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.

We consider the initial value problem for full nonlinear dissipative-dispersive perturbations of multidimensional scalar hyperbolic conservation laws, say generalized KdV-Burgers equations. And, as the perturbations vanish, we analyse the convergence of solutions for such problem to the classical entropy weak solution of the limit hyperbolic conservation laws. This is a step for the proof of a “vanishing viscosity-capillarity method”. We use the setting of DiPerna’s measure-valued solution uniqueness result. The class of equations under consideration have the form of \pa_t u+div f(u)=\eps div b(u,\grad u)+\del div \pa_(\xi) c(u,\grad u), which include generalized Korteweg-de Vries-Burgers equation (when \xi is a space variable) and Benjamin-Bona-Mahony-Burgers equation (when \xi is the time variable), or that of \pa_t u+div f(u)=\del div c(u,\grad \pa_(\xi)u), which can present unexpected dissipative properties.

In this presentation we attempt to stress two points of view on hyperbolic conservation laws: modelization and analytical theory. And, how they are sensitively related. While appliers are concerned with reliability, integrity or failure of solutions, mathematicians are concerned with non uniqueness, selection of physically relevant solutions or entropy criteria. In the modeling process, within simplifications, some “spurious terms” are usually discarded from the equations and so, in order to address uniqueness, a crucial information is lost. We discuss here the relevant dissipative or dispersive effect of some of those small scale terms (zero singular limits).

In this presentation we attempt to stress two points of view on hyperbolic conservation laws: modelization and analytical theory. And, how they are sensitively related. While appliers are concerned with reliability, integrity or failure of solutions, mathematcians are concerned with non uniqueness, selection of physically relevant solutions or entropy criteria. In the modeling process, within simplifications, some “spurious terms” are usually discarded from the equations and so, in order to address uniqueness, a crucial information is lost. We discuss here the relevant dissipative or dispersive effect of some of those small scale terms (zero singular limits).