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Perte de régularité pour les équations d’ondes sur-critiques

Gilles Lebeau (2005)

Bulletin de la Société Mathématique de France

On prouve que le problème de Cauchy local pour l’équation d’onde sur-critique dans d , u + u p = 0 , p impair, avec d 3 et p > ( d + 2 ) / ( d - 2 ) , est mal posé dans H σ pour tout σ ] 1 , σ crit [ , où σ crit = d / 2 - 2 / ( p - 1 ) est l’exposant critique.

Probabilistic well-posedness for the cubic wave equation

Nicolas Burq, Nikolay Tzvetkov (2014)

Journal of the European Mathematical Society

The purpose of this article is to introduce for dispersive partial differential equations with random initial data, the notion of well-posedness (in the Hadamard-probabilistic sense). We restrict the study to one of the simplest examples of such equations: the periodic cubic semi-linear wave equation. Our contributions in this work are twofold: first we break the algebraic rigidity involved in our previous works and allow much more general randomizations (general infinite product measures v.s. Gibbs...

Propagation of singularities for the wave equation on manifolds with corners

András Vasy (2004/2005)

Séminaire Équations aux dérivées partielles

In this talk we describe the propagation of 𝒞 and Sobolev singularities for the wave equation on 𝒞 manifolds with corners M equipped with a Riemannian metric g . That is, for X = M × t , P = D t 2 - Δ M , and u H loc 1 ( X ) solving P u = 0 with homogeneous Dirichlet or Neumann boundary conditions, we show that WF b ( u ) is a union of maximally extended generalized broken bicharacteristics. This result is a 𝒞 counterpart of Lebeau’s results for the propagation of analytic singularities on real analytic manifolds with appropriately stratified boundary,...

Propagation through trapped sets and semiclassical resolvent estimates

Kiril Datchev, András Vasy (2012)

Annales de l’institut Fourier

Motivated by the study of resolvent estimates in the presence of trapping, we prove a semiclassical propagation theorem in a neighborhood of a compact invariant subset of the bicharacteristic flow which is isolated in a suitable sense. Examples include a global trapped set and a single isolated periodic trajectory. This is applied to obtain microlocal resolvent estimates with no loss compared to the nontrapping setting.

Quasilinear waves and trapping: Kerr-de Sitter space

Peter Hintz, András Vasy (2014)

Journées Équations aux dérivées partielles

In these notes, we will describe recent work on globally solving quasilinear wave equations in the presence of trapped rays, on Kerr-de Sitter space, and obtaining the asymptotic behavior of solutions. For the associated linear problem without trapping, one would consider a global, non-elliptic, Fredholm framework; in the presence of trapping the same framework is available for spaces of growing functions only. In order to solve the quasilinear problem we thus combine these frameworks with the normally...

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