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

Abstract

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 u + 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 : D( ) ⊂ 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 formu(t) + u(t) + Au(t) + ∂ϕ(u(t)) + f = 0 starting at u(0) = u 0 . 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].

A variational approach to Navier–Stokes

“…A complete variational resolution has been provided by Ghoussoub within the theory of self-dual Lagrangians [20]. In particular, in [19,21] the Navier-Stokes system is reformulated in terms of a so-called null-minimization principle, namely the attainment the value 0 of a specific nonnegative functional inspired by Fenchel's duality. Such attainment is then ascertained in [19,21], giving rise to a complete existence theory for periodic-in-time solutions.…”

A Self‐Dual Polar Factorization for Vector Fields

“…It is convex-concave and satisfies H L .q; p/ Ä H L .p; q/. In most concrete examples, it is actually antisymmetric [9]. If now A is a maximal monotone operator, then A 1 is also maximal monotone and therefore can be written as A 1 D x @L, where L is a self-dual Lagrangian on X X that can be constructed in the following way: First, let (1.20) N.p; x/ D supfhp; yi C hq; x yi W .y; q/ 2 Graph.A/g be the Fitzpatrick function [5] associated to A.…”

A theory of anti-selfdual Lagrangians: stationary case

“…The case of linear unbounded operators is dealt with in [8]. Non-linear but appropriately defined 'skew-adjoint' operators such as those appearing in the Navier-Stokes and other equations of hydrodynamics will be considered in [7]. This theme also appears in the conjectures of Brezis-Ekeland [4] and of Auchmuty [1] which were resolved in [6] and [8].…”