arXiv subjects

Rongrong Yan

Publications and source records attributed to Rongrong Yan.

2 recordsLinked to original sources

arXiv2026

Derivation and local well-posedness of a relativistic quantum hydrodynamic system on the Heisenberg group

We derive and analyze a relativistic quantum hydrodynamic (RQHD) system on the Heisenberg group. Starting from the Klein--Gordon--Poisson system, we apply the Madelung transformation to obtain a fluid-type model in which the relativistic and quantum parameters are explicitly separated. The Heisenberg-group structure gives rise to an additional geometric term in the momentum equation, reflecting the underlying noncommutative structure. A central analytical difficulty is the possible appearance of vacuum, where the phase function and the quantum potential become singular. To address this issue, we reformulate the RQHD system as an extended hyperbolic--elliptic system with auxiliary variables. For this extended system, we establish uniform higher-order energy estimates on $\mathbb H^1$ by combining the Banach algebra property of sub-elliptic Sobolev spaces with noncommutative Fourier analysis. We then prove that the extended system is equivalent to the original RQHD system at the level of classical solutions. As a consequence, we obtain the local-in-time existence and uniqueness of non-vacuum classical solutions to the RQHD system on $\mathbb H^1$. The result also provides a framework for the study of related singular limits, including the semiclassical and non-relativistic limits on nilpotent Lie groups.

math.AP↗

arXiv2026

A relativistic quantum Euler-Poisson system derived from the Klein-Gordon-Poisson equation: hyperbolic-elliptic structure

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.

math.AP↗