Blow-up of the smooth solution to quantum hydrodynamic models in Rd

In this paper we firstly investigate the local-in-time existence of smooth solution for the quantum hydrodynamic models (QHD) in RdRd. Then we prove that any smooth solution of the QHD model which satisfies suitable conditions will blow up in finite time. The model can be derived from nonlinear Schrödinger equation by a Madelung transformation. The main idea is based on the construction of approximate solution and energy inequality.