Incompressible fluid motion without friction (which would, if friction were considered, come from the billiard effect creating Brownian motion) is governed by the Euler pde (1752-57).

This randomness is still present beneath the surface in the Euler pde because an integral operator which is an inverse to exterior d is used in Euler and the one used is actually described by Brownian motion .

The goal is to derive apriori bounds for Euler and Euler with friction (derived by Navier 1822 & Stokes 1846 ),both incompressible. This , by using Random geometry together with the properties of transport of related geometric structures. For example , the circulation around a null homologous circle moving with the fluid ( Lord Kelvin 1867using Stokes theorem and Helmhotz below )giving fluid motion invariant functions on those components of the free loop space ) . Also the angular momentum exact two form of the fluid motion ( Helmhotz vorticity 1858 , which is transported by the Euler pde). Finally the constant in time , zero curvature of the metric obtained by the motion’s rearranging of the background metric on triply periodic Euclidean three space.