Mixed high field and diffusion asymptotics for the fermionic Boltzmann equation

In a previous work [J. of Hyperbolic Diff. Eq. 4, pp. 679-704 (2007)], the high field asymptotics of the fermionic Boltzmann equation has been proven to lead to a nonlinear conservation law for the particle density. Under symmetry conditions on the collission cross section, the nonlinear limiting flux is parallel to the force field. In the present work, we investigate the orthogonal direction to the electric field, and prove after a suitable rescaling that the behaviour is governed by the original conservation law with an additional nonlinear diffusion in the orthogonal (to the force field) direction. The main tool used in the convergence proof is a new estimate for the dissipation of the family of entropies introduced in the above cited work. While the entropy dissipation is usually estimated by quantities of the type dist $(f, F_{eq})$ representing the distance of the distribution function to the equilibrium set, the new estimate involves a quantity of the form $\int$ dist $(f, F_{eq}(\u)) d\mu(\u)$, where the macroscopic equilibria depend on a velocity variable $\u$ and $\mu$ is a probability measure. This estimate allows to control high velocities, pass to the limit in the diffusion current and shows the convergence to the entropy solution of the limiting equation.