Semiclassical limit of the Gross-Pitaevskii equation in an exterior domain

In this paper, we study the semiclassical limit of the Gross-Pitaevskii equation (a cubic nonlinear Schrödinger equation) with the Neumann boundary condition in an exterior domain. We prove that before the formation of singularities in the limit system, the quantum density and the quantum momentum converge to the unique solution of the compressible Euler equation with the slip boundary condition as the scaling parameter approaches 0.