The relaxation time limit from the quantum Navier–Stokes–Poisson system to the quantum drift–diffusion equation is performed in the framework of finite energy weak solutions. No assumptions on the limiting solution are made. The proof exploits the suitably scaled a priori bounds inferred by the energy and BD entropy estimates. Moreover, it is shown how from those estimates the Fisher entropy and free energy estimates associated to the diffusive evolution are recovered in the limit. As a byproduct, our main result also provides an alternative proof for the existence of finite energy weak solutions to the quantum drift–diffusion equation.

中文翻译：

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从量子纳维-斯托克斯方程到量子漂移-扩散方程的弛豫极限

从量子 Navier-Stokes-Poisson 系统到量子漂移-扩散方程的弛豫时间限制是在有限能量弱解的框架内进行的。没有对极限解做任何假设。该证明利用了由能量和 BD 熵估计推断出的适当缩放的先验界限。此外，还显示了如何从这些估计中恢复与扩散演化相关的 Fisher 熵和自由能估计值。作为副产品，我们的主要结果还为量子漂移-扩散方程的有限能量弱解的存在提供了另一种证明。