Ill-posedness of nonlocal Burgers equations

Benzoni-Gavage, Sylvie; Coulombel, Jean-François; Tzvetkov, Nikolay

Document type :

Compte-rendu et recension critique d'ouvrage

DOI :

10.1016/j.aim.2011.04.017

Title :

Ill-posedness of nonlocal Burgers equations

Author(s) :

Benzoni-Gavage, Sylvie [Auteur correspondant]
Institut Camille Jordan [ICJ]
Coulombel, Jean-François [Auteur]
SImulations and Modeling for PArticles and Fluids [SIMPAF]
Tzvetkov, Nikolay [Auteur]
Analyse, Géométrie et Modélisation [AGM - UMR 8088]

Journal title :

Advances in Mathematics

Pages :

2220-2240

Publisher :

Elsevier

Publication date :

2011-08-20

ISSN :

0001-8708

English keyword(s) :

Elliptic regularity
Evolution equation
Well-posedness
Ill-posedness

HAL domain(s) :

Mathématiques [math]/Equations aux dérivées partielles [math.AP]

English abstract : [en]

Nonlocal generalizations of Burgers equation were derived in earlier work by Hunter [Contemp. Math. 1989], and more recently by Benzoni-Gavage and Rosini [Comput. Math. Appl. 2009], as weakly nonlinear amplitude equations ...
Show more >Nonlocal generalizations of Burgers equation were derived in earlier work by Hunter [Contemp. Math. 1989], and more recently by Benzoni-Gavage and Rosini [Comput. Math. Appl. 2009], as weakly nonlinear amplitude equations for hyperbolic boundary value problems admitting linear surface waves. The local-in-time well-posedness of such equations in Sobolev spaces was proved by Benzoni-Gavage [Diff. Int. Eq. 2009] under an appropriate stability condition originally pointed out by Hunter. In this article, it is shown that the latter condition is not only sufficient for well-posedness in Sobolev spaces but also necessary. The main point of the analysis is to show that when the stability condition is violated, nonlocal Burgers equations reduce to second order elliptic PDEs. The resulting ill-posedness result encompasses various cases previously studied in the literature.Show less >

Language :

Anglais

Popular science :

Non

ANR Project :

Interaction d'ondes compressibles

Collections :