A note on nonlinear KdV-type equations

Abstract

We consider the approximation of the inviscid Burgers equation by nonlinear Korteweg-de Vries (KdV) type equations. It has been conjectured by Brenier and Levy [1] that in some special kind of nonlinear dispersion the behaviour is dissipative, when generally we expect a dispersive behaviour as in the linear case. We provide here a priori estimates enough to establish the first step in a proof of the conjecture above.