Strong and classical solutions of the Hopf equation---an example of functional derivative equation of second order

Abstract

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 in- equality of strong form, we apply Foia§'s argument with slight modifica- tions instead of using Prokhorov's compactness argument.