The zero capillarity limit for the Euler-Korteweg system with no-flux boundary conditions

In this article, we study the small dispersion limit of the Euler-Korteweg system in a bounded domain with no-flux boundary conditions. We exploit a relative entropy approach to study the convergence of finite energy weak solutions towards strong solutions to the compressible Euler system. Since we consider non-trivial boundary conditions, our approach needs a correction for the limiting particle density, due to the appearance of a (weak) boundary layer. We believe this approach can be adapted to study similar singular limits involving non-trivial boundary conditions.