  We study hypocoercivity for a class of linear and linearized BGK models for
discrete and continuous phase spaces. We develop methods for constructing
entropy functionals that prove exponential rates of relaxation to equilibrium.
Our strategies are based on the entropy and spectral methods, adapting
Lyapunov's direct method (even for "infinite matrices" appearing for continuous
phase spaces) to construct appropriate entropy functionals. Finally, we also
prove local asymptotic stability of a nonlinear BGK model.
