Global well-posedness for the Fokker–Planck–Boltzmann equation in Besov–Chemin–Lerner type spaces

In this paper, motivated by [16], we use the Littlewood–Paley theory to establish some estimates on the nonlinear collision term, which enable us to investigate the Cauchy problem of the Fokker–Planck–Boltzmann equation. When the initial data is a small perturbation of the Maxwellian equilibrium state, under the Grad's angular cutoff assumption, the unique global solution for the hard potential case is obtained in the Besov–Chemin–Lerner type spaces C([0,∞);L˜ξ2(B2,rs)) with 1≤r≤21≤r≤2 and s>3/2s>3/2 or s=3/2s=3/2 and r=1r=1. Besides, we also obtain the uniform stability of the dependence on the initial data.