Solvability for perturbations of a class of real vector fields on the two-torus

Abstract

Let L=∂t+a(x)∂x be a real vector field defined on the two-dimensional torus T2, where a is a real-valued and smooth function on T1. We deal with the global solvability of equations in the form Lu+pu=f, where p,f∈C∞(T2). Solvability to the equation Lu=f is well-understood. We show that a perturbation of zero order may affect the global solvability of L; we may maintain, gain or lose solvability by adding a perturbation. This phenomenon is linked to the order of vanishing of the coefficient a of L. We obtained results in the class of smooth functions on T2 and, also, in the space of Schwartz distributions D′(T2).