On the mass transfer in the 3D Pitaevskii model

TL;DR

This paper analyzes a 3D superfluidity model combining nonlinear Schrödinger and Navier-Stokes equations, proving the existence of weak solutions and addressing mass transfer control between superfluid and normal fluid phases.

Contribution

It establishes the existence of weak solutions for the coupled superfluid-normal fluid system with power-type nonlinearity, handling mass transfer challenges.

Findings

Proved existence of weak solutions in 3D torus for small initial data.

Developed methods to control inter-phase mass transfer.

Ensured positivity of normal fluid density over time.

Abstract

We examine a micro-scale model of superfluidity derived by Pitaevskii in 1959 which describes the interacting dynamics between superfluid He-4 and its normal fluid phase. This system consists of the nonlinear Schr\"odinger equation and the incompressible, inhomogeneous Navier-Stokes equations, coupled to each other via a bidirectional nonlinear relaxation mechanism. The coupling permits mass/momentum/energy transfer between the phases, and accounts for the conversion of superfluid into normal fluid. We prove the existence of weak solutions in $\mathbb{T}^3$ for a power-type nonlinearity, beginning from small initial data. The main challenge is to control the inter-phase mass transfer in order to ensure the strict positivity of the normal fluid density, while obtaining time-independent a priori estimates.