On the zero capillarity limit for the Euler-Korteweg system

TL;DR

This paper rigorously analyzes the zero capillarity limit of the Euler-Korteweg system in full space and half space, establishing convergence to Euler equations and exploring boundary layer phenomena.

Contribution

It provides a rigorous proof of the zero capillarity limit in multiple dimensions and introduces a BKW expansion with boundary layers for half-space problems.

Findings

Convergence of Euler-Korteweg to Euler equations as capillarity vanishes

Development of a BKW expansion capturing boundary layer effects

Extension of classical semi-classical limit results to new settings

Abstract

We study the Euler-Korteweg equations with a weak capillarity tensor. It formally converges to the Euler equations in the zero capillarity limit. Our aim is two-fold : first we prove rigorously this limit in R d , d $\ge$ 1, and obtain a more precise BKW expansion of the solution, second we initiate the study of the problem on the half space. In this case we obtain a priori estimates for the solutions that degenerate as the capillary coefficient converges to zero, and we explain this degeneracy with the construction of a (formal) BKW expansion that exhibits boundary layers. The results on the full space extend and improve a classical result of Grenier (1998) on the semi-classical limit of nonlinear Schr{\"o}dinger equations. The analysis on the half space is restricted to the case of quantum fluids with irrotational velocity.