Low-regularity solutions to Fokker-Planck-type systems in the whole space

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 _k ¹ L _T ^∞ L _v ² ⋂ L _k ^p L _T ^⋡ L _v ² . The second part proves the global existence of the mild solutions to the Vlasov-Poisson-Fokker-Planck system in the function space L _k ¹ L _T ^∞ L _v ² , 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 _k ¹ L _T ^∞ L _v ² function space introduced by Duan et al. (Comm Pure Appl Math, 2021, 74: 932–1020), the L _k ¹ ⋂ L _k ^p 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 _k ¹ ⋂ L _k ^p 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).