Injections de Sobolev probabilistes et applications

Burq, Nicolas; Lebeau, Gilles

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 46 (2013) no. 6, pp. 917-962.

On démontre dans cet article des versions probabilistes des injections de Sobolev sur une variété riemannienne compacte, $(M,g)$. Plus précisément on démontre que pour des mesures de probabilité naturelles sur l’espace $L^2\left(M\right)$, presque toute fonction appartient à tous les espaces $L^p\left(M\right)$, $p<+\infty$. On donne ensuite des applications à l’étude des harmoniques sphériques sur la sphère ${𝕊}^d$ : on démontre (encore pour des mesures de probabilité naturelles) que presque toute base hilbertienne de $L^2\left({𝕊}^d\right)$ formée d’harmoniques sphériques a tous ses éléments uniformément bornés dans tous les espaces $L^p\left({𝕊}^d\right),p<+\infty$. On démontre aussi des résultats similaires sur les tores ${𝕋}^d$. On donne aussi une application à l’étude du taux de décroissance de l’équation des ondes amortie dans un cadre où la condition de contrôle géométrique de Bardos, Lebeau et Rauch n’est pas vérifiée. En supposant le flot ergodique, on démontre qu’il existe sur des ensembles de mesure arbitrairement proche de $1$ (dans l’espace des données initiales d’énergie finie), un taux de décroissance uniforme. Finalement, on conclut avec une application à l’étude de l’équation des ondes semi-linéaire $H^1$-surcritique, pour laquelle on démontre que pour presque toute donnée initiale, les solutions faibles sont fortes et uniques (localement en temps).

In this article, we give probabilistic versions of Sobolev embeddings on any Riemannian manifold $(M,g)$. More precisely, we prove that for natural probability measures on $L^2\left(M\right)$, almost every function belongs to all spaces $L^p\left(M\right)$, $p<+\infty$. We then give applications to the study of the growth of the $L^p$ norms of spherical harmonics on spheres ${𝕊}^d$: we prove (again for natural probability measures) that almost every Hilbert base of $L^2\left({𝕊}^d\right)$ made of spherical harmonics has all its elements uniformly bounded in all $L^p\left({𝕊}^d\right),p<+\infty$ spaces. We also prove similar results on tori ${𝕋}^d$. We give then an application to the study of the decay rate of damped wave equations in a framework where the geometric control property of Bardos-Lebeau-Rauch is not satisfied. Assuming that it is violated for a measure $0$ set of trajectories, we prove that there exists almost surely a rate. Finally, we conclude with an application to the study of the $H^1$-supercritical wave equation, for which we prove that for almost all initial data, the weak solutions are strong and unique, locally in time.

Classification : 35PXX, 35J05, 35L05, 46T12, 28C20
Mot clés : concentration de la mesure, formule de Weyl, ondes amorties, ondes non linéaires, fonctions propres
Keywords: measure concentration, Weyl formula, damped wave equations, nonlinear wave equations, eigenfunctions

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