A Steady-State Quantum Euler–Poisson System for Potential Flows
A Steady-State Quantum Euler–Poisson System for Potential Flows
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DOI：
10.1007/s002200050364
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发表时间：
1998-06
影响因子：
2.4
通讯作者：
A. Jüngel
中科院分区：
文献类型：
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作者：
A. Jüngel
给出了粒子密度和电流密度的量子玻姆势流体动力学方程的势流形式。该方程自洽耦合到泊松方程的电势。稳态模型由退化型的非线性椭圆型方程与二次增长的梯度。物理动机的Dirichlet边界条件规定。在电能与热能相比很小的假设下，证明了解的存在性。证明是基于Leray-Schauder的不动点定理和截断方法。主要的困难是找到一个一致的密度下界。对于足够大的电能，存在（简化系统的）广义解，其中密度在某点为零。最后，解决方案的唯一性示出了一个足够大的规模普朗克常数。
A potential flow formulation of the hydrodynamic equations with the quantum Bohm potential for the particle density and the current density is given. The equations are selfconsistently coupled to Poisson's equation for the electric potential. The stationary model consists of nonlinear elliptic equations of degenerate type with a quadratic growth of the gradient. Physically motivated Dirichlet boundary conditions are prescribed. The existence of solutions is proved under the assumption that the electric energy is small compared to the thermal energy. The proof is based on Leray-Schauder's fixed point theorem and a truncation method. The main difficulty is to find a uniform lower bound for the density. For sufficiently large electric energy, there exists a generalized solution (of a simplified system), where the density vanishes at some point. Finally, uniqueness of the solution is shown for a sufficiently large scaled Planck constant.