Anti-symmetric Hamiltonians: Variational resolutions for Navier-Stokes and other nonlinear evolutions

The theory of anti-selfdual (ASD) Lagrangians –introduced in [7]– is developed further to allow for a variational resolution of non-linear PDEs of the form Λu+Au+ ∂φ(u)+ f = 0 where φ is a convex lower-semi-continuous function on a reflexive Banach space X , f ∈ X∗, A : D(A) ⊂ X → X∗ is a positive linear operator and where Λ :D(Λ)⊂X→X∗ is a non-linear operator that satisfies suitable continuity and anti-symmetry properties. ASD Lagrangians on path spaces also yield variational resolutions for nonlinear evolution equations of the form u̇(t)+Λu(t)+Au(t)+∂φ(u(t))+ f = 0 starting at u(0) = u0. In both stationary and dynamic cases, the equations associated to the proposed variational principles are not derived from the fact they are critical points of the action functional, but because they are also zeroes of the Lagrangian itself. For that we establish a general nonlinear variational principle which has many applications, in particular to Navier-Stokes type equations, to generalized Choquard-Pekar Schrödinger equations with non-local terms as well as to complex Ginsburg-Landau type initial-value problems. The case of Navier-Stokes evolutions is more involved and will be dealt with in [9]. The general theory of anti-symmetric Hamiltonians and its applications is developed in detail in the upcoming monograph [8]. c © 2000 Wiley Periodicals, Inc.