Local weak solutions to a Navier-Stokes-nonlinear-Schr\"odinger model of superfluidity

TL;DR

This paper proves the local existence of weak solutions for a coupled Navier-Stokes and nonlinear Schrödinger system modeling superfluidity, providing the first rigorous mathematical analysis of this complex PDE system.

Contribution

It establishes the local existence of weak solutions for a coupled Navier-Stokes and nonlinear Schrödinger model of superfluidity, with new a priori estimates and energy inequalities.

Findings

Proved local existence of weak solutions in 3D bounded domains.

Derived a priori estimates for the coupled PDE system.

Established an energy inequality for the weak solutions.

Abstract

In a 1959 paper by Pitaevskii, a macroscopic model of superfluidity was derived from first principles, to describe the interacting dynamics between the superfluid and normal fluid phases of Helium-4. The model couples two of the most fundamental PDEs in mathematics: the nonlinear Schr\"odinger equation (NLS) and the Navier-Stokes equations (NSE). In this article, we show the local existence of weak solutions to this system (in a smooth bounded domain in 3D), by deriving the required a priori estimates. (We will also establish an energy inequality obeyed by the weak solutions constructed in Kim's 1987 paper for the incompressible, inhomogeneous NSE.) To the best of our knowledge, this is the first rigorous mathematical analysis of a bidirectionally coupled system of the NLS and NSE.