On the Nonlinear Boltzmann Equation of the Carrier Transport in Semiconductors I: Existence and Uniqueness of Solutions

Abstract

There are proved propositions on the existence and uniquenessof solutions of a steady-state, spatially homogeneous nonlinear Boltzmann equation which describes the charge carrier transport in semiconductors. In contrast to more known kinds of the Boltzmann equation (e.g. in radiation transfer theory), the form in question contains $\delta$-functions in the collision integral. Therefore, smooth functions are transformed by the collision operator into discontinuous ones in general. The precise investigation of the properties of the operators describing the Boltzmann equation leads to the construction of suitable aisisotropic Sobolev spaces, in which existence and uniqueness of solutions can be shown.