Existence of Variational Principles for the Navier-Stokes Equation

Fréchet differentials are introduced to decide when a classical variational principle exists for a given nonlinear differential equation. The formalism is applied to the steady-state Navier-Stokes equation and the continuity equation, and no variational principle exists unless u × (∇ × u) = 0 or u· ∇u = 0. The concept of an adjoint equation is extended to nonlinear equations and a variational principle is derived for the Navier-Stokes equation and its adjoint.