Zero limit of dispersive-dissipative perturbed hyperbolic conservation laws

Abstract

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.