THE WELL-POSEDNESS AND ASYMPTOTICS OF MULTI-DIMENSIONAL QUANTUM HYDRODYNAMICS

The multi-dimensional quantum hydrodynamic equations for charge transport in ultra-small electronic devices like semiconductors, where quantum effects （like particle tunnelling through potential barriers and built-up in quantum wells） take place, is considered in the present paper, and the recent progress on well-posedness, stability analysis, and small scaling limits are reviewed.

Direct Relativistic Extension of the Madelung-de-Broglie-Bohm Reformulations of Quantum Mechanics and Quantum Hydrodynamics

The basic equations of the non-relativistic quantum mechanics with trajectories and quantum hydrodynamics are extended to the relativistic domain. This is achieved by using a Schr?dinger-like equation, which describes a particle with mass and spin-0 and with the correct relativistic relation between its linear momentum and kinetic energy. Some simple but instructive free particle examples are discussed.

Numerical Approach of Interactions of Proton Beams and Dense Plasmas with Quantum-Hydrodynamic/Particle-in-Cell Model

A one dimensional quantum-hydrodynamic/particle-in-cell （QHD/PIC） model is used to study the interaction process of an intense proton beam （injection density of 1017 cm-3） with a dense plasma （initial density of -10^21 cm^-3）, with the PIC method for simulating the beam particle dynamics and the QHD model for considering the quantum effects including the quantum statistical and quantum diffraction effects. By means of the QHD theory, the wake electron density and wakefields are calculated, while the proton beam density is calculated by the PIC method and compared to hydrodynamic results to justify that the PIC method is a more suitable way to simulate the beam particle dynamics. The calculation results show that the incident continuous proton beam when propagating in the plasma generates electron perturbations as well as wakefields oscillations with negative valleys and positive peaks where the proton beams are repelled by the positive wakefields and accelerated by the negative wakefields. Moreover, the quantum correction obviously hinders the electron perturbations as well as the wakefields. Therefore, it is necessary to consider the quantum effects in the interaction of a proton beam with cold dense plasmas, such as in the metal films.

LARGE-TIME BEHAVIOR OF SOLUTIONS OF QUANTUM HYDRODYNAMIC MODEL FOR SEMICONDUCTORS

A one-dimensional quantum hydrodynamic model （or quantum Euler-Poisson system） for semiconductors with initial boundary conditions is considered for general pressure-density function. The existence and uniqueness of the classical solution of the corresponding steady-state quantum hydrodynamic equations is proved. Furthermore, the global existence of classical solution, when the initial datum is a perturbation of t he steadystate solution, is obtained. This solution tends to the corresponding steady-state solution exponentially fast as the time tends to infinity.

LARGE TIME BEHAVIOR OF SOLUTIONS TO1-DIMENSIONAL BIPOLAR QUANTUM HYDRODYNAMIC MODEL FOR SEMICONDUCTORS

In this article, we study the 1-dimensional bipolar quantum hydrodynamic model for semiconductors in the form of Euler-Poisson equations, which contains dispersive terms with third order derivations. We deal with this kind of model in one dimensional case for general perturbations by constructing some correction functions to delete the gaps between the original solutions and the diffusion waves in L2-space, and by using a key inequality we prove the stability of diffusion waves. As the same time, the convergence rates are also obtained.

A BLOW-UP CRITERION OF STRONG SOLUTIONS TO THE QUANTUM HYDRODYNAMIC MODEL

In this article,we focus on the short time strong solution to a compressible quantum hydrodynamic model.We establish a blow-up criterion about the solutions of the compressible quantum hydrodynamic model in terms of the gradient of the velocity,the second spacial derivative of the square root of the density,and the first order time derivative and first order spacial derivative of the square root of the density.

Drift vortices in inhomogeneous collisional dusty magnetoplasma

For the sake of investigating the drift coherent vortex structure in an inhomogeneous dense dusty magnetoplasma,using the quantum hydrodynamic model a nonlinear controlling equation is deduced when the collision effect is considered.New vortex solutions of the electrostatic potential are obtained by a special transformation method, and three evolutive cases of monopolar vortex chains with spatial and temporal distribution are analyzed by representative parameters. It is found that the collision frequency, particle density, drift velocity, dust charge number, electron Fermi wavelength, quantum correction,and quantum parameter are all influencing factors of the vortex evolution. Compared to the uniform dusty system, the vortex solutions of the inhomogeneous system present richer spatial evolution and physical meaning. These results may explain corresponding vortex phenomena and support beneficial references for the dense dusty plasma atmosphere.

Dust acoustic waves in collisional uniform dense magnetoplasma

In order to study the characteristics of dust acoustic waves in a uniform dense dusty magnetoplasma system, a nonlinear dynamical equation is deduced using the quantum hydrodynamic model to account for dust–neutral collisions. The linear dispersion relation indicates that the scale lengths of the system are revised by the quantum parameter, and that the wave motion decays gradually leading the system to a stable state eventually. The variations of the dispersion frequency with the dust concentration, collision frequency, and magnetic field strength are discussed. For the coherent nonlinear dust acoustic waves, new analytic solutions are obtained, and it is found that big shock waves and wide explosive waves may be easily produced in the background of high dusty density, strong magnetic field, and weak collision. The relevance of the obtained results is referred to dense dusty astrophysical circumstances.

GLOBAL EXISTENCE OF WEAK SOLUTIONS FOR GENERALIZED QUANTUM MHD EQUATION

We prove the existence of a weak solution for a generalized quantum MHD equation in a 2-dimensional periodic box for large initial data. The existence of a global weak solution is established through a three-level approximation,energy estimates, and weak convergence for the adiabatic exponent γ 〉 1.

EXPONENTIAL DECAY FOR THE VISCOUS BIPOLAR QUANTUM HYDRODYNAMIC MODEL

In this paper, we pay attention to the time-decay rate of the viscous bipolar quantum hydrodynamic （QHD） models for semiconductors. By applying the entropy method, we prove that the solution of the viscous bipolar QHD models tends to the equilibrium state at an exponential decay rate for the multi-dimensional cases. The arguments is based on a series of a priori estimates.

SEMICLASSICAL LIMIT TO THE GENERALIZED NONLINEAR SCHR?DINGER EQUATION

In this paper, we investigate the semiclassical limit of the generalized nonlinear Schrdinger equation for initial data with Sobolev regularity. Also, we will analyze the structure of the fluid dynamical system with quantum effect corresponding to the semiclassical limit of the generalized nonlinear Schrdinger equation.