On well-posedness of the Cauchy problem for the wave equation in static spherically symmetric spacetimes

We give simple conditions implying the well-posedness of the Cauchy problem for the propagation of classical scalar fields in general (n+2)-dimensional static and spherically symmetric spacetimes. They are related to properties of the underlying spatial part of the wave operator, one of which being the standard essentially self-adjointness. However, in many examples the spatial part of the wave operator turns out to be not essentially selfadjoint, but it does satisfy a weaker property that we call here quasi essentially self-adjointness, which is enough to ensure the desired well-posedness. This is why we also characterize this second property. We state abstract results, then general results for a class of operators encompassing many examples in the literature, and we finish with the explicit analysis of some of them.Comment: 36 pages. Final version to appear in Classical and Quantum Gravit

Espaces critiques pour le système des equations de Navier-Stokes incompressibles

No modification to the text. This work was done when the first author was at Université de Picardie.In this work, we exhibit abstract conditions on a functional space E who insure the existence of a global mild solution for small data in E or the existence of a local mild solution in absence of size constraints for a class of semi-linear parabolic equations, which contains the incompressible Navier-Stokes system as a fundamental example. We also give an abstract criterion toward regularity of the obtained solutions. These conditions, given in terms of Littlewood-Paley estimates for products of spectrally localized elements of $E$, are simple to check in all known cases: Lebesgue, Lorents, Besov, Morrey... spaces. These conditions also apply to non-invariant spaces E and we give full details in the case of some 2-microlocal spaces. The following comments did not show on the first version: This article was written around 1998-99 and never published, because at that time, Koch and Tataru announced their result on well-posedness of Navier-stokes equations with initial data in $BMO^{-1}$. We believe though that some results and counterexamples here are of independent interest and we make them available electronically

On well-posedness of the Cauchy problem for the wave equation in static spherically symmetric spacetimes

We give simple conditions implying the well-posedness of the Cauchy problem for the propagation of classical scalar fields in general (n + 2)-dimensional static and spherically symmetric spacetimes. They are related to the properties of the underlying spatial part of the wave operator, one of which being the standard essentially self-adjointness. However, in many examples the spatial part of the wave operator turns out to be not essentially self-adjoint, but it does satisfy a weaker property that we call here quasi-essentially self-adjointness, which is enough to ensure the desired well-posedness. This is why we also characterize this second property. We state abstract results, then general results for a class of operators encompassing many examples in the literature, and we finish with the explicit analysis of some of them.Instituto de Física La Plat

Retrievial of a Time-Dependent Source in an Acoustic Propagation Problem

International audienceConsider two media separated by a plane interface, a time- dependent source F(t) at a point S in the first medium (of the lower celerity), receivers in the second medium, located at a large radial distance from the source. Thanks to a diffusion-like behaviour of the transmitted wave, we are able to retrieve the term F(t), under high frequencies hypothesis in an elementary way