抄録
In this note, we construct, strong and classical solutions of the Hopf equation, a statistical version of the Navier-Stokes equation on a compact Riemannian manifold with or without boundary. Our points are to regard the Hopf equation as a given Functional Derivative Equation (F.D.E. for short) of second order, to derive the Navier-Stokes equation as the characteristic equation of it and to give an exact meaning to the ‘trace’ of the second order functional derivatives which appear in the Hopf equation. To construct a solution of the Hopf-Foias equation with the energy inequality of strong form, we apply Foias's argument with slight modifications instead of using Prokhorov's compactness argument.
