Existence of global solutions for a class of vector fields on the three-dimensional torus

This work deals with global solvability of a class of vector fields of the form L=∂/∂t+(a(x)+ib(x))(∂/∂x+λ∂/∂y), where a,b∈C∞(T1,R) and λ∈R, defined on the three-dimensional torus T3(x,y,t)≃R3/2πZ3. In addition to the interplay between the order of vanishing of the functions a and b, the change of sign of b between two consecutive zeros of a+ib has influence in the global solvability. Also, a Diophantine condition appears in a natural way in our results.