The nonlinear selfdual variational principle established in a preceding paper [N. Ghoussoub, Anti-symmetric Hamiltonians: Variational resolution of Navier–Stokes equations and other nonlinear evolutions, Comm. Pure Appl. Math. 60 (5) (2007) 619–653] – though good enough to be readily applicable in many stationary nonlinear partial differential equations – did not however cover the case of nonlinear evolutions such as the Navier–Stokes equations. One of the reasons is the prohibitive coercivity condition that is not satisfied by the corresponding selfdual functional on the relevant path space. We show here that such a principle still hold for functionals of the formI(u)=∫0T[L(t,u(t),u˙(t)+Λu(t))+〈Λu(t),u(t)〉]dt+ℓ(u(0)−u(T),u(T)+u(0)2) where L (resp., ℓ) is an anti-selfdual Lagrangian on state space (resp., boundary space), and Λ is an appropriate nonlinear operator on path space. As a consequence, we provide a variational formulation and resolution to evolution equations involving nonlinear operators such as the Navier–Stokes equation (in dimensions 2 and 3) with various boundary conditions. In dimension 2, we recover the well-known solutions for the corresponding initial-value problem as well as periodic and anti-periodic ones, while in dimension 3 we get Leray solutions for the initial-value problems, but also solutions satisfying u(0)=αu(T) for any given α in (−1,1). Our approach is quite general and does apply to many other situations.