Solutions for nonlinear Fokker-Planck equations with measures as initial data and McKean-Vlasov equations

One proves the existence and uniqueness of a generalized (mild) solution for the nonlinear Fokker-Planck equation (FPE) \begin{align*} &u_t-\Delta (\beta(u))+{$\mathrm{d}\mathrm{i}\mathrm{v}$}(D(x)b(u)u)=0, \quad t\geq0,\ x\in$\mathbb{R}$^d,
d\ne2, \ &u(0,\cdot)=u_0,\mbox{in }$\mathbb{R}$^d, \end{align*} where $u_0\in L^1($\mathbb{R}$^d)$,$\beta\in C^2($\mathbb{R}$)$isanondecreasingfunction,$b\in C^1$,bounded,$b\ge0$,$D\in {L^\infty}($\mathbb{R}$^d;$\mathbb{R}$^d)$,${\rm div},D\in L^2($\mathbb{R}$^d)+L^\infty($\mathbb{R}$^d),$with${({\rm div}, D)^-}\in L^\infty($\mathbb{R}$^d)$,$\beta$strictlyincreasing,if$b$ is not constant. Moreover, $t\to u(t,u_0)$ is a semigroup of contractions in $L^1({\mathbb{R}}^d)$, which leaves invariant the set of probability density functions in ${\mathbb{R}}^d$. If $\mathrm{d}\mathrm{i}\mathrm{v}D\ge 0$, $\beta’®\ge a|r|^{\alpha-1}$,and$|\beta®|\le C r^\alpha$,$\alpha\ge1,$ $d\ge 3$, then $\mathrm{\mid }u(t){\mathrm{\mid }}_{L^{\mathrm{\infty }}}\le C{t}^{-\frac{d}{d+(\alpha -1)d}}\mathrm{\mid }{u}_0{\mathrm{\mid }}^{\frac{2}{2+(m-1)d}},$ $t>0$, and, if $D\in L^2({\mathbb{R}}^d;{\mathbb{R}}^d)$, the existence extends to initial data $u_0$ in the space ${\mathcal{M}}_b$ of bounded measures in ${\mathbb{R}}^d$. As a consequence for arbitrary initial laws, we obtain weak solutions to a class of McKean-Vlasov SDEs with coefficients which have singular dependence on the time marginal laws.