcal{E}}_\gamma(h(t)+M_1) \leq e^{-t/2820}{\mathcal{E}}_\gamma(h^I+M_1)\ ,\qquad t\ge0\ .$$

2.  \[BGK-decay:b\] Moreover, for all $\gamma>3/2$, there is an explicitly computable $\delta_\gamma>0$ such that, if $f$ is a solution of the nonlinear BGK equation with initial data $f^I$ and $\|f^I-M_1\|_{\mathcal{H}_\gamma} < \delta_\gamma$, then for the same entropy function ${\mathcal{E}}_\gamma$, the following decay estimate holds: $${\mathcal{E}}_\gamma(f(t)) \leq e^{-t/2820}{\mathcal{E}}_\gamma(f^I)\ ,\qquad t\ge0\ .$$

Note that part \[BGK-decay:a\] of this theorem generalizes Theorem \[linBGK-decay\] to the Sobolev-type entropies ${\mathcal{E}}_\gamma(f)$ in the case $d=3$, $L=2\pi$.

This paper is organized as follows: In §\[sec:P-method\] we review from [@AAC16] a Lyapunov-type method for hypocoercive ODEs that yields their sharp exponential decay rate. While this approach requires all eigenvectors of the system matrix, we also develop an approach using simplified Lyapunov functionals. This alternative strategy comes at the price of yielding only a suboptimal decay rate, but it can be extended to infinite dimensional systems 