Anti-symmetric Hamiltonians (II): Variational resolutions for Navier–Stokes and other nonlinear evolutions
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Bibliographic record
Abstract
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 form I(u)\! =\!\! \int \limits_{0}^{T}\!\left[L\left(t,u(t),\.{u}(t)\! +\! \Lambda u(t)\right) +\langle \Lambda u(t),u(t)\rangle \right] dt\! +\! ℓ\! \left(u(0)\!−\!u(T),\frac{u(T)\! +\! u(0)}{2}\right) 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) = \alpha u(T) for any given α in (−1,1) . Our approach is quite general and does apply to many other situations. Résumé Le principe variationnel auto-dual nonlinéaire établi par le premier auteur dans un article antérieur – quoique suffisant pour les équations nonlinéaires stationnaires – ne couvrait pas le cas des équations d'évolution de Navier–Stokes. Celà est dû aux hypothèses de coercivité forte requises, qui sont rarement satisfaites par les fonctionnelles auto-duales une fois définies sur les espaces de trajectoires. Dans cet article, on établit un nouveau principe variationnel qui s'applique à des fonctionnelles de la forme I(u)\! =\!\! \int \limits_{0}^{T}\!\left[L\left(t,u(t),\.{u}(t)\! +\! \Lambda u(t)\right) +\langle \Lambda u(t),u(t)\rangle \right] dt\! +\! ℓ\! \left(u(0)\!−\!u(T),\frac{u(T)\! +\! u(0)}{2}\right) où L (resp., ℓ ) est un Lagrangien anti-autodual sur l'espace des états (resp., sur la frontière), et Λ est un opérateur convenable sur un espace de trajectoires. Comme application, on retrouve variationellement entre autres, les solutions de Leray pour les équations de Navier–Stokes en dimension 2 et 3 avec, soit des conditions initiales, ou soit des conditions au bord de type périodiques. L'approche est assez générale pour s'appliquer à d'autres équations d'évolution non linéaire.
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| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.003 | 0.001 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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