The Milne Problem for High Field Kinetic Equations

Half space problems of the linear Boltzmann equation with a constant driving force are considered. Such problems model boundary layers between kinetic zones and fluid zones described by a high field limit of the Boltzmann equation. Existence, uniqueness, and asymptotic behavior of solutions are studied for positive and negative driving forces. In the positive case, the force field accelerates the particles, and we show that the solution of the half space problem is determined only by the inflow data. In contrast, for negative forces, the behavior at infinity has to be prescribed in order to insure uniqueness. Due to the nonvanishing forces, the problem does not possess any entropy. The existence and uniqueness issues are dealt with by supersolution techniques, while the asymptotic behavior is analyzed by semiexplicit integration of the equations along the characteristics. In the case of relaxation time approximation, a fast numerical method for computing the asymptotic state method is presented and tested.