  An operatorial based approach is used here to prove the existence and
uniqueness of a strong solution $u$ to the time-varying nonlinear
Fokker--Planck equation
  $u_t(t,x)-\Delta(a(t,x,u(t,x))u(t,x))+{\rm div}(b(t,x,u(t,x))u(t,x))=0$ in
$(0,\infty)\times \mathbb{R}$
  $u(0,x)=u_0(x),\ x\in\mathbb{R}^d$ in the Sobolev space
$H^{-1}(\mathbb{R}^d)$, under appropriate conditions on the
  $a:[0,T]\times\mathbb{R}^d\times\mathbb{R}\to\mathbb{R}$ and
$b:[0,T]\times\mathbb{R}^d\times\mathbb{R}\to\mathbb{R}^d.$
  It is proved also that, if $u_0$ is a density of a probability measure, so is
$u(t,\cdot)$ for all $t\ge0$.
  Moreover, we construct a weak solution to the McKean-Vlasov SDE associated
with the Fokker-Planck equation such that $u(t)$ is the density of its time
marginal law.
  MSC: 60H15, 47H05, 47J05.
  Keywords: Fokker--Planck equation, Cauchy problem, stochastic differential
equation, Sobolev space, periodic solution.
