obtain precise information on linearized BGK models. Finally, we also prove local asymptotic stability of a nonlinear BGK model.

**keywords:** kinetic equations, BGK models, hypocoercivity, Lyapunov functionals, perturbation methods for matrix equations

Introduction
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This paper is concerned with the large time behavior of nonlinear BGK models (named after the physicists Bhatnagar-Gross-Krook [@BGK54]) and their linearizations around their Maxwellian steady state. With respect to position, we consider here only models on $\tilde{\mathbf{T}}^d:=\big(\frac{L}{2\pi}{\mathbf{T}}\big)^d$, the $d$-dimensional torus of side length $L$ without confinement potential. Then, the usual BGK model for a phase space density $f(x,v,t)$; $x\in\tilde{\mathbf{T}}^d,\,v\in{{\mathbb{R}}^d}$ satisfies the kinetic evolution equation $$\label{bgk}
  \partial_t f+v\cdot\nabla_x f= {\mathbf{Q}}f := M_f(x,v,t) -f(x,v,t)\ ,\quad t\ge0\ ,$$ where $M_f$ denotes the local Maxwellian corresponding to $f$; i.e., the local Maxwellian with the same hydrodynamic moments as $f$: $$M_f(x,v,t)=\frac{\rho(x,t)}{(2\pi T(x,t))^{\frac{d}2}}\,e^{-\frac{|v-u(x,t)|^2}{2T(x,t)}}
  =\frac{\rho(x,t)^{1+\frac{d}{2}}}{(2\pi P(x,t))^\frac{d}2}\,e^{-\frac{|v-u(x,t)|^2\rho(x,t)}{2P(x,t)}}\ ,$$ with density $$\rho(x,t):={\int_{{\mathbb{R}}^d}}f(x,v,t)\,{\,\operatorname{d}\!v}\ ,$$ mean velocity $$u(x,t):=\frac1{\rho(x,t)}{\int_{{\mathbb{R}}^d}}vf(x,v,t)\,{\,\operatorname{d}\!v}\ ,$$ temperature $$T(x,t):=\frac1{d\rho(x,t)}{\int_{{\mathbb{R}}^d}}|v-u(x,t)|^2 f(x,v,t)\,{\,\operatorname{d}\!v}\ ,$$ and pressure (setting the gas constant $R=1$) $$P(x,t):=T(x,t)\rho(x,t)=\frac1{d}{\int_{{\mathbb{R}}^d}}|v-u(x,t)|^2 f(x,v,t)\,{\,\operatorname{d}\!v}\ .$$

Let ${\,\operatorname{d}\!\tilde x}:=L^{-d}{\,\operatorname{d}\!x}$ denote the normalized Lebesgue measure on $\tilde{\mathbf{T}}^d$, and consider normalized initial data $f^I(x,v)$ such that $$\label{f0-normal}
  \int_{\tilde{\mathbf{T}}^d\times{\mathbb{R}}^d} f^I(x,v){\,\operatorname{d}\!\tilde x} {\,\operatorname{d}\!v} =1\ ,\quad 
  \int_{\tilde{\mathbf{T}}^d\times{\mathbb{R}}^d} vf^I(x,v){\,\operatorname{d}\!\tilde x} {\,\operatorname{d}\!v} =0\ ,\quad 
  \int_{\tilde{\mathbf{T}}^d\times{\mathbb{R}}^d} |v|^2f^I(x,v){\,\operatorname{d}\!\tilde x} {\,\operatorname{d}\!v} =d\ .$$ This means, our system has unit 