# Global weak solutions for quantum isothermal fluids

## TL;DR

This paper establishes the existence of global weak solutions for quantum isothermal fluids, including Korteweg and Navier-Stokes equations, using a reformulation and approximation techniques in unbounded domains.

## Contribution

It introduces a novel reformulation with a time-dependent rescaling and extends the existence theory to include Korteweg and quantum Navier-Stokes equations in three dimensions.

## Key findings

- Global weak solutions constructed for quantum Navier-Stokes equations.
- Existence results for isothermal Korteweg equations with well-prepared initial data.
- Methodology applicable to unbounded domains via torus approximation and renormalized solutions.

## Abstract

We construct global weak solutions to isothermal quantum Navier-Stokes equations, with or without Korteweg term, in the whole space of dimension at most three. Instead of working on the initial set of unknown functions, we consider an equivalent reformulation, based on a time-dependent rescaling, that we introduced in a previous paper to study the large time behavior, and which provides suitable a priori estimates, as opposed to the initial formulation where the potential energy is not signed. We proceed by working on tori whose size eventually becomes infinite. On each fixed torus, we consider the equations in the presence of drag force terms. Such equations are solved by regularization, and the limit where the drag force terms vanish is treated by resuming the notion of renormalized solution developed by I. Lacroix-Violet and A. Vasseur. We also establish global existence of weak solutions for the isothermal Korteweg equation (no viscosity), when initial data are well-prepared, in the sense that they stem from a Madelung transform.