Global weak solutions to the compressible quantum navier-stokes equation and its semi-classical limit

TL;DR

This paper constructs global weak solutions for the quantum Navier-Stokes equations with bounded energy and entropy, and performs the semi-classical limit to classical Navier-Stokes equations, addressing degeneracies and integrability issues.

Contribution

It introduces a novel method to construct renormalized weak solutions that are uniform in the Planck constant, bypassing the need for Mellet-Vasseur inequality.

Findings

Existence of global weak solutions for quantum Navier-Stokes equations.

Successful semi-classical limit to classical Navier-Stokes equations.

Solutions are stable and do not require Mellet-Vasseur inequality.

Abstract

This paper is dedicated to the construction of global weak solutions to the quantum Navier-Stokes equation, for any initial value with bounded energy and entropy. The construction is uniform with respect to the Planck constant. This allows to perform the semi-classical limit to the associated compressible Navier-Stokes equation. One of the difficulty of the problem is to deal with the degenerate viscosity, together with the lack of integrability on the velocity. Our method is based on the construction of weak solutions that are renormalized in the velocity variable. The existence, and stability of these solutions do not need the Mellet-Vasseur inequality.