Finite-Energy Weak Solutions to the Quantum Isothermal Euler System via a Logarithmic Schr\"odinger Approximation

TL;DR

This paper constructs global weak solutions for the quantum isothermal Euler system using a logarithmic Schr"odinger approximation, providing a new framework for quantum hydrodynamics with isothermal internal energy.

Contribution

It introduces a rigorous logarithmic Schr"odinger approximation method to solve the quantum Euler system with isothermal pressure, expanding the analytical tools for quantum hydrodynamics.

Findings

Established existence of global weak solutions via Schr"odinger approximation.

Utilized Madelung transform and compactness to ensure strong convergence.

Provided a robust framework for QHD models with isothermal internal energy.

Abstract

This paper investigates the collisionless quantum hydrodynamic, or quantum Euler, system in \(\mathbb{T}^3\) with the linear pressure law \(P(\rho)=\rho\). Since this pressure is associated with the logarithmic internal energy \(f(\rho)=\rho\log\rho\), the model admits a natural logarithmic Schr\"odinger approximation. By means of a regularized logarithmic Schr\"odinger equation, we rigorously construct global weak solutions to the quantum isothermal Euler system. The proof relies on the Madelung transform, the polar decomposition of the wave functions, and compactness arguments. In particular, an energy identity is used to recover the strong convergence of the hydrodynamic variables. More broadly, the analysis provides a robust Schr\"odinger approximation framework for QHD models whose internal energy contains an isothermal component.