Inertial Forms of Navier-Stokes Equations on Sphere

The Navier--Stokes equations for incompressible flows past a two--dimensional sphere are considered in this article. The existence of an inertial form of the equations is established. Furthermore for the first time for fluid equations, we derive an upper bound on the dimension of the differential system (inertial manifold) which fully reproduces the infinite dimensional dynamics. This bound is expressed in terms of Grashof Numbers.