Semiclassical and relaxation limits of bipolar quantum hydrodynamic   model
  The global in-time semiclassical and relaxation limits of the bipolar quantum hydrodynamic model for semiconductors are investigated in $R^3$. We prove that the unique strong solution converges globally in time to the strong solution of classical bipolar hydrodynamical equation in the process of semiclassical limit and to that of the classical Drift-Diffusion system under the combined relaxation and semiclassical limits. 
Guojing Zhang, Hai-Liang Li, Kaijun Zhang