一维量子Euler-Poisson方程的解的渐近性(英文)

Asymptotic behavior of the solutions of one-dimensional quantum Euler-Poisson equations

讨论了一类一维量子半导体方程,这类方程具有等熵Euler-Poisson方程的形式,并且动量方程有量子势力项和松弛项.当远场动量不一致和远场电场非零时,证明了一维量子Euler-Poisson方程的初值问题的解的渐近性.通过选择适当的修正函数和能量估计的方法,得到了上述初值问题的解在时间足够大时收敛到相应的稳态解.这个结果改进了前人的关于远场动量一致和零远场电场时解的渐近性的结果.

We study the one-dimensional quantum hydrodynamic system for semiconductors. It takes the isentropic Euler-Poisson equations with the quantum potential and momentum relaxation term in the momentum equations. We show the asymptotic behavior of the solutions for the initial value prob- lem to one-dimensional quantum Euler-Poisson equations, when the far field states of the current density are inconsistent and the far field of the electric field is not zero. Choosing proper corrections and using the energy methods, we prove that the solutions of one-dimensional isentropic quantum Euler-Poisson equations decay exponentially fast to the stationary solutions. This result improves previous results in which the current density＇s far fields are equal and the far field of the electric field is zero.