arXiv subjects

Ben Duan

Publications and source records attributed to Ben Duan.

11 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

Global Uniqueness of Subsonic Flows for the Steady Euler-Poisson System

We prove the global uniqueness of multidimensional subsonic flows for the steady Euler--Poisson system in a bounded nozzle in the sense that uniqueness holds without restricting solutions to be small perturbations of a background state. The proof is based on a convexity property of the set of subsonic states and energy estimates.

math.AP↗

arXiv2025

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↗

arXiv2025

Classical solutions to a mixed-type PDE with a Keldysh-type degeneracy and accelerating transonic solutions to the Euler-Poisson system

In this paper, we first prove the existence of classical solutions to a class of Keldysh-type equations. Next, we apply this existence result to prove the structural stability of one-dimensional smooth transonic solutions to the steady Euler-Poisson system. Most importantly, the solutions constructed in this paper are classical solutions to the Euler-Poisson system, thus their sonic interfaces are not weak discontinuities in the sense that all the flow variables, such as density, velocity and pressure, are at least $C^1$ across the interfaces.

math.AP↗

arXiv2023

The steady Euler-Poisson system and accelerating flows with transonic $C^1$-transitions

In this paper, we prove the existence of two-dimensional solutions to the steady Euler-Poisson system with continuous transonic transitions across sonic interfaces of codimension 1. First, we establish the well-posedness of a boundary value problem for a linear second order system that consists of an elliptic-hyperbolic mixed type equation with a degeneracy occurring on an interface of codimension 1, and an elliptic equation weakly coupled together. Then we apply the Schauder fixed point theorem to prove the existence of two-dimensional solutions to the potential flow model of the steady Euler-Poisson system with continuous transonic transitions across sonic interfaces. With the aid of Helmholtz decomposition, established in [6], we extend the existence result to the full Euler-Poisson system for the case of nonzero vorticity. Most importantly, the solutions constructed in this paper are classical solutions to Euler-Poisson system, thus their sonic interfaces are not weak discontinuities in the sense that all the flow variables are $C^1$ across the interfaces.

math.AP↗

arXiv2022

Pricing Stocks with Trading Volumes

The present paper proposes a new framework for describing the stock price dynamics. In the traditional geometric Brownian motion model and its variants, volatility plays a vital role. The modern studies of asset pricing expand around volatility, trying to improve the understanding of it and remove the gap between the theory and market data. Unlike this, we propose to replace volatility with trading volume in stock pricing models. This pricing strategy is based on two hypotheses: a price-volume relation with an idea borrowed from fluid flows and a white-noise hypothesis for the price rate of change (ROC) that is verified via statistic testing on actual market data. The new framework can be easily adopted to local volume and stochastic volume models for the option pricing problem, which will point out a new possible direction for this central problem in quantitative finance.

q-fin.MF↗

arXiv2020

Modeling the Control of COVID-19: Impact of Policy Interventions and Meteorological Factors

In this paper, we propose a dynamical model to describe the transmission of COVID-19, which is spreading in China and many other countries. To avoid a larger outbreak in the worldwide, Chinese government carried out a series of strong strategies to prevent the situation from deteriorating. Home quarantine is the most important one to prevent the spread of COVID-19. In order to estimate the effect of population quarantine, we divide the population into seven categories for simulation. Based on a Least-Squares procedure and officially published data, the estimation of parameters for the proposed model is given. Numerical simulations show that the proposed model can describe the transmission of COVID-19 accurately, the corresponding prediction of the trend of the disease is given. The home quarantine strategy plays an important role in controlling the disease spread and speeding up the decline of COVID-19. The control reproduction number of most provinces in China are analyzed and discussed adequately. We should pay attention to that, though the epidemic is in decline in China, the disease still has high risk of human-to-human transmission continuously. Once the control strategy is removed, COVID-19 may become a normal epidemic disease just like flu. Further control for the disease is still necessary, we focus on the relationship between the spread rate of the virus and the meteorological conditions. A comprehensive meteorological index is introduced to represent the impact of meteorological factors on both high and low migration groups. As the progress on the new vaccine, we design detail vaccination strategies for COVID-19 in different control phases and show the effectiveness of efficient vaccination. Once the vaccine comes into use, the numerical simulation provide a promptly prospective research.

q-bio.PE↗

arXiv2019

Structural stability of Supersonic solutions to the Euler-Poisson system

The well-posedness for the supersonic solutions of the Euler-Poisson system for hydrodynamical model in semiconductor devices and plasmas is studied in this paper. We first reformulate the Euler-Poisson system in the supersonic region into a second order hyperbolic-elliptic coupled system together with several transport equations. One of the key ingredients of the analysis is to obtain the well-posedness of the boundary value problem for the associated linearized hyperbolic-elliptic coupled system, which is achieved via a delicate choice of multiplier to gain energy estimate. The nonlinear structural stability of supersonic solution in the general situation is established by combining the iteration method with the estimate for hyperbolic-elliptic system and the transport equations together.

math.AP↗

arXiv2014

Two dimensional subsonic flows with self-gravitation in bounded domain

We investigate two dimensional steady Euler-Poisson system which describe the motion of compressible self-gravitating flows. The unique existence and stability of subsonic flows in a duct of finite length are obtained when prescribing the entropy at the entrance and the pressure at the exit. After introducing the stream function, the Euler-Poisson system can be decomposed into several transport equations and a second order nonlinear elliptic system. We discover an energy estimate for the associated elliptic system which is a key ingredient to prove the unique existence and stability of subsonic flow.

math.AP↗

arXiv2013

Subsonic solutions for steady Euler-Poisson system in two dimensional nozzles

In this paper, we prove the existence and stability of subsonic flows for steady full Euler-Poisson system in a two dimensional nozzle of finite length when imposing the electric potential difference on non-insulated boundary from a fixed point at the entrance, and prescribing the pressure at the exit of the nozzle. The Euler-Poisson system for subsonic flow is a hyperbolic-elliptic coupled nonlinear system. One of the crucial ingredient of this work is the combination of Helmholtz decomposition for the velocity field and stream function formulation together. In terms of the Helmholtz decomposition, the Euler-Poisson system is rewritten as a second order nonlinear elliptic system of three equations and transport equations for entropy and pseudo-Bernoulli's invariant. The associated elliptic system in a Lipschitz domain with nonlinear boundary conditions is solved with the help of the estimates developed in [2] based on its nice structure. The transport equations are resolved via the flow map induced by the stream function formulation. Furthermore, the delicate estimates for the flow map give the uniqueness of the solutions.

math.AP↗

arXiv2013

Subsonic flow for multidimensional Euler-Poisson system

We establish unique existence and stability of subsonic potential flow for steady Euler-Poisson system in a multidimensional nozzle of a finite length when prescribing the electric potential difference on non-insulated boundary from a fixed point at the exit, and prescribing the pressure at the exit of the nozzle. The Euler-Poisson system for subsonic potential flow can be reduced to a nonlinear elliptic system of second order. In this paper, we develop a technique to achieve a priori $C^{1,\alp}$ estimates of solutions to a quasi-linear second order elliptic system with mixed boundary conditions in a multidimensional domain with Lipschitz continuous boundary. Particularly, we discovered a special structure of the Euler-Poisson system which enables us to obtain $C^{1,\alp}$ estimates of velocity potential and electric potential functions, and this leads us to establish structural stability of subsonic flows for the Euler-Poisson system under perturbations of various data.

math.AP↗