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 ħk has the dimension of a momentum. The variable varies over a bounded subset ß of Rn, called a Brillouin region. In this model, the velocity is a given function of k, which depends on the crystal energySince ß(k) the particles described by this model carry a charge, the 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 equilibria.
KeywordsCharged Particle Boltzmann Equation Hyperbolic System Moment Equation Collision Operator
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