In the Klein-Gordon equation, the quantum and relativistic parameters are intricately coupled, which complicates the direct consideration of quantum fluctuations. In this paper, the so-called Relativistic Quantum Hydrodynamics System is derived from the Klein-Gordon equation with Poisson effects via the Madelung transformation, providing a fresh perspective for analyzing the singular limits, such as the semi-classical limits and non-relativistic limits. The Relativistic Quantum Hydrodynamics System, when the semiclassical limit is taken, formally reduces to the Relativistic Hydrodynamics System. When the relativistic limit is taken, it formally reduces to the Quantum Hydrodynamics System. Additionally, we establish the local classical solutions for the Cauchy problem associated with the Relativistic Quantum Hydrodynamic System. The initial density value is assumed to be a small perturbation of some constant state, but the other initial values do not require this restriction. The key point is that the Relativistic Quantum Hydrodynamic System is reformulated as a hyperbolic-elliptic coupled system.
Duan et al. (Fri,) studied this question.