We verify -after appropriate modifications- an old conjecture of
Brezis-Ekeland  concerning the feasibility of a global and variational approach
to the problems of existence and uniqueness of solutions of non-linear transport equations,
which do not normally fit in an Euler-Lagrange framework. Our method is based on a concept of
"anti-self duality" that seems to be inherent in many problems, including gradient flows of
convex energy functionals treated in  and other parabolic evolution equations ().