Comptes Rendus
no. 4
Calculus of Variations
A theory of anti-selfdual Lagrangians: dynamical case
[Lagrangiens anti-autoduaux : le cas dynamique]

Présenté par : Gilles Pisier

Nassif Ghoussoub1
1 Department of Mathematics, University of British Columbia, Vancouver BC, Canada V6T 1Z2
Comptes Rendus. Mathématique, Volume 340 (2005) no. 4, pp. 325-330.

On considère le cas des lagrangiens anti-autoduaux qui dépendent du paramètre temps. Comme dans le cas stationnaire annoncé dans Ghoussoub [C. R. Acad. Sci. Paris, Ser. I 340 (2005)], cette classe possède des propriétés de permanence remarquables qui permettent une formulation et une résolution variationnelle de plusieurs équations paraboliques dissipatives qui ne sont pas normalement de type Euler–Lagrange.

We consider the class of time-dependent anti-selfdual Lagrangians, which – just like the stationary case announced in Ghoussoub [C. R. Acad. Sci. Paris, Ser. I 340 (2005)] – enjoys remarkable permanence properties and provides variational formulations and resolutions for several initial-value parabolic equations including gradient flows and other dissipative systems. Even though these evolutions do not fit in the standard Euler–Lagrange theory, we show that their solutions can be obtained as minima – but also as zeroes – of action functionals of the form ∫0TL(t,u(t),u˙(t)+Λtu(t)dt) where L is a time-dependent anti-selfdual Lagrangian and where Λt is a flow of skew-adjoint operators.

Reçu le :
Accepté le :
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DOI : 10.1016/j.crma.2004.12.008
Nassif Ghoussoub 1

1 Department of Mathematics, University of British Columbia, Vancouver BC, Canada V6T 1Z2 
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Nassif Ghoussoub. A theory of anti-selfdual Lagrangians: dynamical case. Comptes Rendus. Mathématique, Volume 340 (2005) no. 4, pp. 325-330. doi : 10.1016/j.crma.2004.12.008. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.12.008/

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[6] N. Ghoussoub, A variational principle for non-linear transport equations, Commun. Pure Appl. Anal. (2004), in press, p. 10

[7] N. Ghoussoub, Anti-selfdual Lagrangians: variational resolutions for non self-adjoint equations and dissipative evolutions (2004), submitted for publication

[8] N. Ghoussoub, Anti-selfdual Hamiltonians: variational resolutions for Navier–Stokes equations and other nonlinear evolutions (2004), in preparation

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[10] N. Ghoussoub; L. Tzou A variational principle for gradient flows, Math. Ann., Volume 30 (2004) no. 3, pp. 519-549

[11] N. Ghoussoub, L. Tzou, “Anti-selfdual Lagrangians: unbounded non self-adjoint operators and evolutions (2004), in preparation

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