A relativistic quantum Euler-Poisson system derived from the Klein-Gordon-Poisson equation: hyperbolic-elliptic structure
Ben Duan, Jun Li, Bin Guo, Rongrong Yan
TL;DR
The paper derives a relativistic quantum hydrodynamic model from the self-consistent Klein--Gordon--Poisson system using the Madelung transformation, revealing a coupled hyperbolic-elliptic structure with nonlocal Poisson interaction and explicit quantum ($\varepsilon$) and relativistic ($\upsilon$) corrections. Formal limits show the RQHD reduces to both relativistic Euler–Poisson and quantum Euler–Poisson models in appropriate regimes, illustrating a unified framework for quantum-relativistic fluid dynamics. The main result is a local-in-time well-posedness theory for the nonlinear hyperbolic-elliptic system, established via energy estimates, linearized solvability, and Schauder fixed-point arguments, with a density positivity preservation. These results provide a rigorous basis for studying parameter-dependent limits and connecting quantum and relativistic fluid descriptions in plasmas and semiconductor contexts.
Abstract
In the Klein-Gordon equation, quantum and relativistic parameters are strongly coupled, which poses significant analytical challenges in the derivation and analysis of related classical fluid models. In this paper, starting from the Klein-Gordon-Poisson system, we formally derive a relativistic quantum hydrodynamic (RQHD) system via the Madelung transformation, in which the relativistic and quantum correction terms in the Euler-Poisson framework are clearly exhibited. In particular, at a formal level, the RQHD system reduces to the relativistic hydrodynamics system in the semiclassical regime and to the quantum hydrodynamics system in the non-relativistic regime. These limiting procedures highlight the unified structure of the proposed model and clarify the role played by the coupled relativistic and quantum effects. From an analytical point of view, by reformulating the RQHD system as a coupled hyperbolic-elliptic system with a nonlocal Poisson interaction, we establish the local-in-time existence and uniqueness of classical solutions to the associated Cauchy problem. The initial density is assumed to be a small perturbation of a positive constant state, while the remaining initial data are taken to be general smooth functions. The analysis relies on energy estimates and suitable estimates for the nonlocal terms, and provides a rigorous well-posedness result in the natural energy space.