We discuss analytically the stationary viscous quantum hydrodynamic model including a barrier potential, which is a nonlinear system of partial differential equations of mixed order in the sense of Douglis–Nirenberg. Combining a reformulation by means of an adjusted Fermi level, a variational functional, and a fixed point problem, we prove the existence of a weak solution. There are no assumptions on the size of the given data or their variation. We also provide various estimates of the solution that are independent of the quantum parameters.
- viscous quantum hydrodynamics
- stationary problem
- existence of solutions