From the Vlasov-Poisson-Boltzmann System to the Vlasov-Poisson-Landau System with Coulomb Potential

Abstract

In the celebrated work [Y. Guo., J. Amer. Math. Soc., 25 (2012), pp. 759–812], Guo proved global well-posedness of the Vlasov–Poisson–Landau (VPL) system with Coulomb potential in algebraically weighted Sobolev spaces. Thanks to this breakthrough, there are rich developments on the VPL system and also on the Vlasov–Poisson–Boltzmann (VPB) system with cutoff or noncutoff Boltzmann kernels. In this work, we are interested in the connection between the two important physical systems where particles interact through the Coulomb force and are influenced by their self-consistent electrostatic field. The Coulomb potential yields the famous Rutherford scattering cross section which is too singular in the angular variable for the Boltzmann equation to be meaningful. With an angular cutoff and a proper scaling as in [R. Alexandre and C. Villani, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), pp. 61–95], we prove global well-posedness of VPB in algebraically weighted Sobolev spaces. To this end, we develop a new two-stage energy method and merely rely on the -norm (from the dissipation of the linearized Boltzmann operator) to control the nonlinear terms containing the electrostatic field. Consequently, the same well-posedness result also holds for the VPB system with all the other soft potentials, which removes the technical constraint in [R. Duan and S. Liu, Comm. Math. Phys., 324 (2013), pp. 1–45]. Moreover, when the cutoff parameter tends to zero, convergence of VPB to VPL is established with an explicit rate.