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Xiangsheng Xu : Mixed boundary conditions for a simplified quantum energy-transport model in multi-dimensional domains

In this talk we consider the existence of suitable weak solutions for a quantum energy-transport model for semiconductors. The model is formally derived from the quantum hydrodynamic model by J\"{u}ngel and Mili\v{s}i\'{c} (Nonlinear Anal.: Real World Appl., 12(2011), pp. 1033-1046). It consists of a fourth-order nonlinear parabolic equation for the electron density, an elliptic equation for the electron temperature, and the Poisson equation for the electric potential. Our solution is global in the time variable, while the space variables lie in a bounded Lipschitz domain with a mixed boundary condition. The existence proof is based upon a carefully-constructed approximation scheme which generates a sequence of positive approximate solutions. These solutions are shown to be so regular that they can be used to form a variety of test functions , from which we can derive enough a prior estimates to justify passing to the limit in the approximate problems.

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