  First order semi-linear coupling of scalar hypoelliptic equations of second
order leads to a natural class of incompressible Navier Stokes equation
systems, which encompasses systems with variable viscosity and essentially
Navier Stokes equation systems on manifolds. We introduce a controlled global
solution scheme which is based on a) local contraction results in function
spaces with polynomial decay of some order at spatial infinity related to the
polynomial growth factors of standard a priori estimates of densities and their
derivatives for hypoelliptic diffusions of Hoermander type (cf. [15]), and b)
on a controlled equation system where we discuss variations of the scheme we
considered in [10]. We supplement our notes on global bounds of the Leray
projection term in that paper and related controlled Navier Stokes equation
schemes in [6, 7, 9, 10, 12]. Some arguments concerning linear upper bounds of
the control function are added.
