  The purpose of this paper is to investigate the well-posedness of several
linear and nonlinear equations with a parabolic forward-backward structure, and
to highlight the similarities and differences between them. The epitomal linear
example will be the stationary Kolmogorov equation $y\partial_x u
-\partial_{yy} u=f$ in a rectangle. We first prove that this equation admits a
finite number of singular solutions, of which we provide an explicit
construction. Hence, the solutions to the Kolmogorov equation associated with a
smooth source term are regular if and only if $f$ satisfies a finite number of
orthogonality conditions. This is similar to well-known phenomena in elliptic
problems in polygonal domains.
  We then extend this theory to a Vlasov--Poisson--Fokker--Planck system, and
to two quasilinear equations: the Burgers type equation $u \partial_x u -
\partial_{yy} u = f$ in the vicinity of the linear shear flow, and the Prandtl
system in the vicinity of a recirculating solution, close to the line where the
horizontal velocity changes sign. We therefore revisit part of a recent work by
Iyer and Masmoudi. For the two latter quasilinear equations, we introduce a
geometric change of variables which simplifies the analysis. In these new
variables, the linear differential operator is very close to the Kolmogorov
operator $y\partial_x -\partial_{yy}$. Stepping on the linear theory, we prove
existence and uniqueness of regular solutions for data within a manifold of
finite codimension, corresponding to some nonlinear orthogonality conditions.
