On well-posedness of quantum fluid systems in the class of dissipative solutions

TL;DR

This paper investigates the well-posedness of quantum fluid systems, specifically the quantum Navier-Stokes and Euler equations, establishing existence, uniqueness, and semiflow properties within dissipative solutions.

Contribution

It introduces the concept of dissipative solutions for quantum fluid systems and proves global existence, weak-strong uniqueness, and semiflow selection results.

Findings

Global existence for finite energy initial data

Weak-strong uniqueness principle established

Semiflow selection in dissipative solutions class

Abstract

The main objects of the present work are the quantum Navier-Stokes and quantum Euler systems; for the first one, in particular, we will consider constant viscosity coefficients. We deal with the concept of dissipative solutions, for which we will first prove the weak-strong uniqueness principle and afterwards, we will show the global existence for any finite energy initial data. Finally, we will prove that both systems admit a semiflow selection in the class of dissipative solutions.