The Fokker–Planck–Boltzmann equation in low regularity space. (June 2023) - Fan, Yingzhe; Lei, Yuanjie
Abstract: We prove the global existence and the exponential time decay rates of the mild solutions to the Fokker–Planck–Boltzmann equation near Maxwellians if initial data satisfy some smallness in the function space L k 1 L T ∞ L v 2 . Compared with previous results, there are two features in this manuscript. The first is that we don't need the velocity weight estimates in the work (Duan et al., 2021) with the help of the Fokker–Planck operator. The second is that we only need the low regularity space L k 1 L T ∞ L v 2 rather than H x, v 4 in the work (Xiong et al., 2014). Our analysis is based on L k 1 L T ∞ L v 2 function space introduced by the work of Duan et al. (2021) and the coercivity property of Fokker–Planck operator.