Finite-Energy Weak Solutions to the Quantum Isothermal Euler System via a Logarithmic Schrödinger Approximation

1. Problem Statement

The paper addresses the mathematical challenge of constructing global finite-energy weak solutions to the collisionless quantum hydrodynamic (QHD) system, specifically the quantum isothermal Euler system, in a three-dimensional flat torus ($T^3$).

The system is governed by the equations:
$$\begin{cases} \partial_t \rho + \text{div } J = 0, \\ \partial_t J + \text{div}\left(\frac{J \otimes J}{\rho}\right) + \nabla P(\rho) = \frac{\hbar^2}{2} \rho \nabla \left(\frac{\Delta \sqrt{\rho}}{\sqrt{\rho}}\right), \end{cases}$$
where the pressure law is linear, $P(\rho) = \rho$. This specific pressure law corresponds to a logarithmic internal energy $f(\rho) = \rho \log \rho$.

Key Difficulties:

• Singular Nonlinearity: The logarithmic term $\log |\psi|^2$ in the associated wave equation diverges as the density approaches zero (vacuum), making direct analysis delicate.
• Lack of Compactness: In the absence of viscosity (collisionless case), standard tools like the Bresch–Desjardins (BD) entropy, which provide control over $\nabla \sqrt{\rho}$ in viscous quantum Navier-Stokes systems, are unavailable.
• Failure of Strichartz Estimates: Previous approaches for QHD systems with power-law pressures ($P(\rho) = \rho^\gamma, \gamma > 1$) relied on Strichartz estimates for the wave function gradient. These are not applicable to the logarithmic Schrödinger framework.

2. Methodology

The author employs a regularized logarithmic Schrödinger approximation combined with compactness arguments and the polar decomposition of the wave function.

A. Regularized Approximation

To handle the singularity at vacuum, the author introduces a regularized logarithmic Schrödinger equation with a parameter $\delta > 0$:
$$i\hbar \partial_t \psi_\delta + \frac{\hbar^2}{2} \Delta \psi_\delta = \left(\log(|\psi_\delta|^2 + \delta) + 1\right)\psi_\delta.$$
This regularization removes the singularity while preserving the logarithmic energy structure associated with the isothermal pressure law. The hydrodynamic variables are defined via the Madelung transform:
$$\rho_\delta = |\psi_\delta|^2, \quad J_\delta = \hbar \Im(\psi_\delta \nabla \psi_\delta).$$

B. Polar Decomposition and Quadratic Structure

A central tool is the polar decomposition $\psi = \sqrt{\rho} e^{iS}$ (formally), which leads to the momentum variable $\Lambda = \hbar \sqrt{\rho} \nabla S$.
The analysis relies on the fundamental quadratic identity:
$$\hbar^2 |\nabla \psi|^2 = \hbar^2 |\nabla \sqrt{\rho}|^2 + |\Lambda|^2.$$
This identity decomposes the kinetic energy into a "quantum pressure" term ($\nabla \sqrt{\rho}$) and a "kinetic" term ($\Lambda$). Crucially, the pair $(\nabla \sqrt{\rho}, \Lambda)$ possesses better stability properties than the velocity field itself, which is essential for the compactness argument.

C. Compactness Strategy

1. Global Well-Posedness: For fixed $\delta$, the regularized equation is globally well-posed in $H^1(T^3)$ with conserved mass and energy.
2. Weak Convergence: Using uniform $H^1$ bounds and the Aubin-Lions lemma, a subsequence of $\psi_\delta$ converges weakly to a limit $\psi$.
3. Energy Equality: A critical step involves proving an energy equality for the limiting weak solution of the logarithmic Schrödinger equation. By establishing that the energy is conserved ($E(t) = E(0)$) and utilizing the convergence of the nonlinear energy terms, the author proves the strong convergence of the gradients:
   $$\nabla \psi_\delta \to \nabla \psi \quad \text{strongly in } L^2.$$
4. Hydrodynamic Limit: The strong convergence of $\nabla \psi_\delta$ implies, via the polar decomposition, the strong convergence of the hydrodynamic variables $\nabla \sqrt{\rho_\delta}$ and $\Lambda_\delta$. This strong convergence allows passing to the limit in the nonlinear terms of the Euler system.

3. Key Contributions

• Novel Approximation Scheme: The paper introduces a regularized logarithmic Schrödinger approximation specifically tailored for the isothermal case ($P(\rho)=\rho$), avoiding the need for viscosity or Strichartz estimates.
• Resolution of Compactness Issue: It overcomes the lack of compactness in the collisionless quantum Euler system by leveraging the energy identity and the quadratic structure of the wave function, rather than relying on viscous dissipation mechanisms.
• Rigorous Construction: It provides the first rigorous construction of global finite-energy weak solutions for the collisionless quantum isothermal Euler system with initial data in the natural energy space ($H^1$ with finite logarithmic energy).
• Energy Conservation: The constructed weak solutions satisfy the exact energy equality, a significant result given the singular nature of the nonlinearity.

4. Main Results

Theorem 1: For any initial data $\psi_0 \in H^1(T^3)$ such that $|\psi_0|^2 \log |\psi_0|^2 \in L^1(T^3)$, there exists a global finite-energy weak solution $(\rho, J)$ to the quantum isothermal Euler system on $[0, T) \times T^3$.

• The solution satisfies the weak formulation defined in the paper.
• The solution conserves the total energy: $E(t) = E(0)$ for all $t > 0$.
• The solution is derived as the limit of the regularized Schrödinger approximations.

Theorem 2 (Intermediate): The regularized logarithmic Schrödinger approximations converge to a unique weak solution of the limiting logarithmic Schrödinger equation, which satisfies the energy equality.

5. Significance

This work bridges a critical gap in the mathematical theory of quantum hydrodynamics.

• Physical Relevance: The isothermal pressure law ($P=\rho$) is physically relevant for Bose-Einstein condensates and superfluids at specific temperatures. The logarithmic nonlinearity is a canonical model in quantum mechanics.
• Mathematical Robustness: The paper demonstrates that the Schrödinger approximation framework is robust enough to handle the singular isothermal case without artificial viscosity. This opens the door for analyzing other QHD models where the internal energy contains an isothermal component.
• Methodological Advance: The technique of using the energy identity to recover strong convergence of gradients in the absence of viscosity offers a new pathway for tackling singular nonlinearities in dispersive fluid systems.

In summary, Cheng Yu successfully constructs global weak solutions to a challenging singular quantum fluid system by exploiting the deep connection between the hydrodynamic variables and the underlying wave mechanics, utilizing a carefully designed regularization and compactness argument.