  The long-time behavior of solutions to different versions of Oseen equations
of fluid flow on the 2D torus is analyzed using the concept of hypocoercivity.
The considered models are isotropic Oseen equations where the viscosity acts
uniformly in all directions and anisotropic Oseen-type equations with different
viscosity directions. The hypocoercivity index is determined (if it exists) and
it is shown that similar to the finite dimensional case of ordinary
differential equations and differential-algebraic equations it characterizes
its decay behavior.
