\(\mathcal {M}_1(N)\) : Mappings for Velocity-Scalar and Position-Scalar Pdfs

The CCK mapping method is extended to velocity-scalar Pdfs in the spatial description and to position- scalar Pdfs in the material description. The starting point is the Leray version of the Navier-Stokes pdes assuming Green’s function exists to express the pressure in terms of velocity; the result is a system of three integrodifferential equations for the velocity field. The domain of definition of this system is defined as the velocity-scalar phase space \(v_{\alpha }(\mathbf {x}),\Phi _i(\mathbf {x}),i=1(1)N\) , ad the space spanned by the possible range of values is denoted by \(\Omega _{loc}\) . Local mappings \(\chi :R^{N+3}\rightarrow \Omega _{loc}\) are constructed with the aid of the Jacobian of the mapping thus leading to mapping equations. Single-point and multipoint mapping equations are developed in the spatial and material description.