Well-posedness of the Fractional Fokker-Planck Equation
Ke Chen, Ruilin Hu, Quoc-Hung Nguyen
TL;DR
This work studies the critical well-posedness of the Fractional Fokker-Planck Equation (FFPE) $\partial_t f + v\cdot \nabla_x f + \Lambda_v^{\alpha} f = \nabla_v \cdot (f \nabla_v \Lambda_v^{-\beta} f)$ with $\Lambda_v = (-\Delta_v)^{1/2}$ in the regime $\beta\in(0,1)$, $\alpha\in(1,2)$ and $\alpha+\beta>2$, highlighting its role as a semi-linear analogue of the non-cutoff Boltzmann equation. The authors develop a Schauder-type estimate framework to prove global existence and uniqueness for small initial data in a critical, anisotropic Besov-type space, quantified by a scaling-invariant norm $[f_0]$ and a bound of the form $\sup_{t>0}\sum_{m+n\le A} t^{m+(m+n+\alpha+\beta-2)/\alpha} \|\nabla_x^m \nabla_v^n f(t)\|_{L^{\infty}}$. Central to the approach is a representation formula for the linear problem with kernel $H(t,x,v)$ and a fixed-point map $\mathcal{S}$ on a Banach space $X$, where the nonlinear term is encoded as $F[g]=g\nabla_v \Lambda_v^{-\beta} g$ and controlled via Schauder-type nonlinear estimates. The method not only yields FFPE well-posedness but also provides a blueprint for tackling non-cutoff Boltzmann and Landau equations using adaptations of the same kinetic-Schauder framework, thereby contributing to the broader analytical understanding of fractional kinetic models.
Abstract
In this paper, we employ a Schauder-type estimate method, as developed in \cite{CHN}, to establish critical well-posedness result for the Fractional Fokker-Planck Equation. 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.