Anti-symmetric Hamiltonians (II): Variational resolutions for Navier–Stokes and other nonlinear evolutions

Abstract

The nonlinear selfdual variational principle established in a preceeding paper [8] -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 formwhere 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.

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

“…However, in order to obtain a solution of the equation (5.16), we need to show that the infimum is actually 0. The argument requires a further refinement of Theorem 2.8 and is postponed to a forthcoming paper [9].…”

“…This is illustrated in Section 5 by an application to the complex Ginsburg-Landau initial-value problem with various parameters. Further applications to other models in hydrodynamics and magnetohydrodynamics will follow in a forthcoming paper [9]. The general theory of antisymmetric Hamiltonians is detailed in the upcoming monograph [7].…”

“…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].…”

Hamiltonian systems of PDEs with selfdual boundary conditions

“…on an appropriate path space A 2 H and by showing that the infimum is zero.This point of view has been developed in a series of recent papers [13,14,17,18], and extends considerably the ideas of Brezis and Ekeland [4] and Auchmuty [2,3]. However, several new phenomena emerge while dealing with evolutions of the form (1) and (2), and many useful new techniques are introduced here to selfdual variational calculus.…”

“…in [13,14,18]-to write these evolution equations as u(t) + Au(t) ∈ −∂ L(t, u(t)) u(T )+u(0) 2 ∈ −∂ (u(0) − u(T )), (1) and the Hamiltonian systems as…”