On global schemes for highly degenerate Navier Stokes equation systems

TL;DR

This paper develops a global solution scheme for highly degenerate Navier-Stokes systems, utilizing local contraction in polynomial decay function spaces and controlling the equation system, extending previous work on hypoelliptic diffusions.

Contribution

It introduces a novel controlled global solution scheme for degenerate Navier-Stokes systems, incorporating polynomial decay function spaces and extending prior theoretical frameworks.

Findings

Established local contraction results in polynomial decay spaces.

Developed a controlled scheme for degenerate Navier-Stokes equations.

Provided bounds for the Leray projection term in the scheme.

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.