Kinetic Fokker-Planck Equations with Nonlinear Diffusion

Abstract

We study existence, regularity, and uniqueness for the nonlinear kinetic Fokker--Planck equation $$ \partial_t f=\Delta_v\Psi(f)-v\cdot\nabla_x f, \qquad f|_{t=0}=f_0, $$ on $\mathbb R^{2d}$. In the model case $\Psi(r)=r^s$, this equation couples nonlinear fast-diffusion/porous-medium type diffusion with kinetic transport. A distinctive feature is that the diffusion acts only in the velocity variable $v$, so that compactness in the spatial variable $x$ cannot be obtained from standard elliptic estimates and must instead be recovered through the hypoelliptic structure. Under general structural assumptions on $\Psi$, including the fast-diffusion powers $\Psi(r)=r^s$ with $s\in(0,1)$, we construct nonnegative weak solutions and prove quantitative anisotropic Besov regularity estimates. Under an additional mass-critical growth condition on the fast-diffusion side, the constructed weak solution preserves mass, admits a renormalized kinetic formulation, and is unique in the $L^1$-class of mass-preserving renormalized kinetic solutions. In the power-law case $\Psi(r)=r^s$, this condition is precisely $s\ge 1-1/d$ when $d\ge2$, while in dimension $d=1$ the whole fast-diffusion range $s\in(0,1)$ is covered. The main analytic ingredient is a parameter-dependent smoothing estimate for the kinetic semigroup generated by $$ \Psi'(\zeta)\Delta_v - v\cdot\nabla_x , $$ which quantitatively tracks the dependence on the kinetic level $\zeta$. Combined with the kinetic formulation, this estimate yields compactness in both spatial and velocity variables for the nonlinear hypoelliptic problem. As an application, we also obtain martingale-problem solutions to the associated distributional-density dependent stochastic differential equation.