Injections de Sobolev probabilistes et applications

TL;DR

This paper develops probabilistic Sobolev embeddings on Riemannian manifolds and applies these results to analyze the behavior of eigenfunctions, wave equations, and solutions to supercritical wave equations, revealing almost sure properties.

Contribution

It introduces probabilistic versions of Sobolev embeddings on manifolds and applies them to eigenfunctions, wave equations, and supercritical wave equations, providing new almost sure results.

Findings

Almost every function in $L^2(M)$ belongs to all $L^p(M)$ spaces for $p<+ obreak ext{ extendash}+ obreak ext{ extendash}$

Almost every spherical harmonic basis element is uniformly bounded in all $L^p$ spaces

Existence of almost sure decay rates for damped wave equations under geometric control violations

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.