Exemples d'instabilités pour des équations d'ondes non linéaires

Métivier, Guy

Séminaire Bourbaki : volume 2002/2003, exposés 909-923, Astérisque, no. 294 (2004), Exposé no. 911, pp. 63-75.

Le but de l’exposé est de donner un guide de lecture pour un article de Gilles Lebeau où il est montré que le problème de Cauchy pour l’équation d’onde surcritique $({\partial }_t^2-{\Delta }_x)u+u^p=0$ est mal posé au sens de Hadamard dans l’espace d’énergie, pour $p\ge 7$ en dimension 3. La preuve repose sur des constructions d’optique géométrique et des analyses d’instabilité dans des régimes fortement non linéaires. On donnera les étapes de l’analyse en essayant de les situer dans leur contexte plus général : construction de solutions asymptotiques où équations eikonales et équations d’amplitudes sont liées, mécanismes d’instabilité linéaires et non linéaires par résonances et interactions.

The aim of the lecture is to provide a guide to a paper by Gilles Lebeau where it is shown that the Cauchy problem for the overcritical wave equation $({\partial }_t^2-{\Delta }_x)u+u^p=0$ is ill-posed in the sense of Hadamard in the energy space, for $p\ge 7$ in dimension 3. The proof relies on geometric optics constructions and analyses of instabilities in highly nonlinear regimes. We shall give the steps of the analysis, trying to place them in their more general context: construction of asymptotic solutions where eikonal equations and amplitude equations are linked, linear and nonlinear instability mechanisms by resonance and interactions.

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