Global dissipative solutions of the defocusing isothermal Euler–Langevin–Korteweg equations

Article type: Research Article

Authors: Chauleur, Quentin^{; *}

Affiliations: Univ Rennes, CNRS, IRMAR – UMR 6625, F-35000 Rennes, France. E-mail: quentin.chauleur@ens-rennes.fr

Correspondence: [*] Corresponding author. E-mail: quentin.chauleur@ens-rennes.fr.

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.

Keywords: Euler–Langevin–Korteweg system, Navier–Stokes–Langevin–Korteweg system, relative entropy estimates, dissipative solutions, augmented systems

DOI: 10.3233/ASY-211681

Journal: Asymptotic Analysis, vol. 126, no. 3-4, pp. 255-283, 2022