A STEADY-STATE QUANTUM EULER-POISSON SYSTEM FOR POTENTIAL FLOWS

A potential flow formulation of the hydrodynamic equations with the qu antum Bohm potential for the particle density and the current density is given. The equations are selfconsistently coupled to Poisson's equa tion for the electric potential. The stationary model consists of nonl inear 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 truncatio n method. The main difficulty is to find a uniform lower bound for the density. For sufficiently large electric energy, there exists a gener alized solution (of a simplified system), where the density vanishes a t some point. Finally, uniqueness of the solution is shown for a suffi ciently large scaled Planck constant.