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

• Published in: Communications on pure and applied mathematics 2007-05, Vol.60 (5), p.619-653
• Main Author: Ghoussoub, Nassif
• Format: Article
• Language: English
• Subjects: Exact sciences and technology; Global analysis, analysis on manifolds; Mathematical analysis; Mathematics; Nonlinear equations; Operator theory; Partial differential equations; Sciences and techniques of general use; Theory; Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds
• ISSN: 0010-3640; 1097-0312
• DOI: 10.1002/cpa.20176

The theory of anti‐self‐dual (ASD) Lagrangians, introduced in [6], is developed further to allow for a variational resolution of nonlinear PDEs of the form Λu + Au + ∂φ(u) + f = 0 where φ is a convex lower‐semicontinuous 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 nonlinear operator that satisfies suitable continuity and antisymmetry 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 that 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 that has many applications, in particular to Navier‐Stokes‐type equations, to generalized Choquard‐Pekar Schrödinger equations with nonlocal 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 antisymmetric Hamiltonians and its applications is developed in detail in an upcoming monograph (7). © 2007 Wiley Periodicals, Inc.