Global dissipative solutions of the defocusing isothermal Euler-Langevin-Korteweg equation

TL;DR

This paper establishes the existence of global dissipative solutions for a complex quantum fluid model with isothermal conditions, using approximation and entropy methods, in dimensions up to three.

Contribution

It introduces a novel approach to prove global dissipative solutions for the defocusing isothermal Euler-Langevin-Korteweg system, extending previous results to isothermal quantum fluids.

Findings

Existence of global weak solutions to the Navier-Stokes-Langevin-Korteweg system.

Construction of dissipative solutions for the Euler-Langevin-Korteweg system.

Application of relative entropy and inviscid limit techniques.

Abstract

We construct global dissipative solutions on the torus of dimension at most three of the defocusing isothermal Euler-Langevin-Korteweg system, which corresponds to the Euler-Korteweg system of compressible quantum fluids with an isothermal pressure law and a linear drag term with respect to the velocity. In particular, the isothermal feature prevents the energy and the BD-entropy from being positive. Adapting standard approximation arguments we first show the existence of global weak solutions to the defocusing isothermal Navier-Stokes-Langevin-Korteweg system. Introducing a relative entropy function satisfying a Gronwall-type inequality we then perform the inviscid limit to obtain the existence of dissipative solutions of the Euler-Langevin-Korteweg system.