Semiclassical microlocal normal forms and global solutions of modified one-dimensional KG equations

Delort, Jean-Marc

Annales de l'Institut Fourier, Tome 66 (2016) no. 4, pp. 1451-1528.

La méthode des champs de Klainerman joue un rôle essentiel dans l’étude de l’existence globale de solutions d’équations aux dérivées partielles hyperboliques non-linéaires à données petites, régulières, décroissantes à l’infini. Toutefois, certaines équations issues de la physique, comme l’équation des ondes de gravité en profondeur finie, ne possèdent pas de champ de Klainerman. Le but de cet article est de développer, sur une équation modèle, un substitut à la méthode des champs de Klainerman, qui permette d’obtenir des résultats d’existence globale, même dans le cas critique pour lequel il n’y a pas diffusion linéaire à l’infini. L’idée essentielle est d’utiliser des opérateurs pseudo-différentiels semi-classiques au lieu de champs de vecteurs, combinés avec une méthode de formes locales microlocale, afin de réduire la non-linéarité à des expressions pour lesquelles une règle de Leibniz est valable pour de tels opérateurs.

The method of Klainerman vector fields plays an essential role in the study of global existence of solutions of nonlinear hyperbolic PDEs, with small, smooth, decaying Cauchy data. Nevertheless, it turns out that some equations of physics, like the one dimensional water waves equation with finite depth, do not possess any Klainerman vector field. The goal of this paper is to design, on a model equation, a substitute to the Klainerman vector fields method, that allows one to get global existence results, even in the critical case for which linear scattering does not hold at infinity. The main idea is to use semiclassical pseudodifferential operators instead of vector fields, combined with microlocal normal forms, to reduce the nonlinearity to expressions for which a Leibniz rule holds for these operators.

Classification : 35L71, 35A01, 35B40
Keywords: Global solution of Klein-Gordon equations, Klainerman vector fields, Microlocal normal forms, Semiclassical analysis
Mot clés : Solutions globales d’équations de Klein-Gordon, champs de Klainerman, formes normales microlocales, analyse semi-classique

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