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An intrinsically hydrodynamic approach to multidimensional QHD systems

Paolo Antonelli, Pierangelo Marcati, Hao Zheng

TL;DR

This work presents an intrinsically hydrodynamic treatment of multidimensional Quantum Hydrodynamics (QHD), extending the one-dimensional theory to higher dimensions and addressing two key solution families: 2D data with countable, isolated point vortices and multi-dimensional data with spherical symmetry. The authors prove global existence of finite-energy weak solutions via a wave function lifting that connects QHD to nonlinear Schrödinger dynamics, thereby handling vacuum regions and quantized vorticity through a generalized irrotationality condition. They derive intrinsically hydrodynamic dispersive (Morawetz-type) estimates that apply to Euler–Korteweg-type systems and establish stability for sequences of weak solutions under positivity or symmetry assumptions, using a higher-order functional $I(t)$ linked to a generalized chemical potential. Together, these results extend the Madelung framework to multidimensional settings, provide robust dispersive controls, and contribute a rigorous compactness theory for finite-energy QHD, with potential relevance to Bose–Einstein condensates, quantum plasmas, and related diffusive-capillary fluids. Mathematical notation is preserved through all results, emphasizing $\rho$, $J$, $\Lambda$, $\sqrt{\rho}$, $\psi$, and the quantified vorticity structure.

Abstract

In this paper we consider the multi-dimensional Quantum Hydrodynamics (QHD) system, by adopting an intrinsically hydrodynamic approach. The present work continues the analysis initiated in [6] where the one dimensional case was studied. Here we extend the analysis to the multi-dimensional problem, in particular by considering two physically relevant classes of solutions. First of all we consider two-dimensional initial data endowed with point vortices; by assuming the continuity of the mass density and a quantization rule for the vorticity we are able to study the Cauchy problem and provide global finite energy weak solutions. The same result can be obtained also by considering spherically symmetric initial data in the multi-dimensional setting. For rough solutions with finite energy, we are able to provide suitable dispersive estimates, which also apply to a more general class of Euler-Korteweg equations. Moreover we are also able to show the sequential stability of weak solutions with positive density. Analogously to the one dimensional case this is achieved through the a priori bounds given by a new functional first introduced in [6].

An intrinsically hydrodynamic approach to multidimensional QHD systems

TL;DR

This work presents an intrinsically hydrodynamic treatment of multidimensional Quantum Hydrodynamics (QHD), extending the one-dimensional theory to higher dimensions and addressing two key solution families: 2D data with countable, isolated point vortices and multi-dimensional data with spherical symmetry. The authors prove global existence of finite-energy weak solutions via a wave function lifting that connects QHD to nonlinear Schrödinger dynamics, thereby handling vacuum regions and quantized vorticity through a generalized irrotationality condition. They derive intrinsically hydrodynamic dispersive (Morawetz-type) estimates that apply to Euler–Korteweg-type systems and establish stability for sequences of weak solutions under positivity or symmetry assumptions, using a higher-order functional linked to a generalized chemical potential. Together, these results extend the Madelung framework to multidimensional settings, provide robust dispersive controls, and contribute a rigorous compactness theory for finite-energy QHD, with potential relevance to Bose–Einstein condensates, quantum plasmas, and related diffusive-capillary fluids. Mathematical notation is preserved through all results, emphasizing , , , , , and the quantified vorticity structure.

Abstract

In this paper we consider the multi-dimensional Quantum Hydrodynamics (QHD) system, by adopting an intrinsically hydrodynamic approach. The present work continues the analysis initiated in [6] where the one dimensional case was studied. Here we extend the analysis to the multi-dimensional problem, in particular by considering two physically relevant classes of solutions. First of all we consider two-dimensional initial data endowed with point vortices; by assuming the continuity of the mass density and a quantization rule for the vorticity we are able to study the Cauchy problem and provide global finite energy weak solutions. The same result can be obtained also by considering spherically symmetric initial data in the multi-dimensional setting. For rough solutions with finite energy, we are able to provide suitable dispersive estimates, which also apply to a more general class of Euler-Korteweg equations. Moreover we are also able to show the sequential stability of weak solutions with positive density. Analogously to the one dimensional case this is achieved through the a priori bounds given by a new functional first introduced in [6].

Paper Structure

This paper contains 13 sections, 29 theorems, 304 equations.

Table of Contents

  1. Introduction
  2. Global existence of weak solutions for initial data with point vortices
  3. Global existence of spherically symmetric weak solutions
  4. Dispersive estimates
  5. Stability
  6. Notations and Preliminaries
  7. Wave function lifting: hydrodynamic data with points vacuum
  8. Extension to 3D hydrodynamic data: straight vortex lines
  9. Wave function lifting: spherical symmetric data
  10. Spherically symmetric QHD system
  11. Wave function lifting for spherically symmetric data
  12. Dispersive estimates and bounds on $I(t)$
  13. Stability of weak solutions with positive density

Key Result

Theorem 1

Let us consider a hydrodynamic state $(\sqrt{\rho_0}, \Lambda_0)$ satisfying the bound for some $M_1>0$ and let us further assume that $\rho_0$ is continuous, with isolated vacuum points, namely there exists an at most countable index set $\mathcal{A}$, such that and $v_0=J_0/\rho_0$ satisfies the quantized vorticity condition Then there exists a global in time finite energy weak solution to th

Theorems & Definitions (63)

  • Theorem 1: Global Existence of finite energy weak solutions
  • Remark 2
  • Theorem 3: Global existence of regular weak solutions
  • Remark 4
  • Remark 5
  • Theorem 6
  • Theorem 7
  • Theorem 8
  • Theorem 9
  • Theorem 10: Morawetz-type and dispersive estimates
  • ...and 53 more