On Solvability in the Sense of Sequences for some Non-Fredholm Operators with Drift and Anomalous Diffusion

Abstract

We study the solvability of certain linear nonhomogeneous elliptic equations and establish that, under some technical assumptions, the L2-convergence of the right-hand sides yields the existence and convergence of solutions in an appropriate Sobolev space. The problems involve differential operators with or without Fredholm property, in particular, the one-dimensional negative Laplacian in a fractional power, on the whole real line or on a finite interval with periodic boundary conditions. We prove that the presence of the transport term in these equations provides regularization of the solutions.