{\,\operatorname{d}\!v}\right)^2\nonumber\\
&=& \| f(t) - f^\infty\|^2_{L^1(\tilde{\mathbf{T}}\times{\mathbb{R}}^d,{\,\operatorname{d}\!\tilde x}{\,\operatorname{d}\!v})}\ .\end{aligned}$$

Many hypocoercive equations have been studied in recent years [@ViH06; @He06; @DoMoScH09; @DoMoScH10; @ArEr14], including BGK models in §1.4 and §3.1 of [@DoMoScH10], but sharp decay rates were rarely an issue there. In our earlier work [@AAC16], we established hypocoercivity for such models in 1D by an approach that yields explicit – and quite reasonable – values for $c$ and $\lambda$. To this end, our main tools have been variants of the *entropy–entropy production method*.

The articles [@AAC16] and [@DoMoScH10] only consider BGK models with conserved mass, and partly with also conserved energy. But the tools presented there did not apply to BGK equations that also conserve momentum. This is in fact an important structural restriction that we shall formalize in §\[sec-hypo-index\] with the notion *hypocoercivity index*. The common feature of all models analyzed in [@AAC16] as well as in [@DoMoScH10] is that their hypocoercivity index is 1. The main goal of this paper is to extend the methods from [@AAC16] (i.e. constructing feasible Lyapunov functionals) to models with higher hypocoercivity index. Applied to BGK equations this then also includes models with conserved momentum.

The existence of global solutions for the Cauchy 