Solvability in the sense of sequences for certain non-Fredholm operators with a drift and Laplace and bi-Laplace operators

We study the solvability of some linear nonhomogeneous elliptic problems and establish that under certain technical assumptions the convergence in $L^2$ of their right-hand sides yields the existence and the convergence in $H^4$ of the solutions. The equations contain fourth order differential operators with or without the Fredholm property, in particular the second and the fourth derivative operators, on the whole real line or on a finite interval with periodic boundary conditions. We establish that the transport term involved in these problems provides the regularization of the solutions.