Behaviour of some 3rd order pde’s

Abstract

We are concerned with zero limits for nonlinear hyperbolic conservation laws since dissipative and dispersive small scale effects of diffusion and capillarity are under consideration. We consider the behaviour and selection of both the right models and the physical solutions. We intend to focus on technical difficulties we need to overcome to prove a general “vanishing viscosity-­‐capillarity method” and comment on the analytical proof of such a convergence for nonlinear generalized Korteweg-­‐de Vries equations. This was first conjectured by Y. Brenier and D. Levy, based upon by numerical evidence [Dissipative behavior of some fully non-­‐linear KdV-­‐type equations, Physica D 137 (2000)].