On global schemes for highly degenerate Navier Stokes equation systems

Abstract

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.