Existence and Ergodicity for Nonlinear Fokker-Planck Equations for Probability Density Functions

For a class of nonlinear Fokker-Planck equations for probability densities, we construct distributional solutions by using an approximation scheme with nonlinear Neumann problems in balls. Moreover, we introduce nonlinear functional inequalities to estimate the convergence rates of the distributional solutions to the stationary solution, with respect to the $p$-variance for $p\in [1,2]$ including the relative entropy $(p=1)$ and the variance $(p=2)$. These nonlinear functional inequalities are then established in both non-degenerate and degenerate settings, so that the main results are applied to a number of typical models with various convergence rates.