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

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. Contents 1. Introduction 1 2. The isothermal Navier-Stokes-Langevin-Korteweg system 5 2.1. Regularized NSLK system 8 2.2. NSLK with drag forces 12 2.3. The limit r 0 , r 1 → 0 13 3. The isothermal Euler-Langevin-Korteweg system 15 3.1. A Gronwall inequality 17 3.2. The viscous limit ν → 0 21 Appendix A. Definition of the operators and technical lemmas 23 References 24