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A convergence result for finite volume schemes on Riemannian manifolds

Jan Giesselmann (2009)

ESAIM: Mathematical Modelling and Numerical Analysis

This paper studies a family of finite volume schemes for the hyperbolic scalar conservation law u t + ∇ g · f ( x , u ) = 0 on a closed Riemannian manifold M. For an initial value in BV(M) we will show that these schemes converge with a h 1 4 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 h 1 2 .

Almost global solutions for non hamiltonian semi-linear Klein-Gordon equations on compact revolution hypersurfaces

Jean-Marc Delort, Jérémie Szeftel (2005)

Journées Équations aux dérivées partielles

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.

Complétude asymptotique pour l'équation des ondes dans une classe d'espaces-temps stationnaires et asymptotiquement plats

Dietrich Häfner (2001)

Annales de l’institut Fourier

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.

Counterexamples to the Strichartz inequalities for the wave equation in general domains with boundary

Oana Ivanovici (2012)

Journal of the European Mathematical Society

In this paper we consider a smooth and bounded domain Ω ⊂ ℝ d of dimension d ≥ 2 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...

Differential invariants of generic hyperbolic Monge-Ampère equations

Michal Marvan, Alexandre Vinogradov, Valery Yumaguzhin (2007)

Open Mathematics

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.

Geometric renormalization of large energy wave maps

Terence Tao (2004)

Journées Équations aux dérivées partielles

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...

Huygens’ principle and equipartition of energy for the modified wave equation associated to a generalized radial Laplacian

Jamel El Kamel, Chokri Yacoub (2005)

Annales mathématiques Blaise Pascal

In this paper we consider the modified wave equation associated with a class of radial Laplacians L 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 c -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...

Hyperbolic inverse mean curvature flow

Jing Mao, Chuan-Xi Wu, Zhe Zhou (2020)

Czechoslovak Mathematical Journal

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 ℝ n + 1 ( n ≥ 2 ) 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...

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