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

TL;DR

This paper investigates the small dispersion limit of the Euler-Korteweg system with no-flux boundary conditions, using a relative energy approach to analyze convergence towards the Euler system without requiring additional conditions.

Contribution

It introduces a novel analysis method that accounts for boundary layers in the zero capillarity limit without assuming absence of anomalous energy concentration.

Findings

Proves convergence of weak solutions to strong solutions in the small dispersion limit.

Identifies a boundary layer correction necessary due to boundary conditions.

Shows the boundary layer is weaker than in vanishing viscosity scenarios.

Abstract

In this article, we study the small dispersion limit of the Euler-Korteweg system in a domain with a smooth boundary and no-flux boundary conditions. We exploit a relative energy approach to study the convergence of finite energy weak solutions towards strong solutions to the compressible Euler system. Given the boundary conditions under consideration, our approach requires a correction for the limiting particle density, due to the appearance of a boundary layer. Unlike conditional result on the vanishing viscosity limit, our analysis does not require additional conditions on the lack of anomalous concentration of capillary energy. This is due to the fact that the boundary layer appearing in our context is weaker than the one formed in the vanishing viscosity limit. We believe this approach can be adapted to study similar singular limits involving non-trivial boundary conditions.