Logarithmic Schr{\"o}dinger equation and isothermal fluids

TL;DR

This paper explores the large-time behavior of logarithmic Schrödinger equations and various isothermal fluid models, highlighting connections that provide insights into their dynamics and long-term properties.

Contribution

It establishes links between logarithmic Schrödinger equations and isothermal fluid models, offering heuristic insights and a unified perspective on their asymptotic behaviors.

Findings

Connections between Schrödinger and fluid equations elucidated

Heuristic arguments provide intuition for long-term dynamics

Insights applicable to multiple models and equations

Abstract

We consider the large time behavior in two types of equations, posed on the whole space R^d: the Schr{\"o}dinger equation with a logarithmic nonlinearity on the one hand; compressible, isothermal, Euler, Korteweg and quantum Navier-Stokes equations on the other hand. We explain some connections between the two families of equations, and show how these connections may help having an insight in all cases. We insist on some specific aspects only, and refer to the cited articles for more details, and more complete statements. We try to give a general picture of the results, and present some heuristical arguments that can help the intuition, which are not necessarily found in the mentioned articles.