The ergodicity of nonlinear Fokker-Planck flows in $L^1({\mathbb{R}}^d)$

One proves in this work that the nonlinear semigroup $S(t)$ in $L^1({\mathbb{R}}^d)$, $d\ge 3$, associated with the nonlinear Fokker-Planck equation $u_t-\mathrm{\Delta }\beta (u)+\text{div}(Db(u)u)=0$, $u(0)=u_0$ in $(0,\mathrm{\infty })\times {\mathbb{R}}^d$, under suitable conditions on the coefficients $\beta :\mathbb{R}\to \mathbb{R}$, $D:{\mathbb{R}}^d\to {\mathbb{R}}^d$ and $b:\mathbb{R}\to \mathbb{R}$, is mean ergodic. In particular, this implies the mean ergodicity of the time marginal laws of the solutions to the corresponding McKean-Vlasov stochastic differential equation. This completes the results established in [7] on the nature of the corresponding omega-set $\omega (u_0)$ for $S(t)$ in the case where the flow $S(t)$ in $L^1({\mathbb{R}}^d)$ has not a fixed point and so the corresponding stationary Fokker-Planck equation has no solutions.