On the Landau approximation in plasma physics

This paper studies the approximation of the Boltzmann equation by the Landau equation in a regime when grazing collisions prevail. While all previous results in the subject were limited to the spatially homogeneous case, here we manage to cover the general, space-dependent situation, assuming only basic physical estimates of finite mass, energy, entropy and entropy production. The proofs are based on the recent results and methods introduced previously in [R. Alexandre, C. Villani, Comm. Pure Appl. Math. 55 (1) (2002) 30-70] by both authors, and the entropy production smoothing effects established in [R. Alexandre et al., Arch. Rational Mech. Anal. 152 (4) (2000) 327-355]. We are able to treat realistic singularities of Coulomb type, and approximations of the Debye cut. However, our method only works for finite-time intervals, while the Landau equation is supposed to describe long-time corrections to the Vlasov-Poisson equation. If the mean-field interaction is neglected, then our results apply to physically relevant situations after a time rescaling. (C) 2003 Elsevier SAS. All rights reserved.