  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{ div}}(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 $u_0\in
L^1(\mathbb{R}^d)$, $\beta\in C^2(\mathbb{R})$ is a nondecreasing function,
$b\in C^1$, 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 ${({\rm div}\,
D)^-}\in L^\infty(\mathbb{R}^d)$, $\beta$ strictly increasing, 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 ${\rm div}\,D\ge0$, $\beta'(r)\ge
a|r|^{\alpha-1}$, and $|\beta(r)|\le C r^\alpha$, $\alpha\ge1,$ $d\ge3$, then
$|u(t)|_{L^\infty}\le Ct^{-\frac d{d+(\alpha-1)d}}\ |u_0|^{\frac2{2+(m-1)d}},$
$t>0$, 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.
