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- W1569101238 abstract "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 $L^2(M)$, almost every function belong to all spaces $L^p(M)$, $p<+infty$. We then give applications to the study of the growth of the $L^p$ norms of spherical harmonics on spheres $mathbb{S}^d$: we prove (again for natural probability measures) that almost every Hilbert base of $L^2(mathbb{S}^d)$ made of spherical harmonics has all its elements uniformly bounded in all $L^p(mathbb{S}^d), p<+infty$ spaces. We also prove similar results on tori $mathbb{T}^d$. We give then an application to the study of the decay rate of damped wave equations in a frame-work where the geometric control property on 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." @default.
- W1569101238 created "2016-06-24" @default.
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- W1569101238 date "2011-11-30" @default.
- W1569101238 modified "2023-09-27" @default.
- W1569101238 title "Injections de Sobolev probabilistes et applications" @default.
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