The purpose of this memoir 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. The authors 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.

The authors then extend this theory to a Vlasov–Poisson–Fokker–Planck system, and to two quasilinear equations: the Burgers-type equation \(u\partial_x u - \partial_y u = f\) 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. The authors therefore revisit part of a recent work by Iyer and Masmoudi. For the two latter quasilinear equations, they 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, the authors prove the existence and uniqueness of regular solutions for data within a manifold of finite codimension, corresponding to some nonlinear orthogonality conditions.