Abstract: 
BGK equations are kinetic transport equations with a relaxation operator that drives the phase space distribution towards the spatially local equilibrium, a Gaussian with the same macroscopic parameters. Due to the absence of dissipation w.r.t. the spatial direction, convergence to the global equilibrium is only possible thanks to the transport term that mixes various positions. Hence, such models are hypocoercive. We shall prove exponential convergence towards the equilibrium with explicit rates for several linear, space periodic BGKmodels in dimension 1 and 2. Their BGKoperators differ by the number of conserved macroscopic quantities (like mass, momentum, energy), and hence their hypocoercivity index. Our discussion includes also discrete velocity models, and the local exponential stability of a nonlinear BGKmodel. The proof is based, first, on a Fourier decomposition in space and Hermite function decomposition in velocity. Then, the crucial step is to construct a problem adapted Lyapunov functional, by introducing equivalent norms for each mode.
References
[1] F. Achleitner, A. Arnold, E.A. Carlen, On linear hypocoercive BGK models, in Springer Proceedings in Mathematics & Statistics, Vol. 126, 137 (2016)
[2] F. Achleitner, A. Arnold, E.A. Carlen, On multidimensional hypocoercive BGK models, Kinetic Rel. Models 11, no. 4, (2018). 
