Asymptotic behaviors of incompressible Schr\"{o}dinger flow for small data in three dimensions

TL;DR

This paper proves the global regularity and describes the long-term behavior of small-data solutions to the incompressible Schrödinger flow in three dimensions, a model capturing vortex dynamics in quantum fluids.

Contribution

It establishes the global existence and asymptotic properties of solutions for the incompressible Schrödinger flow with small initial data in three dimensions.

Findings

Global regularity for small initial data

Asymptotic decay of solutions

Use of Fourier analysis and vector fields method

Abstract

The incompressible Schr\"odinger flow is a Madelung's hydrodynamical form of quantum mechanics, which can simulate classical fluids with particular advantage in its simplicity and its ability of capturing thin vortex dynamics. This model enables robust simulation of intricate phenomena such as vortical wakes and interacting vortex filaments. In this article, we prove the global regularity and asymptotic behaviors for incompressible Schr\"odinger flow with small and localized data in three dimensions. We choose a suitable gauge to rewrite the system, and then use Fourier analysis and vector fields method to prove global existence and asymptotic behaviors.