The existence of stationary subsonic solutions and their stability for 3-D hydrodynamic model of unipolar semiconductors with the Ohmic contact boundary have been open for long time due to some technical reason, as we know. In this paper, we consider 3-D radial solutions to the system in a hollow ball, and prove that the 3-D radial subsonic stationary solutions uniquely exist and are asymptotically stable, when the initial perturbations around the subsonic steady-state are small enough. Different from the existing studies on the radial solutions for fluid dynamics where the inner boundary of the hollow ball must be far away from the singular origin, here we may allow the chosen inner boundary arbitrarily close to the singular origin and reveal the relationship between the inner boundary and the large time behavior of the radial solution. This partially answers the open question of the stability of stationary waves subjected to the Ohmic contact boundary conditions in the multiple dimensional space. We also prove the existence of non-flat stationary subsonic solution, which essentially improve and develop the previous studies in this subject. The proof is based on the technical energy estimates in certain weighted Sobolev spaces, where the weight functions are artfully selected to be the distance of the targeted spatial location and the singular point.

我们知道，由于某种技术原因，具有欧姆接触边界的单极半导体的3-D流体动力学模型的稳态亚音速解的存在及其稳定性已经开放了很长时间。在本文中，我们考虑了空心球中系统的3D径向解，并证明了当亚音速稳态周围的初始扰动足够小时，3D径向亚音速平稳解是唯一存在的，并且渐近稳定。与现有的关于流体动力学径向解的研究不同，在径向解中空心球的内边界必须远离奇异点，这里我们可以允许选定的内边界任意靠近奇异点，并揭示内部之间的关系。边界和大时间行为的径向解。这部分回答了在多维空间中经受欧姆接触边界条件的驻波稳定性的未解决问题。我们还证明了非平面平稳亚音速解决方案的存在，从根本上改进和发展了该领域的先前研究。证明是基于某些加权Sobolev空间中的技术能量估计，其中权重函数被巧妙地选择为目标空间位置和奇异点的距离。