One proves the existence and uniqueness of a generalized (mild) solution for
the nonlinear Fokker--Planck equation (FPE) egin{align*} &u_t-Delta
(eta(u))+{mathrm{ div}}(D(x)b(u)u)=0, quad tgeq0, xinmathbb{R}^d,
d
e2, \ &u(0,cdot)=u_0,mbox{in }mathbb{R}^d, end{align*} where $u_0in
L^1(mathbb{R}^d)$, $etain C^2(mathbb{R})$ is a nondecreasing function,
$bin C^1$, bounded, $bgeq 0$, $Din(L^2cap
L^infty)(mathbb{R}^d;mathbb{R}^d)$ with ${
m div}, Din
L^infty(mathbb{R}^d)$, and ${
m div},Dgeq0$, $eta$ strictly increasing,
if $b$ is not constant. Moreover, $t o u(t,u_0)$ 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 ${
m div},Dgeq0$,
$eta'(r)geq a|r|^{alpha-1}$, and $|eta(r)|leq C r^alpha$,
$alphageq1,$ $alpha>frac{d-2}d$, $dgeq3$, then $|u(t)|_{L^infty}le
Ct^{-frac d{d+(alpha-1)d}} |u_0|^{frac2{2+(m-1)d}},$ $t>0$, and the
existence extends to initial data $u_0$ in the space $mathcal{M}_b$ of bounded
measures in $mathbb{R}^d$. The solution map $mumapsto S(t)mu$, $tgeq0$, is
a Lipschitz contractions on $mathcal{M}_b$ and weakly continuous in
$tin[0,infty)$. 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.