Conditional hypocoercivity for nonlinear kinetic Fokker--Planck equations

We investigate the long-time behaviour of nonlinear kinetic Fokker--Planck equations with porous medium diffusion in a non-perturbative setting. Under a priori conditional bounds on macroscopic quantities, we establish exponential convergence to equilibrium in $L^1$. These bounds are automatically satisfied if the initial data is trapped between two global equilibrium profiles. Our approach combines the entropy-entropy dissipation structure and some techniques from $L^2$-hypocoercivity.