Quantum Euler-Poisson systems: global existence and exponential decay

Abstract

Abstract.A one-dimensional transient quantum Euler-Poisson system for the electron density, the current density, and the electrostatic potential in bounded intervals is considered. The equations include the Bohm potential accounting for quantum mechanical effects and are of dispersive type. They are used, for instance, for the modelling of quantum semiconductor devices. The existence of local-in-time solutions with small initial velocity is proven for general pressure-density functions. If a stability condition related to the subsonic condition for the classical Euler equations is imposed, the local solutions are proven to exist globally in time and tend to the corresponding steady-state solution exponentially fast as the time tends to infinity.

On the blowing up of solutions to quantum hydrodynamic models on bounded domains

“…local in time existence of classical solutions and global existence of small solutions to (1.9)-(1.11) is proved in [10]. We point out that the proof of local existence of smooth solutions to (1.9)-(1.13) remains at this stage of the work an open problem and will be addressed in future work.…”

“…In the recent years problem (1.1), (1.2) has attracted a lot of interest and many papers have been published. In [10] Jüngel and Li proved local-in-time existence of smooth solutions to (1.1), (1.2) for the one dimensional case with Dirichlet and Neumann boundary conditions for the particle density ρ. Moreover, if the initial data are close enough to the steady-state solution, the local-in-time solutions are shown to exist globally in time (so called small solutions).…”

“…It will turn out that the function a(x) must be a concave function with zero value on a part of the boundary and its form will depend on the domain and on the boundary conditions of the problem. This paper is organized as follows: in Section 2 we investigate the finite time blowup of smooth solutions to the one-dimensional quantum hydrodynamic equations (we assume for simplicity ε 2 = 2) 10) with the initial and boundary conditions…”

Diffusive semiconductor moment equations using Fermi–Dirac statistics

“…Boundary conditions satisfying the Shapiro-Lopatinskii criterion have been examined in [12]. Furthermore, in [32,35] Dirichlet-type conditions have been employed in the analyzed, but only for the (simpler) one-dimensional equations.…”

“…First results, e.g. [32,38,44], have been concerned with the local existence of solutions or the global existence of near-equilibrium solutions. For the stationary problem, only the existence of "subsonic" solutions has been achieved so far [29].…”