Well-posedness of the fractional Fokker–Planck equation

In this paper, we employ a Schauder-type estimate method, as developed in the study by Chen et al. [ Well-posedness for local and nonlocal quasilinear evolution equations in fluids and geometry , arXiv:2407.05313, 2024] to establish critical well-posedness result for the Fractional Fokker–Planck Equation (FFPE). This equation serves as a fundamental model in kinetic theory and can be regarded as a semi-linear analogue of the non-cutoff Boltzmann equation. We demonstrate that the techniques introduced in this study are not only effective for the FFPE but also hold promise for broader applications, particularly in addressing the non-cutoff Boltzmann equation and the Landau equation. Our results contribute to a deeper understanding of the analytical framework required for these complex kinetic models.