This paper studies a family of finite volume schemes for the hyperbolic scalar conservation law on a closed Riemannian manifold M. For an initial value in BV(M) we will show that these schemes converge with a convergence rate towards the entropy solution. When M is 1-dimensional the schemes are TVD and we will show that this improves the convergence rate to
This paper is devoted to the proof of almost global existence results for Klein-Gordon equations on compact revolution hypersurfaces with non-Hamiltonian nonlinearities, when the data are smooth, small and radial. The method combines normal forms with the fact that the eigenvalues associated to radial eigenfunctions of the Laplacian on such manifolds are simple and satisfy convenient asymptotic expansions.
En utilisant une méthode dépendante du temps, nous démontrons la complétude asymptotique pour l'équation des ondes dans une classe d'espaces-temps stationnaires et asymptotiquement plats. On introduit l'observable de vitesse asymptotique et on décrit son spectre (sous des hypothèses plus faibles que pour la complétude asymptotique). Les méthodes utilisées sont inspirées par celles de l'analyse du problème à deux corps en mécanique quantique.
In this paper we consider a smooth and bounded domain of dimension with boundary and we construct sequences of solutions to the wave equation with Dirichlet boundary condition which contradict the Strichartz estimates of the free space, providing losses of derivatives at least for a subset of the usual range of indices. This is due to microlocal phenomena such as caustics generated in arbitrarily small time near the boundary. Moreover, the result holds for microlocally strictly convex domains...
In this paper basic differential invariants of generic hyperbolic Monge-Ampère equations with respect to contact transformations are constructed and the equivalence problem for these equations is solved.
There has been much progress in recent years in understanding the existence problem for wave maps with small critical Sobolev norm (in particular for two-dimensional wave maps with small energy); a key aspect in that theory has been a renormalization procedure (either a geometric Coulomb gauge, or a microlocal gauge) which converts the nonlinear term into one closer to that of a semilinear wave equation. However, both of these renormalization procedures encounter difficulty if the energy of the...
In this paper we consider the modified wave equation associated with a class of radial Laplacians generalizing the radial part of the Laplace-Beltrami operator on hyperbolic spaces or Damek-Ricci spaces. We show that the Huygens’ principle and the equipartition of energy hold if the inverse of the Harish-Chandra -function is a polynomial and that these two properties hold asymptotically otherwise. Similar results were established previously by Branson, Olafsson and Schlichtkrull in the case of...
We prove the short-time existence of the hyperbolic inverse (mean) curvature flow (with or without the specified forcing term) under the assumption that the initial compact smooth hypersurface of () is mean convex and star-shaped. Several interesting examples and some hyperbolic evolution equations for geometric quantities of the evolving hypersurfaces are shown. Besides, under different assumptions for the initial velocity, we can get the expansion and the convergence results of a hyperbolic...