Moment equations for charged particles: global existence results

In this chapter we develop a general mathematical theory for the moment equations for charged particles obtained by applying Levermore's method (Maximum Entropy Principle). We will show that the main drawbacks of this method disappear when it is applied to the semiclassical Boltzmann equation for semiconductors. In this case, the phase space is given by a space variable $x$, as usual, and a variable $k$ which accounts for the crystal wave number, so that $\hbar k$ has the dimension of a momentum. The variable $k$ varies over a bounded subset ${\cal B}$ of $\RR^n$, called Brillouin region. In this model, the velocity is a given function of $k$, which depends on the crystal energy ${\cal E}(k)$. Since the particles described by this model carry charge, Boltzmann equation is coupled to a Poisson equation for the electric potential which drives the particles. For this model, we prove a local existence result, and a global existence result of smooth solutions around equilibrium.