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

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.