关键词: global existence, algebraic time decay rates, exponential time decay rates, the whole space, low regularity space

Abstract: In this manuscript, we consider two kinds of the Fokker-Planck-type systems in the whole space. The first part involves proving the global existence and the algebraic time decay rates of the mild solutions to the Fokker-Planck-Boltzmann equation near Maxwellians if initial data satisfies some smallness in the function space $L^1_kL^\infty_TL^2_v\cap L^p_kL^\infty_TL^2_v$. The second part proves the global existence of the mild solutions to the Vlasov-Poisson-Fokker-Planck system in the function space $L^1_kL^\infty_TL^2_v$, and we also obtain the exponential time decay rates, which are different from the algebraic time decay rates of the Fokker-Planck-Boltzmann equation. Our analysis is based on $L^1_kL^\infty_TL^2_v$ function space introduced by Duan $et~ al$. (Comm Pure Appl Math, 2021, 74: 932-1020), the $L^1_k\cap L^p_k$ approach developed by Duan $et~ al$. (SIAM J Math Anal, 2024, 56: 762-800), and the coercivity property of the Fokker-Planck operator. However, it is worth pointing out that the $L^1_k\cap L^p_k$ approach is not required for the Vlasov-Poisson-Fokker-Planck system, due to the influence of the electric field term, which is different from the Fokker-Planck-Boltzmann equation in this paper and in the work of Duan $et~ al$. (SIAM J Math Anal, 2024, 56: 762-800).

Key words: global existence, algebraic time decay rates, exponential time decay rates, the whole space, low regularity space