NILPOTENT AND RECURSIVE FLOWS

In this paper we introduce a class of nonlinear vector fields on infinite dimensional manifolds such that the corresponding evolution equations can be solved with the same method one uses to solve ordinary differential equations with constant coefficients. Mostly, these equations are nonlinear partial differential equations. It is shown that these flows are characterized by a generalization of the 'method of variation of constants' which is widely used for second order problems to find general solutions out of particular ones. Invariant densities are constructed for these flows in a natural way. These invariant densities are providing an essential tool for solving initial value and boundary value problem for the equations under consideration. Many applications are presented.