Small-data global existence of solutions for the Pitaevskii model of superfluidity

TL;DR

This paper proves the global existence of solutions for a coupled superfluidity model involving nonlinear Schrödinger and Navier-Stokes equations, specifically in small-data regimes on a two-dimensional torus.

Contribution

It establishes the first global existence results for the Pitaevskii superfluidity model with small initial data, considering different nonlinearities in the Schrödinger component.

Findings

Global solutions exist for small initial data in the model.

Results cover both strong and weak solution regimes.

The analysis applies to the two-dimensional torus setting.

Abstract

We investigate a micro-scale model of superfluidity derived by Pitaevskii in 1959 to describe the interacting dynamics between the superfluid and normal fluid phases of Helium-4. The model involves the nonlinear Schr\"odinger equation (NLS) and the Navier-Stokes equations (NSE), coupled to each other via a bidirectional nonlinear relaxation mechanism. Depending on the nature of the nonlinearity in the NLS, we prove global/almost global existence of solutions to this system in $\mathbb{T}^2$ -- strong in wavefunction and velocity, and weak in density.