Global analysis of quasilinear wave equations on asymptotically de Sitter spaces

Hintz, Peter

Annales de l'Institut Fourier, Tome 66 (2016) no. 4, pp. 1285-1408.

On établit la resolubilité, sur une classe géométrique d’espaces-temps incluant les espaces asymptotiquement de Sitter, de certaines equations d’ondes et de Klein-Gordon dans des espaces de grande régularité. On obtient ces résultats en prouvant l’inversibilité globale d’opérateurs linéaires à coefficients dans des espaces de grande régularité de type $L^2$, et en utilisant des processus itératifs pour les problèmes non-linéaires. L’analyse linéaire est fait en deux étapes : premièrement, on développe une théorie de la régularité au moyen d’un calcul pour les opérateurs pseudo-différentiels à coefficients peu réguliers sur variétés à bord, similaire à celui développé par Beals et Reed. Deuxièmement, on étudie le comportement asymptotique des solutions des équations linéaires en utilisant des développements en états résonnants, introduite dans ce contexte par Vasy, dans le cadre de la b-analyse de Melrose.

We establish the small data solvability of suitable quasilinear wave and Klein-Gordon equations in high regularity spaces on a geometric class of spacetimes including asymptotically de Sitter spaces. We obtain our results by proving the global invertibility of linear operators with coefficients in high regularity $L^2$-based function spaces and using iterative arguments for the nonlinear problems. The linear analysis is accomplished in two parts: Firstly, a regularity theory is developed by means of a calculus for pseudodifferential operators with non-smooth coefficients, similar to the one developed by Beals and Reed, on manifolds with boundary. Secondly, the asymptotic behavior of solutions to linear equations is studied using resonance expansions, introduced in this context by Vasy using the framework of Melrose’s b-analysis.

Classification : 35L70, 35B40, 35S05, 58J47
Keywords: quasilinear wave equations, asymptotically de Sitter spaces, microlocal analysis, resonance expansions
Mot clés : équations d’ondes quasilinéaires, espaces asymptotiquement de Sitter, analyse microlocale, développements en états résonnants

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