Some results about the quantum Navier-Stokes model

The quantum Navier-Stokes model correspond to the classical Navier-Stokes model in which a quantum correction term called Bohm potential is added. In this talk I will first present some results about the existence of weak solutions which is obtained using renormalized solutions. The results allow us to obtain also the semi-classical limit i.e. the limit when the parameter behind the Bohm potential tends to zero. This limit lead to the classical Navier-Stokes model. In a second part I will speak about the viscous limit of the model for which we use the previous existence result, the notion of dissipative solutions and relative entropy estimates. In this limit we obtain a dissipative solution of the quantum Euler model.