Ill-posedness of nonlocal Burgers equations

Sylvie Benzoni-Gavage (1) , Jean-François Coulombel (2) , Nikolay Tzvetkov (3)

1 ICJ - Institut Camille Jordan
2 SIMPAF - SImulations and Modeling for PArticles and Fluids
3 AGM - UMR 8088 - Analyse, Géométrie et Modélisation

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.

Keywords: Elliptic regularity; Evolution equation; Well-posedness; Ill-posedness

Domains: Analysis of PDEs [math.AP]

Sylvie Benzoni-Gavage, Jean-François Coulombel, Nikolay Tzvetkov. Ill-posedness of nonlocal Burgers equations. Advances in Mathematics, 2011, 227 (6), pp.2220-2240. ⟨10.1016/j.aim.2011.04.017⟩. ⟨hal-00491136⟩